Asset Pricing Teaching Notes

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Asset PricingTeaching Notes

Joao Pedro PereiraFinance Department

ISCTE Business School - Lisbon

[email protected]

www.iscte.pt/

jpsp

September 9, 2013


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Contents

1 Introduction 5

2 Choice theory 7

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Choice under certainty . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Choice under uncertainty . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Interpretation of utility numbers . . . . . . . . . . . . . . . . . . . 11

2.3 Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Measures of risk aversion . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Risk neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Important utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Certainty Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Stochastic dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6.1 First Order Stochastic Dominance . . . . . . . . . . . . . . . . . . 182.6.2 Second Order Stochastic Dominance . . . . . . . . . . . . . . . . . 19

2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Portfolio choice 243.1 Canonical portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Analysis of the optimal portfolio choice . . . . . . . . . . . . . . . . . . . 26

3.2.1 Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Canonical portfolio problem for N > 1 . . . . . . . . . . . . . . . . . . . . 313.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Portfolio choice for Mean-Variance investors 354.1 Mean-Variance preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Quadratic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.2 Normal returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Review: Mean-Variance frontier with 2 stocks . . . . . . . . . . . . . . . . 39

2


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Contents 3

4.3 Setup for general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.2 Brief notions of matrix calculus . . . . . . . . . . . . . . . . . . . . 41

4.4 Frontier with N risky assets . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4.1 Efficient portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4.2 Frontier equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.3 Global minimum variance portfolio . . . . . . . . . . . . . . . . . . 45

4.5 Frontier with N risky assets and 1 risk-free asset . . . . . . . . . . . . . . 454.5.1 Efficient portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5.2 Frontier equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5.3 Tangency portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Optimal portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Additional properties of frontier portfolios . . . . . . . . . . . . . . . . . . 504.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Capital Asset Pricing Model 545.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Capital Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.2 Security Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Other remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Arbitrage Pricing Theory and Factor Models 61

6.1 Factor Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Example of simple factor structure: Market Model . . . . . . . . . . . . . 63

6.2.1 Return generating process . . . . . . . . . . . . . . . . . . . . . . . 636.2.2 Application: the Covariance matrix is simplified . . . . . . . . . . 636.2.3 Implication: Diversification eliminates Specific risk . . . . . . . . . 646.2.4 Another interpretation of the CAPM . . . . . . . . . . . . . . . 65

6.3 Pricing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.1 Exact factor pricing with one factor . . . . . . . . . . . . . . . . . 676.3.2 Exact factor pricing with more than one factor . . . . . . . . . . . 686.3.3 Approximate factor pricing . . . . . . . . . . . . . . . . . . . . . . 70

6.4 How to identify the factors . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.4.2 Fama and French model . . . . . . . . . . . . . . . . . . . . . . . . 71

6.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5.1 Fund performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5.2 Market neutral strategy . . . . . . . . . . . . . . . . . . . . . . . . 74

6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


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Contents 4

7 Pricing in Complete Markets 777.1 Basic and Complex securities . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Computing AD prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3.1 Price of complex securities . . . . . . . . . . . . . . . . . . . . . . . 797.3.2 Quick test for market completeness . . . . . . . . . . . . . . . . . . 80

7.4 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.4.1 Price of complex securities . . . . . . . . . . . . . . . . . . . . . . . 817.4.2 Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8 Consumption-Based Asset Pricing 868.1 The investors problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.2 Fundamental Asset Pricing Equation . . . . . . . . . . . . . . . . . . . . . 888.3 Relation to Arrow-Debreu Securities . . . . . . . . . . . . . . . . . . . . . 898.4 Relation to the Risk-Neutral measure . . . . . . . . . . . . . . . . . . . . 908.5 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.6 Consumption CAPM (CCAPM) . . . . . . . . . . . . . . . . . . . . . . . 928.7 The CAPM reloaded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9 Conclusion 99

Bibliography 100

A Background Review 102A.1 Math Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.1.1 Logarithm and Exponential . . . . . . . . . . . . . . . . . . . . . . 102A.1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.1.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.1.4 Means and Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A.2 Undergraduate Finance Review . . . . . . . . . . . . . . . . . . . . . . . . 107A.2.1 Financial Markets and Instruments . . . . . . . . . . . . . . . . . . 107A.2.2 Time value of money . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.2.3 Risk and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A.2.4 Equilibrium and No Arbitrage . . . . . . . . . . . . . . . . . . . . 111

B Solutions to Problems 112


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Chapter 1

Introduction

These notes follow Danthine and Donaldson (2005) closely, though we will use othersources as needed. We will start by analyzing individual choices and portfolio decisions.Then, we will study the prices that result from the interaction of many individuals inthe market.

To motivate the work to come, consider the following question:

What is the role of financial markets?

Answer: allowing the desynchronization of agents income and consumption. Ex-ample: buy a house now and pay for it during the next 20 years. This is achieved bytrading financial securities with financial institutions.

Preference for smooth consumption

Financial economists see the world in two dimensions. It is useful to understand whyagents want to dissociate consumption and income across these two dimensions.

1. Time Dimension. Most people prefer to smooth their consumption through

their life cycle. Usually, consumption is higher than income during early yearsof life (buy the house), then people save during active life (y > c), finally peopleconsume their savings after retirement (y = 0, c > 0).

2. Risk Dimension. The future is uncertain. At any point in the future, one ofmany states of nature will be realized.1 Most people want to smooth consumption

1A state of nature is a complete description of a possible scenario for the future across all thedimensions relevant for the problem at hand.

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6

across the different possibilities that may arise. Thats why people buy healthinsurance (to be able to consume even if they stop working) or fire insurance forthe new house (avoid low consumption in the burned to the ground state of

nature).

Financial assets serve precisely to move consumption through time and across statesof nature.

Modelling the preference for smoothness

Financial economics builds on the fact that people have a preference for smoothness, asjust mentioned. How to model this preference for smoothness, also called risk aversion?

Consider two assets that offer two different consumption plans:

asset 1 asset 2

time/state 1 4 3time/state 2 4 5

Since investors like smoothness, they must prefer asset 1.2 Let U(c) be the utilityfunction, i.e., it tells us how much the investor likes consumption c. The utility functionmust thus satisfy

U(4) + U(4) > U(3) + U(5)

U(4) > 12

U(3) + 12

U(5)

What shape must U(.) have to satisfy this condition?3 Plot it:

-c

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U(c)

2Suppose your employer offers you the following salary scheme: under scheme 1, you get$4,000 per month; under scheme 2, you get $3,000 if it rains or $5,000 if it is sunny. Whichscheme would you take?

3Answer: It must be strictly concave


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Chapter 2

Choice theory

1. Under certain conditions, investors preferences can be representedby a utility function,

x y E[U(x)] E[U(y)]

2. Typical utility functions:

U(w) = ln(w) [CRRA]U(w) = w1/(1 ) [CRRA]U(w) = exp(w) [CARA]U(w) = aw bw2

2.1 Motivation

We want to find a method to choose between risky assets. Consider the following simpleexample:

Example 2.1.1. There are 3 assets and 2 equally likely possible states ofnature in the future:

t = 0 t = 1state = 1 state = 2

asset 1 -1000 1030 1050asset 2 -1000 1012 1070asset 3 -1000 1030 1100

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2.2. The utility function 8

Which asset would you rather have? In this case, the choice is easy. Asset 3 clearly dominates the other assets, since it pays at least as much in all states ofnature, and strictly more in some states. This is an example of state-by-state

dominance.

State-by-state dominance is the strongest possible form of dominance. We can safelyassume that all rational agents will always prefer asset 3.1

However, the world is not that simple and we will not usually be able to use thisconcept to make choices. (Is it likely we will observe a market like in this example? Whynot?)

Suppose now that asset 3 does not exist. Do you prefer asset 1 or asset 2? The choice is not obvious... To understand the choices people make in the real world we needa better machinery utility theory.

2.2 The utility function

To be able to represent agents preferences by a formal mathematical object like a func-tion, we need to make precise assumptions about how people make choices.2

2.2.1 Choice under certainty

We start by postulating the existence of a preference relation. For two consumptionbundles a and b (two vectors with the amount of consumption of each good), we eithersay that

a b a is strictly preferred to ba b a is indifferent to ba b a is strictly preferred or indifferent to b (a not worse than b)

We make the following economic rationality assumptions:

A1: Every investor possesses a complete preference relation. I.e., he must be able tostate a preference for all a and b.

1More precisely, we are assuming agents to be nonsatiated in consumption (always like moreconsumption)

2People have wasted time thinking about reformulating the canonical portfolio problem justbecause they were not aware of the axioms that lead to an expected utility representation.


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A2: The preference relation satisfies the property of transitivity:

a, b, c, a b and b c a c

A3: The preference relation is continuous.3

Under these circumstances, we can now state the following useful theorem:

Theorem 2.2.1. Assumptions A13 are sufficient to guarantee the existence of a con-tinuous function u : RN R such that, for any consumption bundles a and b,

a b u(a) u(b)This real-valued function u is called a utility function.

Note that the notion of consumption bundle used in the theorem is quite general.Different elements of the bundle may represent the consumption of the same good indifferent time periods or in different states of nature.

