Investment Analysis HW Week 6 Solutions
September 29, 2020
1. (a) According to the Expectations Hypothesis, the 2-year interest rate (or the yield on a 2-year risk-free zero coupon bond) is equal to the total expected return on two 1-year investments. Hence, the two-year interest rate, r2.t, the one-year interest, T1,t, and the expected 1-year rate one year from now, E(ri,t+1), for next year are related via the formula:
(1+ r2,t)2 = (1+r1,+)(1+ E(r1,t+1))
(1)
Rearranging, the expected one-year rate next year is given by
E(r1,+1) = 1+r1
1=.12
1.08
(b) If the investor invests in the one-year bond, we know that she will make the one year interest rate, or 8%. $1 will turn into $1.08 in one year. What about if the investor invests in the two-year bond and pulls out after a year? This is a litte tricky. Let's work it out as we did in class. What is the price of the 2 year risk free zero coupon bond today? We can work this out in two ways. First, we can use the YTM of this bond (which is 10% EAR) and the price is the PV of the promised payments at the YTM. The promised cashflow on this bond is its face value, which we are not told. Suppose we assume it's $100 (it doesn't matter what number we assume, it'll cancel out in the end). So the price today is 100/1.12. Another way to do it is to find its fair value, and say "since this bond is risk free, arbitrage will make its price equal to fair value". Its fair value is the PV of expected cashflows at OCC. The promised cashflow on this bond is its face value of $100. The expected cashflow is also $100, since the bond is risk free. The OCC on this bond, the comparable rate in the marketplace, is 10%: the rate being paid by similar investments - risk free and lasting two years. So its price is 100/1.12. It's not surprising that these two ways of finding the price give the same answer: the bond's YTM is set to equal ro,2. What is the price of the 2 year bond one year from now? Well, at that time there'll be only one year to go on the bond, and the one year interest rate will be
1
T1,2, and so the fair value of the bond will be 100/(1 + r1,2). Standing today, we don't know what this will be, because we don't know today what r1,2 will be. But we know what we expect it to be, we expect it to be E(r1.2). So the expected price
Er1,2 to be 6%, she expects the price to be 100/1.06. Now we are ready to find her HPR from buying the bond today (paying 100/1.12 and selling after one year (making an expected amount of 100/(1 + Er1,2)) Just like HPR is (amount made)/(amount paid), the expected HPR is (expected amount made)/amount paid, or 100 1.06 100