課程名稱：投資學
課程性質︰必修
課程教師︰陳其美
開課學院：管理學院
開課系所︰財金系
考試日期（年月日）︰2010/11/25
考試時限（分鐘）：150分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
Part II, Computations. This is an open-book section. Don't work on this part
until the TA announces that you can start. Solutions must be supported by
explicit computations; an answer without associated computations will not
earn you any credit.
1. Consider a two-period economy with perfect financial markets, where N>=5
risky assets and 1 riskless asset (asset zero) 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 opportunity), but the N risky assets are strictly in
positive net supply. The Sharpe-Lintner CAPM holds in the date-0 equilibrium.
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. In the following, rj, rm and rf stand for respectively the
rate of return on asset j, the rate of return on the market portfolio, and the
riskless rate of interest. Also, we denote the date-1 cash flow generated by
one unit of asset j by xj.
(i) Consider a traded portfolio Z that generates the following date-1 cash
flow:
z = 1 - 5rm + rf + 3r1 + 2r2.
What is the date-0 equilibrium price Pz for portfolio Z? (hint 1.)
The following additional information is relevant for part (ii) and part (iii)
● Investor A's date-0 initial wealth is W(A0) = $1,000,000, and she lends
$200,000 in equilibrium.
● Investor B's date-0 initial wealth is W(B0) = $1,500,000, and she borrows
$300,000 in equilibrium.
● Investor C's equilibrium portfolio consists of only investor A's
equlibrium portfolio and the market portfolio, and the portfolio weights that
investor C assigns to those two portfolios are respectively 1/4 and 3/4.
● Let rA, rB, and rC be the (random) rates of return on the equilibrium
portfolios held by respectively A, B and C. You are told that var(rB)-Var(rA)
= 13/1125 and that E[rC]-rf = 247/3000.
● You are told that E[x3] = 2 = E[x4], cov(x3,rm) = 1/6, cov(x4,rm) = 3/20.
(ii) Suppose that the date-0 equilibrum price for asset 3 is p3 = 10/11.
Compute the date-0 equilibrium price p4 of asset 4.
(iii) Continue with part (ii). Suppose that investor D has initial wealth
W(D0) = $650,000, and the expected rate of return on her date-0 equilibrium
portfolio is 2/15. How much does investor D borrow or lend at date-0?
hint 1: You can solve this problem in at least two ways. First, you can apply
the uncertainty equivalent formula to portfolio Z. Second, you can construct
a portfolio consisting of asset 0, asset 1, asset 2, and the market portfolio,
such that the date-1 cash flow of the latter portfolio is exactly z.
2. This problem is adapted from Example 12 of Lecture 4. In this two-period
economy, the date-0 markets for the N risky assets are perfect. The N risky
assets are in strictly positive net supply. The riskless asset(asset 0),
however, is either in positive net supply or in zero net supply. Regarding
the return on the riskless asset, the lending rate rL differs from the
borrowing rate rB. Recall that r(mvp) denotes the rate of return on the
minimum variance portfolio composed of risky assets only.
Here, the key difference from Example 12 of Lecture 4 is that now we assume
rB = E[rmvp] > rL > 0.
(i) Suppose that we need at least K distinct portfolios to span the portfolio
frontier, where K is a positive integer. Compute K. Draw the portfolio
frontier on the σ-μ space.
(ii) Suppose that we need at least k distinct portfolios to span the efficient
frontier, where k is a positive integer. Compute k. Draw the efficient
frontier on the σ-μ space.
(iii) Suppose that the riskless asset is in zero net supply and that N = 2.
Suppose that in addition to her position in the riskless asset, investor A
spends $600,000 on asset 1 and $120,000 on asset 2 at date 0. Suppose that
investor B spends $2,000,000 on asset 1. How much does investor B spend on
asset 2?
3. This last problem tests your knowledge about stochastic dominance. A
monopolistic firm, M, is about to introduce either product X or product Y to
the market. M can choose to produce X or Y, but not both. Producing X would
incur a fixed cost Fx, and producing U would incur a fixed cost Fy. For
simplicity, producing X or Y does not incur variable costs.
Since X and Y are new products, their demands are random from M's perspective.
The demand for product X would be Qx = Ax(random) - Px, if M choose to produce
X, and the demand for product Y would be Qy = Ay(random) - Py, if M chooses to
produce Y, where Qx, Qy stand for the damand quantities, Px, Py stand for the
product prices chosen by M, and the two strictly positive random variables Ax
, Ay are such that Ax >=(SSD) Ay, with E[Ax]=1, var(Ax) = 2 < 4 = var(Ay).
The timing of relevant events is as follows.
● M must first choose to produce X or to produce Y. At this point, M is risk
neutral and so M seeks to maximized expected profits.
● Suppose that M has chosen to produce product j = X or Y. Then M must spend
Fj.
● Then the realization of Aj is observed by M.
● Then, given the realization of Aj, M chooses Pj to maximize profits.
(i) Suppose that M has chosen to produce product j and has spent Fj. Moreover,
suppose that M has seen the realization a of the random variable Aj. Show that
the optimal product price for M is a/2, and that in this case, M's optimal
profit is [(a^2)/4 - Fj]. (hint 2)
(ii) Show that there exists a real number T > 0 such that, at the very
beginning, before seeing the realizations of Ax and Ay, M prefers producing
X to producing Y if and only if Fy - Fx > T. Compute T.
(iii) Explain how part (ii) is related to Theorem 7 of Lecture 2, Part I.
hint 2: At this moment, M's maximization problem is
max (a-Pj)(Pj-0) - Fj
where 0 is the zero variable cost.
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