Mathematical Economics and Finance Michael Harrison
Patrick Waldron
December 2, 1998
CONTENTS
i
Contents List of T...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Mathematical Economics and Finance Michael Harrison

Patrick Waldron

December 2, 1998

CONTENTS

i

Contents List of Tables

iii

List of Figures

v

PREFACE vii What Is Economics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii What Is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viii NOTATION

ix

I

1

MATHEMATICS

1 LINEAR ALGEBRA 1.1 Introduction . . . . . . . . . . . . . . . . 1.2 Systems of Linear Equations and Matrices 1.3 Matrix Operations . . . . . . . . . . . . . 1.4 Matrix Arithmetic . . . . . . . . . . . . . 1.5 Vectors and Vector Spaces . . . . . . . . 1.6 Linear Independence . . . . . . . . . . . 1.7 Bases and Dimension . . . . . . . . . . . 1.8 Rank . . . . . . . . . . . . . . . . . . . . 1.9 Eigenvalues and Eigenvectors . . . . . . . 1.10 Quadratic Forms . . . . . . . . . . . . . 1.11 Symmetric Matrices . . . . . . . . . . . . 1.12 Definite Matrices . . . . . . . . . . . . .

. . . . . . . . . . . .

3 3 3 7 7 11 12 12 13 14 15 15 15

2 VECTOR CALCULUS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vector-valued Functions and Functions of Several Variables . . .

17 17 17 18

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

Revised: December 2, 1998

ii

CONTENTS 2.4 2.5 2.6 2.7 2.8 2.9

Partial and Total Derivatives . . . . . . . The Chain Rule and Product Rule . . . . The Implicit Function Theorem . . . . . . Directional Derivatives . . . . . . . . . . Taylor’s Theorem: Deterministic Version The Fundamental Theorem of Calculus .

3 CONVEXITY AND OPTIMISATION 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Convexity and Concavity . . . . . . . . . 3.2.1 Definitions . . . . . . . . . . . . 3.2.2 Properties of concave functions . 3.2.3 Convexity and differentiability . . 3.2.4 Variations on the convexity theme 3.3 Unconstrained Optimisation . . . . . . . 3.4 Equality Constrained Optimisation: The Lagrange Multiplier Theorems . . . . 3.5 Inequality Constrained Optimisation: The Kuhn-Tucker Theorems . . . . . . . 3.6 Duality . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

20 21 23 24 25 26

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

27 27 27 27 29 30 34 39

. . . . . . . . . . . . .

43

. . . . . . . . . . . . . 50 . . . . . . . . . . . . . 58

II APPLICATIONS 4 CHOICE UNDER CERTAINTY 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Definitions . . . . . . . . . . . . . . . . . . . . 4.3 Axioms . . . . . . . . . . . . . . . . . . . . . 4.4 Optimal Response Functions: Marshallian and Hicksian Demand . . . . . . . 4.4.1 The consumer’s problem . . . . . . . . 4.4.2 The No Arbitrage Principle . . . . . . . 4.4.3 Other Properties of Marshallian demand 4.4.4 The dual problem . . . . . . . . . . . . 4.4.5 Properties of Hicksian demands . . . . 4.5 Envelope Functions: Indirect Utility and Expenditure . . . . . . . . 4.6 Further Results in Demand Theory . . . . . . . 4.7 General Equilibrium Theory . . . . . . . . . . 4.7.1 Walras’ law . . . . . . . . . . . . . . . 4.7.2 Brouwer’s fixed point theorem . . . . . Revised: December 2, 1998

61 63 . . . . . . . . . . 63 . . . . . . . . . . 63 . . . . . . . . . . 66 . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. 69 . 69 . 70 . 71 . 72 . 73

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

73 75 78 78 78

CONTENTS

4.8

4.9

iii

4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . The Welfare Theorems . . . . . . . . . . . . . . . . . . . . 4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . 4.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . 4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . 4.8.4 The Separating Hyperplane Theorem . . . . . . . . 4.8.5 The Second Welfare Theorem . . . . . . . . . . . . 4.8.6 Complete markets . . . . . . . . . . . . . . . . . . 4.8.7 Other characterizations of Pareto efficient allocations Multi-period General Equilibrium . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

78 78 78 78 79 80 80 82 82 84

5

CHOICE UNDER UNCERTAINTY 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85 5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88 5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88 5.4.1 Completion of markets using options . . . . . . . . . . . 90 5.4.2 Restrictions on security values implied by allocational efficiency and covariance with aggregate consumption . . . 91 5.4.3 Completing markets with options on aggregate consumption 92 5.4.4 Replicating elementary claims with a butterfly spread . . . 93 5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93 5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2 Existence of expected utility functions . . . . . . . . . . . 95 5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97 5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102 5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104

6

PORTFOLIO THEORY 6.1 Introduction . . . . . . . . . . . . . . . . . . . 6.2 Notation and preliminaries . . . . . . . . . . . 6.2.1 Measuring rates of return . . . . . . . . 6.2.2 Notation . . . . . . . . . . . . . . . . 6.3 The Single-period Portfolio Choice Problem . . 6.3.1 The canonical portfolio problem . . . . 6.3.2 Risk aversion and portfolio composition 6.3.3 Mutual fund separation . . . . . . . . . 6.4 Mathematics of the Portfolio Frontier . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

105 105 105 105 108 110 110 112 114 116

Revised: December 2, 1998

iv

CONTENTS The portfolio frontier in 0.

But the first order condition guarantees that the LHS of this inequality is zero (not positive), which is the required contradiction. Q.E.D. Theorem 3.4.3 Uniqueness condition for equality constrained maximisation. If 1. x∗ is a solution, 2. f is strictly quasiconcave, and 3. g j is an affine function (i.e. both convex and concave) for j = 1, . . . , m, then x∗ is the unique (global) maximum. Revised: December 2, 1998

3.4. EQUALITY CONSTRAINED OPTIMISATION: THE LAGRANGE MULTIPLIER THEOREMS

48

Proof The uniqueness result is also proved by contradiction. Note that it does not require any differentiability assumption. • We first show that the feasible set is convex. Suppose x 6= x∗ are two distinct solutions. Consider the convex combination of these two solutions xα ≡ αx+(1 − α) x∗ . Since each g j is affine and g j (x∗ ) = g j (x) = 0, we have g j (xα) = αg j (x) + (1 − α) g j (x∗ ) = 0. In other words, xα also satisfies the constraints. • To complete the proof, we find the required contradiction: Since f is strictly quasiconcave and f (x∗ ) = f (x), it must be the case that f (xα) > f (x∗ ). Q.E.D. The construction of the obvious corollaries for minimisation problems is left as an exercise. We conclude this section with Theorem 3.4.4 (Envelope Theorem.) Consider the modified constrained optimisation problem: max f (x, α) x

subject to g (x, α) = 0,

(3.4.2)

where x ∈ 0

(3.5.7) (3.5.8)

The various sign conditions which we have looked at are summarised in Table 3.1. Theorem 3.5.1 Necessary (first order) conditions for optimisation with inequality constraints. If 1. x∗ solves Problem (3.5.1), with g i (x∗ ) = 0, i = 1, 2, . . . , b and g i (x∗ ) > 0, i = b + 1, . . . , m (in other words, the first b constraints are binding (active) at x∗ and the last n − b are non-binding (inactive) at x∗ , renumbering the constraints if necessary to achieve this), Revised: December 2, 1998

52

3.5. INEQUALITY CONSTRAINED OPTIMISATION: THE KUHN-TUCKER THEOREMS 2. f and g are continuously differentiable, and 3. the b × n submatrix of g 0 (x∗ ),

∂g 1 ∂x1

(x∗ ) . . . .. ... .

∂g 1 ∂xn

∂g b ∂x1

(x∗ ) . . .

∂g b ∂xn

(x∗ ) .. , . (x∗ )

is of full rank b (i.e. there are no redundant binding constraints, both in the sense that there are fewer binding constraints than variables and in the sense that the constraints which are binding are ‘independent’), then ∃λ ∈ <m such that f 0 (x∗ ) + λ> g 0 (x∗ ) = 0, with λi ≥ 0 for i = 1, 2, . . . , m and g i (x∗ ) = 0 if λi > 0. Proof The proof is similar to that of Theorem 3.4.1 for the equality constrained case. It can be broken into seven steps. 1. Suppose x∗ solves Problem (3.5.1). We begin by restricting attention to a neighbourhood B (x∗ ) throughout which the non-binding constraints remain non-binding, i.e. g i (x) > 0 ∀x ∈ B (x∗ ) , i = b + 1, . . ., m.

(3.5.9)

Such a neighbourhood exists since the constraint functions are continuous. Since x∗ solves Problem (3.5.1) by assumption, it also solves the following problem: s.t.

maxx∈B (x∗ ) f (x) g i (x) ≥ 0, i = 1, 2, . . . , b.

(3.5.10)

In other words, since the non-binding constraints are non-binding ∀x ∈ B (x∗ ) by construction, we can ignore them if we confine our search for a maximum to this neighbourhood. We will return to the non-binding constraints in the very last step of this proof, but until then g will be taken to refer to the vector of b binding constraint functions only and λ to the vector of b Kuhn-Tucker multipliers corresponding to these binding constraints. 2. We now introduce slack variables s ≡ (s1 , . . ., sb ), one corresponding to each binding constraint, and consider the following equality constrained maximisation problem: maxx∈B (x∗ ),s∈ = x A+A x 2 and

1 2

(3.5.30) (3.5.31) (3.5.32)

A + A> is always symmetric.

Let G be the m × n matrix whose ith row is gi . G must have full rank if we are to apply the Kuhn-Tucker conditions. The Lagrangean is: x> Ax + λ> (Gx − α) .

(3.5.33)

The first order conditions are: 2x> A + λ> G = 0n

(3.5.34)

or, transposing and multiplying across by 12 A−1 : 1 x = − A−1 G> λ. 2

(3.5.35)

If the constraints are binding, then we will have: 1 α = − GA−1 G> λ. 2

(3.5.36)

Now we need the fact that G (and hence GA−1 G> ) has full rank to solve for the Lagrange multipliers λ:

λ = −2 GA−1 G>

−1

α.

(3.5.37)

Now the sign conditions tell us that each component of λ must be nonnegative. An easy fix is to let the Kuhn-Tucker multipliers be defined by: ∗

−1

> −1

λ ≡ max 0m , −2 GA G

α ,

(3.5.38)

Revised: December 2, 1998

58

3.6. DUALITY where the max operator denotes component-by-component maximisation. The effect of this is to knock out the non-binding constraints (those with negative Lagrange multipliers) from the original problem and the subsequent analysis. We can now find the optimal x by substituting for λ in (3.5.35) the value of λ∗ from (3.5.38). In the case in which all the constraints are binding, the solution is:

x = A−1 G> GA−1 G>

−1

α

(3.5.39)

and the envelope function is given by:

−1

GA−1 AA−1 G> GA−1 G>

−1

α

x> Ax = α> GA−1 G> = α> GA−1 G> 1 = − α> λ. 2

−1

α

(3.5.40)

The applications of this problem will include ordinary least squares and generalised least squares regression and the mean-variance portfolio choice problem in finance. 2. Maximising a Cobb-Douglas utility function subject to a budget constraint and non-negativity constraints. The applications of this problem will include choice under certainty, choice under uncertainty with log utility where the parameters are reinterpreted as probabilities, the extension to Stone-Geary preferences, and intertemporal choice with log utility, where the parameters are reinterpreted as time discount factors. Further exercises consider the duals of each of the forgoing problems, and it is to the question of duality that we will turn in the next section.

3.6 Duality Let X ⊆ x + (1 − λ) p0> x > M, which contradicts the first inequality. It follows that the maximum value of uh (x) on the subset B (pλ) is less than or equal to its maximum value on the superset B (p) ∪ B (p0 ). In terms of the indirect utility function, this says that vh (pλ, M ) ≤ max {vh (p, M ) , vh (p0 , M )} , or that vh is quasiconvex. 4. vh (p, M ) is homogenous of degree zero in p, M , or vh (λp, λM ) = vh (p, M ) . Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

75

The following are interesting properties of the expenditure function: 1. The expenditure function is continuous. 2. The expenditure function itself is non-decreasing in prices, since raising the price of one good while holding the prices of all other goods constant can not reduce the minimum cost of attaining a fixed utility level. 3. The expenditure function is concave in prices. To see this, we just fix two price vectors p and p0 and consider the value of the expenditure function at the convex combination pλ ≡ λp + (1 − λ) p0 . e (pλ, u¯) = (pλ)> h (pλ, u¯)

(4.5.5) 0 >

>

= λp h (pλ, u¯) + (1 − λ) (p ) h (pλ, u¯) 0 >

≥ λp> h (p, u¯) + (1 − λ) (p ) h (p0 , u¯) = λe (p, u¯) + (1 − λ) e (p0 , u¯) ,

(4.5.6) (4.5.7) (4.5.8)

where the inequality follows because the cost of a suboptimal bundle for the given prices must be greater than the cost of the optimal (expenditureminimising) consumption vector for those prices. 4. The expenditure function is homogenous of degree 1 in prices: eh (αp, u¯) = αeh (p, u¯) .

(4.5.9)

4.6 Further Results in Demand Theory In this section, we present four important theorems on demand functions and the corresponding envelope functions. Shephard’s Lemma will allow us to recover Hicksian demands from the expenditure function. Similarly, Roy’s Identity will allow us to recover Marshallian demands from the indirect utility function. The Slutsky symmetry condition and the Slutsky equation provide further insights into the properties of consumer demand. Theorem 4.6.1 (Shephard’s Lemma.) ∂eh ∂ > (p, u ¯ ) = p x + λ (u (x) − u ¯ ) h ∂pn ∂pn = xn

(4.6.1) (4.6.2)

which, when evaluated at the optimum, is just hnh (p, u¯). In other words, the partial derivatives of the expenditure function with respect to prices are the corresponding Hicksian demand functions. Revised: December 2, 1998

76

4.6. FURTHER RESULTS IN DEMAND THEORY

Proof By differentiating the expenditure function with respect to the price of good n and applying the envelope theorem (Theorem 3.4.4), we obtain Shephard’s Lemma: (To apply the envelope theorem, we should be dealing with an equality constrained optimisation problem; however, if we assume local non-satiation, we know that the budget constraint or utility constraint will always be binding, and so the inequality constrained expenditure minimisation problem is essentially and equality constrained problem.) Q.E.D. Theorem 4.6.2 (Roy’s Identity.) Marshallian demands may be recovered from the indirect utility function using: ∂v n

xn (p, M ) = − ∂p ∂v

∂M

(p, M ) (p, M )

.

(4.6.3)

Proof For Roy’s Identity, see ?. It is obtained by differentiating equation (4.4.7) with respect to pn , using the Chain Rule: v (p, e (p, u¯)) = u¯ implies that ∂v ∂v ∂e (p, e (p, u¯)) + (p, e (p, u¯)) n (p, u¯) = 0 n ∂p ∂M ∂p

(4.6.4)

and using Shephard’s Lemma gives: ∂v ∂v (p, e (p, u¯)) + (p, e (p, u¯)) hn (p, u¯) = 0 n ∂p ∂M Hence n

h (p, u¯) = −

∂v ∂pn ∂v ∂M

(p, e (p, u¯)) (p, e (p, u¯))

(4.6.5)

(4.6.6)

and expressing this last equation in terms of the relevant level of income M rather than the corresponding value of utility u¯: n

x (p, M ) = −

∂v ∂pn ∂v ∂M

(p, M ) (p, M )

.

(4.6.7)

Q.E.D. Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

77

Theorem 4.6.3 (Slutsky symmetry condition.) All cross-price substitution effects are symmetric: ∂hnh ∂hm h = . (4.6.8) m ∂p ∂pn Proof From Shephard’s Lemma, we can easily derive the Slutsky symmetry conditions, assuming that the expenditure function is twice continuously differentiable, and hence that ∂ 2 eh ∂ 2 eh = . (4.6.9) ∂pm ∂pn ∂pn ∂pm Since hm h =

∂eh ∂pm

and hnh =

∂eh , ∂pn

and the result follows. Q.E.D.

The next result doesn’t really have a special name of its own. Theorem 4.6.4 Since the expenditure function is concave in prices (see p. 75), the corresponding Hessian matrix is negative semi-definite. In particular, its diagonal entries are non-positive, or ∂ 2 eh ≤ 0, ∂ (pn )2

n = 1, . . . , N.

(4.6.10)

Using Shephard’s Lemma, it follows that ∂hnh ≤ 0, ∂pn

n = 1, . . . , N.

(4.6.11)

In other words, Hicksian demand functions, unlike Marshallian demand functions, are uniformly decreasing in own price. Another way of saying this is that own price substitution effects are always negative. Theorem 4.6.5 (Slutsky equation.) The total effect of a price change on (Marshallian) demand can be decomposed as follows into a substitution effect and an income effect: ∂xm ∂hm ∂xm (p, M ) = (p, u ¯ ) − (p, M ) hn (p, u¯) , n n ∂p ∂p ∂M

(4.6.12)

where u¯ ≡ V (p, M ). Before proving this, let’s consider the signs of the various terms in the Slutsky equation and look at what it means in a two-good example. By Theorem 4.6.4, we know that own price substitution effects are always nonpositive. [This is still on a handwritten sheet.] Revised: December 2, 1998

78

4.7. GENERAL EQUILIBRIUM THEORY

Proof Differentiating both sides of the lth component of (4.4.9) with respect to pn , using the Chain Rule, will yield the so-called Slutsky equation which decomposes the total effect on demand of a price change into an income effect and a substitution effect. Differentiating the RHS of (4.4.9) with respect to pn yields: ∂xm ∂e ∂xm (p, e (p, u ¯ )) + (p, e (p, u¯)) n (p, u¯) . n ∂p ∂M ∂p

(4.6.13)

To complete the proof: 1. set this equal to

∂hm ∂pn

(p, u¯)

2. substitute from Shephard’s Lemma 3. define M ≡ e (p, u¯) (which implies that u¯ ≡ V (p, M ))

Q.E.D.

4.7 General Equilibrium Theory 4.7.1 Walras’ law Walras . . . 3

4.7.2 Brouwer’s fixed point theorem 4.7.3 Existence of equilibrium

4.8 The Welfare Theorems 4.8.1 The Edgeworth box 4.8.2 Pareto efficiency Definition 4.8.1 A feasible allocation X = (x1 , . . . , xH ) is Pareto efficient if there does not exist any feasible way of reallocating the same initial aggregate P endowment, H h=1 xh , which makes one individual better off without making any other worse off. Definition 4.8.2 X is Pareto dominated by X0 = (x01 , . . . , x0H ) if PH 0 0 0 h=1 xh , xh h xh ∀h and xh h xh for at least one h. 3

PH

h=1

xh =

This material still exists only in handwritten form in Alan White’s EC3080 notes from 19912. One thing missing from the handwritten notes is Kakutani’s Fixed Point Theorem which should be quoted from ?.

Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

79

4.8.3 The First Welfare Theorem (See ?.) Theorem 4.8.1 (First Welfare Theorem) If the pair (p, X) is an equilibrium (for given preferences, h , which exibit local non-satiation and given endowments, eh , h = 1, . . . , H), then X is a Pareto efficient allocation. Proof The proof is by contradiction. Suppose that X is an equilibrium allocation which is Pareto dominated by a feasible allocation X0 . If individual h is strictly better off under X0 or x0h h xh , then it follows that individual h cannot afford x0h at the equilibrium prices p or p> x0h > p> xh = p> eh .

(4.8.1)

The latter equality is just the budget constraint, which is binding since we have assumed local non-satiation. Similarly, if individual h is indifferent between X and X0 or x0h ∼h xh , then it follows that (4.8.2) p> x0h ≥ p> xh = p> eh , since if x0h cost strictly less than xh , then by local non-satiation some consumption vector near enough to x0h to also cost less than xh would be strictly preferred to xh and xh would not maximise utility given the budget constraint. Summing (4.8.1) and (4.8.2) over households yields p>

H X

x0h > p>

h=1

H X

xh = p>

h=1

H X

eh ,

(4.8.3)

h=1

(where the equality is essentially Walras’ Law). But since X0 is feasible we must have for each good n H X

h=1

x0n h ≤

H X

enh

h=1

and, hence, multiplying by prices and summing over all goods, p>

H X

h=1

x0h ≤ p>

H X

eh .

(4.8.4)

h=1

But (4.8.4) contradicts the inequality in (4.8.3), so no such Pareto dominant allocation X0h can exist. Q.E.D. Before proceeding to the second welfare theorem, we need to say a little bit about separating hyperplanes. Revised: December 2, 1998

80

4.8. THE WELFARE THEOREMS

4.8.4 The Separating Hyperplane Theorem n

o

Definition 4.8.3 The set z ∈ z = p> z∗ is the hyperplane through z∗ with normal p. Note that any hyperplane divides z∗

and

z ∈ z ≥ p> z∗ .

