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subscript refers to the length of the forward rate/price, and P represents current or spot

prices of various maturities. Similarly, fT,K = (PT/PT+K) â€“ 1. Thus

f1,1 = 100(90.15625/95.3125) = 94.590 ; f1,1 = (95.3125/90.15625) â€“ 1 = 5.719%

The implied 1-year forward rate is larger than the current 1-year spot rate, reflecting the

expectation that interest rates will go up in the future. Hence, for upward-sloping term

structures, the implied forward rate curve lies above the spot rate curve.

17. f1,5 = 100(69.50/95.3125) = 72.918% of par = 72:29 rounded to the nearest 32nd.

72.918 = 100/(1+f1,5)5 ; f1,5 = 6.521%

f3,2 = 100(74.21875/84.90625) = 87.413% of par = 87:13 rounded to the nearest 32nd.

87.413 = 100/(1+f3,2)2 ; f3,2 = 6.958%

[1 + (.05249/2)]4 = [1 + (.04859/2)]2(1+f1,1) + .0030 ; f1,1 = 5.433%

18.

f1,1 = 100/(1.05433) = 94.847% of par = 94:27 rounded to the nearest 32nd.

Intuitively, the maturity premium on 2-year investments makes the future 1-year STRIP

more valuable; hence the forward price is greater and the forward rate lower.

Alternatively, verify that if the forward rate and 1-year spot rate stayed the same as before,

the spot 2-year price would become 89.913% of par and the corresponding yield would be

5.388%; i.e., the longer maturity investment would be less valuable.

FUNDAMENTALS OF INVESTMENTS B-33

P* = 100/[1 + (.04859+.0025) / 2 ]2 = 95.080% of par

19. Feb01 STRIPS:

?% = (95.080 â€“ 95.3125)/95.3125 = â€“ 0.244%

P* = 100/[1 + (.05529+.0025) / 2 ]6 = 84.290% of par

Feb03 STRIPS:

?% = (84.290 â€“ 84.90625)/84.90625 = â€“ 0.726%

P* = 100/[1 + (.06157+.0025) / 2 ]12 = 68.496% of par

Feb06 STRIPS:

?% = (68.496 â€“ 69.5)/69.5 = â€“ 1.445%

For equal changes in yield, the longer the maturity, the greater the percentage price

change. Hence, for parallel yield curve shifts, the price volatility is greater for longer-term

instruments.

95.3125 â€“ .50 = 100/[1+(y*/2)]2 ;

Feb01 STRIPS: y* = 5.398%

? = 5.398 â€“ 4.859 = + 0.539% ; ?% = .539/4.859 = 11.09%

84.90625 â€“ .50 = 100/[1+(y*/2)]6 ;

Feb03 STRIPS: y* = 5.732%

? = 5.732 â€“ 5.529 = + 0.203% ; ?% = .203/5.529 = 3.66%

69.50 â€“ .50 = 100/[1+(y*/2)]12 ;

Feb06 STRIPS: y* = 6.281%

? = 6.281 â€“ 6.157 = + 0.124% ; ?% = .124/6.157 = 2.01%

For equal changes in price, the absolute yield volatility is greater the shorter the maturity;

the effect is magnified for percentage yield volatility when the yield curve is upward

sloping, because yields (the divisor) are smaller for short maturities. Because of this, note

that for sharply downward sloping yield curves, itâ€™ possible for shorter maturity

s

instruments to have less percentage yield volatility, but greater absolute yield volatility,

than slightly longer maturity instruments.

20. Real rate = .05 â€“ .035 = 1.5%. Real interest rates are not observable because they do not

correspond to any traded asset (at least not until very recently in the U.S.); hence, they

must be inferred from nominal interest rates (which do correspond to traded assets), and

from estimated inflation data. Real interest rate estimates are therefore only as good as (1)

the inflation estimates used in the Fisher relation, and (2) the degree to which the Fisher

relation itself actually describes the behavior of economic agents.

B-34 CORRADO AND JORDAN

Chapter 10

Bond Prices and Yields

Answers to Questions and Problems

Core Questions

1. Premium (par, discount) bonds are bonds that sell for more (the same as, less) than their

face or par value.

2. The face value is normally $1,000 per bond. The coupon is expressed as a percentage of

face value (the coupon rate), so the annual dollar coupon is calculated by multiplying the

coupon rate by $1,000. Coupons are paid semi-annually; the semi-annual coupon is equal

to the annual coupon divided by two.

3. The coupon rate is the annual dollar coupon expressed as percentage of face value. The

current yield is the annual dollar coupon divided by the current price. If a bondâ€™ price

s

rises, the coupon rate wonâ€™ change, but the current yield will fall.

t

4. Interest rate risk refers to the fact that bond prices fluctuate as interest rates change.

Lower coupon and longer maturity bonds have greater interest rate risk.

5. For a premium bond, the coupon rate is higher than the yield. The reason is simply that the

bonds sells at a premium because it offers a coupon rate that is high relative to current

market required yields. The reverse is true for a discount bond: it sells at a discount

because its coupon rate is too low.

6. A bondâ€™ promised yield is an indicator of what an investor can expect to earn if (1) all of

s

the bondâ€™ promised payments are made, and (2) market conditions do not change. The

s

realized yield is the actual, after-the-fact return the investor receives. The realized yield is

more relevant, of course, but it is not knowable ahead of time. A bondâ€™ calculated yield

s

to maturity is the promised yield.

7. The yield to maturity is the required rate of return on a bond expressed as a nominal

annual interest rate. For noncallable bonds, the yield to maturity and required rate of

return are interchangeable terms. Unlike YTM and required return, the coupon rate is not

a return used as the interest rate in bond cash flow valuation, but is a fixed percentage of

par over the life of the bond used to set the coupon payment amount. For the example

given, the coupon rate on the bond is still 10 percent, and the YTM is 8 percent.

FUNDAMENTALS OF INVESTMENTS B-35

8. Price and yield move in opposite directions; if interest rates rise, the price of the bond will

fall. This is because the fixed coupon payments determined by the fixed coupon rate are

not as valuable when interest rates riseâ€” hence, the price of the bond decreases.

9. P = $35(PVIFA4.25%,22) + $1000(PVIF4.25%,22) = $894.16

10. P = $1,225 = $51.25(PVIFAr%,28) + $1000(PVIFr%,28) ; r = 3.805%, YTM = 7.61%.

current yield = $102.50/$1,225 = 8.37%

Intermediate Questions

11. P = $860 = $C(PVIFA5%,21) + $1000(PVIF5%,21) ; C=$39.08, coupon rate=2(3.908) =

7.82%

12. P = $43.75(PVIFA7.25%/2,18) + $1000(PVIF7.25%/2,18) = $1,097.91

13. P = $960 = $47.50(PVIFAr%,20) + $1000(PVIFr%,20) ; r = 5.073%; YTM = 10.15%

14. a. Bond price is the present value of the future cash flows from a bond; YTM is the

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