A1 Expected Utility and Risk Aversion. [30]

(a) Under which conditions does the Expected Utility (EU) hypoth-       esis hold within the State-Preference approach?.  Explain why       Expected Utility is a particular case of the State-Preference ap-       proach.               [10]

(b) DeÖne risk aversion, and the coe¢ cients of absolute and relative       risk aversion.                 [10]

(c) Consider the utility function u(W) = ; 7  1: What is the     coe¢ cient of absolute risk aversion at W = 1 and at W = 100?     And what is the coe¢ cient of relative risk aversion at W = 1      and at W = 100?      [5]

(d) Finally consider the utility function u(W) = W _ bW2  : b < 1=(2W): Recalling that the investorís terminal wealth W is de- Öned in terms of the portfolio return rP  as W = (1 + rP )A; de- rive the mean-variance representation of expected utility in terms of the mean and variance of the portfolio return, G(uP ; 7P(2)).     [5]

A2 Martingales and E¢ ciency. [30]

(a) Consider the martingale model of asset prices for a particular asset that trades at a price pt , E(pt+1/ |t ) = (1 + u)pt : Brieáy explain the intuition behind it, and discuss the role of the infor- mation set |t : Interpret u: What does it imply when forecasting

the assetís return, Tt+1 = ?

(b) Expressing the martingale model as Tt+1 = u+et+1  : E(et+1/ |t ) = 0; brieáy discuss the di§erences with (i) a random walk model,   and with (ii) a geometric Brownian motion in terms of the ad-ditional restrictions put on the error term et+1:        [10]

(c) Under which condition would that asset market be strong-form       e¢ cient? (Hint: recall the Grossman-Stiglitz paradox)        [10]

Section B

B1 Capital Asset Pricing Model and Arbitrage Pricing Theory.              [40]

(a) Suppose that an investor with preferences G(uP ; 7P(2)) = uP  _ a7 P(2)  : a > 0 optimally chooses a portfolio composed of only two assets: a single risky asset with expected return uj  and variance of return 7j(2) ; and a risk-free asset which yields r0 . Denote by q the proportion of the portfolio invested in the risky asset, and by uP  = quj + (1 _ q)r0 the expected return on the portfolio. Af- ter deriving the expression for the portfolio risk-return trade-o§, derive the investorís optimal proportion of initial wealth invested in the risky asset.                                                                   [10]

(b) Now assume that the asset market is frictionless and investors can freely borrow or lend at a risk-free rate, but that they are constrained to invest in portfolios that íblendíthe riskless asset and one risky asset.  Suppose that in equilibrium, the only two available risky assets yield uj  = 0:14 with a standard deviation of

7j  = 0:21 and ul = 0:08 with a standard deviation of 7l = 0:06: What is the risk-free rate of interest in this market?   If the investor wishes to hold a portfolio with a standard deviation of

0:11, how should s/he optimally allocate her/his initial wealth?  [10]

(c) If the Beta of asset l is 0.6, what are (i) the Beta of asset j, and (ii) the expected return uM  and standard deviation 7M  of the market portfolio? And (iii) what are the covariances of assets j

and l with the market portfolio?                                              [10]

(d) Suppose now that the matrix of asset payo§s is as follows:

0   j

State 1  5   15

State 2  5   8 Price   p0   pj

Assets

l

10

7

pl

If the probability of states is the same, write the CAPM in terms       of asset prices, and obtain p0 ; pj and pl : At those prices, are there       arbitrage oportunities?                                                            [10]

B2 Testing the Capital Asset Pricing Model (CAPM). [40]