The intersection of these two closed half-spaces is the hyperplane itself. In two dimensions, a hyperplane is just a line; in three dimensions, it is just a plane. The idea behind the separating hyperplane theorem is quite intuitive: if we take any point on the boundary of a convex set, we can find a hyperplane through that point so that the entire convex set lies on one side of that hyperplane. We will essentially be applying this notion to the upper contour sets of quasiconcave utility functions, which are of course convex sets. We will interpret the separating hyperplane as a budget hyperplane, and the normal vector as a price vector, so that at those prices nothing giving higher utility than the cutoff value is affordable. Theorem 4.8.2 (Separating Hyperplane Theorem) If Z is a convex subset of z ∀z ∈ Z, or Z is contained in one of the closed half-spaces associated with the hyperplane through z∗ with normal p∗ . Proof Not given. See ? Q.E.D.

4.8.5 The Second Welfare Theorem (See ?.) We make slightly stronger assumptions than are essential for the proof of this theorem. This allows us to give an easier proof. Theorem 4.8.3 (Second Welfare Theorem) If all individual preferences are strictly convex, continuous and strictly monotonic, and if X∗ is a Pareto efficient allocation such that all households are allocated positive amounts of all goods (x∗g h > 0 ∀g = 1, . . . , N ; h = 1, . . . , H), then a reallocation of the initial aggregate endowment can yield an equilibrium where the allocation is X∗ . Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

81

Proof There are four main steps in the proof. 1. First we construct a set of utility-enhancing endowment perturbations, and use the separating hyperplane theorem to find prices at which no such endowment perturbation is affordable. We need to use the fact (Theorem 3.2.1) that a sum of convex sets, such as X + Y ≡ {x + y : x ∈ X, y ∈ Y } , is also a convex set. ∗ Given an aggregate initial endowment x∗ = H h=1 xh , we interpret any vecPH ∗ tor of the form z = h=1 xh − x as an endowment perturbation. Now consider the set of all ways of changing the aggregate endowment without making anyone worse off:

P

Z ≡(

z∈

0 ≤ p∗> z ∀z ∈ Z. Since preferences are monotonic, the set Z must contain all the standard unit basis vectors ((1, 0, . . . , 0), &c.). This fact can be used to show that all components of p∗ are non-negative, which is essential if it is to be interpreted as an equilibrium price vector. 3. Next, we specify one way of redistributing the initial endowment in order that the desired prices and allocation emerge as a competitive equilibrium. All we need to do is value endowments at the equilibrium prices, and redistribute the aggregate endowment of each good to consumers in proportion to their share in aggregate wealth computed in this way. 4. Finally, we confirm that utility is maximised by the given Pareto efficient allocation, X∗ , at these prices. As usual, the proof is by contradiction: the details are left as an exercise. Q.E.D.

4.8.6 Complete markets The First Welfare Theorem tells us that competitive equilibrium allocations are Pareto optimal if markets are complete. If there are missing markets, then competitive trading may not lead to a Pareto optimal allocation. We can use the Edgeworth Box diagram to illustrate the simplest possible version of this principle.

4.8.7 Other characterizations of Pareto efficient allocations There are a total of five equivalent characterisations of Pareto efficient allocations. Theorem 4.8.4 Each of the following is an equivalent description of the set of allocations which are Pareto efficient: 1. by definition, feasible allocations such that no other allocation strictly increases at least one individual’s utility without decreasing the utility of any other individual; 2. by the Welfare Theorems, equilibrium allocations for all possible distributions of the fixed initial aggregate endowment; 3. in two dimensions, allocations lying on the contract curve in the Edgeworth box; Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY 4. allocations which solve:

83

5

max

{xh :h=1,...,H}

H X

λh [uh (xh )]

(4.8.6)

h=1

subject to the feasibility constraints H X

xh =

h=1

H X

eh

(4.8.7)

h=1

for some non-negative weights {λh }H h=1 . 5. allocations which maximise the utility of a representative agent given by H X

λh [uh (xh )]

(4.8.8)

h=1

where {λh }H h=1 are again any non-negative weights. Proof If an allocation is not Pareto efficient, then the Pareto-dominating allocation gives a higher value of the objective function in the above problem for all possible weights. If an allocation is Pareto efficient, then the relative weights for which the above objective function is maximized are the ratios of the Lagrange multipliers from the problem of maximizing any individual’s utility subject to the constraint that all other individuals’ utilities are unchanged: max u1 (x1 )

(4.8.9)

s.t. uh (xh ) = uh (x∗h ) h = 2, . . . , H

(4.8.10)

since these two problems will have the same necessary and sufficient first order conditions. The absolute weights corresponding to a particular allocation are not unique, as they can be multiplied by any positive constant without affecting the maximum. Different absolute weights (or Lagrange multipliers) arise from fixing different individuals’ utilities in the last problem, but the relative weights will be the same. 5

The solution here would be unique if the underlying utility function were concave, since linear combinations of concave functions with non-negative weights are concave, and the constraints specify a convex set on which the objective function has a unique optimum. This argument can not be used with merely quasiconcave utility functions.

Revised: December 2, 1998

84

4.9. MULTI-PERIOD GENERAL EQUILIBRIUM Q.E.D.

Note that corresponding to each Pareto efficient allocation there is at least one: 1. set of non-negative weights defining (a) the objective function in 4. and (b) the representative agent in 5. and 2. initial allocation leading to the competitive equilibrium in 2.

4.9 Multi-period General Equilibrium In Section 4.2, it was pointed out that the objects of choice can be differentiated not only by their physical characteristics, but also both by the time at which they are consumed and by the state of nature in which they are consumed. These distinctions were suppressed in the intervening sections but are considered again in this section and in Section 5.4 respectively. The multi-period model should probably be introduced at the end of Chapter 4 but could also be left until Chapter 7. For the moment this brief introduction is duplicated in both chapters. Discrete time multi-period investment problems serve as a stepping stone from the single period case to the continuous time case. The main point to be gotten across is the derivation of interest rates from equilibrium prices: spot rates, forward rates, term structure, etc. This is covered in one of the problems, which illustrates the link between prices and interest rates in a multiperiod model.

Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

85

Chapter 5 CHOICE UNDER UNCERTAINTY 5.1 Introduction [To be written.]

5.2 Review of Basic Probability Economic theory has, over the years, used many different, sometimes overlapping, sometimes mutually exclusive, approaches to the analysis of choice under uncertainty. This chapter deals with choice under uncertainty exclusively in a single period context. Trade takes place at the beginning of the period and uncertainty is resolved at the end of the period. This framework is sufficient to illustrate the similarities and differences between the most popular approaches. When we consider consumer choice under uncertainty, consumption plans will have to specify a fixed consumption vector for each possible state of nature or state of the world. This just means that each consumption plan is a random vector. Let us review the associated concepts from basic probability theory: probability space; random variables and vectors; and stochastic processes. Let Ω denote the set of all possible states of the world, called the sample space. A collection of states of the world, A ⊆ Ω, is called an event. Let A be a collection of events in Ω. The function P : A → [0, 1] is a probability function if 1.

(a) Ω ∈ A (b) A ∈ A ⇒ Ω − A ∈ A (c) Ai ∈ A for i = 1, . . . , ∞ ⇒

S∞

i=1

Ai ∈ A

(i.e. A is a sigma-algebra of events) Revised: December 2, 1998

86

5.2. REVIEW OF BASIC PROBABILITY and 2.

(a) P (Ω) = 1 (b) P (Ω − A) = 1 − P (A) ∀A ∈ A (redundant assumption) (c) P ( ∞ i=1 Ai ) = events in A. S

P∞

i=1

P (Ai ) when A1 , A2 , . . . are pairwise disjoint

(Ω, A, P ) is then called a probability space Note that the certainty case we considered already is just the special case of uncertainty in which the set Ω has only one element. We will consider these concepts in more detail when we come to intertemporal models. Suppose we are given such a probability space. The function x˜ : Ω → < is a random variable (r.v.) if ∀x ∈ < {ω ∈ Ω : x˜ (ω) ≤ x} ∈ A, i.e. a function is a random variable if we know the probability that the value of the function is less than or equals any given real number. The function Fx˜ : < → [0, 1] : x 7→ Pr (˜ x ≤ x) ≡ P ({ω ∈ Ω : x˜ (ω) ≤ x}) is known as the cumulative distribution function (c.d.f.) of the random variable x˜. The convention of using a tilde over a letter to denote a random variable is common in financial economics; in other fields capital letters may be reserved for random variables. In either case, small letters usually denote particular real numbers (i.e. particular values of the random variable). A random vector is just a vector of random variables. It can also be thought of as a vector-valued function on the sample space Ω. A stochastic process is a collection of random variables or random vectors indexed by time, e.g. {˜ xt : t ∈ T } or just {˜ xt } if the time interval is clear from the context. For the purposes of this part of the course, we will assume that the index set consists of just a finite number of times i.e. that we are dealing with discrete time stochastic processes. Then a stochastic process whose elements are N -dimensional random vectors is equivalent to an N |T |-dimensional random vector. The (joint) c.d.f. of a random vector or stochastic process is the natural extension of the one-dimensional concept. Random variables can be discrete, continuous or mixed. The expectation (mean, average) of a discrete r.v., x˜, with possible values x1 , x2 , x3 , . . . is given by E [˜ x] ≡

∞ X

xi P r (˜ x = xi ) .

i=1

For a continuous random variable, the summation is replaced by an integral. Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

87

The covariance of two random variables x˜ and y˜ is given by Cov [˜ x, y˜] ≡ E [(˜ x − e [˜ x]) (˜ y − E [˜ y ])] . The covariance of a random variable with itself is called its variance. The expectation of a random vector is just the vector of the expectations of the component random variables. The variance (variance-covariance matrix) of a random vector is the (symmetric, positive semi-definite) matrix of the covariances between the component random variables. Given any two random variables x˜ and y˜, we can define a third random variable˜ by ˜ ≡ y˜ − α − β x˜. (5.2.1) To specify ˜ completely, we can either specify α and β explicitly or fix them implicitly by imposing (two) conditions on˜. We do the latter by insisting 1. ˜ and x˜ are uncorrelated (this is not the same as assuming statistical independence, except in special cases such as bivariate normality) 2. E [˜] = 0 It follows that: β =

Cov [˜ x, y˜] Var [˜ x]

(5.2.2)

and α = E [˜ y ] − βE [˜ x] .

(5.2.3)

But what about the conditional expectation, E [˜ y |˜ x = x]? This is not equal to α + βx, as one might expect, unless E [˜|˜ x = x] = 0. This requires statistical independence rather than the assumed lack of correlation. Again, a sufficient condition is multivariate normality. The notion of the β of y˜ with respect to x˜ as given in (5.2.2) will recur frequently. The final concept required from basic probability theory is the notion of a mixture of random variables. For lotteries which are discrete random variables, with payoffs x1 , x2 , x3 , . . . occuring with probabilities π1 , π2 , π3 , . . . respectively, we will use the notation: π 1 x1 ⊕ π 2 x2 ⊕ π 3 x3 ⊕ . . . Similar notation will be used for compound lotteries (mixtures of random variables) where the payoffs themselves are further lotteries. This might be a good place to talk about the MVN distribution and Stein’s lemma. Revised: December 2, 1998

88

5.3. TAYLOR’S THEOREM: STOCHASTIC VERSION

5.3 Taylor’s Theorem: Stochastic Version We will frequently use the univariate Taylor expansion as applied to a function of a random variable expanded about the mean of the random variable.1 Taking expectations on both sides of the Taylor expansion: f (˜ x) = f (E[˜ x]) +

∞ X

1 (n) f (E[˜ x])(˜ x − E[˜ x])n n! n=1

(5.3.1)

yields: E[f (˜ x)] = f (E[˜ x]) +

∞ X

1 (n) f (E[˜ x])mn (˜ x), n! n=2

(5.3.2)

where mn (˜ x) ≡ E [(˜ x − E[˜ x])n ] .

(5.3.3)

In particular, h

i

(5.3.4)

h

i

(5.3.5)

h

i

(5.3.6)

h

i

(5.3.7)

m1 (˜ x) = E (˜ x − E[˜ x])1 ≡ 0 m2 (˜ x) = E (˜ x − E[˜ x])2 ≡ Var [˜ x] m3 (˜ x) = E (˜ x − E[˜ x])3 ≡ Skew [˜ x] and

m4 (˜ x) = E (˜ x − E[˜ x])4 ≡ Kurt [˜ x] ,

which allows us to start the summation in (5.3.2) at n = 2 rather than n = 1. Indeed, we can rewrite (5.3.2) as 1 1 E[f (˜ x)] = f (E[˜ x]) + f 00 (E[˜ x])Var [˜ x] + f 000 (E[˜ x])Skew [˜ x] 2 6 ∞ X 1 1 + f 0000 (E[˜ x])Kurt [˜ x] + f (n) (E[˜ x])mn (˜ x). (5.3.8) 24 n=5 n!

5.4 Pricing State-Contingent Claims This part of the course draws on ?, ? and ?. The analysis of choice under uncertaintly will begin by reinterpreting the general equilibrium model of Chapter 4 so that goods can be differentiated by the state of nature in which they are consumed. Specifically, it will be assumed that the 1

This section will eventually have to talk separately about kth order and infinite order Taylor expansions.

Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

89

underlying sample space comprises a finite number of states of nature. A more thorough analysis of choice under uncertainty, allowing for infinite and continuous sample spaces and based on additional axioms of choice, follows in Section 5.5. Consider a world with M possible states of nature (distinguished by a first subscript), markets for N securities (distinguished by a second subscript) and H consumers (distinguished by a superscript).2 Definition 5.4.1 A state contingent claim or Arrow-Debreu security is a random variable or lottery which takes the value 1 in one particular state of nature and the value 0 in all other states. Definition 5.4.2 A complex security is a random variable or lottery which can take on arbitrary values. The payoffs of a typical complex security will be represented by a column vector, yj ∈ <M , where yij is the payoff in state i of security j. The set of all complex securities on a given finite sample space is an M -dimensional vector space and the M possible Arrow-Debreu securities constitute the standard basis for this vector space. State contingent claims prices are determined by the market clearing equations in a general equilibrium model: Aggregate consumption in state i = Aggregate endowment in state i. Each individual will have an optimal consumption choice depending on endowments and preferences and conditional on the state of the world. Optimal future consumption is denoted ∗ x1 x∗ x∗ = 2 . (5.4.1) ... x∗N If there are N complex securities, then the investor must find a portfolio w = (w1 , . . . , wN ) whose payoffs satisfy x∗i

=

N X

yij wj .

j=1

Let Y be the M × N matrix3 whose jth column contains the payoffs of the jth complex security in each of the M states of nature, i.e. Y ≡ (y1 , y2 , . . . yN ) . 2 3

(5.4.2)

Check for consistency in subscripting etc in what follows. Or maybe I mean its transpose.

Revised: December 2, 1998

90

5.4. PRICING STATE-CONTINGENT CLAIMS

Theorem 5.4.1 If there are M complex securities (M = N ) and the payoff matrix Y is non-singular, then markets are complete. Proof Suppose the optimal trade for consumer i state j is xij − eij . Then can invert Y to work out optimal trades in terms of complex securities. Q.E.D. An (N + 1)st security would be redundant. Either a singular square matrix or < N complex securities would lead to incomplete markets. So far, we have made no assumptions about the form of the utility function, written purely as u (x0 , x1 , x2 , . . . , xN ) , where x0 represents the quantity consumed at date 0 and xi (i > 0) represents the quantity consumed at date 1 if state i materialises.

5.4.1 Completion of markets using options Assume that there exists a state index portfolio, Y , yielding different non-zero payoffs in each state (i.e. a portfolio with a different payout in each state of nature, possibly one mimicking aggregate consumption). WLOG we can rank the states so that Yi < Yj if i < j. We now present some results, following ?, showing conditions under which trading in a state index portfolio and in options on the state index portfolio can lead to the Pareto optimal complete markets equilibrium allocation. Now consider completion of markets using options on aggregate consumption. In real-world markets, the number of linearly independent corporate securities is probably less than M . However, options on corporate securities may be sufficient to form complete markets, and thereby ensure allocational (Pareto) efficiency for arbitrary preferences. Further assume that ∃ M − 1 European call options on Y with exercise prices Y1 , Y2 , . . . , YM −1 . A European call option with exercise price K is an option to buy a security for K on a fixed date. An American call option is an option to buy on or before the fixed date. A put option is an option to sell. Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

91

Here, the original state index portfolio and the M − 1 European call options yield the payoff matrix: y1 0 0 . ..

0

y2 y2 − y1 0 .. .

y3 y3 − y1 y3 − y2 .. .

0

0

... ... ... .. .

yM yM − y1 yM − y2 .. .

. . . yM − yM −1

security Y call option 1 call option 2 .. .

=

call option M − 1

(5.4.3)

and as this matrix is non-singular, we have constructed a complete market. Instead of assuming a state index portfolio exists, we can assume identical probability beliefs and state-independent utility and complete markets in a similar manner (see below).

5.4.2 Restrictions on security values implied by allocational efficiency and covariance with aggregate consumption Cω = aggregate consumption in state ω Ωk = {ω ∈ Ω : Cω = k} Let φ(k) be the value of the claim with payoffs Let

ykω =

1 if Cω = k 0 otherwise

(5.4.4)

and let the agreed probability of the event Ωk (i.e. of aggregate consumption taking the value k) be: X πω (5.4.5) π(k) = ω∈Ωk

By time-additivity and state-independence of the utility function: φω =

πωu0i (ciω ) u0i0 (ci0 )

∀ω ∈ Ω

(5.4.6)

The no arbitrage condition implies φ(k) = = =

X

φω

ω∈Ωk u0i (fi (k)) X u0i0 (ci0 ) ω∈Ωk u0i (fi (k)) π(k) u0i0 (ci0 )

(5.4.7) πω

(5.4.8) (5.4.9)

where fi (k) denotes the i-th individual’s equilibrium consumption in those states where aggregate consumption equals k. Revised: December 2, 1998

92

5.4. PRICING STATE-CONTINGENT CLAIMS x(0)

x(1)

x(2)

1 2 3 · · · L

0 1 2 · · · L−1

0 0 1 · · · L−2

C˜ = 1 C˜ = 2 C˜ = 3 · · · C˜ = L

Table 5.1: Payoffs for Call Options on the Aggregate Consumption State-independence of the utility function is required for fi (k) to be well-defined. Therefore, an arbitrary security x has value: Sx =

X

φωxω

(5.4.10)

ω∈Ω

= = =

X X

X

(5.4.11) πωxω

πω xω π(k)

X

φ(k)

X

φ(k)E[˜ x|C˜ = k]

k

=

φωxω

k ω∈Ωk X u0i (fi (k)) X u0i0 (ci0 ) ω∈Ωk k

ω∈Ωk

(5.4.12) (5.4.13) (5.4.14)

k

5.4.3 Completing markets with options on aggregate consumption Let x(k) be the vector of payoffs in the various possible states on a European call option on aggregate consumption with one period to maturity and exercise price k. Let {1, 2, . . . , L} be the set of possible values of aggregate consumption C(ω). Then payoffs are as given in Table 5.1. This all assumes 1. identical probability beliefs 2. time-additivity of u 3. state-independent u Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

93

5.4.4 Replicating elementary claims with a butterfly spread Elementary claims against aggregate consumption can be constructed as follows, for example, for state 1, using a butterfly spread: [x(0) − x(1)] − [x(1) − x(2)]

(5.4.15)

yields the payoff:

0 1 0 0 0 1 1 1 2 1 1 0 3 − 2 − 2 − 1 = 1 − 1 . . . . . . . . . . . . . . . . . . 1 L−2 L−1 L−1 L 1 1 0 (5.4.16) = 0. . . 0 i.e. this replicating portfolio pays 1 iff aggregate consumption is 1, and 0 otherwise. The prices of this, and the other elementary claims, must, by no arbitrage, equal the prices of the corresponding replicating portfolios.

