Microeconomics 3 Economics Programme, University of Copenhagen Spring semester 2006 Week 17 Lars Peter Østerdal 1
Today s programme General equilibrium over time and under uncertainty (slides from week 16). Asset pricing 1. Capital Asset Pricing Model (CAPM) 2. Arbitrage Pricing Theory (APT) 2
Assets: basic terminology An asset is a title to receive either physical goods or money in a future period in an amount that may depend on which state occurs. The payo of an asset is called a return. Real assets: The returns are in physical goods. Financial assets: The returns are in money. 3
Two approaches: 1) Pricing of real assets in a general equilibrium model with uncertainty. 2) Pricing of nancial assets. We focus mostly on 2). 4
Pricing of nancial assets Chapter 20 considers a variety of models. General nding: asset prices must satisfy certain rules since otherwise there would be a possibility for riskless arbitrage. 5
Trivial case: only riskless assets - using no-arbitrage, pricing is trivial. General case: risky as well as riskless assets - we will nd that: 1) Expected return of asset a = return on riskless asset + risk premium for asset a. 2) The risk premium depends on how the asset covaries with other assets (to be made precise in each model). 6
CAPM (Capital Asset Pricing Model) The assets: 1 riskless asset and A risky assets. R 0 denotes the total return on a risk-less asset. er a denotes the total return on asset.a (a random variable), a = 1; :::; A. R a denotes the expected return on asset a. (we will argue that there must be a certain relationship between these returns). 7
The investors: An investor s utility of a random variable depends only on its mean and variance. Utility increasing in the mean and decreasing in the variance. Question: compatible with expected utility model? 8
Plan: 1) Show that whenever an investor holds a mean-variance e cient portfolio, there must be a certain relationship between the returns of the assets in the portfolio. 2) Argue that all investors are interested in holding the same portfolio of risky assets. 3) Combine these observation and give it empirical content. 9
Ad 1): mean-variance e ciency and its implications. x a denotes fraction of an investor s wealth invested in asset a. (x 0 ; :::; x A ) portfolio of assets held by an investor. P Aa=0 x a = 1: 10
A portfolio (x 0 ; :::; x A ) is mean-variance e cient if it minimizes the variance given the mean. That is, it solves a programme of the form (see Varian p. 486): subject to and min x 0 ;:::;x A A X a=0 AX a=0 AX a=0 AX b=0 x a R a = R, x a = 1: x a x b ab 11
Lagrange: L = AX AX a=0 b=0 x a x b ab 0 AX @ a=0 x a R a R 1 A 0 AX @ a=0 x a 1 1 A : First-order conditions: a = 0; :::; A. 2 AX b=0 x b ab R a = 0; 12
Let (x e 1 ; :::; xe A ) denote some mean-variance e cient portfolio consisting entirely of risky assets. Suppose that asset e is a "mutual fund" that holds this e cient portfolio. Then the portfolio (0; 0; :::; {z} 1 ; :::; 0; 0) entry number e is mean-variance e cient and satis es the rst-order conditions. 13
The a th rst-order condition: 2 ae R a = 0: Two special cases. a = 0 : R 0 = 0: and a = e : 2 ee R e = 0: 14
Solve these equations for and : and = 2 ee R e R 0 ; = 2 eer 0 R e R 0 : 15
Substituting these values of and in a th rst-order condition 2 ae R a = 0; we get: R a = R 0 + ae ee (R e R 0 ). This equation must hold for any e cient portfolio of risky assets. We now identity one particular e cient portfolio. But rst, some general considerations about the set of e cient portfolios. 16
Two risky assets (make gure): 17
Three risky assets (make gure): 18
All investors want to hold the same portfolio of risky assets (x m 1 ; :::; xm A ). x m a is therefore the fraction of all investors total wealth that is invested in asset a. (x m 1 ; :::; xm A ) is the market portfolio. 19
We can write: R a = R 0 + am (R m mm R 0 ) = R 0 + a (R m R 0 ): 20
Recall from statistics: For linear least square tting, b in is given by xy xx : y = a + bx And the correlation coe cient (Pearson s R 2 test). xy x y is a measure of quality of this tting. 21
