*ONLY ANSWER PART iii WHICH IS THE LAST PART TO THE QUESTION. EVERYTHING ELSE IS ANSWERED AND SHOWN BELOW

1. A risk-free zero coupon bond pays $1,000 at the end of six years.

(a) The risk-free rate is currently 10% effective annual. What should the current price of the bond be?

(b) Suppose the bond currently costs $500. Describe an arbitrage to take advantage of any discrepancy you see. What will you buy? What will you sell? What are your cashflows today and in the future? For the remaining sub-questions, assume that there are no arbitrage opportunities.

(c) Joan buys the bond today after arbitrage by market participants has restored its price to what it should be. She intends to hold it till maturity, and to receive the payment of $1,000. What will her HPR be? What will her annualized HPR (AHPR) be?

(d) So shes bought the bond, and holds it for four years. Then, without warning, her son decides to go to medical school and she has to sell all her investments to pay the fees.

i. Suppose the risk-free rate on the date she sells the bond is 6% effective annual. What is the price she sold the bond for? What was the HPR she made over the four years? What was the AHPR she made?

ii. Suppose instead that the risk-free rate on the date she sells the bond is 10% effective annual. What is the price she sold the bond for? What was the HPR she made? What was the AHPR she made?

iii. Evaluate the truth of this statement: At the time Joan bought the bond, the interest rate in the market was 10%, and the bond was priced so that we will make 10% per year on it. We already know that the payment the bond makes is risk-free. Therefore Joan is guaranteed to make 10% on her investment in this bond. Under what conditions is this statement true? Under what conditions can this statement be false?

ANSWER:

a) The price of the bond should be its fair value: what it costs to buy the bonds cashow(s) in the market. Since the market interest rate is 10%, to get 1,000 in six years you would need 1000/(1+.1)⁶= 564.474

b) The two ways of getting $1,000 in six years are: Buy the ZCB, pay 500 Lend for six years in the marketplace (buy the markets risk-free asset),pay 564.474. Buy low, sell high, means we buy the ZCB and sell the markets risk free asset,i.e., borrow. Buying the ZCB means that we pay $500 now and get $1000 in six years. Sellingthe markets risk-free asset means that we get $564.474 now and pay $1000 in sixyears. In total, we get 64.474 now and pay nothing in six years.

c) She would get 1,000 at the end of six years, and she paid $564.474 now. So her HPR is (1000/564.474)-1 = 1.77156-1 = 77.156% To nd her AHPR, remember that the AHPR asks the question: if shed been earning interest at a constant yearly eective rate for the period she held the bond, what would that yearly rate have been?Suppose that yearly rate wasx. If she had invested a dollar at the start, in oneyear she would have made 1(1 +x). In two years, that times (1 +x), or (1 +x)². In six years she would have made (1 +x)⁶. But we know that her dollar turned into 1.77156, so (1 +x)⁶= 1.77156 Or (1 +x) = 1.77156^(1/6)= 1.099999 So x = 10%

d) (i)On the date she sells it, the bond promises to pay $1,000 for sure in two years. Since the interest rate is 6%, the fair value of the bond on that date is 1000/(1.06)²= 889.996. Since were assuming that theres no arbitrage opportunity,the price of the bond also equals 889.996. Therefore her HPR (889.996/564.474)-1 =1.57668-1 = 57.668% Suppose that yearly rate was x. If she had invested a dollar at the start, in four years she would have made (1+x)⁴. But we know that her dollar turned into 1.57668, so(1 +x)⁴= 1.57668 Or x= 12.056%

(ii) Working as above, the price of the bond on the date she sells it is 1000/1.12=826.4463. Therefore her HPR is (826.4463/564.474) - 1 = 1.46409-1 = 46.409%. To nd her AHPR, suppose that yearly rate was x. If she had invested a dollar at the start, in four years she would have made (1 +x)⁴. But we know that her dollar turned into 1.46409, so (1 +x)⁴= 1.46409 Or x = 10%

iii) ANSWER NEEDED FOR THIS PART