課程名稱︰ 投資學
課程性質︰ 系必修
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）︰ 2012/6/23
考試時限（分鐘）： 170分鐘
是否需發放獎勵金： 是 感謝
（如未明確表示，則不予發放）
試題 :
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 sup-
ported by explicit computations; an answer without associated computations
will not earn you any credit.
1. (15 points) 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 | face value |date-0 price
-----------------------------------------------
1 | date-3 | 40 | 1000 | 796
-----------------------------------------------
2 | date-3 | 20 | 1000 | 748
In addition to bonds 1 and 2, asset X is also traded at date 0, and
it generates sure cash inflows per share at dates t = 1; 2; 3, which are
summarized in the following table.
asset|date-0 price|date-1 cash inflow|date-2 cash inflow|date-3 cash inflow
---------------------------------------------------------------------------
X | 2340 | 100 | x | 3100
Then, it can be shown that no arbitrage opportunities exist if and only
if _ _
x < x < x compute x , x
_ = = _
2. (15 points) Consider an economy with perfect nancial markets that
extends 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 that the true state is an element of
Ω = {ω1; ω2; ω3; ω4}. At t = 1, investors know whether the true state is
an element of E = {ω1; ω2} or an element of Ec = {ω3; ω4}. At t = 2,
investors know exactly which among w1; w2; w3; w4 is the true state. It
is known that markets are dynamically complete over the date-0-date-2
period, and there are many assets traded at date 0, including assets 1
and 2. Asset 1 pays dividends only at date 2, and asset 2 is a money
market account. The cum-dividend prices 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 | 121/84 | 1.1 | 2.2 | 1 | 1.48 | 3.3 | 1.1 |
------------------------------------------------------------------------
2 | 11/14 | 1.1 | 1.1 | 1.32 | 1.32 | 1.1 | 1.1 |
(i) Compute the forward rate f0(1; 2). (Notation follows from Lecture
7. Recall that f0(1; 2) is the one-period interest rate stated in the date-
0 forward contract for a loan to be lent at date 1 and repaid at date 2.)
(ii) Consider the following coupon bond (referred to as bond Z), (C; F; T) =
(100; 1000; 2), traded at date 0. Consider the futures contract signed at
date 0 for 1 unit of bond Z to be delivered at date 1, after the bond has
already paid its date-1 coupon payment. Compute the futures price
H(0) specied in this date-0 contract.
3.(15 points) Consider an economy with perfect nancial markets that
extends 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 that the true state is an element of
Ω = {ω1; ω2; ω3; ω4}. At t = 1, investors know whether the true state is
an element of E = {ω1; ω2} or an element of Ec = {ω3; ω4}. At t = 2,
investors know exactly which among w1; w2; w3; w4 is the true state. It
is known that markets are dynamically complete over the date-0-date-2
period, and there are many assets traded at date 0, including assets 1
and 2. Asset 1 pays dividends only at date 2, and asset 2 is a money
market account. The cum-dividend prices 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 |
Consider two traded bonds A and B at date 0, where bond A is a coupon
bond (C; F; T) = (C; 1000; 2) and bond B is a floating-rate bond that
will mature at date 2 with face value equal to 1000 and that will make
at date t + 1 屬於 {1; 2} the interest payment 1000[0:05 + r~(t; at)] after
event at occurs, where recall that r~(t; at) is the (realized) interest rate
for the date-t-date-(t + 1) period.
At date 0, originally, Mr. A is holding 1 unit of bond A, and Mr. B is
holding 1 unit of bond B. Now, Mr. A and Mr. B decide to swap the
bonds, so that after the trade, Mr. A will hold 1 unit of bond B and
Mr. B will hold 1 unit of bond A. Suppose that trade takes place at
date 0 without money change-hands. Compute C.
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