5.5 The Expected Utility Paradigm 5.5.1 Further axioms The objects of choice with which we are concerned in a world with uncertainty could still be called consumption plans, but we will acknowledge the additional structure now described by terming them lotteries. If there are k physical commodities, a consumption plan must specify a k-dimensional vector, x ∈ 0

(6.3.9)

E[u0 (W0 rf ) (˜ r − rf )] > 0

(6.3.10)

u0 (W0 rf ) E[(˜ r − rf )] > 0

(6.3.11)

⇐⇒

⇐⇒

⇐⇒ E[˜ r] > E[rf ] = rf This is the property of local risk neutrality — a risk averse investor will always prefer a little of a risky asset paying a higher expected return than rf to none of the risky asset. Definition 6.3.1 Let f : X → 0

da dW0

1) • CRRA ⇒ constant proportion of wealth invested in the risky asset (η = 1) • IRRA ⇒ decreasing proportion of wealth invested in the risky asset (η < 1) Theorem 6.3.1 DARA ⇒ RISKY ASSET NORMAL Proof By implicit differentiation of the now familiar first order condition (6.3.3), which can be written:

we have

E[u0 (W0 rf + a(˜ r − rf ))(˜ r − rf )] = 0,

(6.3.16)

˜ )(˜ da E[u00 (W r − rf )]rf = . 00 ˜ dW0 −E[u (W )(˜ r − rf )2 ]

(6.3.17)

Revised: December 2, 1998

114

6.3. THE SINGLE-PERIOD PORTFOLIO CHOICE PROBLEM

By concavity, the denominator is positive. Therefore: ˜ )(˜ sign (da/dW0 ) = sign {E[u00 (W r − rf )]}

(6.3.18)

We will show that both are positive. For decreasing absolute risk aversion:3 ˜ ) < RA (W0 rf ) r˜ > rf ⇒ RA (W ˜ ) ≥ RA (W0 rf ) r˜ ≤ rf ⇒ RA (W ˜ )(˜ Multiplying both sides of each inequality by −u0 (W r − rf ) gives respectively: ˜ )(˜ ˜ )(˜ u00 (W r − rf ) > −RA (W0 rf )u0 (W r − rf )

(6.3.19)

in the event that r˜ > rf , and ˜ )(˜ ˜ )(˜ u00 (W r − rf ) ≥ −RA (W0 rf )u0 (W r − rf )

(6.3.20)

(the same result) in the event that r˜ ≤ rf Integrating over both events implies: ˜ )(˜ ˜ )(˜ E[u00 (W r − rf )] > −RA (W0 rf )E[u0 (W r − rf )],

(6.3.21)

provided that r˜ > rf with positive probability. The RHS of inequality (6.3.21) is 0 at the optimum, hence the LHS is positive as claimed. Q.E.D. The other results are proved similarly (exercise!).

6.3.3 Mutual fund separation Commonly, investors delegate portfolio choice to mutual fund operators or managers. We are interested in conditions under which large groups of investors will agree on portfolio composition. For example, all investors with similar utility functions might choose the same portfolio, or all investors with similar probability beliefs might choose the same portfolio. More realistically, we may be able to define a group of investors whose portfolio choices all lie in a subspace of small dimension (say 2) of the N -dimensional portfolio space. The first such result is due to ?. 3

Think about whether separating out the case of r˜ = rf is necessary.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

115

Theorem 6.3.2 ∃ Two fund monetary separation i.e. Agents with different wealths (but the same increasing, strictly concave, VNM utility) hold the same risky unit cost portfolio, p∗ say, (but may differ in the mix of the riskfree asset and risky portfolio) i.e. ∀ portfolios p, wealths W0 , ∃λ s.t. h

E u W0 rf + λW0 p∗ > (˜r − rf 1)

i

h

i

≥ E u W0 rf + p> (˜r − rf 1)

(6.3.22)

⇐⇒ Risk-tolerance (1/RA (z)) is linear (including constant) i.e. ∃ Hyperbolic Absolute Risk Aversion (HARA, incl. CARA) i.e. the utility function is of one of these types: • Extended power: u(z) =

1 (A (C+1)B

+ Bz)C+1

• Logarithmic: u(z) = ln(A + Bz) A • Negative exponential: u(z) = − B exp{Bz}

where A, B and C are chosen to guarantee u0 > 0, u00 < 0. i.e. marginal utility satisfies u0 (z) = (A + Bz)C

or u0 (z) = A exp{Bz}

(6.3.23)

where A, B and C are again chosen to guarantee u0 > 0, u00 < 0. Proof The proof that these conditions are necessary for two fund separation is difficult and tedious. The interested reader is referred to ?. We will show that u0 (z) = (A + Bz)C is sufficient for two-fund separation. The optimal dollar investments wj are the unique solution to the first order conditions: ˜ ˜ ) δW ] 0 = E[u0 (W δwi ˜ )C (˜ = E[(A + B W ri − rf )] X = E[(A + BW0 rf + Bwj (˜ rj − rf ))C (˜ ri − rf )],

(6.3.24) (6.3.25) (6.3.26)

j

or equivalently to the system of equations E[(1 +

X j

Bwj (˜ rj − rf ))C (˜ ri − rf )] = 0 A + BW0 rf

(6.3.27)

Revised: December 2, 1998

116

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

or E[(1 +

X

xj (˜ rj − rf ))C (˜ ri − rf )] = 0

(6.3.28)

j

where

Bwj . A + BW0 rf The unique solutions for xj are clearly independent of W0 which does not appear in (6.3.28). Since A and B do not appear either, the unique solutions for xj are also independent of those parameters. However, they do depend on C. But the risky portfolio weights are xj =

wi j wj

P

Bwi /(A + BW0 rf ) = P j Bwj /(A + BW0 rf ) xi = P j xj

(6.3.29) (6.3.30)

and so are also independent of initial wealth. Since the dollar investment in the jth risky asset satisfies: wj = xj (

A + W0 rf ) B

(6.3.31)

we also have in this case that the dollar investment in the common risky portfolio is a linear function of the initial wealth. The other sufficiency proofs are similar and are left as exercises. Q.E.D. Some humorous anecdotes about Cass may now follow.

6.4 Mathematics of the Portfolio Frontier 6.4.1 The portfolio frontier in 1 = W0

(6.4.2)

w> e ≥ W1 = µW0 .

(6.4.3)

and

The first constraint is just the budget constraint, while the second constraint states that the expected rate of return on the portfolio is at least the desired mean return µ. The frontier in this case is the set of solutions for all values of W0 and W1 (or µ) to this variance minimisation problem, or to the equivalent maximisation problem: max −w> Vw w

(6.4.4)

subject to the same linear constraints (6.4.2) and (6.4.3). The properties of this two-moment frontier are well known, and can be found, for example, in ? or ?. The notation here follows ?. The derivation of the meanvariance frontier is generally presented in the literature in terms of portfolio weight vectors or, equivalently, assuming that initial wealth, W0 , equals 1. This assumption is not essential and will be avoided. Revised: December 2, 1998

118

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

The solution The inequality constrained maximisation problem (6.4.4) is just a special case of the canonical quadratic programming problem considered at the end of Section 3.5, except that it has explicitly one equality constraint and one inequality constraint. To avoid degeneracies, we require: 1. that not every portfolio has the same expected return, i.e. e 6= E[˜ r1 ]1,

(6.4.5)

and in particular that N > 1. 2. that the variance-covariance matrix, V, is (strictly) positive definite. We already know from (1.12.4) that V must be positive semi-definite, but we require this slightly stronger condition. To see why, suppose ∃w 6= 0N s.t. w> Vw = 0

(6.4.6)

Then ∃ a portfolio whose return w>˜r = r˜w has zero variance. This implies that r˜w = r0 (say) w.p.1 or, essentially, that this portfolio is riskless. Arbitrage will force the returns on all riskless assets to be equal in equilibrium, so this situation is equivalent economically to the introduction of a riskless asset later. In the portfolio problem, the place of the matrix A in the canonical quadratic programming problem is taken by the (symmetric) negative definite matrix, −V, which is just the negative of the variance-covariance matrix of asset returns; g1 = 1> and α1 = W0 ; and g2 = e> and α2 = W1 . (6.4.5) guarantees that the 2 × N matrix G is of full rank 2. The parallels are a little fuzzy in the case of the budget constraint since it is really an equality constraint. (3.5.39) says that the optimal w is a linear combination of the two columns of the N × 2 matrix −1 V−1 G> GV−1 G> , with columns weighted by initial wealth W0 and expected final wealth, W1 . We will call these columns g and h and write the solution as w = W0 g + W1 h = W0 (g + µh) . Revised: December 2, 1998

(6.4.7)

CHAPTER 6. PORTFOLIO THEORY

119

The components of g and h are functions of the means and variances of security returns. Thus the vector of optimal portfolio proportions, 1 w = g + µh, W0

(6.4.8)

is independent of the initial wealth W0 . It is easy to see the economic interpretation of g and h: • g is the frontier portfolio corresponding to W0 = 1 and W1 = 0. In other words, it is the normal portfolio which would be held by an investor whose objective was to (just) go bankrupt with minimum variance. • Similarly, h is the frontier portfolio corresponding to W0 = 0 and W1 = 1. In other words, it is the hedge portfolio which would be purchased by a variance-minimising investor in order to increase his expected final wealth by one unit. Alternatively, (3.5.35) says that the optimal w is a linear combination of the two columns of the N × 2 matrix 1 −1 > 1 −1 V G = 2V 1 2

1 −1 V e 2

,

with columns weighted by the Lagrange multipliers corresponding to the two constraints. We will call the Lagrange multipliers 2γ/C and 2λ/A respectively, where we define: A ≡ 1> V−1 e = e> V−1 1 B ≡ e> V−1 e > 0 C ≡ 1> V−1 1 > 0

(6.4.9) (6.4.10) (6.4.11)

D ≡ BC − A2

(6.4.12)

and and the inequalities follow from the fact that V−1 (like V) is positive definite. This allows the solution to be written as: w=

γ λ (V−1 1) + (V−1 e). C A

(6.4.13)

1 (V−1 1) C

and A1 (V−1 e) are both unit portfolios, so γ + λ = W0 . We know that for the portfolio which minimises variance for a given initial wealth, regardless of expected final wealth, the corresponding Lagrange multiplier, λ = 0. Thus γ (V−1 1) is the global minimum variance portfolio with cost W0 (which in fact C Revised: December 2, 1998

120

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

equals γ in this case) and C1 (V−1 1) is the global minimum variance unit cost portfolio, which we will denote wMVP . In fact, we can combine (6.4.7) and (6.4.13) and write the solution as: w = W0

A wMVP + µ − h . C

(6.4.14)

The details are left as an exercise.4 The set of solutions to this quadratic programming problem for all possible (W0 , W1 ) combinations (including negative W0 ) is the vector subspace of the portfolio space, which is generated either by the vectors g and h or by the vectors wMVP and h (or by any pair of linearly independent frontier portfolios). In Vg =

(6.4.17) (6.4.18)

from which it follows that D > 0. Orthogonal decomposition of portfolios At this stage, we must introduce a scalar product on the portfolio space, namely that based on the variance-covariance matrix V. Since V is a non-singular, positive definite matrix, it defines a well behaved scalar product and all the standard results on orthogonal projection (&c.) from linear algebra are valid. 4

At least for now.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

121

Two portfolios w1 and w2 are orthogonal with respect to this scalar product ⇐⇒ w1> Vw2 = 0 ⇐⇒h i Cov w1>˜r, w2>˜r = 0 ⇐⇒ the random variables representing the returns on the portfolios are uncorrelated. Thus, the terms ‘orthogonal’ and ‘uncorrelated’ may legitimately, and shall, be applied interchangeably to pairs of portfolios. Furthermore, the squared length of a weight vector corresponds to the variance of its returns. Note that wMVP and h are orthogonal vectors in this sense. In fact, we have the following theorem: Theorem 6.4.1 If w is a frontier portfolio and u is a zero mean hedge portfolio, then w and u are uncorrelated. Proof There is probably a full version of this proof lost somewhere but the following can be sorted out. Since wMVP is collinear with V−1 1, it is orthogonal to all portfolios w for which w> VV−1 1 = 0 or in other words to all portfolios for which w> 1 = 0. But these are precisely all hedge portfolios, including h. Similarly, any portfolio collinear with V−1 e is orthogonal to all portfolios with zero expected return, since w> VV−1 e = 0 or in other words w> e = 0. Q.E.D. Some pictures are in order at this stage. For N = 3, in the set of portfolios costing W0 (the W0 simplex), the iso-variance curves are concentric ellipses, the iso-mean curves are parallel lines, and the solutions for different µs (or W1 s) are the tangency points between these ellipses and lines, which themselves lie on a line orthogonal (in the sense defined above) to the iso-mean lines. The centre of the concentric ellipses is at the global minimum variance portfolio corresponding to W0 , W0 wMVP . A similar geometric interpretation can be applied in higher dimensions. ? has some nice pictures of the frontier in portfolio space, as opposed to meanvariance space. At this stage, recall the definition of β in (5.2.2). We will now derive an orthogonal decomposition of a portfolio q into two frontier portfolios and a zero-mean zero-cost portfolio and prove that the coefficients on the two frontier portfolios are the βs of q with respect to those portfolios and sum to unity. We can always choose an orthogonal basis for the portfolio frontier. Revised: December 2, 1998

122

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

For any frontier portfolio p 6= wMVP , there is a unique unit cost frontier portfolio zp which is orthogonal to p. Another rp ] and h i important exercise is to figure out the relationship between E [˜ E r˜zp . Any two frontier portfolios span the frontier, in particular any unit cost p 6= wMVP and zp (or the original basis, wMVP and h). Any (frontier or non-frontier) portfolio q with non-zero cost W0 can be written in the form fq + uq where fq ≡ W0 (g + E[˜ rq ]h) = W0 (βqp p + (1 − βqp )zp ) (say)

(6.4.19) (6.4.20)

is the frontier portfolio with expected return E[˜ rq ] and cost W0 and uq is a hedge portfolio with zero expected return. Geometrically, this decomposition is equivalent to the orthogonal projection of q onto the frontier. Theorem 6.4.1 shown that any portfolio sharing these properties of uq is uncorrelated with all frontier portfolios.5 If p is a unit cost frontier portfolio (i.e. the vector of portfolio proportions) and q is an arbitrary unit cost portfolio, then the following decomposition therefore holds: q = fq + uq = βqp p + (1 − βqp ) zp + uq (6.4.21) where the three components (i.e. the vectors p, zp and uq ) are mutually orthogonal. We can extend this decomposition to cover 1. portfolio proportions (orthogonal vectors) 2. portfolio proportions (scalars/components) 3. returns (uncorrelated random variables) 4. expected returns (numbers) Note again the parallel between orthogonal portfolio vectors and uncorrelated portfolio returns/payoffs. We will now derive the relation: E[˜ rq ] − E[˜ rzp ] = βqp (E[˜ rp ] − E[˜ rzp ]) 5

(6.4.22)

Aside: For the frontier portfolio fq to second degree stochastically dominate the arbitrary portfolio q, we will need zero conditional expected return on uq , and will have to show that Cov r˜uq , r˜fq = 0 =⇒ E[˜ ruq |˜ rfq ] = 0 The normal distribution is the only case where this is true.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

123

which may be familiar from earlier courses in financial economics and which is quite general and neither requires asset returns to be normally distributed nor any assumptions h about ipreferences. h i Since Cov r˜uq , r˜p = Cov r˜zp , r˜p = 0, taking covariances with r˜p in (6.4.21) gives: h i Cov [˜ rq , r˜p ] = Cov r˜fq , r˜p = βqp Var[˜ rp ] (6.4.23) or βqp =

Cov [˜ rq , r˜p ] Var[˜ rp ]

(6.4.24)

Thus β in (6.4.21) has its usual definition from probability theory, given by (5.2.2).6 Reversing the roles of p and zp , it can be seen that βqzp = 1 − βqp

(6.4.25)

Taking expected returns in (6.4.21) yields again: E[˜ rq ] = βqp E[˜ rp ] + (1 − βqp )E[˜ rzp ],

(6.4.26)

which can be rearranged to obtain (6.4.22). The Global Minimum Variance Portfolio Var[˜ rg+µh ] = g> Vg + 2µ(g> Vh) + µ2 (h> Vh)

(6.4.27)

which has its minimum at

g> Vh (6.4.28) h> Vh The latter expression reduces to A/C and the minimum value of the variance is 1/C. The global minimum variance portfolio is denoted MVP. µ=−

g> Vh Cov [˜ rh , r˜MVP ] = h V g − > h h Vh g> Vh.h> Vh =0 = h> Vg − h> Vh >

!

(6.4.29) (6.4.30)

i.e. the returns on the portfolio with weights h and the minimum variance portfolio are uncorrelated. 6

Assign some problems involving the construction of portfolio proportions for various desired βs. Also problems working from prices for state contingent claims to returns on assets and portfolios in both single period and multi-period worlds.

Revised: December 2, 1998

124

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

Further, if p is any portfolio, the itself and p, i.e. a = 0 solves:

MVP

is the minimum variance combination of

1 min Var[˜ rap+(1−a)MVP ] a 2

(6.4.31)

which has necessary and sufficient first order condition: aVar[˜ rp ] + (1 − 2a)Cov [˜ rp , r˜MVP ] − (1 − a)Var[˜ rMVP ] = 0

(6.4.32)

Hence, setting a = 0: Cov [˜ rp , r˜MVP ] − Var[˜ rMVP ] = 0

(6.4.33)

and the covariance of any portfolio with MVP is 1/C.

6.4.2 The portfolio frontier in mean-variance space: risky assets only The portfolio frontier in mean-variance and in mean-standard deviation space We now move on to consider the mean-variance relationship along the portfolio frontier. The mean, µ, and variance, σ 2 , of the rate of return associated with each point on the frontier are related by the quadratic equation: (σ 2 − Var[w> r]) = φ(µ − E[w> r])2 , MVP ˜ MVP ˜

(6.4.34)

where the shape parameter φ = C/D represents the variance of the (gross) return on the hedge portfolio, h. The two-moment frontier is generally presented as the graph in mean-variance space of this parabola, showing the most desirable distributions attainable, but the frontier can also be thought of as a plane in portfolio space or as a line in portfolio weight space. The latter interpretations are far more useful when it comes to extending the analysis to higher moments. The equations of the frontier in mean-variance and mean-standard deviation space can be derived heuristically using the following stylized diagram illustrating the portfolio decomposition. Figure 3A goes here.

Applying Pythagoras’ theorem to the triangle with vertices at 0, p and MVP yields: A σ = Var[˜ rp ] = Var[˜ rMVP ] + µ − C 2

Revised: December 2, 1998

2

Var[˜ rh ]

(6.4.35)

CHAPTER 6. PORTFOLIO THEORY

125

Recall from the coordinate geometry of conic sections that Var[˜ rp ] = Var[˜ rMVP ] + (µ − E[˜ rMVP ])2 Var[˜ rh ] or V (µ) =

1 C A + µ− C D C

2

(6.4.36)

(6.4.37)

is a quadratic equation in µ. i.e. the equation of the parabola with vertex at 1 C A µ = E[˜ rMVP ] = C

Var[˜ rp ] = Var[˜ rMVP ] =

(6.4.38) (6.4.39)

Thus in mean-variance space, the frontier is a parabola. Figure 3.11.2 goes here: indicate position of g on figure.

Similarly, in mean-standard deviation space, the frontier is a hyperbola. To see this, recall that: A 2 2 σ = Var[˜ rMVP ] + µ − Var[˜ rh ] (6.4.40) C is the equation of the hyperbola with vertex at σ =

q

µ =

A C

Var[˜ rMVP ] =

s

1 C

(6.4.41) (6.4.42)

centre at σ = 0, µ = A/C and asymptotes as indicated. Figure 3.11.1 goes here: indicate position of g on figure.

The other half of the hyperbola (σ < 0) has no economic meaning. Recall two other types of conic sections: Var[˜ rh ] < 0 (impossible) gives a circle with center (1/C, A/C). Var[˜ rMVP ] = 0 (the presence of a riskless asset) allows the square root to be taken on both sides: A q σ =± µ− Var[˜ rh ] (6.4.43) C i.e. the conic section becomes the pair of lines which are its asymptotes otherwise. Revised: December 2, 1998

126

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

Portfolios on which the expected return, µ, exceeds w> MVP e are termed efficient, since they maximise expected return given variance; other frontier portfolios minimise expected return given variance and are inefficient. A frontier portfolio is said to be an efficient portfolio iff its expected return exceeds the minimum variance expected return A/C = E[˜ rMVP ]. N The set of efficient portfolios in < (or efficient frontier) is the half-line emanating from MVP in the direction of h, and hence is also a convex set. Convex combinations (but not all linear combinations with weights summing to 1) of efficient portfolios are efficient. We now consider zero-covariance (zero-beta) portfolios. In portfolio weight space, can easily construct a frontier portfolio having zero covariance with any given frontier portfolio: Figure 3B goes here.

Algebraically, the expected return µ0 on the zero-covariance frontier portfolio of a frontier portfolio with expected return µ solves: Cov [˜ rMVP + (µ − E[˜ rMVP ])˜ rh , r˜MVP + (µ0 − E[˜ rMVP ])˜ rh ] = 0

(6.4.44)

or, since r˜h and r˜MVP are uncorrelated: Var[˜ rMVP ] + (µ − E[˜ rMVP ])(µ0 − E[˜ rMVP ])Var[˜ rh ] = 0

(6.4.45)

To make this true, we must have (µ − E[˜ rMVP ])(µ0 − E[˜ rMVP ]) < 0

(6.4.46)

or µ and µ0 on opposite sides of E[˜ rMVP ] as shown. There is a neat trick which allows zero-covariance portfolios to be plotted in meanstandard deviation space. Implicit differentiation of the µ − σ relationship (6.4.35) along the frontier yields: dµ σ = dσ (µ − E[˜ rMVP ])Var[˜ rh ]

(6.4.47)

so the tangent at (σ, µ) intercepts the µ axis at µ−σ

dµ σ2 = µ− (6.4.48) dσ (µ − E[˜ rMVP ])Var[˜ rh ] Var[˜ rMVP ] = µ− − (µ − E[˜ rMVP ]) (6.4.49) (µ − E[˜ rMVP ])Var[˜ rh ] Var[˜ rMVP ] = E[˜ rMVP ] − (6.4.50) (µ − E[˜ rMVP ])Var[˜ rh ]

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

127

where we substituted for σ 2 from the definition of the frontier. A little rearrangement shows that expression (6.4.50) satisfies the equation (6.4.45) defining the return on the zero-covariance portfolio. In mean-standard deviation space the picture is like this: Figure 3.15.1 goes here.

