A1 Expected Utility and Portfolio Choice. [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) Suppose that the agent has initial wealth A = 1 to invest in two Önancial assets, one riskless and one risky. The price of the riskless asset is 1 and its return is 2, and short-selling on this asset is allowed. The price of the risky asset is 1 and its return is r with probability distribution:

r = 1 with r = 2 with r = 3 with

probability p1

probability p2

probability p3

Short-selling the risky asset is not allowed. If the agent invests a in the risky asset and 1 l a in the riskless asset, Önd the support of the probability distribution u = (u1 ; u2 ; u3 ) of the agentís portfolio return rP : If the agent maximises a von Neumann- Morgestern utility function u(W) over Önal wealth W , show that the optimal choice of a is positive if and only if the expectation of r is greater than 2. [Hint: Önd the Örst derivative of u(:) and

calculate its value when a = 0:]                                               [5]

(d) Finally, Önd a when u(W) = 1 l exp(lbW); b > 0 and when u(W) =  ( ) W1l3 ; 0 < 3 < 1 for the case in which u3  >

u 1 :                                                                                                    [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+1l 2t ) = (1 + e)pt : Brieáy explain the intuition behind it, and discuss the role of the infor- mation set 2t : Interpret e: What does it imply when forecasting

the assetís return, Tt+1 = ?

(b) Expressing the martingale model as Tt+1 = e+et+1  : E(et+1l 2t ) = 0; brieáy discuss the di§erences with (i) a random walk model,   and (ii) a geometric Brownian motion in terms of the additional 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(eP ; aP(2)) = eP  l aa P(2)  : a > 0 optimally chooses a portfolio composed of only two assets: a single risky asset with expected return ej  and variance of return aj(2) ; and a risk-free asset which yields r0 . Denote by q the proportion of the portfolio invested in the risky asset, and by eP  = qej + (1 l 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 ej  = 0:11 with a standard deviation of aj  = 0:18 and el = 0:06 with a standard deviation of al = 0:08:

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 0.8, what are (i) the Beta of asset j, and (ii) the expected return eM  and standard deviation aM  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]

(a) Consider the following linear relationship between the return to an asset j , rj ; and the return to the market portfolio m, rm :

ajm

a m(2)

where rf  denotes the return to the risk free asset, ajm denotes the covariance between the returns to asset j and the market portfolio m, and am(2) denotes the variance of the market portfolio return.  Provide an interpretation of the above linear relation- ship.  In particular, what is the risk premium of asset j? Why? What is the Beta of asset j?  What is its relationship to non- diversiÖable market risk? Support your answers with the help of a graph, where you clearly identify the Capital Market Line and

the E¢

ciency

Frontier.

[10]

(b) Make all necessary assumptions to derive the CAPM principal

estimating equation:

rj l rf = aj + 8j (rm l rf ) + ∈j

where ∈j  is i.i.d. N(0; a): In particular, show that 8j =  :    [10]

(c) Suppose that you estimated the above relationship by OLS, and that you Önd that your coe¢ cient estimate of aj  is signiÖcant and statistically di§erent from zero.  Would this invalidate the CAPM predictions?  Under which assumption(s) would you ex- pect the constant coe¢ cient estimate to be positive?  Why? [10]

(d) Finally, describe a statistical procedure that would allow you to       test the validity of the CAPM. (Hint:  consider the Arbitrage       Pricing Model.  Does it nest the CAPM? Under which null hy-       pothesis?)                    [10]