2.2.2 Choice under uncertainty

Even thought the previous thm is quite general, we want to extend it in a way thatcaptures uncertainty explicitly and separates utility from probabilities.

Definition (Lottery). The simple lottery (x,y ,) is a gamble that offers payoff x withprobability and payoff y with probability 1 .

This notion of lottery is quite general. The payoffs x and y can represent monetaryor consumption amounts. If there is no uncertainty, we can write

(x,y, 1) = x

The payoffs can themselves be other lotteries, leading to compound lotteries. For exam-ple, if y = (y1, y2, ), we will have

(x,y ,) = (x, (y1, y2, ), )

We assume that the agent is able to work out the probability tree and only cares aboutthe final outcomes.4

Assume the following axioms:

3Technical assumption. See Danthine and Donaldson (2005) for details on this and Huangand Litzenberger (1988) for further technical details.

4A lottery is the simplest example of a random variable. Stock prices are random variables,so you can see where we are going.


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B1: There exists a preference relation , defined on lotteries, which is complete, tran-sitive, and continuous.

Since the consumption bundles in theorem 2.2.1 where general enough to includeconsumption in different states of nature, it can be applied here to ensure that thereexists a utility function U() defined on lotteries. To get an expected utility representationof preferences, we need the following crucial axiom:

B2: Independence of irrelevant alternatives. Let (x,y ,) and (x,z ,) be any twolotteries. Then,

y z (x,y ,) (x,z ,)

In other words, x is irrelevant; including it does not change the investors preferences

about y and z.

This axiom is not trivial and has been strongly contested. One well know violation isthe Allais Paradox.5 This and other violations have lead to the exploration of alternativesto the expected utility framework, namely to the growing field of Behavioral Finance.Despite this, recall that the goal of financial economics is to understand the aggregatemarket behavior and not individual behavior. At this point, expected utility is the mostuseful framework.

We now get to the punchline:

Theorem 2.2.2 (Expected Utility Theorem). If axioms B12 hold, then there exists areal-valued function U, defined on the space of lotteries, such that the preference relationcan be represented as an expected utility, that is, for any lotteries x and y,

x y E[U(x)] E[U(y)]

The function U(), defined over lotteries, is called a von Neumann-Morgenstern (vNM)utility function.6

5Allais Paradox. Given the four lotteries defined below, most people show the followingpreferences:

L1 = ($10000, $0, 0.10)

L2 = ($15000, $0, 0.09)

andL3 = ($10000, $0, 1.00) L4 = ($15000, $0, 0.90)

However, given that L1 = (L3, $0, 0.1) and L2 = (L4, $0, 0.1), with $0 the irrelevant alternative,the independence axiom would imply L3 L4 L1 L2 !

6This designation is sometimes confusing. Some people define U := E[U()] and call this Uthe vNM utility function, while others call vNM to the u() defined on sure things. Nonetheless,it is always used in the context of preferences that have an expected utility representation theorem 2.2.2


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Note that x and y can be lotteries with multiple outcomes. Denoting by xs theoutcome in state s that occurs with probability s,

7 we have

E[U(x)] =

s U(xs)s x is a discrete r.v.s U(xs)s ds x is a continuous r.v.

Example 2.2.1. Let U(x) =

x. Choose between assets 1 and 2 in example2.1.1.

2.2.3 Interpretation of utility numbers

The numbers returned by the utility function do not have any meaning per se, as thefollowing proposition makes clear.

Proposition 2.2.1. If U(x) is a vNM utility function for a given preference relation,then V(x) = aU(x) + b, a > 0, is also a vNM utility function for the same preferencerelation, that is,

E[U(x)] E[U(y)] E[V(x)] E[V(y)]

Proof.

E[U(x)] E[U(y)] aE[U(x)] + b aE[U(y)] + b, since a > 0 E[aU(x) + b] E[aU(y) + b] E[V(x)] E[V(y)]

Example 2.2.2. Suppose a different investor has utility V(x) = 1+2

x. Hischoice between assets 1 and 2 (from example 2.1.1) will be the same as thechoice of the investor with U(x) =

x. (Check it!)

Hence, the utility function serves only to rank the choices under consideration. Theprecise magnitude of the number does not have any meaning. It is in this sense thatutility is said to be cardinal.

7More often, especially in probability classes, the state of nature is denoted by , andthe probability measure by P().


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2.3. Risk aversion 12

2.3 Risk aversion

2.3.1 Concepts

Consider an investor with wealth Y. Consider also the fair gamble, or lottery, L =(+h, h, 1/2). Definition (Risk aversion). An investor displays risk aversion if he wishes to avoid afair gamble, i.e., Y Y + L.

This implies that the utility function of a risk-averse agent must satisfy

E[U(Y)] > E[U(Y + L)]

U(Y) > 12

U(Y + h) + 12

U(Y h)

This inequality is satisfied for all wealth levels if the utility function is strictly con-cave.8 Plot it:

-Y

6

U(Y)

For twice differentiable utility functions, the sufficient condition for concavity isthat U(Y) < 0. This means that U(Y) is decreasing in wealth. This importanteconomic concept is called decreasing marginal utility. As wealth increases, the utility

from additional consumption decreases. When I am starving, a sandwich tastes great,while when I am almost satiated I dont care about another sandwich.

8This is formally justified by Jensens inequality: E[g(X)] g(E[X]), for concave g. If g isstrictly concave, the inequality is strict. For the utility function in particular, E[U(Y + L)] 0

Check that ARA = 0 and RRA = 0, which implies (Y, h) = (Y, ) = 1/2. Hence,

risk neutral investors are indifferent to fair games (i.e., symmetrical games with 5050chances).

They will always choose the asset with highest expected payoff, regardless of its risk.

2.4 Important utility functions

The most common utility functions are the following:

Name U(Y) = Restrictions ARA RRAon parameters

Log ln(Y) na

Power Y1/(1 )

Exponential exp(Y)

Quadratic aY bY2


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2.5. Certainty Equivalent 16

Complete the table. In particular, define the restrictions on parameters s.t. the functions are proper utility functions, i.e., U > 0 and U < 0. Note that the quadraticutility function also needs a restriction on the domain (Y < . . . ). Also, compute the

ARA and RRA functions, and classify the corresponding utility as increasing, decreasing,or constant ARA/RRA.

As mentioned above, the power (and log) utility are considered good utility func-tions. Typical values for the degree of risk-aversion are = 1, 2, 3, 5. The other twoutility functions are not so good descriptors of human behavior (as you can see by theARA and RRA functions you got). As we will see in later sections, the exponential utilityis used because it simplifies the calculations when asset returns are normally distributed,and the quadratic utility simplifies them even further for any distribution.

2.5 Certainty Equivalent

How much is an investor willing to pay for a risky asset? Consider an investor withinitial wealth Y. Consider a gamble Z = (Z1, Z2, ).

Definition (Certainty Equivalent). CE(Y, Z), the certainty equivalent of the risky in-vestment Z, is the certain amount of money which provides the same utility as thegamble, i.e.,

E[U(Y + Z)] = U(Y + CE)

The investor is indifferent between receiving CE(Y, Z) for sure and playing the gam-ble Z. In other words, if the investor owns the asset, he is willing to sell it at a priceequal to the certainty equivalent. The CE is useful to compare different assets in moreintuitive terms (money, instead of utility numbers).

Note that a risk-averse agent will always value an asset at something less than itsexpected payoff: CE < E[Z].13

12Thinking about the cross section of assets, note that (2.3) allows different assets to havedifferent expected returns: increases with , and thus the expected return also increases with. Does this make sense? Think about risk!

13Let Z be any random variable. Since U is strictly concave (U < 0), from Jensens inequality,

E[U(Y + Z)] < U(E[Y + Z]) = U(Y + E[Z])

Hence, from the definition of CE,

U(Y + CE) < U(Y + E[Z])

Since U is increasing (U > 0), we must have

Y + CE < Y + E[Z] CE < E[Z]


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2.5. Certainty Equivalent 17

Example 2.5.1. The investor has log utility and initial wealth Y = 1000. Therisky investment is Z = (200, 0, 0.5). Compute the CE:

E[U(Y + Z)] = U(Y + CE) . . .CE = 95.45

Why is the investor willing to accept less than the expected value of the gamble,ie, why is CE = 95.45 < E[Z] = 100? Risk aversion.

Plot the utility function, marking the points Y + Z1, Y + Z2, Y + EZ,Y + CE.

-Y

6

U(Y)

Consider now a fair gamble:

Example 2.5.2. The investor has log utility and initial wealth Y = 100. Therisky prospect is Z = (20, 20, 0.5). We get:

E[U(Y + Z)] = U(Y + CE)

1/2 ln(120) + 1/2 ln(80) = ln(100 + CE)CE = 2.02

What does it mean the CE to be negative? Plot the utility function, marking the points Y + Z1, Y + Z2, Y + EZ,Y + CE.


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2.6. Stochastic dominance 18

2.6 Stochastic dominance

We now reverse gears and look for circumstances where the ranking among randomvariables is preference free, that is, where we do not need to specify a utility function. Wewill develop two concepts of dominance that are weaker, thus more broadly applicable,than state-by-state dominance.