Hence, = am mm is the theoretical regression coe cient from regression of e R a on e R m. Can be estimated from observable data: Expected return on market portfolio of risky assets (OMXC20? stock market? world market?). Entire a (last 12 months? last 12 years?). 22
Example: Asset market: Copenhagen Stock Exchange. Data: Monthly returns. Observation period: between 20 and 60 months. 23
Stock Beta Genmab Common Stock 1,90 SAS AB Common Stock. 1,82. Novo Nordisk Common Stock 1,04 H.Lundbeck A/S Common Stock. 1,00. Bonusbanken A/S Common Stock 0,00 Gudme Raaschou V Common Stock 0,00 Alm Brand PANT Common Stock. 0,00. Viborg Haandbold Common Stock -0,31 Dan Ejendomme Common Stock -0,32 SIS International Common Stock -0,40 Source: Danske Bank / Reuters. Data available at: http://www.aktieugebrevet.dk/ ler/beta.xls. 24
Section 20.6 "Expected utility". CAPM tells us that there must be a certain relationship between asset returns. - but nothing about how much an investor should consumer now and much he should invest. For this, we need a model of investor s intertemporal utility as in Chapter 19. Such a model is outlined in Section 20.6. 25
From consumer s utility maximization (the rst-order conditions) we get: Conditions for intertemporal optimization - as in chapter 19. The relationship between asset returns as previously derived in section 20.4 (assuming that returns are normally distributed and only one future period). 26
APT (Arbitrage Pricing Theory) Model: 1 riskless asset (return R 0 ) and A risky assets. er a = b 0a + b 1a e f 1 + b 2a e f 2 + ::: + e a, a = 1; :::; A. ef 1 ; e f 2 ; ::: are stochastic variables that in uence all asset returns. Suppose that they have zero mean and are mutually independent. b 0a ; b 1a ; ::: are parameters - the "sensitivity". 27
If there are "many" assets, asset-speci c risk is not important: By Law of Large Numbers the risk of a highly diversi ed portfolio must involve very little asset-speci c risk. 28
No asset-speci c risk, one risk factor er a = b 0a + b 1a e f 1, a = 0; 1; :::; A. The riskless asset: b 10 = 0, b 00 = R 0. R a = b 0a. 29
We can construct a riskless portfolio: hold x in asset a hold 1 x in asset b (NB: short-selling possible). Return: x e R a + (1 x) e R b = [xb 0a + (1 x)b 0b ] + [xb 1a + (1 x)b 1b ] {z } choose x such this term = 0 ef 1 : 30
x b 1a + (1 x )b 1b = 0 if we assume b 1b 6= b 1a : x = b 1b b 1b b 1a : (*) Since we have constructed a risk-less portfolio, by no-arbitrage we must have: x b 0a + (1 x )b 0b = R 0 ; or x (b 0a b 0b ) = R 0 b 0b : 31
Subst. (*) : b 1b b 1b b 1a (b 0a b 0b ) = R 0 b 0b : Rearranging: Interchanging a & b: b 0b R 0 b 1b = b 0b b 0a b 1b b 1a : b 0a R 0 = b 0a b 0b b 1a b 1a b 1b Note that right-hand sides are identical. 32
Hence: b 0a R 0 b 1a = 1 ; for some constant 1 and for all assets a. Since R a = b 0a we get: R a = R 0 + b 1a 1 : Interpretation of 1 : risk premium paid for portfolio that has sensitivity one. 33
No asset-speci c risk, two risk factors er a = b 0a + b 1a e f 1 + b 2a e f 2 Take three risky assets a; b; c. Construct portfolio (x a ; x b ; x c ) such that: x a b 1a + x b b 1b + x c b 1c = 0 (eliminate risk factor 1) x a b 2a + x b b 2b + x c b 2c = 0 (eliminate risk factor 1) x a + x b + x c = 1: 34
Portfolio has zero risk. By non-arbitrage we must have: x a b 0a + x b b 0b + x c b 0c = R 0. In matrix form a "homogenous equation system": 0 B @ b 0a R 0 b 0b R 0 b 0c R 0 b 1a b 1b b 1c b 2a b 2b b 2c 1 0 1 0 x a C B C B A @ x b A = @ x c Since we have a solution where x a + x b + x c = 1 (i.e. determinant of is zero. 0 B @ b 0a R 0 b 0b R 0 b 0c R 0 b 1a b 1b b 1c b 2a b 2b b 2c 1 C A 0 0 0 1 C A : not all zero) the 35
Hence the rst row is a linear combination of the last two rows. (assuming that the last two rows are not collinear). That is, there are 1 and 2, such that: all a = 1; :::; A. R a R 0 = b 1a 1 + b 2a 2 ; If we add asset speci c risk, the same relation should hold approximately, since asset-speci c risk not important in a highly diversi ed portfolio. 36
Section 20.8 "Pure arbitrage" Minimum of assumptions. Shows that from "no arbitrage" we get that R a = R 0 R 0 cov( e Z; e R a ); where e Z is a random variable de ned in a certain way. Point: Risk premium depends on the covariance of asset return with a single random variable - the same for all assets. 37