Investment Analysis HW Week 5 solutions
September 23,2020
1. (a) The definition of the YTM is: it is that interest rate, which, when used to take the PV of the promised cashflows, gives you the price of the bond. This is a zero coupon bond, so it only pays its face value. Suppose its face value is F, and the bond has t years to maturity. Suppose P is the price you buy the bond at. For ZCBs, we want the YTM to be an APR with annual compounding, (which is the same as an EAR). So when we take the PV of the promised payments to get the price, it will look like this: Then:
F
(1+YTM)t
Plug in the values we know: F = 1000,P = 800,t = 5, to get:
1000
800 = (1+YTM)5 1000 (1+YTM)5 800 (1+YTM)5 = 1.25 (1+YTM) = 1.251/5 (1+YTM) = 1.0456 YTM 4.56%
(b) I bought the bond for $800. What price do I sell it for? On the date I sold it, I know that its YTM is 7%, and the bond has 4 years until it pays off 1000. From the definition of YTM, the bond's price is the PV of the promised cashflows at the YTM,or
1000 Price = 762.90 (1+.07)4
1
Okay, so you bought the bond at 800 and sold it at 762.9. What's the HPR? The HPR is the dollars you made per dollar invested (minus one), or 800/762.9 -- 1 = 0.0464. What's the AHPR? The AHPR asks: if I had earned a constant rate across all the years I held that bond, what would that rate be? Well, we held the bond for only one year, so the AHPR is the same as the HPR: -4.64%. Your return is less than the YTM because yields rose and you sold the bond at a lower price. In fact, its price fell so far you actually lost money. (c) Working similarly as in the previous question, the selling price after 2 years is:
$1000 = $816.30 (1 +.07)3
Your HPR over the two years is 816.3/800 - 1 == 1.02375 -1 = 2.0375%. What's the AHPR? The AHPR asks: if I had earned a constant rate across all the years I held that bond, what would that rate be? Suppose the rate you made per year is x. This means that in the first year, a dollar would have turned into 1+ c, over two years it would have turned to (1 + x)2. But you know that a dollar turned to 1.02375 in two years, so
(1 + x)2 = 1.02375
or x = 1.01%. Note that even though rates are still 7%, your annual return over the two years is positive because the selling price of the bond is now higher than after one year As time passes the bond price rises: it "gets pulled to par". (d) Since the yield to maturity on comparable zeros is now 3% and there are two year