Question 1:

Treasury bonds paying an 8% coupon rate with semiannual payments currently sell at par value.

What coupon rate would they have to pay in order to sell at par if these bonds instead paid their

coupons annually?

Question 2:

Two bonds have identical times to maturity and coupons rates. One is callable at 105, the other

at 110. Which should have the higher yield to maturity? Why?

Question 3:

Consider a newly issued bond that pays its coupon once annually, and whose coupon rate is 5%;

the maturity is 20 years, and yield to maturity is 8%.

(a) Assuming there are no taxes, find the holding period return for a one-year investment period

if the bond is selling at a yield to maturity of 7% by the end of the year.

(b) (NOT EXAM MATERIAL) If you sell the bond after one year, what taxes will you owe if

the tax rate on interest income is 40% and the tax rate on capital gains income is 30%? The

bond is subject to original- issue discount tax treatment.

(c) (NOT EXAM MATERIAL) What is the after-tax holding period return on the bond?

Question 4:

Assume you have a one-year investment horizon and are trying to choose among three bonds.

All have the same degree of default risk and mature in 10 years. The first bond is a zero-coupon

bond that pays $1,000 at maturity. The second one has an 8% coupon rate and pays the $80

coupon once per year. The third bond has a 10% coupon rate and pays the $100 coupon once per

year. For parts (a) and (b), assume that there are no taxes.

(a) If all three bonds are now priced to yield 8% to maturity, what are their prices?

(b) If you expect their yields to maturity to be 8% at the beginning of next year, what will their

prices be then? What is your before-tax holding period return on each bond?

(c) (NOT EXAM MATERIAL) If your tax bracket is 30% on ordinary income and 20% on

capital gains income, what will your after-tax rate of return be on each bond?

Hint: In computing taxes, assume that the 10% coupon bond was issued at par and that the

drop in price, when the bond is sold at year-end, is treated as a capital loss (and not as

an offset to ordinary income).

Question 5:

You have the following information about a convertible bond issue:

Burroughs Corporation

7 ¼% Due 8-1-2010

————————————————————————–

Agency rating (Moody’s/S&P)

A3/A-

Conversion ratio

12.882

Market price of convertible

102

Market price of common stock

$ 66.00

Dividend per share-common

$ 2.60

Call price (first call: 8-1-2000)

106

Estimated floor price

$ 66.50

————————————————————————-

Using the information above, calculate the following values and show calculations:

(a)

(b)

(c)

(d)

Market conversion value.

Conversion premium per common share.

Current yield-convertible.

Dividend yield-common.

Question 6:

The yield to maturity on one-year zero-coupon bonds is currently 7%, and the yield to maturity

on two-year zeros is 8%. The Treasury plans to issue a two-year maturity coupon bond, paying

coupons once per year with a coupon rate of 9%. The face value of the bond is $100.

(a) At what price will the bond sell?

(b) What will the yield to maturity on the bond be?

(c) If the expectations theory of the yield curve is correct, what is the market expectation of the

price that the bond will sell for next year?

(d) Recalculate your answer to (c) if you believe in the liquidity preference theory, and that the

liquidity premium is 1%.

Question 7:

U.S. Treasuries represent a significant holding in many pension portfolios. You decide to analyze

the yield curve for U.S. Treasury Notes.

(a) Using the data in the table below, calculate the five- year spot and forward rates assuming

annual compounding. Show your calculations.

(Hint: the spot rates are yields to maturity for zero-coupon bonds; yields to maturity for

coupon bonds selling at par will typically differ).

U.S. Treasury Note Yield Curve Data

————————————————————————————————————

Years to

Par Coupon

Calculated

Calculated

Maturity

Yield-to-Maturity

Spot Rates

Forward Rates

———————————————————————————————————–

1

5.00

5.00

5.00

2

5.20

5.21

5.42

3

6.00

6.05

7.75

4

7.00

7.16

10.56

5

7.00

–

–

————————————————————————————————————

(b) Based on the above yield curve analysis, calculate both the expected yield to maturity and the

price of a 4- year zero. Show your calculations.

Question 8:

The yield to maturity on one-year- maturity zero coupon bonds is 5% and the yield to maturity on

two-year- maturity zero coupon bonds is 6%. The yield to maturity on two-year- maturity coupon

bonds with coupon rates of 12% (paid annually) is 5.8%. What arbitrage opportunity is available

for an investment banking firm? What is the profit on the activity?

Practice Set #2: Solutions.

The effective annual yield on the semiannual coupon bonds is 8.16% = (1+8%/2)2 – 1. If the

annual coupon bonds are to sell at par, then they must offer the same yield, which will require an

annual coupon rare of 8.16%.

The bond callable at 105 (110) requires the issuing firm to pay bondholders 105% (110%) of the

bond’s face value if the firm decides to call the bond. The first bond should therefore sell for a

lower price because the call provision is more valuable to the firm that issued it. Therefore, that

bond’s yield to maturity should be higher than that of the bond callable at 110.

