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）︰99.05.06
考試時限（分鐘）：15:50-18:20 (150mins)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
每大題各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.(加分題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.
編註:綠字表下標 黃字表隨機變數 紫字表矩陣 紅字表腳註
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※ 編輯: vincent7977 來自: 61.230.112.119 (06/30 19:30)