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）︰99.06.24
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 :
Open-book.
1.(15points) At date 0, two coupon bonds are traded, and their data are
summarized in the following table (where the date-0 bond prices are obtained
after the date-0 coupon payments are made):
bond│maturity│coupon payment│face value│date-0 price
──┼────┼───────┼─────┼──────
1│ date 3│ 30│ 1000│ 669
──┼────┼───────┼─────┼──────
2│ date 3│ 50│ 1000│ 715
In addition to bonds 1 and 2, assets X and Y are also traded at date 0, and
they both generate sure cash inflows per share at dates t=1,2,3, which are
summarized in the following table.
cash inflow
asset│date-0 price│date-1│date-2│date-3
───┼──────┼───┼───┼───
X│ 600│ 100│ x│ 100
───┼──────┼───┼───┼───
Y│ 350│ 200│ 400│ y
(i) Suppose that x=300. Show that one can obtain a sure stream of cash
inflows {Ct;t=0,1,2,3}, with C0>0, C1>0, C2=0, and C3=0, by holding a1 units
of bond 1, a2 units of bond 2, and ax units of asset X from date 0 to
date 3, where |ax|=1, and where, for j=1,2, one must sell bond j short if
aj<0. Find a1,a2,ax,C0,C1,C2,and C3.
(ii) Suppose that y=100. Show that one can obtain a sure stream of cash
inflows {C't;t=0,1,2,3}, with C'0>0, C'1=0, C'2>0, and C'3=0, by holding b1
units of bond 1, b2 units of bond 2, and by units of asset Y from date 0 to
date 3, where |by|=1, and where, for j=1,2, one must sell bond j short if
bj<0. Find b1,b2,bx,C'0,C'1,C'2,and C'3.
2.(15 points) Consider an economy with perfect financial market that extend
for three dates (t=0,1,2) with 4 date-2 states of nature (ω1,ω2,ω3,ω4).
The common information structure for investors is as follows. At t=0,
investors know whether the ture state is an element of Ω={ω1,ω2,ω3,ω4}.
At t=1, investors know whether the ture state is an element of E={ω1,ω2}
or Ec={ω3,ω4}. At t=2, investors know exactly which among ω1,ω2,ω3,ω4
is the ture state. It is known that markets are dynamically complete, and
there are many assets traded at date 0, including asset 1, forward and
futures contracts written on 1 unit of asset 1 to be delivered at date 2,
zero-coupon bonds maturing at date 2, and so on. The following table gives
some (but not all) information about the cum-dividend prie of asset 1 (which
pays dividend only at date 2) and the futures price for the delivery of
1 unit of asset 1 at date 2 (before asset 1 pays dividends):
asset│(0,Ω)│ (1,E)│(1,Ec)│ (2,ω1)│ (2,ω2)│ (2,ω3)│ (2,ω4)
────┼───┼───┼───┼────┼────┼────┼────
p1(t,at)│ 19│ 20│ 18│ ?│ ?│ ?│ ?
────┼───┼───┼───┼────┼────┼────┼────
H(t,at)│ 45/2│ 25│ 20│ ?│ ?│ ?│ ?
Suppose that at date 0 Mr.A is endowed with 17,000 units od asset 1 and a
long of 1,000 futures contracts described above. Suppose that Mr.A wishes to
consume onlt at date 2, and he insists on consuming the same amount in the 4
date-2 states. Find the optimal amount of date-2 consumption for Mr.A.
3.(15 points) (15 points) Consider an economy with perfect financial market that extend
for three dates (t=0,1,2) with 4 date-2 states of nature (ω1,ω2,ω3,ω4).
The common information structure for investors is as follows. At t=0,
investors know whether the ture state is an element of Ω={ω1,ω2,ω3,ω4}.
At t=1, investors know whether the ture state is an element of E={ω1,ω2}
or Ec={ω3,ω4}. At t=2, investors know exactly which among ω1,ω2,ω3,ω4
is the ture state. It is known that markets are dynamically complete, and
there are many assets traded at date 0, including the equity and the debt
issued by firm X, asset 1 and 2, and so on. Asset 1 pays dividends only at
date 2, and asset 2 is a money market account. The cum-dividend price of
assets 1 and 2 at each time-event node (t,at) are summarized in the
following table:
asset│(0,Ω)│ (1,E)│(1,Ec)│ (2,ω1)│ (2,ω2)│ (2,ω3)│ (2,ω4)
───┼───┼───┼───┼────┼────┼────┼────
1│ 7/4│ 1.1│ 2.2│ 1│ 1.48│ 3.3│ 1.1
───┼───┼───┼───┼────┼────┼────┼────
2│ 1│ 1.1│ 1.1│ 1.32│ 1.32│ 1.1│ 1.1
The following information will be useful.
‧At date 0, firm X's assets consist nothing but 1,000 units of asset 1.
‧The firm has a single shareholder Mr.B, and besides equity, it has issued
some debt before date 0, and the debt will mature at date 2 with a face
value equal to F. The firm is protected by limited liability, and no
coupon payments will be made to distributed will be made to the debtholder
before date 2.
‧The firm has committed that no dividends will be distributed at date 0 and
date 1.
‧At date 1, if event E occurs, then the controlling shareholder (Mr.B) can
decide to or not to spend $1,100 immediately on an investment project,
which will generate a date-2 cash inflow C(2,ω1)=0 in state ω1 and
C(2,ω2)=$1,500 in state ω2. (Note that before making the new investment,
the firm's assets inplace at date1 in event E are worth exactly $1,100.)
‧Debtholder, according to debt contract, are not allowed to intervene with
Mr.B's investment decision at date 1.
(i) Suppose that F=0; i.e., at date 0 the firm has issued only only equity.
Then the firm's date-0 equity value is v. Find v.
(ii) Suppose that F=$1,200. Then the firm's date-0 equity value and date-0
dept value are respectively e and d. Find e and d. Which one between v and
e+d is larger?
編註:綠字為下標 藍字為上標
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