To find zp in mean-variance space, note that the line joining (σ 2 , µ) to the intercepts the µ axis at: µ − σ2

µ − E[˜ rMVP ] µ − E[˜ rMVP ] = µ − σ2 2 σ − Var[˜ rMVP ] (µ − E[˜ rMVP )2 Var[˜ rh ]

MVP

(6.4.51)

After cancellation, this is exactly the first expression (6.4.48) for the zero-covariance return we had on the previous page. Figure 3.15.2 goes here.

Alternative derivations The treatment of the portfolio frontier with risky assets only concludes with some alternative derivations following closely ?. They should probably be omitted altogether at this stage. 1. The variance minimisation solution from first principles. It can be seen that w is the solution to: 1 min L = w> Vw + λ(µ − w> e) + γ(W0 − w> 1) {w, , } 2

(6.4.52)

which has necessary and sufficient first order conditions: ∂L = Vw − λe − γ1 = 0 ∂w ∂L = µ − w> e = 0 ∂λ ∂L = W0 − w> 1 = 0 ∂γ

(6.4.53) (6.4.54) (6.4.55)

The solution can be found by premultiplying the FOC (6.4.13) in turn by e> and 1> and using the constraints yields: µ = λ(e> V−1 e) + γ(e> V−1 1) 1 = λ(1> V−1 e) + γ(1> V−1 1)

(6.4.56) (6.4.57)

Revised: December 2, 1998

128

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER The solutions for λ and γ are: Cµ − A D B − Aµ γ = D

λ =

(6.4.58) (6.4.59)

2. Derivation of (6.4.22). If we only have frontier portfolio p and interior portfolio q, we get a frontier (in µ-σ space) entirely within the previous frontier and tangent to it at p. The frontiers must have the same slope at p: Figure 3C goes here. E[˜ r −˜ r ] We already saw that the outer frontier has slope √ p zp . Var[˜ rp ]

At the point on the inner frontier with wq invested in q and (1 − wq ) in p, µ = E[˜ rp ] + wq (E[˜ rq − r˜p ]) (6.4.60) 2 2 σ = wq Var[˜ rq ] +2wq (1 − wq )Cov [˜ rp , r˜q ] + (1 − wq )2 Var[˜ rp ] (6.4.61) Differentiating these w.r.t. wq : dµ = E[˜ rq − r˜p ] (6.4.62) dwq dσ = 2wq Var[˜ rq ] 2σ dwq +2(1 − 2wq )Cov [˜ rp , r˜q ] − 2(1 − wq )Var[˜ rp(]6.4.63) Taking the ratio and setting wq = 0 gives the slope of the inner frontier at p: dµ E[˜ rq − r˜p ] = 2Cov[˜rp ,˜rq ]−2Var[˜rp ] (6.4.64) dσ √ 2

Var[˜ rp ]

Equating this to the slope of the outer frontier, setting βqp =

Cov [˜ rp , r˜q ] Var[˜ rp ]

(6.4.65)

and rearranging yields: E[˜ rq ] − E[˜ rzp ] = βqp (E[˜ rp ] − E[˜ rzp ]) Revised: December 2, 1998

(6.4.66)

CHAPTER 6. PORTFOLIO THEORY

129

6.4.3 The portfolio frontier in w Vw 2

(6.4.67)

w> e + (1 − w> 1)rf = µ

s.t.

(6.4.68)

There is no longer a restriction on portfolio weights, and whatever is not invested in the N risky assets is assumed to be invested in the riskless asset. The solution (which can be left as an exercise) is by a similar method to the case where all assets were risky: wp = V−1 (e − rf 1)

µ − rf H

(6.4.69)

where H = (e − 1rf )> V−1 (e − 1rf ) = B − 2Arf + Crf 2 > 0 ∀rf

(6.4.70)

Along the frontier, we have: µ−r √ f

H σ = µ−r −√ f H

if µ ≥ rf , if µ < rf ,

(6.4.71)

6.4.4 The portfolio frontier in mean-variance space: riskfree and risky assets We can now establish the shape of the mean-standard deviation frontier with a riskless asset. Graphically, in mean-standard deviation space, combining any portfolio p with the riskless asset in proportions a and (1 − a) gives a portfolio with expected return aE[˜ rp ] + (1 − a)rf = rf + a(E[˜ rp ] − rf ) q

and standard deviation of returns a Var[˜ rp ]. i.e. these portfolios trace out the ray in σ-µ space emanating from (0, rf ) and passing through p. For each σ the highest return attainable is along the ray from rf which is tangent to the frontier generated by the risky assets. Revised: December 2, 1998

130

6.5. MARKET EQUILIBRIUM AND THE CAPM

On this ray, the riskless asset is held in combination with the tangency portfolio t. This only makes sense for rf < A/C = E[˜ rmvp ]. Above t, there is a negative weight on the riskless asset — i.e. borrowing. Figure 3D goes here.

Limited borrowing Unlimited borrowing as allowed in the preceding analysis is unrealistic. Consider what happens 1. with margin constraints on borrowing: Figure 3E goes here.

The frontier is the envelope of all the finite rays through risky portfolios, extending as far as the borrowing constraint allows.

2. with differential borrowing and lending rates: Figure 3F goes here.

There is a range of expected returns over which a pure risky strategy provides minimum variance; lower expected returns are achieved by riskless lending; and higher expected returns are achieved by riskless borrowing.

6.5 Market Equilibrium and the Capital Asset Pricing Model 6.5.1 Pricing assets and predicting security returns Need more waffle here about prediction and the difficulties thereof and the properties of equilibrium prices and returns. We are looking for assumptions concerning probability distributions that lead to useful and parsimonious asset pricing models. The CAPM restrictions are the best known. At a very basic level, they can be expressed by saying that every Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

131

investor has mean-variance preferences. This can be achieved either by restricting preferences to be quadratic or the probability distribution of asset returns to be normal. CAPM is basically a single-period model, but can be extended by assuming that return distributions are stable over time. ? and ? have generalised the distributional conditions. Recall also the limiting behaviour of the variance of the return on an equally weighted portfolio as the number of securities included goes to infinity. If securities are added in such a way that the average of the variance terms and the average of the covariance terms are stable, then the portfolio variance approaches the average covariance as a lower bound.

6.5.2 Properties of the market portfolio Let mj = weight of security j in the market portfolio m 0) = individual i’s initial wealth wij = proportion of individual i’s initial wealth invested in j-th security Then total wealth is defined by W0i (>

Wm0 ≡

I X

W0i

(6.5.1)

i=1

and in equilibrium the relation I X

wij W0i = mj Wm0

∀j

(6.5.2)

i=1

must hold. Dividing by Wm0 yields: I X

W0i wij = mj Wm0 i=1

∀j

(6.5.3)

and thus in equilibrium the market portfolio is a convex combination of individual portfolios.

6.5.3 The zero-beta CAPM Theorem 6.5.1 (Zero-beta CAPM theorem) If every investor holds a mean-variance frontier portfolio, then the market portfolio, m, is a mean-variance frontier portfolio, and hence, ∀q, the CAPM equation E [˜ rq ] = (1 − βqm ) E [˜ rzm ] + βqm E [˜ rm ]

(6.5.4)

holds. Revised: December 2, 1998

132

6.5. MARKET EQUILIBRIUM AND THE CAPM

Theorem 6.5.2 All strictly risk-averse investors hold frontier portfolios if and only if i h (6.5.5) rfq = 0 ∀q E r˜uq |˜ Note the subtle distinction between uncorrelated returns (in the definition of the decomposition) and independent returns (in this theorem). They are the same only for the normal distribution and related distributions. We can view the market portfolio as a frontier portfolio under two fund separation. If p is a frontier portfolio, then we showed earlier that for purely mathematical reasons in the definition of a frontier portfolio: E[˜ rq ] = (1 − βqp )E[˜ rzp ] + βqp E[˜ rp ]

(6.5.6)

If two fund separation holds, then individuals hold frontier portfolios. Since the market portfolio is then on the frontier, it follows that: E[˜ rq ] = (1 − βqm )E[˜ rzm ] + βqm E[˜ rm ]

(6.5.7)

N X

mj r˜j

(6.5.8)

Cov [˜ rq , r˜m ] Var[˜ rm ]

(6.5.9)

where r˜m =

j=1

βqm =

This implies for any particular security, from the economic assumptions of equilibrium and two fund separation: E[˜ rj ] = (1 − βjm )E[˜ rzm ] + βjm E[˜ rm ]

(6.5.10)

This relation is the ? Zero-Beta version of the Capital Asset Pricing Model (CAPM).

6.5.4 The traditional CAPM Now we add the risk free asset, which will allow us to determine the tangency portfolio, t, and to talk about Capital Market Line (return v. standard deviation) and the Security Market Line (return v. β). Normally in equilibrium there is zero aggregate supply of the riskfree asset. Recommended reading for this part of the course is ?, ?, ? and ?. Now we can derive the traditional CAPM. Note that by construction rf = E [˜ rzt ] . Revised: December 2, 1998

(6.5.11)

CHAPTER 6. PORTFOLIO THEORY

133

Theorem 6.5.3 (Separation Theorem) The risky asset holdings of all investors who hold mean-variance frontier portfolios are in the proportions given by the tangency portfolio, t. Theorem 6.5.4 (Traditional CAPM Theorem) If every investor holds a meanvariance frontier portfolio, then the market portfolio of risky assets, m, is the tangency portfolio, t, and hence, ∀q, the traditional CAPM equation E [˜ rq ] = (1 − βqm ) rf + βqm E [˜ rm ]

(6.5.12)

holds. Theorem 6.5.4 is sometimes known as the Sharpe-Lintner Theorem. The riskless rate is unique by the No Arbitrage Principle, since otherwise a greedy investor would borrow an infinite amount at the lower rate and invest it at the higher rate, which is impossible in equilibrium. We can also think about what happens the CAPM if there are different riskless borrowing and lending rates (see ?). If all individuals face this situation in equilibrium, realism demands that both riskless assets are in zero aggregate supply and hence that all investors hold risky assets only. Note that the No Arbitrage Principle also allows us to rule out correlation matrices for risky assets which permit the construction of portfolios with zero return variance, i.e. synthetic riskless assets. Assume that a riskless asset exists, with return rf < E[˜ rmvp ]. If the distributional conditions for two fund separation are satisfied, then the tangency portfolio, t, must be the market portfolio of risky assets in equilibrium. We know then that for any portfolio q (with or without a riskless component): E[˜ rq ] − rf = βqm (E[˜ rm ] − rf )

(6.5.13)

This is the traditional Sharpe-Lintner version of the CAPM. Figure 4A goes here.

The next theorem relates to the mean-variance efficiency of the market portfolio. Theorem 6.5.5 If 1. the distributional conditions for two fund separation are satisfied; 2. risky assets are in strictly positive supply; and 3. investors have strictly increasing (concave) utility functions then the market/tangency portfolio is efficient. Revised: December 2, 1998

134

6.5. MARKET EQUILIBRIUM AND THE CAPM

Proof By Jensen’s inequality and monotonicity, the riskless asset dominates any portfolio with E[˜ r] < rf (6.5.14) for E[u(W0 (1 + r˜))] ≤ u(E[W0 (1 + r˜)]) < u(W0 (1 + rf ))

(6.5.15) (6.5.16)

Hence the expected returns on all individuals’ portfolios exceed rf . It follows that the expected return on the market portfolio must exceed rf .

Q.E.D.

Now we can calculate the risk premium of the market portfolio. CAPM gives a relation between the risk premia on individual assets and the risk premium on the market portfolio. The risk premium on the market portfolio must adjust in equilibrium to give market-clearing. In some situations, the risk premium on the market portfolio can be written in terms of investors’ utility functions. Assume there is a riskless asset and returns are multivariate normal (MVN). Recall the first order conditions for the canonical portfolio choice problem: ˜ i )(˜ 0 = E[u0i (W rj − rf )] ∀ i, j h i ˜ i )]E[˜ ˜ i ), r˜j = E[u0i (W rj − rf ] + Cov u0i (W h

˜ i , r˜j ˜ i )]E[˜ ˜ i )]Cov W = E[u0i (W rj − rf ] + E[u00i (W

(6.5.17) (6.5.18) i

(6.5.19)

using the definition of covariance and Stein’s lemma for MVN distributions. Rearranging: h i E[˜ rj − rf ] ˜ i , r˜j = Cov W (6.5.20) θi where ˜ i )] −E[u00i (W θi ≡ (6.5.21) ˜ i )] E[u0i (W is the i-th investor’s global absolute risk aversion. Since ˜ i = W0i (1 + rf + W

N X

wik (˜ rk − rf ))

(6.5.22)

k=1

we have, dropping non-stochastic terms, h

i

"

˜ i , r˜j = Cov W0i Cov W

N X

k=1

Revised: December 2, 1998

wik r˜k , r˜j

#

(6.5.23)

CHAPTER 6. PORTFOLIO THEORY

135

Hence, N X E[˜ rj − rf ] = Cov W0i wik r˜k , r˜j θi k=1

"

Summing over i, this gives (since we have P clearing and k wmk r˜k = r˜m by definition): I X

E[˜ rj − rf ](

P

i

#

(6.5.24)

W0i wik = W0m wmk by market-

θi−1 ) = Wm0 Cov [˜ rm , r˜j ]

(6.5.25)

i=1

E[˜ rj − rf ] = (

or

I X

θi−1 )−1 Wm0 Cov [˜ rm , r˜j ]

(6.5.26)

i=1

i.e., in equilibrium, the risk premium on the j-th asset is the product of the aggregate relative risk aversion of the economy and the covariance between the return on the j-th asset and the return on the market. Now take the average over j weighted by market portfolio weights: I X

E[˜ rm − rf ] = (

rm ) θi−1 )−1 Wm0 Var[˜(]˜

(6.5.27)

i=1

i.e., in equilibrium, the risk premium on the market is the product of the aggregate relative risk aversion of the economy and the variance of the return on the market. Equivalently, the return to variability of the market equals the aggregate relative risk aversion. We conclude with some examples. 1. Negative exponential utility: ui (z) = −

1 exp{−ai z} ai

ai > 0

(6.5.28)

implies: I X

(

I X

θi−1 )−1 = (

i=1

−1 a−1 >0 i )

(6.5.29)

i=1

and hence the market portfolio is efficient. 2. Quadratic utility: ui (z) = ai z −

bi 2 z 2

ai , b i > 0

(6.5.30)

implies: I X

(

i=1

θi−1 )−1

=

I X ai i=1

bi

!−1

˜ i] − E[W

(6.5.31)

This result can also be derived without assuming MVN and using Stein’s lemma. Revised: December 2, 1998

136

Revised: December 2, 1998

6.5. MARKET EQUILIBRIUM AND THE CAPM

CHAPTER 7. INVESTMENT ANALYSIS

137

Chapter 7 INVESTMENT ANALYSIS 7.1 Introduction [To be written.]

7.2 Arbitrage and the Pricing of Derivative Securities 7.2.1 The binomial option pricing model This still has to be typed up. It follows very naturally from the stuff in Section 5.4.

7.2.2 The Black-Scholes option pricing model Fischer Black died in 1995. In 1997, Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics ‘for a new method to determine the value of derivatives.’ See http://www.nobel.se/announcement-97/economy97.html Black and Scholes considered a world in which there are three assets: a stock, whose price, S˜t , follows the stochastic differential equation: dS˜t = µS˜t dt + σ S˜t d˜ zt , where {˜ zt }Tt=0 is a Brownian motion process; a bond, whose price, Bt , follows the differential equation: dBt = rBt dt; and a call option on the stock with strike price X and maturity date T . Revised: December 2, 1998

138

7.2. ARBITRAGE AND PRICING DERIVATIVE SECURITIES

They showed how to construct an instantaneously riskless portfolio of stocks and options, and hence, assuming that the principle of no arbitrage holds, derived the Black-Scholes partial differential equation which must be satisfied by the option price. The option pays (ST − X)+ ≡ max {ST − X, 0} at maturity. Let the price of the call at time t be C˜t . Guess that C˜t = C(S˜t , t). By Ito’s lemma: dC˜t =

∂C ∂C ˜ 1 ∂ 2 C 2 ˜2 ∂C ˜ + µSt + σ St dt + σ St d˜ zt 2 ∂t ∂S 2 ∂S ∂S !

The no arbitrage principle yields the partial differential equation: ∂C 1 ∂ 2 C 2 2 ∂C + σ S + rS − rC = 0 ∂t 2 ∂S 2 ∂S subject to the boundary condition C(S, T ) = (S − X)+ . Let τ = T − t be the time to maturity. Then we claim that the solution to the Black-Scholes equation is: √ C(S, t) = SN (d (S, τ )) − Xe−rτ N d (S, τ ) − σ τ ,

where N (·) is the cumulative distribution function of the standard normal distribution and

ln XS + r − 12 σ 2 τ √ √ d (S, τ ) = +σ τ σ τ ln XS + r + 12 σ 2 τ √ = . σ τ

(7.2.1) (7.2.2)

We can check that this is indeed the solution by calculating the various partial derivatives and substituting them in the original equation. Note first that Z z 1 2 1 √ e− 2 t dt N (z) ≡ −∞ 2π and hence by the fundamental theorem of calculus 1 2 1 N 0 (z) ≡ √ e− 2 z , 2π

Revised: December 2, 1998

CHAPTER 7. INVESTMENT ANALYSIS

139

which of course is the corresponding probability density function. For the last step in this proof, we will need the partials of d (S, τ ) with respect to S and t, which are:

r + 12 σ 2 ∂d (S, τ ) ∂d (S, τ ) √ =− = − ∂t ∂τ 2σ τ and

∂d (S, τ ) 1 √ . = ∂S Sσ τ

(7.2.3)

(7.2.4)

Note also that √ √ 1 2 N 0 d (S, τ ) − σ τ = e− 2 σ τ ed(S,τ )σ τ N 0 (d (S, τ )) 1

= e− 2 σ =

2τ

S (r+ 12 σ2 )τ e N 0 (d (S, τ )) X

S rτ 0 e N (d (S, τ )) . X

(7.2.5) (7.2.6) (7.2.7)

Using these facts and the chain rule, we have: ∂C ∂t

∂d (S, τ ) − Xe−rτ × ∂t ! ! ∂d (S, τ ) √ √ σ N 0 d (S, τ ) − σ τ − √ + rN d (S, τ ) − σ τ ∂t 2 τ (7.2.8) √ σ = −SN 0 (d (S, τ )) √ − Xe−rτ rN d (S, τ ) − σ τ (7.2.9) 2 τ ∂C ∂S ∂d (S, τ ) = SN 0 (d (S, τ )) + N (d (S, τ )) ∂S √ ∂d (S, τ ) −Xe−rτ N 0 d (S, τ ) − σ τ (7.2.10) ∂S = N (d (S, τ )) (7.2.11) 2 ∂ C ∂S 2 ∂d (S, τ ) = N 0 (d (S, τ )) . (7.2.12) ∂S = SN 0 (d (S, τ ))

Substituting these expressions in the original partial differential equation yields: ∂C 1 ∂ 2 C 2 2 ∂C + σ S + rS − rC ∂t 2 ∂S 2 ∂S Revised: December 2, 1998

140

7.3. MULTI-PERIOD INVESTMENT PROBLEMS !

−Sσ 1 ∂d (S, τ ) = N (d (S, τ )) (rS − rS) + N (d (S, τ )) √ + σ 2 S 2 2 τ 2 ∂S √ −rτ −rτ +N d (S, τ ) − σ τ −Xe r + rXe (7.2.13) = 0. (7.2.14) 0

The boundary condition should also be checked. As τ → 0, d (S, τ ) → ±∞ according as S > X or S < X. In the former case, C (S, T ) = S − X; and in the latter case, C (S, T ) = 0, so the boundary condition is indeed satisfied.

7.3 Multi-period Investment Problems In Section 4.2, it was pointed out that the objects of choice can be differentiated not only by their physical characteristics, but also both by the time at which they are consumed and by the state of nature in which they are consumed. These distinctions were suppressed in the intervening sections but are considered again in this section and in Section 5.4 respectively. The multi-period model should probably be introduced at the end of Chapter 4 but could also be left until Chapter 7. For the moment this brief introduction is duplicated in both chapters. Discrete time multi-period investment problems serve as a stepping stone from the single period case to the continuous time case. The main point to be gotten across is the derivation of interest rates from equilibrium prices: spot rates, forward rates, term structure, etc. This is covered in one of the problems, which illustrates the link between prices and interest rates in a multiperiod model.