2.6.1 First Order Stochastic Dominance

Consider two assets, X1, X2, with the following payoffs:

Payoff

State (s) Prob(s) X1 X21 0.4 10 102 0.4 100 1003 0.2 100 2000

Clearly, all rational investors prefer X2: it at least matches X1 and has a positiveprobability of exceeding it.

To formalize this intuition, let Fi(x) denote the cumulative distribution function ofXi, that is, Fi(x) = Prob[Xi x].Definition (1SD). Fa(x) 1SD Fb(x) Fa(x) Fb(x), x

Plot the two distribution functions in the example and check that F2(x) F1(x), x.Note that if the distribution of X2 is always below X1, then the probability of X2 exceeding a given payoff is always larger, that is,

F2(x) F1(x) 1 F2(x) 1 F1(x) Prob[X2 x] Prob[X1 x], x

The usefulness of this concept comes from the following theorem:

Theorem 2.6.1.Fa(x) 1SD Fb(x)

Ea[U(x)] Eb[U(x)] for all nondecreasing Uwhere Ei is the expectation under the distribution of i, Ei[U(x)] =

U(x) dFi(x) =

U(x)fi(x) dx.

Hence, all nonsatiable investors prefer asset X2.

Note that 1SD is not the same as state-by-state dominance. See exercise 4.8 inDanthine and Donaldson (2005).


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2.6. Stochastic dominance 19

2.6.2 Second Order Stochastic Dominance

1SD is still a very strong condition, thus not applicable to most situations. If we addthe assumption of risk aversion, we get the much more useful concept of Second OrderStochastic Dominance (2SD).

Consider the following investments:

X3 X4Payoff Prob Payoff Prob

4 0.25 1 0.335 0.50 6 0.339 0.25 8 0.33

Plot the two distribution functions. Even though no investment 1SD the other, intuitively X3 looks better. To make this precise:

Definition (2SD). Fa(x) 2SD Fb(x) x Fa(s) ds

x Fb(s) ds, x x

[Fb(s) Fa(s)] ds 0, xThat is, at any point the accumulated difference between Fb and Fa must be positive.

Note that 1SD implies 2SD, but the converse is not true.

In the plot of the previous example, this basically means that the area of the differencewhere F3 > F4 is small. To make this a bit more precise, we can compute the integrals at all relevant jump points.

x F3(x) x0 F3(s)ds F4(x) x0 F4(s)ds x0 F4(s)ds x0 F3(s)ds1 0.00 0 1/3 0 0 04 0.25 0 1/3 1 1 05 0.75 0.25 1/3 4/3 13/12 06 0.75 1.00 2/3 5/3 2/3 08 0.75 2.50 3/3 3 0.50 09 1.00 3.25 3/3 4 0.75 0

The last columns shows thatx[F4(s) F3(s)] ds 0, x. (After x = 9, the difference

between the two integrals will always be 0.75 0.)All risk averse investors will prefer X3, as the following theorem shows.

Theorem 2.6.2.Fa(x) 2SD Fb(x)

Ea[U(x)] Eb[U(x)] for all nondecreasing and concave U

Note that risk aversion is enough, i.e., we do not have to assume a specific utilityfunction.


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2.7. Exercises 20

Mean preserving spread. The concept of 2SD is even more useful to understandthe tradeoff between risk and return.

Definition. Suppose there exists a random variable Z s.t. Xb = Xa+Z, with E[Z|Xa] =0 for all values of Xa. Then, we say that Xb is a mean preserving spread of Xa. (Or Fbor fb is a m.p.s. of Fa or fa).

Note that Xb has the same mean as Xa, but it is more noisy, i.e., risky. Intuitively, allrisk averse investors should prefer the payoff with less risk, Xa. The following theorem

justifies this intuition:

Theorem 2.6.3. LetFa(x) andFb(x) be two distribution functions with identical means.Then,

Fa(x) 2SD Fb(x)

Fb is a mean preserving spread of Fa

Mean-Variance criterion. This popular investment criterion states that: (i) fortwo investments with the same mean, investors prefer the one with smaller variance; (ii)for two investments with the same variance, investors prefer the one with higher mean.We will discuss later the exact conditions for this criterion to be true. For now, notethat theorem 2.6.3 helps to explain part (i).

2.7 Exercises

Ex. 1 (This is problem 3.1. in Danthine and Donaldson (2005))Utility function. Under certainty, any increasing monotone transformation of a utilityfunction is also a utility function representing the same preferences. Under uncertainty,we must restrict this statement to linear transformations if we are to keep the samepreference representation.Check it with this example. Assume an initial utility function attributes the followingvalues to 3 perspectives:

B u(B) = 100M u(M) = 10P u(P) = 50

a. Check that with this initial utility function, the lottery L = (B,M, 0.50) P.b. The proposed transformations are f(x) = a + bx,a 0, b > 0 and g(x) = ln(x).Check that under f, L P, but that under g, P L.


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2.7. Exercises 21

Ex. 2 (This is problem 3.3. in Danthine and Donaldson (2005))Inter-temporal consumption. Consider a two-date economy and an agent with utilityfunction over consumption:

U(c) = c1

1 , > 0

at each period. Define the inter-temporal utility function as V(c1, c2) = U(c1) + U(c2).Show that the agent will always prefer a smooth consumption stream to a more variableone with the same mean, that is,

U(c) + U(c) > U(c1) + U(c2), if c =c1 + c2

2

1. Start by showing that the utility function U is concave.

2. Then, show the required relation geometrically.3. Finally, do the proof formally.

Hint: use the following definition of a concave function. A function f : RN R1is concave if

f(ax + (1 a)y) af(x) + (1 a)f(y), x, y RN and a [0, 1]

Ex. 3 An agent with wealth = 100 is faced with the following game: with probability1/2 his wealth will increase to 200; with probability 1/2 it will decrease to 0. Completethe following sentence:If the agent is a risk- he is willing to pay some money to play

this game, whereas if he is risk- he is willing to pay some moneyto avoid the game.

Ex. 4 The ARA and RRA measures have the first derivative of the utility functionin the denominator. Why? Hint: read Danthine and Donaldson (2005)

Ex. 5 Prove equation (2.2).

Ex. 6 Complete the table in section 2.4 and plot the utility functions.

Ex. 7 The CRRA utility function is usually presented as

U(W) =ln(W) , = 1

W1/(1 ) , > 1

because ln(W) is almost the limiting case as 1. More precisely, the true limit islim1

W111 = ln(W).

1. Explain why U1(W) =W1

1 and U2(W) =W11

1 represent exactly the samepreferences.


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2.7. Exercises 22

2. Prove that

lim1

W1 11

= ln(W)

Hint: LHopitals rule.

Ex. 8 Consider the utility function U(Y) = 5 + 10Y2. What does it imply in termsof risk-taking behavior? Would it be economically reasonable to model an investorsbehavior with this utility function?

Ex. 9 An investor has an initial wealth of Y = 10. To play a game where he couldwin or loose 5% of his wealth, he demands = 0.6, where is the probability of thefavorable outcome (winning 5%). Nonetheless, if his wealth were Y = 1000, he wouldstill demand the same = 0.6 to play the game.

1. What can you say about the risk characteristics of this investor? (One sentenceanswer).

2. Give an example of an utility function consistent with this behavior.

Ex. 10 The risk-aversion characteristics of an investor can be described by twofunctions: ARA and RRA.

1. Give a very brief definition in words of these two measures.

2. What does it mean to say that an investor has increasing ARA? Does it makeintuitive sense? Give an example of an utility function with this characteristic.

3. Give an example of an utility function with constant RRA (compute the actualcoefficient of RRA).

Ex. 11 An investor with initial wealth Y0 = 100 is faced with the following lottery:win 20 with 0.3 probability; loose 20 with 0.7 probability. The utility function is U(W) =ln(W). What is the Certainty Equivalent of this lottery? What does this number mean?

Ex. 12 Consider the following risky investment: Z = (100, 0, 0.5). The investor haslog utility, U = ln(Y).

1. If the initial wealth is Y = 100, what is the certainty equivalent of the gamble?

2. If the initial wealth is Y = 1, what is the certainty equivalent of the gamble?

3. Explain in simple terms the change in CE.

Ex. 13 Exercise 4.5 in Danthine and Donaldson (2005, p.354)

Ex. 14 Exercise 4.7 in Danthine and Donaldson (2005, p.355). They meant to referto table 4.2.

Ex. 15 Exercise 4.8 in Danthine and Donaldson (2005, p.355). Be careful in distin-guishing between states of nature and distributions defined over payoffs.


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2.7. Exercises 23

Ex. 16 Consider two assets with returns ra N(0.1, 0.2) and rb N(0.1, 0.3). Aninvestor has the utility function U(W) = exp(W). Which asset does the investorprefer?


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Chapter 3

Portfolio choice

1. The investors typical problem is

maximizea

E[U(Y)]

2. It can be solved explicitly if we assume either:1. Quadratic utility, or2. CARA utility and normal returns.

3.1 Canonical portfolio problem

This section analyzes the problem of an investor that must decide how much to investin a risky asset. Consider the following notation1

a amount (in $) to invest in a risky portfolior uncertain rate of return on the risky portfolio

rf risk-free (certain) rate of returnY0 initial wealthY1 terminal wealth

= a(1 + r) + (Y0

a)(1 + rf

) = Y0(1 + rf

) + a(r

rf

)

The investors problem is

maximizea

E[U(Y1)] (3.1)

1Tildes denote random variables. Well drop them when it is clear which variables are random.

