课程名称︰投资学 课程性质︰必修 课程教师︰陈其美 开课学院：管理学院 开课系所︰财金系 考试日期（年月日）︰99.05.06 考试时限（分钟）：15:50-18:20 (150mins) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 每大题各20分 1.Consider a two-period economy with perfect financial markets, where two risky assets (asset 1 and 2) and 1 riskless asset (asset 0) are traded at date 0, and cash flows are generated at date 1. The riskless asset is in zero net supply (a lending and borrowing oppotunity), but the two risky assets are in strickly positive net supply. For j=1,2 let xj denote the date-1 random cash flow generated by one unit of asset j, and let pj denote the date-0 equilibrium price of asset j. The following information is important for solving this problem. ‧The Sharpe-Lintner CAPM holds in the date-0 equilibrium. ‧There are only 3 investors (i=1,2,3) in the date-0 financial markets. They only want to consume at date 1, and hence each of them will invest all the initial wealth in the date-0 traded assets. Investor i is endowed with initial wealth Wi0>0, together with a mean variance utility function Ui(E[Wi],var[Wi]), where Ui(-,-) is increasing in its first argument and decreasing in its second argument, and Wi is investor i's date-1 (random) terminal wealth. ‧Investor 3's date-0 initial wealth is W30=$400,000. ‧In the date-0 equilibrium, the riskless rate rf is known to be lower than E[rmvp], which is the expected rate of return on the minimum varience portfolio (or MVP) generated by assets 1 and 2. ‧The date-0 equilibrium price of asset 1 is p1=10. ‧In the date-0 equilibrium, investor 1, who neither lends nor borrows, holds 6000 units of asset 1 and 1000 units of asset 2. ‧In the date-0 equilibrium, investor 2, in addition to her holdings of assets 1 and 2, lends $500,000. ‧In the date-0 equilibrium, the portfolio weight of asset 2 in investor 3's equilibrium portfolio is 3/4. ┌x1┐ ‧The first two moment of│ │are summarized as follows: └x2┘ ┌var[x1] cov[x1,x2]┐ ┌ 4 -3┐ ┌E[x1]┐ ┌12.4┐ │ │=│ │, │ │=│ │. └cov[x1,x2] var[x2]┘ └-3 9┘ └E[x2]┘ └32.4┘ (i)Compute the first two moments e and V of equilibrium rates of return on respectively aseet 1 and asset 2,1 where ┌var[r1] cov[r1,r2]┐ ┌E[r1]┐ V2x2=│ │, e2x1=│ │. └cov[r1,r2] var[r2]┘ └E[r2]┘ (ii)Suppose that, in addition to her position in asset 0 and asset 2, investor 2 holds 10,000 units of asset 1 in the date-0 equilibrium. Compute the expected rate of return on investor 2's equilibrium portfolio. (iii)Continue with part (ii). Suppose that investor 1's mean-varience utility function takes the linear form: E[W1]-ρvar[W1]. Compute the positive constant ρ.2 (iv)Now ignore part (ii). Assume instead that investor 2's equilibrium portfolio has a beta equal to 1/5. What is the date-0 aggregate wealth Wm=W10+W20+W30?3 2.This problem is adapted from Example 12 of Lecture 4.4 Everything is the same as in Example 12 (read the preceeding footnote), except that now we have E[rmvp]>rB>rL. Suppose that we need at least k distinct portfolio to span the efficient frontier, and we need at least K distinct portfolio to span the portfolio frontier, where k,K are both positive integers. Draw the portfolio frontier on the σ-μ space. Compute k+K. 3.This last problem test your knowledge about stochastic dominance. (i)(First-order stochastic dominance) Mr.A wishes to buy 1 unit of product X from a drugstore, called B. Before arriving at the store, Mr.A is not sure about the price p of product X chosen by B. Mr.A only knows the distribution function of p, which is F(p;x,y)= ┌0 , p＜2/(1+y) │(1+1/x)[1-2/(1+y)p], 2/(1+y)≦p＜1 │(1+1/x)[1-2/(1+y)] , 1≦p＜2 └1 , 2≦p In the above, x>0 and y>1 are two constants. Mr.A's welfare is represented by the expected consumer surplus generated by the transaction with B, which is 2-E[p], where 2 is the gross utility that Mr.A obtains from consuming 1 unit of product X. (i-a)Other things equal, does an increase in y make Mr.A better off or worse off?5 (i-b)Other things equal, does an increase in x make Mr.A better off or worse off?6 (ii)(Second-order stochastic dominance) Mr.C wishes to buy 1 unit of product Y at date 1 from either firm D or firm E. Before date 1, Mr.C has already known the price that firm E choose for product Y, which is 10 dollars, but Mr.C does not know the price P chosen by firm D for product Y. The realization of P will become known to Mr.C at the beginning of date 1, and Mr.C will purchase product Y from the firm that charges a lower price for product Y. (Assume for simplicity that Mr.C will definitely purchase 1 unit of product Y, either from firm D or from firm E.) Mr.C's welfare before date 1 is represented by the expected consumer surplus 100-E[s], where s is tha actual amount of money that Mr.C spends for product Y at date 1, and 100 is the gross utility that Mr.C obtains from consuming 1 unit of product Y. Suppose that P=z+e, where e has zero expected value, and moveover, e and z are independent. (ii-a) Write s=f(P). What is the function f(-)? Is f(-) concave or convex? (ii-b) Would Mr.C become better off or worse off, if firm D's price for product Y were z rather than P?7 4.(加分题20分，可加到期末考) Initially, we assume that only two risky assets in positive net supply are traded in perfect financial market at date 0. It is known that there is only one investor in this economy, and he holds a mean-variance efficient portfolio at date 0. In date-0 equilibrium, the first two monent of the rates of return on the two risky assets are summarized as follows: ┌0.1┐ ┌0.01 0.02┐ e=│ │, V=│ │. └0.2┘ └0.02 0.09┘ Now, suppose that, actually, in addition to the above two risky assets, a riskless asset in zero net supply (a lending and borrowing oppotunity) is also available for trading at date 0. Show that in the date-0 equilibrium with 3 traded assets, rf must satisfy rD<rf<rU. Find rD and rU.8 Footnotes: 1.Apparently, you need to find p2 first. Recall that every investor must hold a portfolio composed only of the riskless asset and the market portfolio in equilibrium. 2.Hint:Investor 1's equlibrium portfolio must maximize the above mean-varience utility function subject to the constraint that investor 1's equilibrium portfolio lies on the CML. 3.Hint:Investor 2's equlibrium portfolio must be a portfolio composed only of the riskless asset and the market portfolio. 4.Suppose that in the two-period economy, the date-0 markets for the N risky assets are perfect. However, regarding the riskless asset, the lending rate rL differs from the borrowing rate rB, with E[rmvp]>rB>rL, where rmvp is the rates of return on the minimum variance portfolio composed of risky assets only. 5.Hint:Consider y'>y>1. Compare F(p;x,y) and F(p;x,y') on each of the following intervals: [0,2/1+y'), [2/1+y',2/1+y), [2/1+y,1), [1,2), and [2,+∞). Which one between the two following lies above the other one? 6.Hint:For x'>x>0. Compare F(p;x',y) and F(p;x,y) as in the preceding hint. 7.Hint:Given a realization z of z, what is E[e|z=z]? 8.Hint:The market for the riskless asset must clear in the new equilibrium. 编注:绿字表下标 黄字表随机变数 紫字表矩阵 红字表脚注 --

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