7.4 Continuous Time Investment Problems ?

Revised: December 2, 1998

Patrick Waldron

December 2, 1998

CONTENTS

i

Contents List of Tables

iii

List of Figures

v

PREFACE vii What Is Economics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii What Is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viii NOTATION

ix

I

1

MATHEMATICS

1 LINEAR ALGEBRA 1.1 Introduction . . . . . . . . . . . . . . . . 1.2 Systems of Linear Equations and Matrices 1.3 Matrix Operations . . . . . . . . . . . . . 1.4 Matrix Arithmetic . . . . . . . . . . . . . 1.5 Vectors and Vector Spaces . . . . . . . . 1.6 Linear Independence . . . . . . . . . . . 1.7 Bases and Dimension . . . . . . . . . . . 1.8 Rank . . . . . . . . . . . . . . . . . . . . 1.9 Eigenvalues and Eigenvectors . . . . . . . 1.10 Quadratic Forms . . . . . . . . . . . . . 1.11 Symmetric Matrices . . . . . . . . . . . . 1.12 Definite Matrices . . . . . . . . . . . . .

. . . . . . . . . . . .

3 3 3 7 7 11 12 12 13 14 15 15 15

2 VECTOR CALCULUS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vector-valued Functions and Functions of Several Variables . . .

17 17 17 18

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

Revised: December 2, 1998

ii

CONTENTS 2.4 2.5 2.6 2.7 2.8 2.9

Partial and Total Derivatives . . . . . . . The Chain Rule and Product Rule . . . . The Implicit Function Theorem . . . . . . Directional Derivatives . . . . . . . . . . Taylor’s Theorem: Deterministic Version The Fundamental Theorem of Calculus .

3 CONVEXITY AND OPTIMISATION 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Convexity and Concavity . . . . . . . . . 3.2.1 Definitions . . . . . . . . . . . . 3.2.2 Properties of concave functions . 3.2.3 Convexity and differentiability . . 3.2.4 Variations on the convexity theme 3.3 Unconstrained Optimisation . . . . . . . 3.4 Equality Constrained Optimisation: The Lagrange Multiplier Theorems . . . . 3.5 Inequality Constrained Optimisation: The Kuhn-Tucker Theorems . . . . . . . 3.6 Duality . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

20 21 23 24 25 26

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

27 27 27 27 29 30 34 39

. . . . . . . . . . . . .

43

. . . . . . . . . . . . . 50 . . . . . . . . . . . . . 58

II APPLICATIONS 4 CHOICE UNDER CERTAINTY 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Definitions . . . . . . . . . . . . . . . . . . . . 4.3 Axioms . . . . . . . . . . . . . . . . . . . . . 4.4 Optimal Response Functions: Marshallian and Hicksian Demand . . . . . . . 4.4.1 The consumer’s problem . . . . . . . . 4.4.2 The No Arbitrage Principle . . . . . . . 4.4.3 Other Properties of Marshallian demand 4.4.4 The dual problem . . . . . . . . . . . . 4.4.5 Properties of Hicksian demands . . . . 4.5 Envelope Functions: Indirect Utility and Expenditure . . . . . . . . 4.6 Further Results in Demand Theory . . . . . . . 4.7 General Equilibrium Theory . . . . . . . . . . 4.7.1 Walras’ law . . . . . . . . . . . . . . . 4.7.2 Brouwer’s fixed point theorem . . . . . Revised: December 2, 1998

61 63 . . . . . . . . . . 63 . . . . . . . . . . 63 . . . . . . . . . . 66 . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. 69 . 69 . 70 . 71 . 72 . 73

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

73 75 78 78 78

CONTENTS

4.8

4.9

iii

4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . The Welfare Theorems . . . . . . . . . . . . . . . . . . . . 4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . 4.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . 4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . 4.8.4 The Separating Hyperplane Theorem . . . . . . . . 4.8.5 The Second Welfare Theorem . . . . . . . . . . . . 4.8.6 Complete markets . . . . . . . . . . . . . . . . . . 4.8.7 Other characterizations of Pareto efficient allocations Multi-period General Equilibrium . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

78 78 78 78 79 80 80 82 82 84

5

CHOICE UNDER UNCERTAINTY 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85 5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88 5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88 5.4.1 Completion of markets using options . . . . . . . . . . . 90 5.4.2 Restrictions on security values implied by allocational efficiency and covariance with aggregate consumption . . . 91 5.4.3 Completing markets with options on aggregate consumption 92 5.4.4 Replicating elementary claims with a butterfly spread . . . 93 5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93 5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2 Existence of expected utility functions . . . . . . . . . . . 95 5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97 5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102 5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104

6

PORTFOLIO THEORY 6.1 Introduction . . . . . . . . . . . . . . . . . . . 6.2 Notation and preliminaries . . . . . . . . . . . 6.2.1 Measuring rates of return . . . . . . . . 6.2.2 Notation . . . . . . . . . . . . . . . . 6.3 The Single-period Portfolio Choice Problem . . 6.3.1 The canonical portfolio problem . . . . 6.3.2 Risk aversion and portfolio composition 6.3.3 Mutual fund separation . . . . . . . . . 6.4 Mathematics of the Portfolio Frontier . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

105 105 105 105 108 110 110 112 114 116

Revised: December 2, 1998

iv

CONTENTS The portfolio frontier in 0.

But the first order condition guarantees that the LHS of this inequality is zero (not positive), which is the required contradiction. Q.E.D. Theorem 3.4.3 Uniqueness condition for equality constrained maximisation. If 1. x∗ is a solution, 2. f is strictly quasiconcave, and 3. g j is an affine function (i.e. both convex and concave) for j = 1, . . . , m, then x∗ is the unique (global) maximum. Revised: December 2, 1998

3.4. EQUALITY CONSTRAINED OPTIMISATION: THE LAGRANGE MULTIPLIER THEOREMS

48

Proof The uniqueness result is also proved by contradiction. Note that it does not require any differentiability assumption. • We first show that the feasible set is convex. Suppose x 6= x∗ are two distinct solutions. Consider the convex combination of these two solutions xα ≡ αx+(1 − α) x∗ . Since each g j is affine and g j (x∗ ) = g j (x) = 0, we have g j (xα) = αg j (x) + (1 − α) g j (x∗ ) = 0. In other words, xα also satisfies the constraints. • To complete the proof, we find the required contradiction: Since f is strictly quasiconcave and f (x∗ ) = f (x), it must be the case that f (xα) > f (x∗ ). Q.E.D. The construction of the obvious corollaries for minimisation problems is left as an exercise. We conclude this section with Theorem 3.4.4 (Envelope Theorem.) Consider the modified constrained optimisation problem: max f (x, α) x

subject to g (x, α) = 0,

(3.4.2)

where x ∈ 0

(3.5.7) (3.5.8)

The various sign conditions which we have looked at are summarised in Table 3.1. Theorem 3.5.1 Necessary (first order) conditions for optimisation with inequality constraints. If 1. x∗ solves Problem (3.5.1), with g i (x∗ ) = 0, i = 1, 2, . . . , b and g i (x∗ ) > 0, i = b + 1, . . . , m (in other words, the first b constraints are binding (active) at x∗ and the last n − b are non-binding (inactive) at x∗ , renumbering the constraints if necessary to achieve this), Revised: December 2, 1998

52

3.5. INEQUALITY CONSTRAINED OPTIMISATION: THE KUHN-TUCKER THEOREMS 2. f and g are continuously differentiable, and 3. the b × n submatrix of g 0 (x∗ ),

∂g 1 ∂x1

(x∗ ) . . . .. ... .

∂g 1 ∂xn

∂g b ∂x1

(x∗ ) . . .

∂g b ∂xn

(x∗ ) .. , . (x∗ )

is of full rank b (i.e. there are no redundant binding constraints, both in the sense that there are fewer binding constraints than variables and in the sense that the constraints which are binding are ‘independent’), then ∃λ ∈ <m such that f 0 (x∗ ) + λ> g 0 (x∗ ) = 0, with λi ≥ 0 for i = 1, 2, . . . , m and g i (x∗ ) = 0 if λi > 0. Proof The proof is similar to that of Theorem 3.4.1 for the equality constrained case. It can be broken into seven steps. 1. Suppose x∗ solves Problem (3.5.1). We begin by restricting attention to a neighbourhood B (x∗ ) throughout which the non-binding constraints remain non-binding, i.e. g i (x) > 0 ∀x ∈ B (x∗ ) , i = b + 1, . . ., m.

(3.5.9)

Such a neighbourhood exists since the constraint functions are continuous. Since x∗ solves Problem (3.5.1) by assumption, it also solves the following problem: s.t.

maxx∈B (x∗ ) f (x) g i (x) ≥ 0, i = 1, 2, . . . , b.

(3.5.10)

In other words, since the non-binding constraints are non-binding ∀x ∈ B (x∗ ) by construction, we can ignore them if we confine our search for a maximum to this neighbourhood. We will return to the non-binding constraints in the very last step of this proof, but until then g will be taken to refer to the vector of b binding constraint functions only and λ to the vector of b Kuhn-Tucker multipliers corresponding to these binding constraints. 2. We now introduce slack variables s ≡ (s1 , . . ., sb ), one corresponding to each binding constraint, and consider the following equality constrained maximisation problem: maxx∈B (x∗ ),s∈ = x A+A x 2 and

1 2

(3.5.30) (3.5.31) (3.5.32)

A + A> is always symmetric.

Let G be the m × n matrix whose ith row is gi . G must have full rank if we are to apply the Kuhn-Tucker conditions. The Lagrangean is: x> Ax + λ> (Gx − α) .

(3.5.33)

The first order conditions are: 2x> A + λ> G = 0n

(3.5.34)

or, transposing and multiplying across by 12 A−1 : 1 x = − A−1 G> λ. 2

(3.5.35)

If the constraints are binding, then we will have: 1 α = − GA−1 G> λ. 2

(3.5.36)

Now we need the fact that G (and hence GA−1 G> ) has full rank to solve for the Lagrange multipliers λ:

λ = −2 GA−1 G>

−1

α.

(3.5.37)

Now the sign conditions tell us that each component of λ must be nonnegative. An easy fix is to let the Kuhn-Tucker multipliers be defined by: ∗

−1

> −1

λ ≡ max 0m , −2 GA G

α ,

(3.5.38)

Revised: December 2, 1998

58

3.6. DUALITY where the max operator denotes component-by-component maximisation. The effect of this is to knock out the non-binding constraints (those with negative Lagrange multipliers) from the original problem and the subsequent analysis. We can now find the optimal x by substituting for λ in (3.5.35) the value of λ∗ from (3.5.38). In the case in which all the constraints are binding, the solution is:

x = A−1 G> GA−1 G>

−1

α

(3.5.39)

and the envelope function is given by:

−1

GA−1 AA−1 G> GA−1 G>

−1

α

x> Ax = α> GA−1 G> = α> GA−1 G> 1 = − α> λ. 2

−1

α

(3.5.40)

The applications of this problem will include ordinary least squares and generalised least squares regression and the mean-variance portfolio choice problem in finance. 2. Maximising a Cobb-Douglas utility function subject to a budget constraint and non-negativity constraints. The applications of this problem will include choice under certainty, choice under uncertainty with log utility where the parameters are reinterpreted as probabilities, the extension to Stone-Geary preferences, and intertemporal choice with log utility, where the parameters are reinterpreted as time discount factors. Further exercises consider the duals of each of the forgoing problems, and it is to the question of duality that we will turn in the next section.

3.6 Duality Let X ⊆ x + (1 − λ) p0> x > M, which contradicts the first inequality. It follows that the maximum value of uh (x) on the subset B (pλ) is less than or equal to its maximum value on the superset B (p) ∪ B (p0 ). In terms of the indirect utility function, this says that vh (pλ, M ) ≤ max {vh (p, M ) , vh (p0 , M )} , or that vh is quasiconvex. 4. vh (p, M ) is homogenous of degree zero in p, M , or vh (λp, λM ) = vh (p, M ) . Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

75

The following are interesting properties of the expenditure function: 1. The expenditure function is continuous. 2. The expenditure function itself is non-decreasing in prices, since raising the price of one good while holding the prices of all other goods constant can not reduce the minimum cost of attaining a fixed utility level. 3. The expenditure function is concave in prices. To see this, we just fix two price vectors p and p0 and consider the value of the expenditure function at the convex combination pλ ≡ λp + (1 − λ) p0 . e (pλ, u¯) = (pλ)> h (pλ, u¯)

(4.5.5) 0 >

>

= λp h (pλ, u¯) + (1 − λ) (p ) h (pλ, u¯) 0 >

≥ λp> h (p, u¯) + (1 − λ) (p ) h (p0 , u¯) = λe (p, u¯) + (1 − λ) e (p0 , u¯) ,

(4.5.6) (4.5.7) (4.5.8)

where the inequality follows because the cost of a suboptimal bundle for the given prices must be greater than the cost of the optimal (expenditureminimising) consumption vector for those prices. 4. The expenditure function is homogenous of degree 1 in prices: eh (αp, u¯) = αeh (p, u¯) .

(4.5.9)

4.6 Further Results in Demand Theory In this section, we present four important theorems on demand functions and the corresponding envelope functions. Shephard’s Lemma will allow us to recover Hicksian demands from the expenditure function. Similarly, Roy’s Identity will allow us to recover Marshallian demands from the indirect utility function. The Slutsky symmetry condition and the Slutsky equation provide further insights into the properties of consumer demand. Theorem 4.6.1 (Shephard’s Lemma.) ∂eh ∂ > (p, u ¯ ) = p x + λ (u (x) − u ¯ ) h ∂pn ∂pn = xn

(4.6.1) (4.6.2)

which, when evaluated at the optimum, is just hnh (p, u¯). In other words, the partial derivatives of the expenditure function with respect to prices are the corresponding Hicksian demand functions. Revised: December 2, 1998

76

4.6. FURTHER RESULTS IN DEMAND THEORY

Proof By differentiating the expenditure function with respect to the price of good n and applying the envelope theorem (Theorem 3.4.4), we obtain Shephard’s Lemma: (To apply the envelope theorem, we should be dealing with an equality constrained optimisation problem; however, if we assume local non-satiation, we know that the budget constraint or utility constraint will always be binding, and so the inequality constrained expenditure minimisation problem is essentially and equality constrained problem.) Q.E.D. Theorem 4.6.2 (Roy’s Identity.) Marshallian demands may be recovered from the indirect utility function using: ∂v n

xn (p, M ) = − ∂p ∂v

∂M

(p, M ) (p, M )

.

(4.6.3)

Proof For Roy’s Identity, see ?. It is obtained by differentiating equation (4.4.7) with respect to pn , using the Chain Rule: v (p, e (p, u¯)) = u¯ implies that ∂v ∂v ∂e (p, e (p, u¯)) + (p, e (p, u¯)) n (p, u¯) = 0 n ∂p ∂M ∂p

(4.6.4)

and using Shephard’s Lemma gives: ∂v ∂v (p, e (p, u¯)) + (p, e (p, u¯)) hn (p, u¯) = 0 n ∂p ∂M Hence n

h (p, u¯) = −

∂v ∂pn ∂v ∂M

(p, e (p, u¯)) (p, e (p, u¯))

(4.6.5)

(4.6.6)

and expressing this last equation in terms of the relevant level of income M rather than the corresponding value of utility u¯: n

x (p, M ) = −

∂v ∂pn ∂v ∂M

(p, M ) (p, M )

.

(4.6.7)

Q.E.D. Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

77

Theorem 4.6.3 (Slutsky symmetry condition.) All cross-price substitution effects are symmetric: ∂hnh ∂hm h = . (4.6.8) m ∂p ∂pn Proof From Shephard’s Lemma, we can easily derive the Slutsky symmetry conditions, assuming that the expenditure function is twice continuously differentiable, and hence that ∂ 2 eh ∂ 2 eh = . (4.6.9) ∂pm ∂pn ∂pn ∂pm Since hm h =

∂eh ∂pm

and hnh =

∂eh , ∂pn

and the result follows. Q.E.D.

The next result doesn’t really have a special name of its own. Theorem 4.6.4 Since the expenditure function is concave in prices (see p. 75), the corresponding Hessian matrix is negative semi-definite. In particular, its diagonal entries are non-positive, or ∂ 2 eh ≤ 0, ∂ (pn )2

n = 1, . . . , N.

(4.6.10)

Using Shephard’s Lemma, it follows that ∂hnh ≤ 0, ∂pn

n = 1, . . . , N.

(4.6.11)

In other words, Hicksian demand functions, unlike Marshallian demand functions, are uniformly decreasing in own price. Another way of saying this is that own price substitution effects are always negative. Theorem 4.6.5 (Slutsky equation.) The total effect of a price change on (Marshallian) demand can be decomposed as follows into a substitution effect and an income effect: ∂xm ∂hm ∂xm (p, M ) = (p, u ¯ ) − (p, M ) hn (p, u¯) , n n ∂p ∂p ∂M

(4.6.12)

where u¯ ≡ V (p, M ). Before proving this, let’s consider the signs of the various terms in the Slutsky equation and look at what it means in a two-good example. By Theorem 4.6.4, we know that own price substitution effects are always nonpositive. [This is still on a handwritten sheet.] Revised: December 2, 1998

78

4.7. GENERAL EQUILIBRIUM THEORY

Proof Differentiating both sides of the lth component of (4.4.9) with respect to pn , using the Chain Rule, will yield the so-called Slutsky equation which decomposes the total effect on demand of a price change into an income effect and a substitution effect. Differentiating the RHS of (4.4.9) with respect to pn yields: ∂xm ∂e ∂xm (p, e (p, u ¯ )) + (p, e (p, u¯)) n (p, u¯) . n ∂p ∂M ∂p

(4.6.13)

To complete the proof: 1. set this equal to

∂hm ∂pn

(p, u¯)

2. substitute from Shephard’s Lemma 3. define M ≡ e (p, u¯) (which implies that u¯ ≡ V (p, M ))

Q.E.D.

4.7 General Equilibrium Theory 4.7.1 Walras’ law Walras . . . 3

4.7.2 Brouwer’s fixed point theorem 4.7.3 Existence of equilibrium

4.8 The Welfare Theorems 4.8.1 The Edgeworth box 4.8.2 Pareto efficiency Definition 4.8.1 A feasible allocation X = (x1 , . . . , xH ) is Pareto efficient if there does not exist any feasible way of reallocating the same initial aggregate P endowment, H h=1 xh , which makes one individual better off without making any other worse off. Definition 4.8.2 X is Pareto dominated by X0 = (x01 , . . . , x0H ) if PH 0 0 0 h=1 xh , xh h xh ∀h and xh h xh for at least one h. 3

PH

h=1

xh =

This material still exists only in handwritten form in Alan White’s EC3080 notes from 19912. One thing missing from the handwritten notes is Kakutani’s Fixed Point Theorem which should be quoted from ?.

Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

79

4.8.3 The First Welfare Theorem (See ?.) Theorem 4.8.1 (First Welfare Theorem) If the pair (p, X) is an equilibrium (for given preferences, h , which exibit local non-satiation and given endowments, eh , h = 1, . . . , H), then X is a Pareto efficient allocation. Proof The proof is by contradiction. Suppose that X is an equilibrium allocation which is Pareto dominated by a feasible allocation X0 . If individual h is strictly better off under X0 or x0h h xh , then it follows that individual h cannot afford x0h at the equilibrium prices p or p> x0h > p> xh = p> eh .

(4.8.1)

The latter equality is just the budget constraint, which is binding since we have assumed local non-satiation. Similarly, if individual h is indifferent between X and X0 or x0h ∼h xh , then it follows that (4.8.2) p> x0h ≥ p> xh = p> eh , since if x0h cost strictly less than xh , then by local non-satiation some consumption vector near enough to x0h to also cost less than xh would be strictly preferred to xh and xh would not maximise utility given the budget constraint. Summing (4.8.1) and (4.8.2) over households yields p>

H X

x0h > p>

h=1

H X

xh = p>

h=1

H X

eh ,

(4.8.3)

h=1

(where the equality is essentially Walras’ Law). But since X0 is feasible we must have for each good n H X

h=1

x0n h ≤

H X

enh

h=1

and, hence, multiplying by prices and summing over all goods, p>

H X

h=1

x0h ≤ p>

H X

eh .

(4.8.4)

h=1

But (4.8.4) contradicts the inequality in (4.8.3), so no such Pareto dominant allocation X0h can exist. Q.E.D. Before proceeding to the second welfare theorem, we need to say a little bit about separating hyperplanes. Revised: December 2, 1998

80

4.8. THE WELFARE THEOREMS

4.8.4 The Separating Hyperplane Theorem n

o

Definition 4.8.3 The set z ∈ z = p> z∗ is the hyperplane through z∗ with normal p. Note that any hyperplane divides z∗

and

z ∈ z ≥ p> z∗ .