24


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3.1. Canonical portfolio problem 25

The (necessary) first order condition for a maximum is

foc:d

da

E[U(Y1)] = 0

E dU(.)dY1 (r rf) = 0

and the (sufficient) second order condition is

soc:d2

da2E[U(Y1)] < 0 E

d2U(.)

dY21(r rf)2

< 0

which is true if the investor is risk averse (U < 0).

Example 3.1.1. Assume U = 11Y 5Y2, with Y0 = $1. Let rf = 0,E[r] = 0.1, Var[r] = 0.22. Recall Var[x] = E[x2] E[x]2. Use the foc to get

the optimal amount invested in the risky asset:foc:

... a = $0.2

(For more real-life numbers, suppose the initial wealth was one million dollars.

Then, the optimal amount to invest in the risky assets would be $200 000.)Use the soc to check that this is indeed a maximum:

soc:

The analysis of the optimality conditions produces the following important theorem:

Theorem 3.1.1. Let a denote the solution to problem (3.1) and assume the investor isnonsatiable (U > 0) and risk-averse (U < 0). Then

a > 0 E[r] > rfa = 0 E[r] = rfa < 0 E[r] < rf


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3.2. Analysis of the optimal portfolio choice 26

The theorem says that a risk-averse investor will only invest in the risky asset (stocks)if its expected return is higher than the risk-free rate. Conversely, if this is the case( E[r] > rf), then the investor will always participate in the stock market (even if with

just a tiny amount of money).

Example 3.1.2. Suppose U(Y) = ln(Y). For simplicity, assume the riskyreturn is the simple lottery (r2, r1, ). Further assume r2 > rf > r1 (why?). The problem is thus

maximizea

E[ln(Y1)]

The foc is

E

r rf

Y0(1 + rf) + a(r rf)

= 0

or, given the two possible states,

r2 rf

Y0(1 + rf) + a(r2 rf) + (1 )r1 rf

Y0(1 + rf) + a(r1 rf) = 0

which after some algebra is

a

Y0=

(1 + rf)(E[r] rf)(r1 rf)(r2 rf)

Check that the sign of the rhs depends on the sign of E[r] rf. In particular,if E[r] rf > 0, we get a/Y0 > 0, as in theorem 3.1.1. Note also the followingintuitive results:1) The fraction of wealth invested in the risky asset (a/Y0) increases with the

return premium ( E[r] rf);2) The fraction of wealth invested in the risky asset ( a/Y0) decreases with thereturn dispersion around rf, ((r1 rf)(r2 rf)).Lastly, note that the fraction of wealth invested in the risky asset ( a/Y0) doesnot depend on the level of wealth (there is no Y0 on the rhs). This result isspecific to the CRRA utility function as described in a theorem below.2

3.2 Analysis of the optimal portfolio choice

3.2.1 Risk aversion

We now relate the portfolio decision to the risk aversion of the investor.

The follwoing theorem states, quite intuitively, that a more risk averse individualwill invest less in the stock market:

2See the numerical examples in Danthine and Donaldson (2005) for further interpretation.


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Theorem 3.2.1. Let a denote the solution to problem (3.1).

Y > 0, ARAinv1(Y) > ARAinv2(Y) =

ainv1 < ainv2

Furthermore, since ARAinv1(Y) > ARAinv2(Y) RRAinv1(Y) > RRAinv2(Y), we alsohave

Y > 0, RRAinv1(Y) > RRAinv2(Y) = ainv1 < ainv2

Lets check this result:

Example 3.2.1. Assume rf = 0.05 and r = (r2 = 0.4, r1 = 0.2, 1/2). ForU(Y) = ln(Y), we can use the results in the last example to get

a/Y0 = 0.6

Now consider the power utility function U(Y) = Y1/(1 ), with = 3.Note that it has both higher RRA (3 > 1) and ARA (3/Y > 1/Y). Check(end-of-chapter exercise 18) that the optimal portfolio decision for this utilityfunction is

a/Y0 = 0.198

Hence, this more risk-averse agent invests a smaller percentage of his wealth inthe risky asset. The initial wealth (Y0) is the same for both investors, so themoney invested (a) is also smaller, as the theorem stated.

3.2.2 Wealth

We now analyze the portfolio decision as the initial wealth changes. We might expectwealthier investors to put more money in the stock market. However, the result is notso simple; it depends on the characteristics of the specific utility function.

Absolute Risk Aversion

Theorem 3.2.2. Let a = a(Y0) denote the solution to problem (3.1). Then,

(Decreasing ARA) ARA(Y) < 0 a(Y0) > 0(Constant ARA) ARA(Y) = 0 a(Y0) = 0

(Increasing ARA) ARA(Y) > 0 a(Y0) < 0


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DARA. If the investor has decreasing absolute risk aversion (DARA), he is willing toput more money at risk as he becomes wealthier. Recall that power utility has DARA(ARA(Y) = / Y). (Is this reasonable behavior?)

CARA. The second case, constant absolute risk aversion (CARA) is also importantbecause the exponential utility satisfies this condition. Recall that

U(Y) = exp(Y) ARA(Y) = ARA(Y) = 0

The theorem states that this investor will put the same amount of money in the riskyasset regardless of how much wealth he has. (Is this a reasonable description of investorsbehavior?)

Illustration: solving the problem for CARA

Lets verify the CARA case of the theorem. The portfolio problem is

maximizea

{E[ exp(Y1)]} (3.2)

with Y1 = Y0(1 + rf) + a(r rf). The foc is

E [(r rf) exp(Y1)] = 0 (3.3)

which cannot be solved explicitly for a without further assumptions! To proceed, weconsider two alternatives.

1. Implicit Function Theorem

Even though we cannot explicitly solve the problem, we can still describe theoptimal solution using a very useful trick in economics: the Implicity FunctionTheorem.3 Intuitively, this theorem says the following. Suppose the (implicity)function y = y(x) is the solution to some equation, that is, f(x, y) = 0. More

3Implicit Function Theorem. Consider the equation f(y, x1, . . . , xm) = 0 and the solution(y, x1, . . . , xm). If f(y, x)/y = 0, then there exists an implicit function y = y(x1, . . . , xm)that satisfies the equation for every (x1, . . . , xm) in the neighborhood of (x1, . . . , xm), i.e.,f(y(x1, . . . , xm), x1, . . . , xm) = 0. Furthermore, the partial derivatives are given by

y(x1, . . . , xm)

xi= f(y, x1, . . . , xm)/xi

f(y, x1, . . . , xm)/y


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precisely, as we change x, y(x) adjusts to keep f at 0, f(x, y) 0. We can thusconclude that f does not change, ie, its total differential is zero. Therefore,

df(x, y) = 0

fx

dx +f

ydy = 0

dydx

= f/xf/y

Going back to the maximization problem, a = a(Y0) is the implicit function thatguarantees that the lhs of (3.3) is always zero. We can thus take the total differ-ential of the foc and get

da(Y0)

dY0 = E[. . . ]/Y0

E[. . . ]/a

= (1 + rf)=0 (foc)

E[(r rf)eY1 ]E[2(r rf)2eY1

>0

]

= 0

Hence, the amount invested in the risky asset does not change with the investorswealth, as the theorem claimed. Furthermore, the implicit function theorem al-lowed us to check this without solving the maximization problem explicitly.

2. Normal returns

To get an explicit closed-form solution to problem (3.2) we need an additionalassumption. It is this assumption that justifies the wide use of exponential utility.Assume the return on the risky asset is normally distributed, r N(, 2). Then,next periods wealth is also normally distributed, Y1 N(Y0(1 + rf) + a( rf), a

22). Using the moment generating function for the normal distribution4,we can simplify the portfolio problem:

maxa

{E[ exp(Y1)]} = maxa

{ exp

([Y0(1 + rf) + a( rf)] + 1/22a22

)}that is, the rhs does not have E[.]. We can thus solve the maximization problemand get a closed-form solution for a. Exercise 24 asks you to do these final steps.Check that the final expression for a does not depend on Y0, as the theoremstated. To summarize, even though the exponential utility is not the best intuitivedescription of human behavior, it is very useful if we assume that returns arenormally distributed.

4If X N(m, s2), then E[eX ] = exp (m + 122s2), for any .


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Relative Risk Aversion

We can also characterize the optimal portfolio choice in terms of the relative risk aversionmeasure, RRA. Define w a/Y0, the optimal proportion of wealth invested in the riskyasset, or the optimal portfolio weight in the risky asset.