(a) You can use Excel or a financial calculator to compute the following:

The initial price is: P0 = $705.46, for [n = 20; PMT = 50; FV = 1000; i = 8]

The next year’s price is: P1 = $793.29, for [n = 19; PMT = 50; FV = 1000; i = 7]

Thus, the holding period return (HPR) is given by:

HPR = [$50 + ($793.29 – $705.46)]/$705.46 => HPR = 0.195 = 19.5%

(b) Using OID tax rules, the price path of the bond under the constant yield method is obtained

by discounting at an 8% yield, and reducing maturity by one year at a time:

Constant yield prices:

P0 = $705.46

P1 = $711.89 (implies implicit interest over first year = $6.43)

P2 = $718.84 (implies implicit interest over second year = $6.95)

• Tax on explicit plus implicit interest in the first year = 0.40 x ($50 + $6.43) = $22.57

• Capital gain in the first year = actual price – constant yield price

= $793.29 – 711.89 = $81.40

• Tax on capital gain = 0.30 x $81.40 = $24.42

• Total taxes = $22.57 + $24.42 = $46.99

(d) The after-tax HPR = [$50 + ($793.29 – $705.46) – $46.99]/$705.46

= 0.129 = 12.9%

(a) Using Excel (for example) we get:

Zero coupon

$463.19

8% coupon

$1,000.00

Zero coupon

8% coupon

10% coupon

Price one year from now

$500.25

$1,000.00

$1,124.94

Price increase

$37.06

$0.00

-$9.26

Coupon income

$0.00

$80.00

$100.00

Pre-tax income

$37.06

$80.00

$90.74

Pre-tax rate of return

8.00%

8.00%

8.00%

Taxes*

$11.12

$24.00

$28.15

After-tax income

$25.94

$56.00

$62.59

After-tax return

5.60%

5.60%

5.52%

* In computing taxes, we have assumed that the 10% coupon bond was issued at par and that the

drop in price, when the bond is sold at year-end, is treated as a capital loss (tax rate = 20%) and

not as an offset to ordinary income (tax rate = 30%).

(a) Market conversion price = value if converted into stock

= market price of common stock x conversion ratio

= 12.882 x $66 = $850.21

(b) Conversion premium = Bond price – value if converted into stock

= $1020 – (12.882 x $66) = $1020 – $850.21 = $169.79

Thus, the conversion premium per share = ($169.79/12.882) = $13.18

(c) Current yield = (coupon/price) = ($72.50/$1020) = 0.0711 = 7.11%

(d) Dividend yield on common = (dividend per share/price) = ($2.60/$66) = 3.94%

(a) P = [(9/1.07) + (109/1.082 )] = $101.86

(b) YTM = 7.958%, which is the solution to: 9 PA(y,2) + 100 PF(y,2) = 101.86

(c) The forward rate for next year, derived from the zero-coupon yield curve, is approximately

9%:

1 + f2 = [1.082 /1.07] = 1.0901, which implies f2 = 9.01%. Therefore, using an expected rate

for next year of r2 = 9%, we can find that the forecast bond price is

(d) If the liquidity premium is 1%, then the forecast interest rate is:

E[r2 ] = f2 – liquidity premium = 9% – 1% = 8%, and you forecast the bond to sell at:

(109/1.08) = $100.93.

1000 = [70/(1 + y1 )] + [70/(1 + y2 )2 ] + [70/(1 + y3 )3 ] + [70/(1 + y4 )4 ] + [1070/(1 + y5 )5 ]

1000 = [70/1.05] + [70/1.05212 ] + [70/1.06053 ] + [70/1.07164 ] + [1070/(1 + y5 )5 ]

1000 = $66.67 + $63.24 + $58.69 + $53.08 + [$1070/(1 + y5 )5 ]

$1000 – $241.68 = [$1070/(1 + y5 )5 ]

(1 + y5 )5 = $1070/$758.32

=> $758.32 = [$1070/(1 + y5 )5 ]

(1 + y5 ) = (1.411)1/5

=>

[1.07135 /1.07164 ] –1 = [1.411/1.3187] –1 = 7%

(b) The spot rate at 4 years is 7.16%. Therefore, 7.16% is the theoretical yield to maturity for the

zero coupon U.S. Treasury note. The price of the zero coupon at 7.16% is the present value

of $1000 to be received in 4 years.

• Annual compounding: PV = [1000/1.07164 ) = $758.35

• With semi-annual compounding, we would have: PV = [1000/(1 + (0.0716/2))8 ] = $754.73

The price of the coupon bond, based on its yield to maturity, is:

120 PA(5.8%, 2) + 1000 PF(5.8%, 2) = $1,113.99.

If the coupons were stripped and sold separately as zeros, then based on the yield to maturity of

zeros with maturities of one and two years, the coupon payments could be sold separately for

[120/1.05] + [1,120/1.062 ] = $1,111.08.

The arbitrage strategy is to buy zeros with face values of $120 and $1,120 and respective

maturities of one and two years, and simultaneously sell the coupon bond. The profit equals

$2.91 on each bond.