The intersection of these two closed half-spaces is the hyperplane itself. In two dimensions, a hyperplane is just a line; in three dimensions, it is just a plane. The idea behind the separating hyperplane theorem is quite intuitive: if we take any point on the boundary of a convex set, we can find a hyperplane through that point so that the entire convex set lies on one side of that hyperplane. We will essentially be applying this notion to the upper contour sets of quasiconcave utility functions, which are of course convex sets. We will interpret the separating hyperplane as a budget hyperplane, and the normal vector as a price vector, so that at those prices nothing giving higher utility than the cutoff value is affordable. Theorem 4.8.2 (Separating Hyperplane Theorem) If Z is a convex subset of z ∀z ∈ Z, or Z is contained in one of the closed half-spaces associated with the hyperplane through z∗ with normal p∗ . Proof Not given. See ? Q.E.D.

4.8.5 The Second Welfare Theorem (See ?.) We make slightly stronger assumptions than are essential for the proof of this theorem. This allows us to give an easier proof. Theorem 4.8.3 (Second Welfare Theorem) If all individual preferences are strictly convex, continuous and strictly monotonic, and if X∗ is a Pareto efficient allocation such that all households are allocated positive amounts of all goods (x∗g h > 0 ∀g = 1, . . . , N ; h = 1, . . . , H), then a reallocation of the initial aggregate endowment can yield an equilibrium where the allocation is X∗ . Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY

81

Proof There are four main steps in the proof. 1. First we construct a set of utility-enhancing endowment perturbations, and use the separating hyperplane theorem to find prices at which no such endowment perturbation is affordable. We need to use the fact (Theorem 3.2.1) that a sum of convex sets, such as X + Y ≡ {x + y : x ∈ X, y ∈ Y } , is also a convex set. ∗ Given an aggregate initial endowment x∗ = H h=1 xh , we interpret any vecPH ∗ tor of the form z = h=1 xh − x as an endowment perturbation. Now consider the set of all ways of changing the aggregate endowment without making anyone worse off:

P

Z ≡(

z∈

0 ≤ p∗> z ∀z ∈ Z. Since preferences are monotonic, the set Z must contain all the standard unit basis vectors ((1, 0, . . . , 0), &c.). This fact can be used to show that all components of p∗ are non-negative, which is essential if it is to be interpreted as an equilibrium price vector. 3. Next, we specify one way of redistributing the initial endowment in order that the desired prices and allocation emerge as a competitive equilibrium. All we need to do is value endowments at the equilibrium prices, and redistribute the aggregate endowment of each good to consumers in proportion to their share in aggregate wealth computed in this way. 4. Finally, we confirm that utility is maximised by the given Pareto efficient allocation, X∗ , at these prices. As usual, the proof is by contradiction: the details are left as an exercise. Q.E.D.

4.8.6 Complete markets The First Welfare Theorem tells us that competitive equilibrium allocations are Pareto optimal if markets are complete. If there are missing markets, then competitive trading may not lead to a Pareto optimal allocation. We can use the Edgeworth Box diagram to illustrate the simplest possible version of this principle.

4.8.7 Other characterizations of Pareto efficient allocations There are a total of five equivalent characterisations of Pareto efficient allocations. Theorem 4.8.4 Each of the following is an equivalent description of the set of allocations which are Pareto efficient: 1. by definition, feasible allocations such that no other allocation strictly increases at least one individual’s utility without decreasing the utility of any other individual; 2. by the Welfare Theorems, equilibrium allocations for all possible distributions of the fixed initial aggregate endowment; 3. in two dimensions, allocations lying on the contract curve in the Edgeworth box; Revised: December 2, 1998

CHAPTER 4. CHOICE UNDER CERTAINTY 4. allocations which solve:

83

5

max

{xh :h=1,...,H}

H X

λh [uh (xh )]

(4.8.6)

h=1

subject to the feasibility constraints H X

xh =

h=1

H X

eh

(4.8.7)

h=1

for some non-negative weights {λh }H h=1 . 5. allocations which maximise the utility of a representative agent given by H X

λh [uh (xh )]

(4.8.8)

h=1

where {λh }H h=1 are again any non-negative weights. Proof If an allocation is not Pareto efficient, then the Pareto-dominating allocation gives a higher value of the objective function in the above problem for all possible weights. If an allocation is Pareto efficient, then the relative weights for which the above objective function is maximized are the ratios of the Lagrange multipliers from the problem of maximizing any individual’s utility subject to the constraint that all other individuals’ utilities are unchanged: max u1 (x1 )

(4.8.9)

s.t. uh (xh ) = uh (x∗h ) h = 2, . . . , H

(4.8.10)

since these two problems will have the same necessary and sufficient first order conditions. The absolute weights corresponding to a particular allocation are not unique, as they can be multiplied by any positive constant without affecting the maximum. Different absolute weights (or Lagrange multipliers) arise from fixing different individuals’ utilities in the last problem, but the relative weights will be the same. 5

The solution here would be unique if the underlying utility function were concave, since linear combinations of concave functions with non-negative weights are concave, and the constraints specify a convex set on which the objective function has a unique optimum. This argument can not be used with merely quasiconcave utility functions.

Revised: December 2, 1998

84

4.9. MULTI-PERIOD GENERAL EQUILIBRIUM Q.E.D.

Note that corresponding to each Pareto efficient allocation there is at least one: 1. set of non-negative weights defining (a) the objective function in 4. and (b) the representative agent in 5. and 2. initial allocation leading to the competitive equilibrium in 2.

4.9 Multi-period General Equilibrium In Section 4.2, it was pointed out that the objects of choice can be differentiated not only by their physical characteristics, but also both by the time at which they are consumed and by the state of nature in which they are consumed. These distinctions were suppressed in the intervening sections but are considered again in this section and in Section 5.4 respectively. The multi-period model should probably be introduced at the end of Chapter 4 but could also be left until Chapter 7. For the moment this brief introduction is duplicated in both chapters. Discrete time multi-period investment problems serve as a stepping stone from the single period case to the continuous time case. The main point to be gotten across is the derivation of interest rates from equilibrium prices: spot rates, forward rates, term structure, etc. This is covered in one of the problems, which illustrates the link between prices and interest rates in a multiperiod model.

Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

85

Chapter 5 CHOICE UNDER UNCERTAINTY 5.1 Introduction [To be written.]

5.2 Review of Basic Probability Economic theory has, over the years, used many different, sometimes overlapping, sometimes mutually exclusive, approaches to the analysis of choice under uncertainty. This chapter deals with choice under uncertainty exclusively in a single period context. Trade takes place at the beginning of the period and uncertainty is resolved at the end of the period. This framework is sufficient to illustrate the similarities and differences between the most popular approaches. When we consider consumer choice under uncertainty, consumption plans will have to specify a fixed consumption vector for each possible state of nature or state of the world. This just means that each consumption plan is a random vector. Let us review the associated concepts from basic probability theory: probability space; random variables and vectors; and stochastic processes. Let Ω denote the set of all possible states of the world, called the sample space. A collection of states of the world, A ⊆ Ω, is called an event. Let A be a collection of events in Ω. The function P : A → [0, 1] is a probability function if 1.

(a) Ω ∈ A (b) A ∈ A ⇒ Ω − A ∈ A (c) Ai ∈ A for i = 1, . . . , ∞ ⇒

S∞

i=1

Ai ∈ A

(i.e. A is a sigma-algebra of events) Revised: December 2, 1998

86

5.2. REVIEW OF BASIC PROBABILITY and 2.

(a) P (Ω) = 1 (b) P (Ω − A) = 1 − P (A) ∀A ∈ A (redundant assumption) (c) P ( ∞ i=1 Ai ) = events in A. S

P∞

i=1

P (Ai ) when A1 , A2 , . . . are pairwise disjoint

(Ω, A, P ) is then called a probability space Note that the certainty case we considered already is just the special case of uncertainty in which the set Ω has only one element. We will consider these concepts in more detail when we come to intertemporal models. Suppose we are given such a probability space. The function x˜ : Ω → < is a random variable (r.v.) if ∀x ∈ < {ω ∈ Ω : x˜ (ω) ≤ x} ∈ A, i.e. a function is a random variable if we know the probability that the value of the function is less than or equals any given real number. The function Fx˜ : < → [0, 1] : x 7→ Pr (˜ x ≤ x) ≡ P ({ω ∈ Ω : x˜ (ω) ≤ x}) is known as the cumulative distribution function (c.d.f.) of the random variable x˜. The convention of using a tilde over a letter to denote a random variable is common in financial economics; in other fields capital letters may be reserved for random variables. In either case, small letters usually denote particular real numbers (i.e. particular values of the random variable). A random vector is just a vector of random variables. It can also be thought of as a vector-valued function on the sample space Ω. A stochastic process is a collection of random variables or random vectors indexed by time, e.g. {˜ xt : t ∈ T } or just {˜ xt } if the time interval is clear from the context. For the purposes of this part of the course, we will assume that the index set consists of just a finite number of times i.e. that we are dealing with discrete time stochastic processes. Then a stochastic process whose elements are N -dimensional random vectors is equivalent to an N |T |-dimensional random vector. The (joint) c.d.f. of a random vector or stochastic process is the natural extension of the one-dimensional concept. Random variables can be discrete, continuous or mixed. The expectation (mean, average) of a discrete r.v., x˜, with possible values x1 , x2 , x3 , . . . is given by E [˜ x] ≡

∞ X

xi P r (˜ x = xi ) .

i=1

For a continuous random variable, the summation is replaced by an integral. Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

87

The covariance of two random variables x˜ and y˜ is given by Cov [˜ x, y˜] ≡ E [(˜ x − e [˜ x]) (˜ y − E [˜ y ])] . The covariance of a random variable with itself is called its variance. The expectation of a random vector is just the vector of the expectations of the component random variables. The variance (variance-covariance matrix) of a random vector is the (symmetric, positive semi-definite) matrix of the covariances between the component random variables. Given any two random variables x˜ and y˜, we can define a third random variable˜ by ˜ ≡ y˜ − α − β x˜. (5.2.1) To specify ˜ completely, we can either specify α and β explicitly or fix them implicitly by imposing (two) conditions on˜. We do the latter by insisting 1. ˜ and x˜ are uncorrelated (this is not the same as assuming statistical independence, except in special cases such as bivariate normality) 2. E [˜] = 0 It follows that: β =

Cov [˜ x, y˜] Var [˜ x]

(5.2.2)

and α = E [˜ y ] − βE [˜ x] .

(5.2.3)

But what about the conditional expectation, E [˜ y |˜ x = x]? This is not equal to α + βx, as one might expect, unless E [˜|˜ x = x] = 0. This requires statistical independence rather than the assumed lack of correlation. Again, a sufficient condition is multivariate normality. The notion of the β of y˜ with respect to x˜ as given in (5.2.2) will recur frequently. The final concept required from basic probability theory is the notion of a mixture of random variables. For lotteries which are discrete random variables, with payoffs x1 , x2 , x3 , . . . occuring with probabilities π1 , π2 , π3 , . . . respectively, we will use the notation: π 1 x1 ⊕ π 2 x2 ⊕ π 3 x3 ⊕ . . . Similar notation will be used for compound lotteries (mixtures of random variables) where the payoffs themselves are further lotteries. This might be a good place to talk about the MVN distribution and Stein’s lemma. Revised: December 2, 1998

88

5.3. TAYLOR’S THEOREM: STOCHASTIC VERSION

5.3 Taylor’s Theorem: Stochastic Version We will frequently use the univariate Taylor expansion as applied to a function of a random variable expanded about the mean of the random variable.1 Taking expectations on both sides of the Taylor expansion: f (˜ x) = f (E[˜ x]) +

∞ X

1 (n) f (E[˜ x])(˜ x − E[˜ x])n n! n=1

(5.3.1)

yields: E[f (˜ x)] = f (E[˜ x]) +

∞ X

1 (n) f (E[˜ x])mn (˜ x), n! n=2

(5.3.2)

where mn (˜ x) ≡ E [(˜ x − E[˜ x])n ] .

(5.3.3)

In particular, h

i

(5.3.4)

h

i

(5.3.5)

h

i

(5.3.6)

h

i

(5.3.7)

m1 (˜ x) = E (˜ x − E[˜ x])1 ≡ 0 m2 (˜ x) = E (˜ x − E[˜ x])2 ≡ Var [˜ x] m3 (˜ x) = E (˜ x − E[˜ x])3 ≡ Skew [˜ x] and

m4 (˜ x) = E (˜ x − E[˜ x])4 ≡ Kurt [˜ x] ,

which allows us to start the summation in (5.3.2) at n = 2 rather than n = 1. Indeed, we can rewrite (5.3.2) as 1 1 E[f (˜ x)] = f (E[˜ x]) + f 00 (E[˜ x])Var [˜ x] + f 000 (E[˜ x])Skew [˜ x] 2 6 ∞ X 1 1 + f 0000 (E[˜ x])Kurt [˜ x] + f (n) (E[˜ x])mn (˜ x). (5.3.8) 24 n=5 n!

5.4 Pricing State-Contingent Claims This part of the course draws on ?, ? and ?. The analysis of choice under uncertaintly will begin by reinterpreting the general equilibrium model of Chapter 4 so that goods can be differentiated by the state of nature in which they are consumed. Specifically, it will be assumed that the 1

This section will eventually have to talk separately about kth order and infinite order Taylor expansions.

Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

89

underlying sample space comprises a finite number of states of nature. A more thorough analysis of choice under uncertainty, allowing for infinite and continuous sample spaces and based on additional axioms of choice, follows in Section 5.5. Consider a world with M possible states of nature (distinguished by a first subscript), markets for N securities (distinguished by a second subscript) and H consumers (distinguished by a superscript).2 Definition 5.4.1 A state contingent claim or Arrow-Debreu security is a random variable or lottery which takes the value 1 in one particular state of nature and the value 0 in all other states. Definition 5.4.2 A complex security is a random variable or lottery which can take on arbitrary values. The payoffs of a typical complex security will be represented by a column vector, yj ∈ <M , where yij is the payoff in state i of security j. The set of all complex securities on a given finite sample space is an M -dimensional vector space and the M possible Arrow-Debreu securities constitute the standard basis for this vector space. State contingent claims prices are determined by the market clearing equations in a general equilibrium model: Aggregate consumption in state i = Aggregate endowment in state i. Each individual will have an optimal consumption choice depending on endowments and preferences and conditional on the state of the world. Optimal future consumption is denoted ∗ x1 x∗ x∗ = 2 . (5.4.1) ... x∗N If there are N complex securities, then the investor must find a portfolio w = (w1 , . . . , wN ) whose payoffs satisfy x∗i

=

N X

yij wj .

j=1

Let Y be the M × N matrix3 whose jth column contains the payoffs of the jth complex security in each of the M states of nature, i.e. Y ≡ (y1 , y2 , . . . yN ) . 2 3

(5.4.2)

Check for consistency in subscripting etc in what follows. Or maybe I mean its transpose.

Revised: December 2, 1998

90

5.4. PRICING STATE-CONTINGENT CLAIMS

Theorem 5.4.1 If there are M complex securities (M = N ) and the payoff matrix Y is non-singular, then markets are complete. Proof Suppose the optimal trade for consumer i state j is xij − eij . Then can invert Y to work out optimal trades in terms of complex securities. Q.E.D. An (N + 1)st security would be redundant. Either a singular square matrix or < N complex securities would lead to incomplete markets. So far, we have made no assumptions about the form of the utility function, written purely as u (x0 , x1 , x2 , . . . , xN ) , where x0 represents the quantity consumed at date 0 and xi (i > 0) represents the quantity consumed at date 1 if state i materialises.

5.4.1 Completion of markets using options Assume that there exists a state index portfolio, Y , yielding different non-zero payoffs in each state (i.e. a portfolio with a different payout in each state of nature, possibly one mimicking aggregate consumption). WLOG we can rank the states so that Yi < Yj if i < j. We now present some results, following ?, showing conditions under which trading in a state index portfolio and in options on the state index portfolio can lead to the Pareto optimal complete markets equilibrium allocation. Now consider completion of markets using options on aggregate consumption. In real-world markets, the number of linearly independent corporate securities is probably less than M . However, options on corporate securities may be sufficient to form complete markets, and thereby ensure allocational (Pareto) efficiency for arbitrary preferences. Further assume that ∃ M − 1 European call options on Y with exercise prices Y1 , Y2 , . . . , YM −1 . A European call option with exercise price K is an option to buy a security for K on a fixed date. An American call option is an option to buy on or before the fixed date. A put option is an option to sell. Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

91

Here, the original state index portfolio and the M − 1 European call options yield the payoff matrix: y1 0 0 . ..

0

y2 y2 − y1 0 .. .

y3 y3 − y1 y3 − y2 .. .

0

0

... ... ... .. .

yM yM − y1 yM − y2 .. .

. . . yM − yM −1

security Y call option 1 call option 2 .. .

=

call option M − 1

(5.4.3)

and as this matrix is non-singular, we have constructed a complete market. Instead of assuming a state index portfolio exists, we can assume identical probability beliefs and state-independent utility and complete markets in a similar manner (see below).

5.4.2 Restrictions on security values implied by allocational efficiency and covariance with aggregate consumption Cω = aggregate consumption in state ω Ωk = {ω ∈ Ω : Cω = k} Let φ(k) be the value of the claim with payoffs Let

ykω =

1 if Cω = k 0 otherwise

(5.4.4)

and let the agreed probability of the event Ωk (i.e. of aggregate consumption taking the value k) be: X πω (5.4.5) π(k) = ω∈Ωk

By time-additivity and state-independence of the utility function: φω =

πωu0i (ciω ) u0i0 (ci0 )

∀ω ∈ Ω

(5.4.6)

The no arbitrage condition implies φ(k) = = =

X

φω

ω∈Ωk u0i (fi (k)) X u0i0 (ci0 ) ω∈Ωk u0i (fi (k)) π(k) u0i0 (ci0 )

(5.4.7) πω

(5.4.8) (5.4.9)

where fi (k) denotes the i-th individual’s equilibrium consumption in those states where aggregate consumption equals k. Revised: December 2, 1998

92

5.4. PRICING STATE-CONTINGENT CLAIMS x(0)

x(1)

x(2)

1 2 3 · · · L

0 1 2 · · · L−1

0 0 1 · · · L−2

C˜ = 1 C˜ = 2 C˜ = 3 · · · C˜ = L

Table 5.1: Payoffs for Call Options on the Aggregate Consumption State-independence of the utility function is required for fi (k) to be well-defined. Therefore, an arbitrary security x has value: Sx =

X

φωxω

(5.4.10)

ω∈Ω

= = =

X X

X

(5.4.11) πωxω

πω xω π(k)

X

φ(k)

X

φ(k)E[˜ x|C˜ = k]

k

=

φωxω

k ω∈Ωk X u0i (fi (k)) X u0i0 (ci0 ) ω∈Ωk k

ω∈Ωk

(5.4.12) (5.4.13) (5.4.14)

k

5.4.3 Completing markets with options on aggregate consumption Let x(k) be the vector of payoffs in the various possible states on a European call option on aggregate consumption with one period to maturity and exercise price k. Let {1, 2, . . . , L} be the set of possible values of aggregate consumption C(ω). Then payoffs are as given in Table 5.1. This all assumes 1. identical probability beliefs 2. time-additivity of u 3. state-independent u Revised: December 2, 1998

CHAPTER 5. CHOICE UNDER UNCERTAINTY

93

5.4.4 Replicating elementary claims with a butterfly spread Elementary claims against aggregate consumption can be constructed as follows, for example, for state 1, using a butterfly spread: [x(0) − x(1)] − [x(1) − x(2)]

(5.4.15)

yields the payoff:

0 1 0 0 0 1 1 1 2 1 1 0 3 − 2 − 2 − 1 = 1 − 1 . . . . . . . . . . . . . . . . . . 1 L−2 L−1 L−1 L 1 1 0 (5.4.16) = 0. . . 0 i.e. this replicating portfolio pays 1 iff aggregate consumption is 1, and 0 otherwise. The prices of this, and the other elementary claims, must, by no arbitrage, equal the prices of the corresponding replicating portfolios.