Theorem 3.2.3. Express the solution to problem (3.1) as a fraction of wealth, w(Y0) a(Y0)/Y0. Then,

(Decreasing RRA) RRA(Y) < 0 w(Y0) > 0(Constant RRA) RRA(Y) = 0 w(Y0) = 0

(Increasing RRA) RRA(Y) > 0 w(Y0) < 0

For example, if the investor has decreasing RRA, he will invest a higher proportionof wealth in the risk asset as he becomes wealthier. The most interesting case is perhapsthe constant relative risk aversion (CRRA) case, as it characterizes the power and logutility functions. These investors always invest the same fraction of their wealth in thestock market, regardless of their initial wealth.5

Example 3.2.2. Consider U = ln(Y). Define w a/Y0, the fraction ofwealth invested in the risky asset. The investors problem is to

maximizew

E[ln(Y1)]

with Y1 = Y0(1 + rf) + wY0(r rf). Writing the foc and using the implicitfunction theorem, we can show that (end-of-chapter exercise 19)

dw

dY0= 0

That is, the optimal fraction does not change with wealth.

5This theorem can also be expressed in terms of da/adY0/Y0 , the wealth elasticity of theinvestment in the risky asset:

(Decreasing RRA) RRA(Y) < 0 > 1

(Constant RRA) RRA

(Y) = 0 = 1(Increasing RRA) RRA(Y) > 0 < 1

To see that increasing w(Y0) a(Y0)/Y0 is the same as > 1, noted

dY0[w(Y0)] =

d

dY0

a(Y0)

Y0

> 0 da

dY0

1

Y0 a/Y20 > 0 da/ dY0 > a/Y0

da/a

dY0/Y0> 1

and similarly for the other cases.


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3.3. Canonical portfolio problem for N > 1 31

3.3 Canonical portfolio problem for N > 1

Now we generalize the portfolio choice problem. There are N risky assets and 1 risk-freeasset. Terminal wealth is

Y1 = Y0(1 + rf) +Ni=1

ai(ri rf)

The investors problem is thus

maximize{a1,...,aN}

E

U

Y0(1 + rf) +

Ni=1

ai(ri rf)

It will be convenient to choose weights instead of $ values. We thus define wi ai/Y0and write Y1 = Y0(1 + rf) +

Ni=1 wiY0(ri rf). The investors problem can thus be

rewritten as

maximize{w1,...,wN}

E

U

Y0

(1 + rf) +

Ni=1

wi(ri rf)

Define rp to be the return on the portfolio:

rp := wfrf+N

i=1wiri

Imposing the constraint that the weights must add up to one, we have that

rp =

1

Ni=1

wi

rf+

Ni=1

wiri = rf+Ni=1

wi(ri rf)

Hence, the portfolio problem can also be written as

maximize{w1,...,wN}

E [ U(Y0(1 + rp))]

Unfortunately, this problem is hard to solve without some simplifying assumptions.

3.4 Exercises

Ex. 17 State the investors problem (expression 3.1) in words.


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3.4. Exercises 32

Ex. 18 Check the results in example 3.2.1. The final expression is in the book; youjust need to do the intermediate calculations. Caution: the expression in the book iscorrect, but the number is not (at least I get a different answer: a/Y = 0.198 instead of

0.24).

Ex. 19 Check the results in example 3.2.2, ie, do the intermediate computations.

Ex. 20 Consider the standard portfolio choice between a risk-free asset and a riskystock. An investor with initial wealth $1000 makes an optimal choice to allocate $400to the stock. We know that if the same investor had an initial wealth larger than $1000,he would allocate more than $400 to the stock.

1. This investor has (decreasing / con-stant / increasing) ARA.

2. Give an example of a utility function consistent with this behavior.

Ex. 21 Consider the utility function U(Y) = egY , where g is a constant param-eter.

1. Compute the ARA and RRA coefficients.

2. Interpret in words the result obtained for ARA (relate it to a simple lottery andto the portfolio choice problem).

Ex. 22 Consider the canonical portfolio choice problem with 1 risky asset (withrandom return r) and 1 risk-free asset (with return rf). The investor chooses the amountof money (a) to invest in the risky asset.

1. Write the problem explicitly for an investor with U(Y) = exp(Y), where Yis the wealth.

2. If the risk-free rate increases, what should happen to the amount invested in therisky asset? Explain intuitively (5 lines).

3. Show it explicitly. Hint: compute dadrf and determine its sign.

Ex. 23 There is a risk-free and a risky asset. The investor chooses the amountinvested in the risky asset, a, to maximizeaEU(Y1), where Y1 is next periods wealth.Assume a regular utility function (U > 0, U < 0).

1. In general, what can you say about the sign ofda/dY0?

2. Assume U(Y) = eY. Compute da/dY0.

Ex. 24 Consider the standard portfolio choice problem

maximizea

E[ exp(Y1)]

where next-periods wealth is Y1 = Y0(1 + rf) + a(r rf), and the return on the riskyasset is normally distributed, r N(, 2). Compute the explicit optimal amount to


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3.4. Exercises 33

invest in the risky asset (a).Hint. Use the following property of the normal distribution (called moment generatingfunction): If X

N(m, s2), then E[eX] = exp (m +

12

2s2), for any .Ex. 25 Computing returns with dividends.Consider the following daily closing prices and dividends (D) for two stocks (in $):

Stock A Stock Bday t Pt Dt Pt Dtfri 0 10 10 mon 1 11 11 tue 2 10 10 wed 3 11 11 1.1thu 4 9 9

fri 5 12 12

Note that when a stock pays dividends, the return should be computed as rt =Pt+DtPt1

1.1. Compute daily returns for these two stocks. Compute also the weekly returns

assuming that the dividends are reinvested in the stock. This is a standard as-sumption, so use the standard formula, 1+r0,T = (1+r0,1)(1+r1,2) . . . (1+rT1,T).Note: this is usually called Holding Period Return in databases such as CRSPor DataStream.

2. Suppose you invested $4,000 in A and $6,000 in B in the beginning of the week.Compute the portfolio return over this week. (Use the weekly returns alreadycomputed and apply the standard formula for the portfolio return).

3. Since we assume that dividends are reinvested in the stock, we may end up withmore shares than we started with. How many shares of each stock do you haveat the beginning of the week? How many shares do you have at the end of theweek?Note: to check that you have the right answer, compute the terminal value of theportfolio by doing V5 = PA,5NA,5 + PB,5NB,5, where N is the number of sharesthat you got. It should imply the same weekly return as in the previous question.

4. Again, the way weekly returns were computed assumes that dividends are rein-vested in the stock. Hence, while for the stock without dividends (A) we have

rA,week = P5/P0

1

0.2 = 12/10 1

the same is no longer true for the dividend-paying stock (stock B)

rB,week = P5/P0 10.32 = 12/10 1


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3.4. Exercises 34

Hence, databases usually also show and adjusted price, Pa, that can be used tocompute returns without having to know the dividends. The true return frommarket closes plus dividends must equal the return with adjusted closes:

Pt + DtPt1

1 = Pat

Pat1 1

Fix the last price Pa5 = P5 = 12. Compute the adjusted prices for the previousdays for both stocks.(Check my website for an exercise with data from finance.yahoo.com)


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Chapter 4

Portfolio choice for

Mean-Variance investors

1. Quadratic utility or Normal returns imply mean-variance prefer-ences, E[U] = f(p,

2p).

2. The optimal investment opportunities are described by the mean-variance frontier.

3. The investors portfolio choice problem with N > 1 risky assets

can be solved explicitly.

These concepts were developed by Harry Markowitz in 1952 and they are still thebenchmark for optimal portfolio allocation.

4.1 Mean-Variance preferences

The general portfolio problem (N > 1) is hard to solve unless we make one of the

simplifying assumptions below. Either one of these assumptions will lead to mean-variance preferences, that is, to investors that care only about the first two moments ofY1 or rp.

1

Expand U(Y1) around E(Y1). To simplify the notation, let Y Y1.

U(Y) = U( EY) + U( EY) (Y EY) + 1/2 U( EY) (Y EY)2 + remainder1Note that the two are related: E[Y1] = Y0(1 + E[rp]) and Var[Y1] = Y20 Var[rp].

35


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4.1. Mean-Variance preferences 36

Taking expectations on both sides we get

EU(Y) = U( EY)+ U( EY)

E[(Y

EY)]+1/2

U( EY)

E[(Y

EY)2]+ E[remainder]

or, simplifying,

EU(Y) = U( EY) + 1/2 U( EY) Var(Y) + E[remainder]

Note that this expression, EU(Y), is what the investor maximizes in his portfolio prob-lem. The question is thus under what conditions can we say that E[remainder] = 0, orat least that E[remainder] itself depends only on the first two moments of wealth?

4.1.1 Quadratic utility

If the utility function is quadratic, U = aY bY2, all derivatives of order higher than 2are null, thus remainder = 0. Therefore, we have an exact expression:

EU(Y) = U( EY) + 1/2 U( EY) Var(Y) (4.1)

and the portfolio problem becomes quite simple to solve.

Drawbacks of quadratic utility. Quadratic utility has IARA, which is not veryreasonable. Furthermore, in practical applications we have to be careful defining the

parameters a and b such that we only use the range of wealth where U is increasing.

4.1.2 Normal returns

Alternatively, we can assume that stock returns are normally distributed. Note that ifrp N, then the wealth is also normally distributed, Y Y0(1 + rp) N.

