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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