5.5 The Expected Utility Paradigm 5.5.1 Further axioms The objects of choice with which we are concerned in a world with uncertainty could still be called consumption plans, but we will acknowledge the additional structure now described by terming them lotteries. If there are k physical commodities, a consumption plan must specify a k-dimensional vector, x ∈ 0

(6.3.9)

E[u0 (W0 rf ) (˜ r − rf )] > 0

(6.3.10)

u0 (W0 rf ) E[(˜ r − rf )] > 0

(6.3.11)

⇐⇒

⇐⇒

⇐⇒ E[˜ r] > E[rf ] = rf This is the property of local risk neutrality — a risk averse investor will always prefer a little of a risky asset paying a higher expected return than rf to none of the risky asset. Definition 6.3.1 Let f : X → 0

da dW0

1) • CRRA ⇒ constant proportion of wealth invested in the risky asset (η = 1) • IRRA ⇒ decreasing proportion of wealth invested in the risky asset (η < 1) Theorem 6.3.1 DARA ⇒ RISKY ASSET NORMAL Proof By implicit differentiation of the now familiar first order condition (6.3.3), which can be written:

we have

E[u0 (W0 rf + a(˜ r − rf ))(˜ r − rf )] = 0,

(6.3.16)

˜ )(˜ da E[u00 (W r − rf )]rf = . 00 ˜ dW0 −E[u (W )(˜ r − rf )2 ]

(6.3.17)

Revised: December 2, 1998

114

6.3. THE SINGLE-PERIOD PORTFOLIO CHOICE PROBLEM

By concavity, the denominator is positive. Therefore: ˜ )(˜ sign (da/dW0 ) = sign {E[u00 (W r − rf )]}

(6.3.18)

We will show that both are positive. For decreasing absolute risk aversion:3 ˜ ) < RA (W0 rf ) r˜ > rf ⇒ RA (W ˜ ) ≥ RA (W0 rf ) r˜ ≤ rf ⇒ RA (W ˜ )(˜ Multiplying both sides of each inequality by −u0 (W r − rf ) gives respectively: ˜ )(˜ ˜ )(˜ u00 (W r − rf ) > −RA (W0 rf )u0 (W r − rf )

(6.3.19)

in the event that r˜ > rf , and ˜ )(˜ ˜ )(˜ u00 (W r − rf ) ≥ −RA (W0 rf )u0 (W r − rf )

(6.3.20)

(the same result) in the event that r˜ ≤ rf Integrating over both events implies: ˜ )(˜ ˜ )(˜ E[u00 (W r − rf )] > −RA (W0 rf )E[u0 (W r − rf )],

(6.3.21)

provided that r˜ > rf with positive probability. The RHS of inequality (6.3.21) is 0 at the optimum, hence the LHS is positive as claimed. Q.E.D. The other results are proved similarly (exercise!).

6.3.3 Mutual fund separation Commonly, investors delegate portfolio choice to mutual fund operators or managers. We are interested in conditions under which large groups of investors will agree on portfolio composition. For example, all investors with similar utility functions might choose the same portfolio, or all investors with similar probability beliefs might choose the same portfolio. More realistically, we may be able to define a group of investors whose portfolio choices all lie in a subspace of small dimension (say 2) of the N -dimensional portfolio space. The first such result is due to ?. 3

Think about whether separating out the case of r˜ = rf is necessary.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

115

Theorem 6.3.2 ∃ Two fund monetary separation i.e. Agents with different wealths (but the same increasing, strictly concave, VNM utility) hold the same risky unit cost portfolio, p∗ say, (but may differ in the mix of the riskfree asset and risky portfolio) i.e. ∀ portfolios p, wealths W0 , ∃λ s.t. h

E u W0 rf + λW0 p∗ > (˜r − rf 1)

i

h

i

≥ E u W0 rf + p> (˜r − rf 1)

(6.3.22)

⇐⇒ Risk-tolerance (1/RA (z)) is linear (including constant) i.e. ∃ Hyperbolic Absolute Risk Aversion (HARA, incl. CARA) i.e. the utility function is of one of these types: • Extended power: u(z) =

1 (A (C+1)B

+ Bz)C+1

• Logarithmic: u(z) = ln(A + Bz) A • Negative exponential: u(z) = − B exp{Bz}

where A, B and C are chosen to guarantee u0 > 0, u00 < 0. i.e. marginal utility satisfies u0 (z) = (A + Bz)C

or u0 (z) = A exp{Bz}

(6.3.23)

where A, B and C are again chosen to guarantee u0 > 0, u00 < 0. Proof The proof that these conditions are necessary for two fund separation is difficult and tedious. The interested reader is referred to ?. We will show that u0 (z) = (A + Bz)C is sufficient for two-fund separation. The optimal dollar investments wj are the unique solution to the first order conditions: ˜ ˜ ) δW ] 0 = E[u0 (W δwi ˜ )C (˜ = E[(A + B W ri − rf )] X = E[(A + BW0 rf + Bwj (˜ rj − rf ))C (˜ ri − rf )],

(6.3.24) (6.3.25) (6.3.26)

j

or equivalently to the system of equations E[(1 +

X j

Bwj (˜ rj − rf ))C (˜ ri − rf )] = 0 A + BW0 rf

(6.3.27)

Revised: December 2, 1998

116

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

or E[(1 +

X

xj (˜ rj − rf ))C (˜ ri − rf )] = 0

(6.3.28)

j

where

Bwj . A + BW0 rf The unique solutions for xj are clearly independent of W0 which does not appear in (6.3.28). Since A and B do not appear either, the unique solutions for xj are also independent of those parameters. However, they do depend on C. But the risky portfolio weights are xj =

wi j wj

P

Bwi /(A + BW0 rf ) = P j Bwj /(A + BW0 rf ) xi = P j xj

(6.3.29) (6.3.30)

and so are also independent of initial wealth. Since the dollar investment in the jth risky asset satisfies: wj = xj (

A + W0 rf ) B

(6.3.31)

we also have in this case that the dollar investment in the common risky portfolio is a linear function of the initial wealth. The other sufficiency proofs are similar and are left as exercises. Q.E.D. Some humorous anecdotes about Cass may now follow.

6.4 Mathematics of the Portfolio Frontier 6.4.1 The portfolio frontier in 1 = W0

(6.4.2)

w> e ≥ W1 = µW0 .

(6.4.3)

and

The first constraint is just the budget constraint, while the second constraint states that the expected rate of return on the portfolio is at least the desired mean return µ. The frontier in this case is the set of solutions for all values of W0 and W1 (or µ) to this variance minimisation problem, or to the equivalent maximisation problem: max −w> Vw w

(6.4.4)

subject to the same linear constraints (6.4.2) and (6.4.3). The properties of this two-moment frontier are well known, and can be found, for example, in ? or ?. The notation here follows ?. The derivation of the meanvariance frontier is generally presented in the literature in terms of portfolio weight vectors or, equivalently, assuming that initial wealth, W0 , equals 1. This assumption is not essential and will be avoided. Revised: December 2, 1998

118

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

The solution The inequality constrained maximisation problem (6.4.4) is just a special case of the canonical quadratic programming problem considered at the end of Section 3.5, except that it has explicitly one equality constraint and one inequality constraint. To avoid degeneracies, we require: 1. that not every portfolio has the same expected return, i.e. e 6= E[˜ r1 ]1,

(6.4.5)

and in particular that N > 1. 2. that the variance-covariance matrix, V, is (strictly) positive definite. We already know from (1.12.4) that V must be positive semi-definite, but we require this slightly stronger condition. To see why, suppose ∃w 6= 0N s.t. w> Vw = 0

(6.4.6)

Then ∃ a portfolio whose return w>˜r = r˜w has zero variance. This implies that r˜w = r0 (say) w.p.1 or, essentially, that this portfolio is riskless. Arbitrage will force the returns on all riskless assets to be equal in equilibrium, so this situation is equivalent economically to the introduction of a riskless asset later. In the portfolio problem, the place of the matrix A in the canonical quadratic programming problem is taken by the (symmetric) negative definite matrix, −V, which is just the negative of the variance-covariance matrix of asset returns; g1 = 1> and α1 = W0 ; and g2 = e> and α2 = W1 . (6.4.5) guarantees that the 2 × N matrix G is of full rank 2. The parallels are a little fuzzy in the case of the budget constraint since it is really an equality constraint. (3.5.39) says that the optimal w is a linear combination of the two columns of the N × 2 matrix −1 V−1 G> GV−1 G> , with columns weighted by initial wealth W0 and expected final wealth, W1 . We will call these columns g and h and write the solution as w = W0 g + W1 h = W0 (g + µh) . Revised: December 2, 1998

(6.4.7)

CHAPTER 6. PORTFOLIO THEORY

119

The components of g and h are functions of the means and variances of security returns. Thus the vector of optimal portfolio proportions, 1 w = g + µh, W0

(6.4.8)

is independent of the initial wealth W0 . It is easy to see the economic interpretation of g and h: • g is the frontier portfolio corresponding to W0 = 1 and W1 = 0. In other words, it is the normal portfolio which would be held by an investor whose objective was to (just) go bankrupt with minimum variance. • Similarly, h is the frontier portfolio corresponding to W0 = 0 and W1 = 1. In other words, it is the hedge portfolio which would be purchased by a variance-minimising investor in order to increase his expected final wealth by one unit. Alternatively, (3.5.35) says that the optimal w is a linear combination of the two columns of the N × 2 matrix 1 −1 > 1 −1 V G = 2V 1 2

1 −1 V e 2

,

with columns weighted by the Lagrange multipliers corresponding to the two constraints. We will call the Lagrange multipliers 2γ/C and 2λ/A respectively, where we define: A ≡ 1> V−1 e = e> V−1 1 B ≡ e> V−1 e > 0 C ≡ 1> V−1 1 > 0

(6.4.9) (6.4.10) (6.4.11)

D ≡ BC − A2

(6.4.12)

and and the inequalities follow from the fact that V−1 (like V) is positive definite. This allows the solution to be written as: w=

γ λ (V−1 1) + (V−1 e). C A

(6.4.13)

1 (V−1 1) C

and A1 (V−1 e) are both unit portfolios, so γ + λ = W0 . We know that for the portfolio which minimises variance for a given initial wealth, regardless of expected final wealth, the corresponding Lagrange multiplier, λ = 0. Thus γ (V−1 1) is the global minimum variance portfolio with cost W0 (which in fact C Revised: December 2, 1998

120

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

equals γ in this case) and C1 (V−1 1) is the global minimum variance unit cost portfolio, which we will denote wMVP . In fact, we can combine (6.4.7) and (6.4.13) and write the solution as: w = W0

A wMVP + µ − h . C

(6.4.14)

The details are left as an exercise.4 The set of solutions to this quadratic programming problem for all possible (W0 , W1 ) combinations (including negative W0 ) is the vector subspace of the portfolio space, which is generated either by the vectors g and h or by the vectors wMVP and h (or by any pair of linearly independent frontier portfolios). In Vg =

(6.4.17) (6.4.18)

from which it follows that D > 0. Orthogonal decomposition of portfolios At this stage, we must introduce a scalar product on the portfolio space, namely that based on the variance-covariance matrix V. Since V is a non-singular, positive definite matrix, it defines a well behaved scalar product and all the standard results on orthogonal projection (&c.) from linear algebra are valid. 4

At least for now.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

121

Two portfolios w1 and w2 are orthogonal with respect to this scalar product ⇐⇒ w1> Vw2 = 0 ⇐⇒h i Cov w1>˜r, w2>˜r = 0 ⇐⇒ the random variables representing the returns on the portfolios are uncorrelated. Thus, the terms ‘orthogonal’ and ‘uncorrelated’ may legitimately, and shall, be applied interchangeably to pairs of portfolios. Furthermore, the squared length of a weight vector corresponds to the variance of its returns. Note that wMVP and h are orthogonal vectors in this sense. In fact, we have the following theorem: Theorem 6.4.1 If w is a frontier portfolio and u is a zero mean hedge portfolio, then w and u are uncorrelated. Proof There is probably a full version of this proof lost somewhere but the following can be sorted out. Since wMVP is collinear with V−1 1, it is orthogonal to all portfolios w for which w> VV−1 1 = 0 or in other words to all portfolios for which w> 1 = 0. But these are precisely all hedge portfolios, including h. Similarly, any portfolio collinear with V−1 e is orthogonal to all portfolios with zero expected return, since w> VV−1 e = 0 or in other words w> e = 0. Q.E.D. Some pictures are in order at this stage. For N = 3, in the set of portfolios costing W0 (the W0 simplex), the iso-variance curves are concentric ellipses, the iso-mean curves are parallel lines, and the solutions for different µs (or W1 s) are the tangency points between these ellipses and lines, which themselves lie on a line orthogonal (in the sense defined above) to the iso-mean lines. The centre of the concentric ellipses is at the global minimum variance portfolio corresponding to W0 , W0 wMVP . A similar geometric interpretation can be applied in higher dimensions. ? has some nice pictures of the frontier in portfolio space, as opposed to meanvariance space. At this stage, recall the definition of β in (5.2.2). We will now derive an orthogonal decomposition of a portfolio q into two frontier portfolios and a zero-mean zero-cost portfolio and prove that the coefficients on the two frontier portfolios are the βs of q with respect to those portfolios and sum to unity. We can always choose an orthogonal basis for the portfolio frontier. Revised: December 2, 1998

122

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

For any frontier portfolio p 6= wMVP , there is a unique unit cost frontier portfolio zp which is orthogonal to p. Another rp ] and h i important exercise is to figure out the relationship between E [˜ E r˜zp . Any two frontier portfolios span the frontier, in particular any unit cost p 6= wMVP and zp (or the original basis, wMVP and h). Any (frontier or non-frontier) portfolio q with non-zero cost W0 can be written in the form fq + uq where fq ≡ W0 (g + E[˜ rq ]h) = W0 (βqp p + (1 − βqp )zp ) (say)

(6.4.19) (6.4.20)

is the frontier portfolio with expected return E[˜ rq ] and cost W0 and uq is a hedge portfolio with zero expected return. Geometrically, this decomposition is equivalent to the orthogonal projection of q onto the frontier. Theorem 6.4.1 shown that any portfolio sharing these properties of uq is uncorrelated with all frontier portfolios.5 If p is a unit cost frontier portfolio (i.e. the vector of portfolio proportions) and q is an arbitrary unit cost portfolio, then the following decomposition therefore holds: q = fq + uq = βqp p + (1 − βqp ) zp + uq (6.4.21) where the three components (i.e. the vectors p, zp and uq ) are mutually orthogonal. We can extend this decomposition to cover 1. portfolio proportions (orthogonal vectors) 2. portfolio proportions (scalars/components) 3. returns (uncorrelated random variables) 4. expected returns (numbers) Note again the parallel between orthogonal portfolio vectors and uncorrelated portfolio returns/payoffs. We will now derive the relation: E[˜ rq ] − E[˜ rzp ] = βqp (E[˜ rp ] − E[˜ rzp ]) 5

(6.4.22)

Aside: For the frontier portfolio fq to second degree stochastically dominate the arbitrary portfolio q, we will need zero conditional expected return on uq , and will have to show that Cov r˜uq , r˜fq = 0 =⇒ E[˜ ruq |˜ rfq ] = 0 The normal distribution is the only case where this is true.

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

123

which may be familiar from earlier courses in financial economics and which is quite general and neither requires asset returns to be normally distributed nor any assumptions h about ipreferences. h i Since Cov r˜uq , r˜p = Cov r˜zp , r˜p = 0, taking covariances with r˜p in (6.4.21) gives: h i Cov [˜ rq , r˜p ] = Cov r˜fq , r˜p = βqp Var[˜ rp ] (6.4.23) or βqp =

Cov [˜ rq , r˜p ] Var[˜ rp ]

(6.4.24)

Thus β in (6.4.21) has its usual definition from probability theory, given by (5.2.2).6 Reversing the roles of p and zp , it can be seen that βqzp = 1 − βqp

(6.4.25)

Taking expected returns in (6.4.21) yields again: E[˜ rq ] = βqp E[˜ rp ] + (1 − βqp )E[˜ rzp ],

(6.4.26)

which can be rearranged to obtain (6.4.22). The Global Minimum Variance Portfolio Var[˜ rg+µh ] = g> Vg + 2µ(g> Vh) + µ2 (h> Vh)

(6.4.27)

which has its minimum at

g> Vh (6.4.28) h> Vh The latter expression reduces to A/C and the minimum value of the variance is 1/C. The global minimum variance portfolio is denoted MVP. µ=−

g> Vh Cov [˜ rh , r˜MVP ] = h V g − > h h Vh g> Vh.h> Vh =0 = h> Vg − h> Vh >

!

(6.4.29) (6.4.30)

i.e. the returns on the portfolio with weights h and the minimum variance portfolio are uncorrelated. 6

Assign some problems involving the construction of portfolio proportions for various desired βs. Also problems working from prices for state contingent claims to returns on assets and portfolios in both single period and multi-period worlds.

Revised: December 2, 1998

124

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

Further, if p is any portfolio, the itself and p, i.e. a = 0 solves:

MVP

is the minimum variance combination of

1 min Var[˜ rap+(1−a)MVP ] a 2

(6.4.31)

which has necessary and sufficient first order condition: aVar[˜ rp ] + (1 − 2a)Cov [˜ rp , r˜MVP ] − (1 − a)Var[˜ rMVP ] = 0

(6.4.32)

Hence, setting a = 0: Cov [˜ rp , r˜MVP ] − Var[˜ rMVP ] = 0

(6.4.33)

and the covariance of any portfolio with MVP is 1/C.

6.4.2 The portfolio frontier in mean-variance space: risky assets only The portfolio frontier in mean-variance and in mean-standard deviation space We now move on to consider the mean-variance relationship along the portfolio frontier. The mean, µ, and variance, σ 2 , of the rate of return associated with each point on the frontier are related by the quadratic equation: (σ 2 − Var[w> r]) = φ(µ − E[w> r])2 , MVP ˜ MVP ˜

(6.4.34)

where the shape parameter φ = C/D represents the variance of the (gross) return on the hedge portfolio, h. The two-moment frontier is generally presented as the graph in mean-variance space of this parabola, showing the most desirable distributions attainable, but the frontier can also be thought of as a plane in portfolio space or as a line in portfolio weight space. The latter interpretations are far more useful when it comes to extending the analysis to higher moments. The equations of the frontier in mean-variance and mean-standard deviation space can be derived heuristically using the following stylized diagram illustrating the portfolio decomposition. Figure 3A goes here.

Applying Pythagoras’ theorem to the triangle with vertices at 0, p and MVP yields: A σ = Var[˜ rp ] = Var[˜ rMVP ] + µ − C 2

Revised: December 2, 1998

2

Var[˜ rh ]

(6.4.35)

CHAPTER 6. PORTFOLIO THEORY

125

Recall from the coordinate geometry of conic sections that Var[˜ rp ] = Var[˜ rMVP ] + (µ − E[˜ rMVP ])2 Var[˜ rh ] or V (µ) =

1 C A + µ− C D C

2

(6.4.36)

(6.4.37)

is a quadratic equation in µ. i.e. the equation of the parabola with vertex at 1 C A µ = E[˜ rMVP ] = C

Var[˜ rp ] = Var[˜ rMVP ] =

(6.4.38) (6.4.39)

Thus in mean-variance space, the frontier is a parabola. Figure 3.11.2 goes here: indicate position of g on figure.

Similarly, in mean-standard deviation space, the frontier is a hyperbola. To see this, recall that: A 2 2 σ = Var[˜ rMVP ] + µ − Var[˜ rh ] (6.4.40) C is the equation of the hyperbola with vertex at σ =

q

µ =

A C

Var[˜ rMVP ] =

s

1 C

(6.4.41) (6.4.42)

centre at σ = 0, µ = A/C and asymptotes as indicated. Figure 3.11.1 goes here: indicate position of g on figure.

The other half of the hyperbola (σ < 0) has no economic meaning. Recall two other types of conic sections: Var[˜ rh ] < 0 (impossible) gives a circle with center (1/C, A/C). Var[˜ rMVP ] = 0 (the presence of a riskless asset) allows the square root to be taken on both sides: A q σ =± µ− Var[˜ rh ] (6.4.43) C i.e. the conic section becomes the pair of lines which are its asymptotes otherwise. Revised: December 2, 1998

126

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER

Portfolios on which the expected return, µ, exceeds w> MVP e are termed efficient, since they maximise expected return given variance; other frontier portfolios minimise expected return given variance and are inefficient. A frontier portfolio is said to be an efficient portfolio iff its expected return exceeds the minimum variance expected return A/C = E[˜ rMVP ]. N The set of efficient portfolios in < (or efficient frontier) is the half-line emanating from MVP in the direction of h, and hence is also a convex set. Convex combinations (but not all linear combinations with weights summing to 1) of efficient portfolios are efficient. We now consider zero-covariance (zero-beta) portfolios. In portfolio weight space, can easily construct a frontier portfolio having zero covariance with any given frontier portfolio: Figure 3B goes here.

Algebraically, the expected return µ0 on the zero-covariance frontier portfolio of a frontier portfolio with expected return µ solves: Cov [˜ rMVP + (µ − E[˜ rMVP ])˜ rh , r˜MVP + (µ0 − E[˜ rMVP ])˜ rh ] = 0

(6.4.44)

or, since r˜h and r˜MVP are uncorrelated: Var[˜ rMVP ] + (µ − E[˜ rMVP ])(µ0 − E[˜ rMVP ])Var[˜ rh ] = 0

(6.4.45)

To make this true, we must have (µ − E[˜ rMVP ])(µ0 − E[˜ rMVP ]) < 0

(6.4.46)

or µ and µ0 on opposite sides of E[˜ rMVP ] as shown. There is a neat trick which allows zero-covariance portfolios to be plotted in meanstandard deviation space. Implicit differentiation of the µ − σ relationship (6.4.35) along the frontier yields: dµ σ = dσ (µ − E[˜ rMVP ])Var[˜ rh ]

(6.4.47)

so the tangent at (σ, µ) intercepts the µ axis at µ−σ

dµ σ2 = µ− (6.4.48) dσ (µ − E[˜ rMVP ])Var[˜ rh ] Var[˜ rMVP ] = µ− − (µ − E[˜ rMVP ]) (6.4.49) (µ − E[˜ rMVP ])Var[˜ rh ] Var[˜ rMVP ] = E[˜ rMVP ] − (6.4.50) (µ − E[˜ rMVP ])Var[˜ rh ]

Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

127

where we substituted for σ 2 from the definition of the frontier. A little rearrangement shows that expression (6.4.50) satisfies the equation (6.4.45) defining the return on the zero-covariance portfolio. In mean-standard deviation space the picture is like this: Figure 3.15.1 goes here.