For a normal distribution, all higher-order central moments are either zero or afunction of the variance:

E[(Y EY)n] = 0, n oddn!(n/2)! (

12 Var[Y])

n/2, n even

These are the terms in E[remainder]. Hence,

EU(Y) = U( EY) + 1/2 U( EY) Var(Y) + f( VarY)

that is, investors objective function depends only on the first two moments.


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Advantages of normality

We are considering the case where the investor can combine several assets into a portfolio.If we start by assuming that the return on individual assets is determined by their meansand variances, we need to make sure that the return on any combination of these assets(portfolio) is also determined by the mean and variance only. The Normal distributionsatisfies this additivity requirement (in fact, it is the only distribution with finite variancethat does so).

To see this, let V denote the value of a portfolio with N assets, and wi denote thepercentage of wealth invested in each asset. The portfolio return is just the weighted-average of individual returns:

rp := V1/V0

1

=Ni=1

ai(1 + ri)/V0 1 =Ni=1

wi(1 + ri) Ni=1

wi =Ni=1

wiri

Since the sum of normally distributed random variables also follows a normal distribu-tion, if we assume that each stock has a normal distribution, then the portfolio returnis also normally distributed: ri N rp N.

Drawbacks of normality

The returns we are considering here are discrete returns, defined as:

r : P1 = P0(1 + r)

Since the Normal distribution has R support, saying that r N is the same as sayingthat prices can be negative. This is an unrealistic description for assets with limitedliability, such as stocks and bonds, where the worst that can happen is bankruptcy, inwhich case P1 = 0 and r = 100%.2

2We can go around this issue by using instead continuously-compounded returns:

z : P1 = P0ez z = ln(P1/P0)

This guarantees P1 > 0, z R. We can thus safely assume z N. Continuous returnsare very convenient for time-series aggregation in multiperiod settings. If short-horizon returnsare normally distributed, then the long-horizon return, z0,T, is also normally distributed: z0,T =

ln(PT/P0) = ln(PTPT1

PT1PT2

. . . P2P1 P1P0 ) = z0,1+z1,2+ +zT2,T1+zT1,T N. For cross-sectionaggregation, the expression is a bit more cumbersome: zp := ln(V1/V0) = ln

Ni=1 aie

zi/V0

=

lnN

i=1 wiezi

or, ezp =N

i=1 wiezi . Normality is not preserved.


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Empirical evidence

It is an empirical question whether normality is a reasonable first approximation tosecurity returns. The answer is yes, the normal distribution is a useful approximation,particularly for returns measured over long horizons, such as one year.

If we were interested in high-frequency returns, then the normal assumption wouldbe more questionable, due to the following empirical facts:

1. Short-term daily returns have fat tails, that is, empirical returns have more kurtosisthan the normal distribution.

2. Short-term daily returns (especially for stock indices) are skewed to the left, thatis, extremely bad returns are more likely than under a true normal distribution.

Fortunately, these problems are less severe at longer horizons, say monthly or yearly.Hence, since the portfolio problem we are considering here typically has a long horizon,normality is a reasonable assumption.

Note that, despite the caveats above, the normal distribution is still the bench-mark and the work-worse in finance. For instance, J.P.Morgan/Reuters RiskMetricssystem (outputs Value-at-Risk estimates) assumes that even daily returns are normallydistributed (see J.P. Morgan, 1996).

4.1.3 Conclusion

Either assuming quadratic utility or normal returns, we conclude that the investor max-imizes a function of the mean ( := E[r]) and variance ( := Var[r]) of the return onthe portfolio:

maximize E[U(Y)] f(p,2p)

Quite intuitively, it can be shown that the objective function increases with theexpected return, df / dp > 0, and decreases with the standard-deviation, df / dp < 0.

3

This leads to two important results.3For quadratic utility, this follows directly from taking derivatives of (4.1). For normal returns,

standardize the portfolio returns: sp =rp N(0, 1). Then, the fn to be maximized is

f := E[U(rp)] =

U(r)p(r)dr =

U(s + )p(s)ds, where p(.) is the Normal pdf. Thus,df/d =

U(.)p(s)ds > 0, since U > 0. Also, df/d =

U(.)sp(s)ds < 0, since U < 0 means

that U is decreasing, which implies that for each s pair the negative s gets more weight. Seeappendix 6.1 in Danthine and Donaldson (2005) for illustrations. To be precise, the investormaximizes EU(Y1), not EU(rp), but the derivatives have the same sign since Y1 = Y0(1 + rp).


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4.2. Review: Mean-Variance frontier with 2 stocks 39

Mean-Variance dominance. Asset a mean-variance dominates asset b iff:

a

b and a < b

or a > b and a b

All mean-variance investors prefer asset a. This implies that, for a fixed given level ofvariance(mean), all mean-variance investors prefer the portfolio with the largest(smallest)return(risk).

Optimal portfolio. It can be shown that a mean-variance investor will choose hisportfolio through

maximize{w1,...,wN}

p

g

22p

That is, his objective function trades-off mean against variance. The parameter g deter-mines how much the investor dislikes variance, i.e., how risk-averse he is.

4.2 Review: Mean-Variance frontier with 2 stocks

This section analyzes the investment opportunity set for an investor with mean variancepreferences (by one of the two possible assumptions in section 4.1). The goal is to

develop intuition for the diversification effect with just two stocks. The following sectionsconsider the portfolio problem in full generality.

Suppose there are just two risky assets (stocks). The investor only cares about themean and variance of the return on the portfolio formed by these two assets:

p E

2i=1

wiri

= w11 + (1 w1)2and

2p Var 2i=1

wiri

= w21

21 + (1 w1)222 + 2w1(1 w1)12

where is the correlation coefficient between r1 and r2 (recall 1 +1). Theopportunity set depends critically on this correlation.


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4.2. Review: Mean-Variance frontier with 2 stocks 40

The main point we want to illustrate in this section is the diversification effect.Whereas the expected return on the portfolio is the weighted average of expected returnson the individual assets, the same is not true for the risk. In fact, the standard-deviation

of the portfolio is typically less than the weighted average of the individual standard-deviations. This is the gain from diversifying the portfolio. The smaller the correlationcoefficient, the greater the benefits from diversification.

Perfect positive correlation ( = 1). There is no gain from diversification sincethe assets are essentially identical (the return on one asset is a linear function of theother). The portfolio standard-deviation is equal to the weighted average of the twostandard-deviations

p = w11 + (1 w1)2

which means that all the possible portfolio lie on the straight line between the two assets(in , - space) see figure 6.2 in Danthine and Donaldson (2005).

Imperfect correlation (1 < < +1). Now we have the diversification benefit.At each level of p, the corresponding p is less than in the = 1 case. This is because2p increases in (

2p/ = 2w1w212 > 0). See figure 6.3 in Danthine and Donaldson

(2005) and appendix 6.2 for a formal proof.

Note that only the portfolios on the upper part of the curve are efficient, that is, they(mean-variance) dominate the ones on the lower part of the curve.

Perfect negative correlation ( = 1). For this (theoretical) case we would beable to construct a risk-free asset. See figure 6.4 in the book.

1 Risk-free and 1 risky asset. If one asset is risk-free (1 = 0), we have 12 = 0and p = w22. The opportunity set is again linear figure 6.5 in the book.

Extension to N risky assets. Intuitively, this analysis can be generalized to 3 riskyassets by taking one of the possible previous portfolios and a new 3rd asset. Proceeding

with these iterations, we could get to N risky assets. The minimum variance frontierwill have the shape in figure 6.6 in the book. We will derive this carefully in the nextsection.


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4.3. Setup for general case 41

Extension to N risky assets plus 1 risk-free asset. The investor will pick onparticular portfolio on the mean-variance frontier (the tangency portfolio) to combinewith the risk-free asset. The straight line going through rf and T is the efficient

frontier. See figure 6.6. Again, this will be derived below.

The fact that all investors will invest in the same two assets (the risk-free and thetangency portfolio), even though in different proportions, is known as the two fundtheorem or the separation theorem.

4.3 Setup for general case

4.3.1 Notation

Let r be the (N.1) vector of returns on the N risky assets. Define the vector of expectedreturns:

r := E[r] =

E[r1]...

E[rN]

Let the covariance matrix be

V := Cov(r) = ...

. . . ij . . ....

Let 1 be a (N.1) vector of ones. Let (scalar) be the required return on the portfolio.The choice variable is the vector of portfolio weights:

w =

w1...

wN

4.3.2 Brief notions of matrix calculus

For a scalar-valued function f(x1, . . . , xn), the gradient is

f(x)

x=

f/x1...

f/xn


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4.4. Frontier with N risky assets 42

Let a be a (n.1) vector of constants and A a (n.n) symmetric matrix of constants. Someuseful rules are:

d(ax)/dx = a

andd xAx

1.n.n.1

/dx = 2Ax

To check the second rule, consider

A =

1 33 4

Note that xAx = x21 + 4x22 + 6x1x2. Thus,

d(xAx)

dx =2x1 + 6x26x1 + 8x2 = 2 66 8 x1x2 = 2Ax

4.4 Frontier with N risky assets

4.4.1 Efficient portfolio

The variance of the return on a portfolio (rp = wr) is given by

Var[w

r] = w

V w

The program to find the minimum-variance portfolio, for a given expected return ,is thus:

minimizew

1

2wV w (4.2)

s.t. wr =

w1 = 1

This is a constrained optimization problem. To solve it, define the Lagrangian

L =1

2wV w + ( wr) + (1 w1)

where the scalars and are Lagrange multipliers.


