SAMPLE PROBLEMS- BONDS-FINANCE 327-FALL 2009
For problems 1 and 2 below, consider a 5-year 6% coupon bond.
1. Assuming that the bond is currently priced at $ 848.35, what is its yield to maturity ?
Using the approximate yield formula, or from your calculator,
60 + (1000 - 848.35)/5
30.33
YTM = ---------------------------- = ---------- = 9.77%
(1000 + 848.35)/2
924.175
2. (a) Assuming that the YTM on the bond is 8%, what is its current price ?
Price = 60 PVIFA (5,8%) + 1000 PVIF (5,8%) = $ 920.162
(b) Suppose now, that in 1 year, interest rates on comparable bonds rise to 9%. What is the price
of the bond at that time?
Price = 60 PVIFA (4,9%) + 1000 PVIF (4,9%) = $ 902.78
(c) If you buy the bond at $ 920.16 and sell it 1 year later (after the interest rate change and first
coupon), then what is your rate of return ?
Rate of return = (902.78 + 60 - 920.162)/920.162 = 4.63 %
(d) Suppose that in 1 year, instead of increasing to 9%. interest rates actually fall to 5%, what is
the price of the bond and the rate of return if you sell it?
Price = 60 PVIFA (4, 5%) + 1000 PVIF (4, 5%) = $ 1035.46
Return = (1035.46 + 60 - 920.162)/920.162 = 19.05%
Assume further that you invest these proceeds at the going rate of 5% per year for the next 4
years. Your five year rate of return is: (1.1905) (1.05) (1.05) (1.05) (1.05) = 1.44706 and the
geometric average annual return is (1.44706)(0.2) = 7.67%
(e) Suppose that instead of selling the bond (when rates dropped to 5%), you decide to hold the
bond until maturity. Interest rates stay at 5%. What is your realized compound yield (RCY) ?
|----------|-------|------|------|------|
-920.162 60 60 60 60 60
| |
|
|
1000
| |
|
|-----63
(60 x CVIF(1,5%)
| |
|------------- 66.15
(60 x CVIF(2,5%)
| |-------------------- 69.46
(60 x CVIF(3,5%)
|------------------------- 72.93
(60 x CVIF(4,5%)
1331.54
Thus 920.162 compounds to 1331.54 in 5 years. The RCY is
920.162 (1 + x)(0.2) = 1331.54, x = 7.67%
Notice that from part(d) above, if you had actually sold the bond at 1035.46 and reinvested the
cash flows for the next 4 years at the prevailing rate of 5% per year, you would still have earned
an average return of 7.67% per year which is identical to the RCY. Your decision to sell must
therefore be based upon your confidence in being able to earn more than 5% per year with the
capital.
3. For this problem, consider three 10% coupon bonds with 1,2 and 3 years to maturity
respectively. The bonds are identical in risk. The yield curve is such that 1-year bonds earn 5%,
2-year bonds earn 6% and 3-year bonds earn 7%.
(a) Consider a strategy of buying the 3-year bond and holding it for one year. Assuming that the
yield curve stays the same over this one year period, what is the rate of return to this strategy ?
Price of the 3-year bond now = 100 PVIFA(3,7%) + 1000 PVIF(3,7%) = 1078.73
If the yield curve stays the same over the next year, then 2-year bonds will yield 6% per year
(like they do now). The price of this bond is:
100 PVIFA (2,6%) + 1000 PVIF(2,6%) = 1073.34
NOTE: Strictly speaking the bond's price must be calculated using the upward sloped yield
curve. This implies that one of the two coupons above will be discounted at 5%, while the other
coupon and the 1000 maturity payment will be discounted at 6%. This technicality is ignored for
this problem.
Rate of return from buying the bond at 1078.73, selling it at 1073.34 and collecting the $ 100
coupon is:
(1073.34 + 100 - 1078.73)/1078.73 = 0.877 or 8.77%
This is clearly better than the strategy of buying a 1-year bond at the yield of 5%.
(b) Calculate the 1-year and 2-year forward rates implied by this yield curve.
Let the 1-year forward rate be x. Then, (1 + 0.05) (1 + x) = (1+ 0.06)2 or x = 7.01%
Let the 2-year forward rate be y. Then (1 + 0.05) (1 + y)2 = (1 + 0.07)3 = 8.01%
In other words, under the pure expectations hypothesis, the market expects that one year from
now, 1-year maturity bonds will yield 7% while 2-year maturity bonds will yield 8%. This is
equivalent to an upward shift of 1% in the yield curve.
(c) Suppose the yield curve does in fact shift according to the pure expectations hypothesis.
Interest rates 1 year later are such that 1-year bonds earn 7% and 2-year bonds earn 8%. What is
the rate of return to the strategy developed in part(a)?
The price of the bond is now calculated at the 2-year yield of 8% under the new (shifted) yield
curve as 100 PVIFA (2,8%) + 1000 PVIF(2,8%) = 1035.63
Note that the same caution as part (a) in pricing the bond still applies.
Rate of return from buying the bond at 1078.73, selling it at 1035.63 and collecting the $ 100
coupon is:
(1035.63 + 100 - 1078.73)/1078.73 = 0.05275 or 5.28% now only
slightly better than buying the 1-year maturity bond.
(d) How much should the yield curve shift for the strategy of buying the 3-year bond to generate
the same rate of return as the 1-year bond (i.e 5%) ?
This requires working backwards,
1) to derive the price (one year later) on the 3-year bond that will generate a 5% return.
2) at this price, to find the YTM on the bond.
For a 5% return over 1-year, assume the 3-year bond is priced at X one year after purchase.
Then, (X + 100 - 1078.73)/1078.73 = 0.05, X = 1032.66
For a 2-year 10% bond to be priced at 1032.66, the YTM is:8.23%
1.C onsider a 5-year, 10% coupon bond priced currently at par. You are considering purchase of
this bond, with the intent of selling it at the end of your 3-year investment horizon. The spot rate
(yield) on 3-year bonds is 6%, and you project an increase in yield to 8% on comparable bonds
after your horizon.
(a)
Assuming you buy the bond now and sell it at the end of your horizon, what price can you
expect to sell it for? What will be your realized compound yield? Reinvestment rate is 6%.
In 3 years, the bond’s price will reflect a 2-year maturity and an 8% yield. Or,
Selling Price = 100 * PVIFA (2, 8%) + 1000 * PVIF (2, 8%) = $ 1035.67.
You will receive $100 in annual coupons for each of the 3 years you hold the bond,
and these will be reinvested at 6% per year. The future value of reinvested coupons is
100 * FVIFA (3, 6% ) = 318.36
Total future value at sale is 1035.67 + 318.36 = 1354.03.
Your RCY is the rate at which an initial $1000 (par) investment grows to $ 1354.,03
in 3 years, or 1000 * FVIF (3, x%) = 1354.03, x = 10.63%
(b)
What forward rates are expected to prevail in the 2 years after your investment horizon?
Since 5-year bonds earn 10% per year and 3-year bonds earn 6% per year in the spot
market, the 2-year forward rate for the years after the investment horizon is given by
[(1.10)5 / (1.06)3] - 1 = 35.22%. The average annual rate in years 4 and 5 is
(1.3522)0.5 - 1 = 16.28%
( c) How would your answers in (a) be affected if the actual spot rate in years 4 and 5 are equal
to the forward rate calculated in (b). No answers are needed, just a discussion.
The sale price of the bond after 3 years will be much less than the estimated
$ 1035.67 above. The Realized Compound yield will be much smaller.