To find zp in mean-variance space, note that the line joining (σ 2 , µ) to the intercepts the µ axis at: µ − σ2

µ − E[˜ rMVP ] µ − E[˜ rMVP ] = µ − σ2 2 σ − Var[˜ rMVP ] (µ − E[˜ rMVP )2 Var[˜ rh ]

MVP

(6.4.51)

After cancellation, this is exactly the first expression (6.4.48) for the zero-covariance return we had on the previous page. Figure 3.15.2 goes here.

Alternative derivations The treatment of the portfolio frontier with risky assets only concludes with some alternative derivations following closely ?. They should probably be omitted altogether at this stage. 1. The variance minimisation solution from first principles. It can be seen that w is the solution to: 1 min L = w> Vw + λ(µ − w> e) + γ(W0 − w> 1) {w, , } 2

(6.4.52)

which has necessary and sufficient first order conditions: ∂L = Vw − λe − γ1 = 0 ∂w ∂L = µ − w> e = 0 ∂λ ∂L = W0 − w> 1 = 0 ∂γ

(6.4.53) (6.4.54) (6.4.55)

The solution can be found by premultiplying the FOC (6.4.13) in turn by e> and 1> and using the constraints yields: µ = λ(e> V−1 e) + γ(e> V−1 1) 1 = λ(1> V−1 e) + γ(1> V−1 1)

(6.4.56) (6.4.57)

Revised: December 2, 1998

128

6.4. MATHEMATICS OF THE PORTFOLIO FRONTIER The solutions for λ and γ are: Cµ − A D B − Aµ γ = D

λ =

(6.4.58) (6.4.59)

2. Derivation of (6.4.22). If we only have frontier portfolio p and interior portfolio q, we get a frontier (in µ-σ space) entirely within the previous frontier and tangent to it at p. The frontiers must have the same slope at p: Figure 3C goes here. E[˜ r −˜ r ] We already saw that the outer frontier has slope √ p zp . Var[˜ rp ]

At the point on the inner frontier with wq invested in q and (1 − wq ) in p, µ = E[˜ rp ] + wq (E[˜ rq − r˜p ]) (6.4.60) 2 2 σ = wq Var[˜ rq ] +2wq (1 − wq )Cov [˜ rp , r˜q ] + (1 − wq )2 Var[˜ rp ] (6.4.61) Differentiating these w.r.t. wq : dµ = E[˜ rq − r˜p ] (6.4.62) dwq dσ = 2wq Var[˜ rq ] 2σ dwq +2(1 − 2wq )Cov [˜ rp , r˜q ] − 2(1 − wq )Var[˜ rp(]6.4.63) Taking the ratio and setting wq = 0 gives the slope of the inner frontier at p: dµ E[˜ rq − r˜p ] = 2Cov[˜rp ,˜rq ]−2Var[˜rp ] (6.4.64) dσ √ 2

Var[˜ rp ]

Equating this to the slope of the outer frontier, setting βqp =

Cov [˜ rp , r˜q ] Var[˜ rp ]

(6.4.65)

and rearranging yields: E[˜ rq ] − E[˜ rzp ] = βqp (E[˜ rp ] − E[˜ rzp ]) Revised: December 2, 1998

(6.4.66)

CHAPTER 6. PORTFOLIO THEORY

129

6.4.3 The portfolio frontier in w Vw 2

(6.4.67)

w> e + (1 − w> 1)rf = µ

s.t.

(6.4.68)

There is no longer a restriction on portfolio weights, and whatever is not invested in the N risky assets is assumed to be invested in the riskless asset. The solution (which can be left as an exercise) is by a similar method to the case where all assets were risky: wp = V−1 (e − rf 1)

µ − rf H

(6.4.69)

where H = (e − 1rf )> V−1 (e − 1rf ) = B − 2Arf + Crf 2 > 0 ∀rf

(6.4.70)

Along the frontier, we have: µ−r √ f

H σ = µ−r −√ f H

if µ ≥ rf , if µ < rf ,

(6.4.71)

6.4.4 The portfolio frontier in mean-variance space: riskfree and risky assets We can now establish the shape of the mean-standard deviation frontier with a riskless asset. Graphically, in mean-standard deviation space, combining any portfolio p with the riskless asset in proportions a and (1 − a) gives a portfolio with expected return aE[˜ rp ] + (1 − a)rf = rf + a(E[˜ rp ] − rf ) q

and standard deviation of returns a Var[˜ rp ]. i.e. these portfolios trace out the ray in σ-µ space emanating from (0, rf ) and passing through p. For each σ the highest return attainable is along the ray from rf which is tangent to the frontier generated by the risky assets. Revised: December 2, 1998

130

6.5. MARKET EQUILIBRIUM AND THE CAPM

On this ray, the riskless asset is held in combination with the tangency portfolio t. This only makes sense for rf < A/C = E[˜ rmvp ]. Above t, there is a negative weight on the riskless asset — i.e. borrowing. Figure 3D goes here.

Limited borrowing Unlimited borrowing as allowed in the preceding analysis is unrealistic. Consider what happens 1. with margin constraints on borrowing: Figure 3E goes here.

The frontier is the envelope of all the finite rays through risky portfolios, extending as far as the borrowing constraint allows.

2. with differential borrowing and lending rates: Figure 3F goes here.

There is a range of expected returns over which a pure risky strategy provides minimum variance; lower expected returns are achieved by riskless lending; and higher expected returns are achieved by riskless borrowing.

6.5 Market Equilibrium and the Capital Asset Pricing Model 6.5.1 Pricing assets and predicting security returns Need more waffle here about prediction and the difficulties thereof and the properties of equilibrium prices and returns. We are looking for assumptions concerning probability distributions that lead to useful and parsimonious asset pricing models. The CAPM restrictions are the best known. At a very basic level, they can be expressed by saying that every Revised: December 2, 1998

CHAPTER 6. PORTFOLIO THEORY

131

investor has mean-variance preferences. This can be achieved either by restricting preferences to be quadratic or the probability distribution of asset returns to be normal. CAPM is basically a single-period model, but can be extended by assuming that return distributions are stable over time. ? and ? have generalised the distributional conditions. Recall also the limiting behaviour of the variance of the return on an equally weighted portfolio as the number of securities included goes to infinity. If securities are added in such a way that the average of the variance terms and the average of the covariance terms are stable, then the portfolio variance approaches the average covariance as a lower bound.

6.5.2 Properties of the market portfolio Let mj = weight of security j in the market portfolio m 0) = individual i’s initial wealth wij = proportion of individual i’s initial wealth invested in j-th security Then total wealth is defined by W0i (>

Wm0 ≡

I X

W0i

(6.5.1)

i=1

and in equilibrium the relation I X

wij W0i = mj Wm0

∀j

(6.5.2)

i=1

must hold. Dividing by Wm0 yields: I X

W0i wij = mj Wm0 i=1

∀j

(6.5.3)

and thus in equilibrium the market portfolio is a convex combination of individual portfolios.

6.5.3 The zero-beta CAPM Theorem 6.5.1 (Zero-beta CAPM theorem) If every investor holds a mean-variance frontier portfolio, then the market portfolio, m, is a mean-variance frontier portfolio, and hence, ∀q, the CAPM equation E [˜ rq ] = (1 − βqm ) E [˜ rzm ] + βqm E [˜ rm ]

(6.5.4)

holds. Revised: December 2, 1998

132

6.5. MARKET EQUILIBRIUM AND THE CAPM

Theorem 6.5.2 All strictly risk-averse investors hold frontier portfolios if and only if i h (6.5.5) rfq = 0 ∀q E r˜uq |˜ Note the subtle distinction between uncorrelated returns (in the definition of the decomposition) and independent returns (in this theorem). They are the same only for the normal distribution and related distributions. We can view the market portfolio as a frontier portfolio under two fund separation. If p is a frontier portfolio, then we showed earlier that for purely mathematical reasons in the definition of a frontier portfolio: E[˜ rq ] = (1 − βqp )E[˜ rzp ] + βqp E[˜ rp ]

(6.5.6)

If two fund separation holds, then individuals hold frontier portfolios. Since the market portfolio is then on the frontier, it follows that: E[˜ rq ] = (1 − βqm )E[˜ rzm ] + βqm E[˜ rm ]

(6.5.7)

N X

mj r˜j

(6.5.8)

Cov [˜ rq , r˜m ] Var[˜ rm ]

(6.5.9)

where r˜m =

j=1

βqm =

This implies for any particular security, from the economic assumptions of equilibrium and two fund separation: E[˜ rj ] = (1 − βjm )E[˜ rzm ] + βjm E[˜ rm ]

(6.5.10)

This relation is the ? Zero-Beta version of the Capital Asset Pricing Model (CAPM).

6.5.4 The traditional CAPM Now we add the risk free asset, which will allow us to determine the tangency portfolio, t, and to talk about Capital Market Line (return v. standard deviation) and the Security Market Line (return v. β). Normally in equilibrium there is zero aggregate supply of the riskfree asset. Recommended reading for this part of the course is ?, ?, ? and ?. Now we can derive the traditional CAPM. Note that by construction rf = E [˜ rzt ] . Revised: December 2, 1998

(6.5.11)

CHAPTER 6. PORTFOLIO THEORY

133

Theorem 6.5.3 (Separation Theorem) The risky asset holdings of all investors who hold mean-variance frontier portfolios are in the proportions given by the tangency portfolio, t. Theorem 6.5.4 (Traditional CAPM Theorem) If every investor holds a meanvariance frontier portfolio, then the market portfolio of risky assets, m, is the tangency portfolio, t, and hence, ∀q, the traditional CAPM equation E [˜ rq ] = (1 − βqm ) rf + βqm E [˜ rm ]

(6.5.12)

holds. Theorem 6.5.4 is sometimes known as the Sharpe-Lintner Theorem. The riskless rate is unique by the No Arbitrage Principle, since otherwise a greedy investor would borrow an infinite amount at the lower rate and invest it at the higher rate, which is impossible in equilibrium. We can also think about what happens the CAPM if there are different riskless borrowing and lending rates (see ?). If all individuals face this situation in equilibrium, realism demands that both riskless assets are in zero aggregate supply and hence that all investors hold risky assets only. Note that the No Arbitrage Principle also allows us to rule out correlation matrices for risky assets which permit the construction of portfolios with zero return variance, i.e. synthetic riskless assets. Assume that a riskless asset exists, with return rf < E[˜ rmvp ]. If the distributional conditions for two fund separation are satisfied, then the tangency portfolio, t, must be the market portfolio of risky assets in equilibrium. We know then that for any portfolio q (with or without a riskless component): E[˜ rq ] − rf = βqm (E[˜ rm ] − rf )

(6.5.13)

This is the traditional Sharpe-Lintner version of the CAPM. Figure 4A goes here.

The next theorem relates to the mean-variance efficiency of the market portfolio. Theorem 6.5.5 If 1. the distributional conditions for two fund separation are satisfied; 2. risky assets are in strictly positive supply; and 3. investors have strictly increasing (concave) utility functions then the market/tangency portfolio is efficient. Revised: December 2, 1998

134

6.5. MARKET EQUILIBRIUM AND THE CAPM

Proof By Jensen’s inequality and monotonicity, the riskless asset dominates any portfolio with E[˜ r] < rf (6.5.14) for E[u(W0 (1 + r˜))] ≤ u(E[W0 (1 + r˜)]) < u(W0 (1 + rf ))

(6.5.15) (6.5.16)

Hence the expected returns on all individuals’ portfolios exceed rf . It follows that the expected return on the market portfolio must exceed rf .

Q.E.D.

Now we can calculate the risk premium of the market portfolio. CAPM gives a relation between the risk premia on individual assets and the risk premium on the market portfolio. The risk premium on the market portfolio must adjust in equilibrium to give market-clearing. In some situations, the risk premium on the market portfolio can be written in terms of investors’ utility functions. Assume there is a riskless asset and returns are multivariate normal (MVN). Recall the first order conditions for the canonical portfolio choice problem: ˜ i )(˜ 0 = E[u0i (W rj − rf )] ∀ i, j h i ˜ i )]E[˜ ˜ i ), r˜j = E[u0i (W rj − rf ] + Cov u0i (W h

˜ i , r˜j ˜ i )]E[˜ ˜ i )]Cov W = E[u0i (W rj − rf ] + E[u00i (W

(6.5.17) (6.5.18) i

(6.5.19)

using the definition of covariance and Stein’s lemma for MVN distributions. Rearranging: h i E[˜ rj − rf ] ˜ i , r˜j = Cov W (6.5.20) θi where ˜ i )] −E[u00i (W θi ≡ (6.5.21) ˜ i )] E[u0i (W is the i-th investor’s global absolute risk aversion. Since ˜ i = W0i (1 + rf + W

N X

wik (˜ rk − rf ))

(6.5.22)

k=1

we have, dropping non-stochastic terms, h

i

"

˜ i , r˜j = Cov W0i Cov W

N X

k=1

Revised: December 2, 1998

wik r˜k , r˜j

#

(6.5.23)

CHAPTER 6. PORTFOLIO THEORY

135

Hence, N X E[˜ rj − rf ] = Cov W0i wik r˜k , r˜j θi k=1

"

Summing over i, this gives (since we have P clearing and k wmk r˜k = r˜m by definition): I X

E[˜ rj − rf ](

P

i

#

(6.5.24)

W0i wik = W0m wmk by market-

θi−1 ) = Wm0 Cov [˜ rm , r˜j ]

(6.5.25)

i=1

E[˜ rj − rf ] = (

or

I X

θi−1 )−1 Wm0 Cov [˜ rm , r˜j ]

(6.5.26)

i=1

i.e., in equilibrium, the risk premium on the j-th asset is the product of the aggregate relative risk aversion of the economy and the covariance between the return on the j-th asset and the return on the market. Now take the average over j weighted by market portfolio weights: I X

E[˜ rm − rf ] = (

rm ) θi−1 )−1 Wm0 Var[˜(]˜

(6.5.27)

i=1

i.e., in equilibrium, the risk premium on the market is the product of the aggregate relative risk aversion of the economy and the variance of the return on the market. Equivalently, the return to variability of the market equals the aggregate relative risk aversion. We conclude with some examples. 1. Negative exponential utility: ui (z) = −

1 exp{−ai z} ai

ai > 0

(6.5.28)

implies: I X

(

I X

θi−1 )−1 = (

i=1

−1 a−1 >0 i )

(6.5.29)

i=1

and hence the market portfolio is efficient. 2. Quadratic utility: ui (z) = ai z −

bi 2 z 2

ai , b i > 0

(6.5.30)

implies: I X

(

i=1

θi−1 )−1

=

I X ai i=1

bi

!−1

˜ i] − E[W

(6.5.31)

This result can also be derived without assuming MVN and using Stein’s lemma. Revised: December 2, 1998

136

Revised: December 2, 1998

6.5. MARKET EQUILIBRIUM AND THE CAPM

CHAPTER 7. INVESTMENT ANALYSIS

137

Chapter 7 INVESTMENT ANALYSIS 7.1 Introduction [To be written.]

7.2 Arbitrage and the Pricing of Derivative Securities 7.2.1 The binomial option pricing model This still has to be typed up. It follows very naturally from the stuff in Section 5.4.

7.2.2 The Black-Scholes option pricing model Fischer Black died in 1995. In 1997, Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics ‘for a new method to determine the value of derivatives.’ See http://www.nobel.se/announcement-97/economy97.html Black and Scholes considered a world in which there are three assets: a stock, whose price, S˜t , follows the stochastic differential equation: dS˜t = µS˜t dt + σ S˜t d˜ zt , where {˜ zt }Tt=0 is a Brownian motion process; a bond, whose price, Bt , follows the differential equation: dBt = rBt dt; and a call option on the stock with strike price X and maturity date T . Revised: December 2, 1998

138

7.2. ARBITRAGE AND PRICING DERIVATIVE SECURITIES

They showed how to construct an instantaneously riskless portfolio of stocks and options, and hence, assuming that the principle of no arbitrage holds, derived the Black-Scholes partial differential equation which must be satisfied by the option price. The option pays (ST − X)+ ≡ max {ST − X, 0} at maturity. Let the price of the call at time t be C˜t . Guess that C˜t = C(S˜t , t). By Ito’s lemma: dC˜t =

∂C ∂C ˜ 1 ∂ 2 C 2 ˜2 ∂C ˜ + µSt + σ St dt + σ St d˜ zt 2 ∂t ∂S 2 ∂S ∂S !

The no arbitrage principle yields the partial differential equation: ∂C 1 ∂ 2 C 2 2 ∂C + σ S + rS − rC = 0 ∂t 2 ∂S 2 ∂S subject to the boundary condition C(S, T ) = (S − X)+ . Let τ = T − t be the time to maturity. Then we claim that the solution to the Black-Scholes equation is: √ C(S, t) = SN (d (S, τ )) − Xe−rτ N d (S, τ ) − σ τ ,

where N (·) is the cumulative distribution function of the standard normal distribution and

ln XS + r − 12 σ 2 τ √ √ d (S, τ ) = +σ τ σ τ ln XS + r + 12 σ 2 τ √ = . σ τ

(7.2.1) (7.2.2)

We can check that this is indeed the solution by calculating the various partial derivatives and substituting them in the original equation. Note first that Z z 1 2 1 √ e− 2 t dt N (z) ≡ −∞ 2π and hence by the fundamental theorem of calculus 1 2 1 N 0 (z) ≡ √ e− 2 z , 2π

Revised: December 2, 1998

CHAPTER 7. INVESTMENT ANALYSIS

139

which of course is the corresponding probability density function. For the last step in this proof, we will need the partials of d (S, τ ) with respect to S and t, which are:

r + 12 σ 2 ∂d (S, τ ) ∂d (S, τ ) √ =− = − ∂t ∂τ 2σ τ and

∂d (S, τ ) 1 √ . = ∂S Sσ τ

(7.2.3)

(7.2.4)

Note also that √ √ 1 2 N 0 d (S, τ ) − σ τ = e− 2 σ τ ed(S,τ )σ τ N 0 (d (S, τ )) 1

= e− 2 σ =

2τ

S (r+ 12 σ2 )τ e N 0 (d (S, τ )) X

S rτ 0 e N (d (S, τ )) . X

(7.2.5) (7.2.6) (7.2.7)

Using these facts and the chain rule, we have: ∂C ∂t

∂d (S, τ ) − Xe−rτ × ∂t ! ! ∂d (S, τ ) √ √ σ N 0 d (S, τ ) − σ τ − √ + rN d (S, τ ) − σ τ ∂t 2 τ (7.2.8) √ σ = −SN 0 (d (S, τ )) √ − Xe−rτ rN d (S, τ ) − σ τ (7.2.9) 2 τ ∂C ∂S ∂d (S, τ ) = SN 0 (d (S, τ )) + N (d (S, τ )) ∂S √ ∂d (S, τ ) −Xe−rτ N 0 d (S, τ ) − σ τ (7.2.10) ∂S = N (d (S, τ )) (7.2.11) 2 ∂ C ∂S 2 ∂d (S, τ ) = N 0 (d (S, τ )) . (7.2.12) ∂S = SN 0 (d (S, τ ))

Substituting these expressions in the original partial differential equation yields: ∂C 1 ∂ 2 C 2 2 ∂C + σ S + rS − rC ∂t 2 ∂S 2 ∂S Revised: December 2, 1998

140

7.3. MULTI-PERIOD INVESTMENT PROBLEMS !

−Sσ 1 ∂d (S, τ ) = N (d (S, τ )) (rS − rS) + N (d (S, τ )) √ + σ 2 S 2 2 τ 2 ∂S √ −rτ −rτ +N d (S, τ ) − σ τ −Xe r + rXe (7.2.13) = 0. (7.2.14) 0

The boundary condition should also be checked. As τ → 0, d (S, τ ) → ±∞ according as S > X or S < X. In the former case, C (S, T ) = S − X; and in the latter case, C (S, T ) = 0, so the boundary condition is indeed satisfied.

7.3 Multi-period Investment Problems In Section 4.2, it was pointed out that the objects of choice can be differentiated not only by their physical characteristics, but also both by the time at which they are consumed and by the state of nature in which they are consumed. These distinctions were suppressed in the intervening sections but are considered again in this section and in Section 5.4 respectively. The multi-period model should probably be introduced at the end of Chapter 4 but could also be left until Chapter 7. For the moment this brief introduction is duplicated in both chapters. Discrete time multi-period investment problems serve as a stepping stone from the single period case to the continuous time case. The main point to be gotten across is the derivation of interest rates from equilibrium prices: spot rates, forward rates, term structure, etc. This is covered in one of the problems, which illustrates the link between prices and interest rates in a multiperiod model.

7.4 Continuous Time Investment Problems ?

Revised: December 2, 1998

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close