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The first-order conditions are:

dL

dw

= V w

r

1 = 0 (N eqns) (4.3)

dL

d= wr = 0 (1 eqn) (4.4)

dL

d= 1 w1 = 0 (1 eqn) (4.5)

The foc for w can be rewritten as

V w = r + 1

V1V w = V1(r + 1)

w = V1r + V11 (4.6)

But this is not over yet because we dont know the value of the multipliers.

Pre-multiplying (4.6) by r and using the foc for we get

rw = (rV1r) + (rV11)

= (rV1r) + (rV11) (4.7)

Pre-multiplying again (4.6) by 1 and using the foc for we get

1

w = (1

V1

r) + (1

V1

1) 1 = (1V1r) + (1V11) (4.8)

Equations (4.7) and (4.8) form a system of two (scalar) equations that can be solvedfor the two unknown lagrange multipliers:

= B + A

1 = A + C

= BAD = CAD

where we defined the scalars A := 1V1r, B := rV1r, C := 1V11, and D :=

BC A2

. Since the matrix of covariances (V) is positive definite and thus also V1

, wehave that B > 0 and C > 0.4 It can also be shown that D > 0.

4We say that the matrix A is positive (semi)definite if xAx > 0 () for all nonzero x. Thecovariance matrix is PD because the variance of a portfolio must be positive, Var[wr] = wV w >0. In general, a covariance matrix need only be PSD, but this would mean that we might be ableto construct a risk-free portfolio using only stocks, Var[wr] = wV w = 0. This is typically notthe case, so we assume that V is PD.


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Plugging these numbers back into (4.6), we get the final answer:

w =C A

D

V1r +B A

D

V11 (4.9)

This equation is a closed formula for the efficient portfolio with return , that is,for the portfolio with smallest variance between all portfolios with return . You candouble check that we do indeed get the required return, i.e. E[rp] w

r = . Theportfolio variance can be computed as Var[rp] Var(wr) = wV w. By varying and computing the respective w and Var[rp], we can plot the frontier of the investmentopportunity set.

Example 4.4.1. Assume that there are only 2 risky assets with E[r1] = 15%,1 = 25%, E[r2] = 10%, 2 = 20%, and zero correlation. First, check that

A = 4.9

B = 0.61

C = 41

D = 1

Hint: see the formula sheet for an easy way to invert a diagonal matrix.

Second, if we require say an expected return of = 0.14, the optimal portfoliofrom the formula above is

w = . . . =

0.80.2

We can check that E[rp] w r = 0.14. The risk of the portfolio isVar[rp] = w

V w = 0.0416 p = 0.204

4.4.2 Frontier equation

If we work out the Var[rp] = wV w expression, we arrive at the following equation for

the mean-variance frontier:

Var[rp] =C

D AC

2

+1

C

which is a parabola in ( Var[rp], E[rp])-space.5

Example 4.4.2. Continuing the previous example, check that we get the sameVar[rp] for = 0.14

5The frontier is an hyperbola in (p, E[rp])-space.


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4.5. Frontier with N risky assets and 1 risk-free asset 45

4.4.3 Global minimum variance portfolio

From this equation, we can immediately identify the global minimum variance portfolio:

E[rmvp] = A/C

Var[rmvp] = 1/C

The set of portfolios located on the mean-variance frontier with E[rp] > A/C is calledthe efficient frontier.6

Example 4.4.3. For the previous example, check that

E[rmvp

] = 0.1195

Var[rmvp] = 0.0244 mvp = 0.1562

4.5 Frontier with N risky assets and 1 risk-freeasset

4.5.1 Efficient portfolio

In addition to the N risky assets of the previous section, we now consider one additionalrisk-free asset with (known) return rf. Let w be the (N.1) vector of weights in the riskyassets as defined before. The proportion of wealth invested in the risk-free asset is thuswhat is left, wf = 1 w1. Therefore, the expected return on a given portfolio is

E[rp] = wr + wfrf

= wr + (1 w1)rfNote that the second equation already imposes that the weights add up to 1.

The program to find the minimum-variance portfolio, for a given expected return ,is now

minimizew

1

2wV w (4.10)

s.t. wr + (1 w1)rf = 6Different people call slightly different names to all these frontiers. So make sure you

understand the concepts well (what dominates what).


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The solution is:

w = rf

HV1(r rf1) (4.11)

where the scalar H := (r rf1)V1(r rf1) = B 2Arf + Cr2f > 0. The scalarsA,B,C are as defined above.

Example 4.5.1. Continuing the previous two-stock example, further assumerf = 0.04. First, check that

H = 0.2836

Second, if we require an expected return of = 0.14, the optimal portfoliofrom the formula above is

w = . . . = 0.6206

0.5289We can check that E[rp] w r + wfrf = 0.14. The risk of the portfolio isVar[rp] = w

V w = 0.0353 p = 0.1878

4.5.2 Frontier equation

To plot the mean-variance frontier, we can again compute w and the respective Var[rp]for different values of. Alternatively, we can compute an explicit expression for Var[rp]:

Var[rp]

Var(wr) = wV w

=

rf

H

2 [V1(r rf1)

]V[

V1(r rf1)]

=

rf

H

2(r rf1)(V1)(r rf1)

=H

Note that V is symmetric, thus (V1) = (V)1 = V1. Finally, the mean-variancefrontier with a risk-free asset is:

Var[rp] =( rf)2

H(4.12)

This draws two straight lines in (p, rp)-space (an exercise will ask you to check thiswith real data). The one that goes through rf and the tangency portfolio (ie, the set ofportfolios with E[rp] > rf) is the efficient frontier:

7

= rf+ p

H (4.13)

7Equation (4.12) implies

p = rf

Hor p = rf

H = rf + p

H or = rf p

H


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Example 4.5.2. Check that we get the same Var[rp] for = 0.14

4.5.3 Tangency portfolio

We can compute the precise coordinates of the tangency portfolio T by noting that itis the only frontier portfolio composed only by risky assets, i.e. 1wT = 1. We can use(4.11) to find the corresponding expected return (T):

1wT = 1

T rfH

1V1(r rf1) = 1

T =

H

A Crf+ rf

Plugging back into (4.11) we obtain an explicit expression for the weights in the tangencyportfolio:

wT =V1(r rf1)

A Crf

Example 4.5.3. Continuing the previous example, check that

wT = . . . = 0.53990.4601Thus,

E[rT] = 0.1270

T = 0.1634

Example 4.5.4. Two-fund separation: Find the linear combination of T andrf that will give E[rp] = 0.14.

Check that the weights in the two stocks are equal to the ones obtained aboveusing (4.11)

We are interested in the line with positive slope, = rf+ p

H, which under normal circum-stances will be the tangent line. More precisely, the tangency portfolio is located on the upperlimb of the hyperbola if rf < E[rmvp] = A/C. If the reverse is true, the tangency portfolio islocated on the lower limb. Further, if rf = A/C, there is no finite point of tangency. However,note that from theorem 3.1.1, the equilibrium case under the CAPM model (section 5) must berf < E[rmvp] (otherwise, there would be no demand for the risky assets). Hence, in equilibrium

the frontier is given by = rf + p

H. See Huang and Litzenberger (1988) or Ingersoll (1987)for details.


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4.6. Optimal portfolio 48

4.6 Optimal portfolio

The particular portfolio on the efficient frontier that the investor picks depends on hislevel of risk aversion. Given that the investor has mean-variance preferences, he chooseshis optimal portfolio weights by

maximize E[rp] g2

V ar[rp]

where g is a constant parameter and rp denotes the return on the portfolio.

Assuming that there are N risky assets plus one risk-free asset, the problem in matrixnotation is

maximizew

wr + wfrf

g

2wV w

s.t. 1 = w1 + wf

ormaximize

wwr + (1 w1)rf g

2wV w

The foc isr rf1 gV w = 0

which implies the solution

w =1

gV1(r rf1) (4.14)

Example 4.6.1. Assume that g = 5, rf = 4%, and that there are only 2risky assets with E[r1] = 15%, 1 = 25%, E[r2] = 10%, 2 = 20%, andzero correlation. Compute the exact expected return and standard-deviation ofthe optimal portfolio. Hint: see the formula sheet for an easy way to invert adiagonal matrix.

Solution:

Using the formula above, w = . . . =

0.3520.300and wf = 0.348.

Thus,

E[rp] = wr + (1 w1)rf = 9.67%

V ar[rp] = wV w = 0.0113 p = 10.65%


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4.6. Optimal portfolio 49

We can verify that this portfolio is efficient, ie, that it actually lies on the mean-variance frontier. Write

E[rp] = wr + (1 w1)rf = (r rf1)w + rfFor this investors optimal portfolio, using w from (4.14),

E[rp] = (r rf1)1

gV1(r rf1) + rf = 1

gH + rf

where we also used H := (r rf1)V1(r rf1).

There are two alternatives now:

1. Plug E[rp] in the formula for frontier portfolios (4.11) and show that the portfolio

is the same one as the investor chose:

w = rf

HV1(r rf1)

=

1gH+ rf rf

HV1(r rf1)

=1

gV1(r rf1)

which is indeed the same as (4.14).

2. Alternatively, we can show that the investors portfolio verifies the equation for

the efficient frontier (4.13). Start by computing the portfolio variance, using w

from (4.14),

Var[rp] = (w)

V w =1

g2H

Then, plug this variance into (4.13):

= rf+ p

H

= rf+

1

g2H

H

= rf+1

gH

which is indeed the expected return on the investors portfolio. Note that thissecond alternative is a bit more correct, since it explicitly shows that the investorportfolio lies on the upper part of the mean-variance frontier, ie, that it is efficient.


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4.7. Additional properties of frontier portfolios 50

4.7 Additional properties of frontier portfolios

We now derive a relation that will be used to prove the CAPM in the next chapter. Wewant to do the derivation right now to stress that the part done here is just math, noteconomics. In other words, it does not depend on any model of market equilibrium.

Define:p a frontier portfolio (still assume there is a risk-free asset)a any portfolio, not necessarily on the frontier (eventually a single asset), but

without the risk-free asset.

The covariance between the two portfolios is given by (exercise 30 at the end showsthis):

Cov(ra, rp) = w

a

V wp

Since p is a frontier portfolio, wp is given by (4.11). Hence,

Cov(ra, rp) = waV

rf

HV1(r rf1)

=

rfH

wa(r rf1)

=E[rp] rf

H( E[ra] rf)

since = E[rp], war = E[ra], and w

a1 = 1. Solving for E[ra] rf and using (4.12) for

H,

E[ra] rf = HCov(ra, rp)E[rp] rf

=Cov(ra, rp)

Var[rp]( E[rp] rf) (4.15)

Note that all we did so far was to characterize the relation between a frontier portfolio(p) and any other asset (a). Since, p can be any frontier portfolio, the previous relationapplies in particular to the Tangency portfolio:

E[ra]

rf =

Cov(ra, rT)

Var[rT

]( E[rT]

rf) (4.16)

4.8 Exercises

Ex. 26 Consider the quadratic utility function U(W) = a + bW + cW2, where Wis the terminal wealth and a,b,c are constants. Assume that W = W0(1 + rp), whereW0 is the initial wealth and the rate of return on the portfolio is normally distributed,


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4.8. Exercises 51

rp N(, 2). (Note that the normality assumption is a bit of an overkill; we onlyneed quadratic utility). Show that the investor only cares about the first two momentsof returns, i.e., write E[U(W)] as an explicit function of and (and the constants

a,b,c,W0).

Ex. 27 Normal returns for PSI20. Download the file PSI20.xls from my website.It has daily and monthly closing prices for the Portuguese Stock Index 20.Note: If you do this in Matlab, you may want to use my DescStats.m function (alsoposted on the website).

1. Compute daily continuously-compounded returns. Compute the mean, variance,skewness, and kurtosis of the distribution. Does it look normal?

2. Do the same for monthly returns.

Ex. 28 Mean-Variance Frontier. Assume there are N risky assets and that thereis no risk-free asset. Formulate the problem of finding the minimum-variance portfoliofor a given level of return. State in words what the objective and the restrictions mean.Solve for the optimal portfolio weights. Note: The goal of this exercise is for you to gothrough all the intermediate calculations in detail.

Ex. 29 Solve problem (4.10), ie, show the intermediate steps that lead to (4.11).

Ex. 30 Let rp and rq denote the returns on two portfolios. By definition, thecovariance between these returns is given by Cov(rp, rq) := E[(rp E[rp])(rq E[rq])].Starting from this definition, show that the covariance can also be computed as wpV wq,where wi is the N by 1 vector of weights in portfolio i and V := Cov(r) = E[(r

Er)(r Er)] is the N by N covariance matrix of individual stock returns. Hint: writeri = w

ir, i = p, q.

Ex. 31 An investor has mean-variance preferences and thus chooses his optimalportfolio weights (w, an N by 1 vector) by solving:

maximizew

E[rp] g2

V ar[rp]

s.t. w1 = 1

where rp is the return on the portfolio, g is a constant parameter, and 1 a vector of ones.There is no risk-free asset. To simplify the notation, denote byV := Cov(r), the covariance matrix, andr := E[r], the vector of expected returns,where r is the random vector of returns on the N risky assets.

1. Solve for the optimal w.Hints: First write E[rp] and V ar[rp] in matrix notation, ie, using w, r, and V.To simplify the notation, use the scalars A, B, and C (as defined section 4.4)along the calculations whenever possible.


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4.8. Exercises 52

2. The rest of the exercise will help you to show that the portfolio just found ismean-variance efficient. Start by computing its expected return.

3. Now look at the solution for an efficient portfolio (equation 4.6):

w =C A

DV1r +

B AD

V11

Plug in the expected return found in part (2.) and verify that the resulting w

is identical to the one found in part (1.). (This shows that the solution to theinitial problem is indeed mean-variance efficient.)

Ex. 32 There are N risky assets and 1 risk-free asset. Consider the standard port-folio choice problem, maximizew E[U(Y1)], where the terminal wealth is Y1 = Y0(1 + rp).All risky assets follow a normal distribution and thus the return on the portfolio isalso normally distributed, rp

N(E[rp], V a r[rp]). The utility function is U(Y) =

exp(b.Y), where b is a constant parameter. Compute the optimal weights in therisky assets, w (an N by 1 vector).Hint: Start by writing E[rp] and V ar[rp] in matrix notation. Then, write the distributionof Y1. Then, use the moment generating function to simplify the objective function.

Ex. 33 Frontier with Industry Portfolios. Download the file 10_Industry_Portfolios.xlsfrom my website. It has monthly returns on 10 industry portfolios (from K. Frenchswebsite, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html).

1. Ignoring the risk-free asset, draw the frontier in mean-std space.

2. Now consider a risk-free rate ofrf = 0.4% (this the 1-month TBill rate at the end

of the sample, as you can check on Frenchs website). Draw the efficient frontier(do it on the same figure as 1; you should get something like fig 6.6 in Danthineand Donaldson (2005)).

3. Compute the tangency portfolio (weights, expected return, standard deviation)and plot it in the figure.

4. An investor has mean-variance preferences and thus chooses his optimal portfolioweights by

maximize E[rp] g2

V ar[rp]

where g is a constant parameter and rp denotes the return on the portfolio. Thesolution is

w =1

g V1(r rf1)Assume that g = 8 and that the investor has $1 Million to invest. Computethe amount of money that the investor should put in each of the 10 industryportfolios and in the risk-free asset. Plot the optimal portfolio in the same figureas the previous questions.

5. Find the value of the parameter g that would make the investor optimally choosethe Tangency portfolio.


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4.8. Exercises 53

Ex. 34 No short selling. Use the same data as in the previous exercise. Consider thesame investor as in question 4, ie, mean-variance preferences with g = 8. Assume thatthe investor cannot short sell any of the stock portfolios. Compute the optimal amount

of money that the investor should put in each of the 10 industry portfolios and in therisk-free asset. Compute the expected return and standard-deviation of the optimalportfolio. Plot the new optimal portfolio in the same figure as the other questions in theprevious exercise.Hint: There is no closed form solution. Look for ways to solve the problem numerically.Matlab and other software (like EXCEL) do this.


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Chapter 5

Capital Asset Pricing Model

The CAPM states that the market portfolio is mean-variance efficient.For any asset,

E[rj ] = rf+ j ( E[rM] rf)

5.1 Introduction

Our goal is to understand why different assets have different average returns. The CAPMproposes a very precise answer to this question.

The value of any asset is the present value, or discounted value, of its future cashflows. The CAPM gives us a formula for the discount rate. Hence, it is used everydayby corporations and investors to price investment projects, stocks, mutual funds, etc.

The CAPM is an equilibrium model that results directly from assuming that allinvestors are mean-variance optimizers. It was developed simultaneously in three papersby Sharpe in 1964, Lintner in 1965, and Mossin in 1966.

5.2 Derivation

We make the following assumptions:

A1: All investors have mean-variance preferences.

A2: There is a risk-free asset with return rf.

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A3: Investors have homogeneous expectations. This means that everybody has thesame beliefs about the return distribution of every asset.

These assumptions immediately imply the following results:

1. The efficient frontier (namely, the straight line through rf and T) is the same forevery investor.

2. Two fund separation: every investor allocates his wealth between two portfolios:the risk-free asset and the Tangency portfolio.

3. In equilibrium, all risky assets must belong to T.

To see this, suppose that IBM is not in T (wTIBM = 0). Then, there would beno demand for this stock, (wiIBM = w

TIBM = 0, for every investor i). We would

thus have Demand = Supply, which is not equilibrium. Therefore, in equilibrium,wTj > 0, asset j.4. Furthermore, for every asset, the weight in T must be the same as in the whole

market:

wTj =Market Capjj Market Capj

=: wMj , asset j

If we all put 2% of our (risky) money into IBM stock, then IBM will have 2% ofall money invested in the stock market, meaning that the market capitalizationof IBM will be worth 2% of the whole market

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Asset Pricing Teaching Notes