## Solution Manual For Corporate Finance 7th Canadian Edition By Ross – Westerfield

Chapter 6: How to Value Bonds and Stocks

6.1 The price of a pure discount (zero coupon) bond is the present value of the par.

a. PV = $1,000/(1+0.05)15 = $481.02

b. PV = $1,000/(1+0.10)15 = $239.39

c. PV = $1,000/(1+0.15)15 = $122.89

6.2 The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes a semiannual coupon. The price of the bond at each YTM will be:

a. P = $35({1 – [1/(1 + 0.035)30] } / 0.035) + $1,000[1 / (1 + 0.035)30]

P = $1,000.00

When the YTM and the coupon rate are equal, the bond will sell at par.

b. P = $35({1 – [1/(1 + 0.045)30] } / 0.045) + $1,000[1 / (1 + 0.045)30]

P = $837.11

When the YTM is greater than the coupon rate, the bond will sell at a discount.

c. P = $35({1 – [1/(1 + 0.025)30] } / 0.025) + $1,000[1 / (1 + 0.025)30]

P = $1,209.30

When the YTM is less than the coupon rate, the bond will sell at a premium.

We would like to introduce shorthand notation here. Rather than write (or type, as the case may be) the entire equation for the PV of an annuity , it is common to abbreviate the equations as:

= ({1 – [1/(1 + r) t] }/r )

which stands for Present Value Interest Factor of an Annuity

This abbreviation is short hand notation for the equations in which the interest rate and the number of periods are substituted into the equation and solved. We will use this shorthand notation in the remainder of the solutions key.

6.3 Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:

PV = $1,050 = $39 + $1,000/(1+r)20

Since we cannot solve the equation directly for r, using a spreadsheet, a financial calculator, or trial and error, we find:

r = 3.547%

Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so:

YTM = 2 3.547% = 7.09%

6.4 Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows:

P = $1,060 = C + $1,000/(1+0.038)23

Solving for the coupon payment, we get:

C = $41.96

Since this is the semiannual payment, the annual coupon payment is:

2 × $41.96 = $83.92

And the coupon rate is the annual coupon payment divided by par value, so:

Coupon rate = $83.92 / $1,000 = 0.0839, or 8.39%

6.5 The price of any bond is the PV of the interest payment, plus the PV of the par value. The fact that the bond is denominated in euros is irrelevant. Notice this problem assumes an annual coupon. The price of the bond will be:

PV = €84 + €1,000/(1 + 0.076)15

PV = €1,070.18

6.6 Here we are finding the YTM of an annual coupon bond. The fact that the bond is denominated in yen is irrelevant. The bond price equation is:

P = ¥92,000 = ¥2,800 + ¥100,000/(1+r) 21

Since we cannot solve the equation directly for r, using a spreadsheet, a financial calculator, or trial and error, we find:

r = 3.34%

Since the coupon payments are annual, this is the yield to maturity.

6.7 Here we are finding the present value of the bonds for various maturity lengths. The bond price equation is:

P = C + $1,000/(1+r)t

Miller Corporation bond:

P1 = $40 + $1,000/(1+0.03)24 = $1,169.36

P3 = $40 + $1,000/(1+0.03)20 = $1,148.77

P8 = $40 + $1,000/(1+0.03)10 = $1,085.30

P12 = $40 + $1,000/(1+0.03)2 = $1,019.13

P13 = $1,000

Modigliani Company bond:

P1 = $30 + $1,000/(1+0.04)24 = $847.53

P3 = $30 + $1,000/(1+0.04)20 = $864.10

P8 = $30 + $1,000/(1+0.04)10 = $918.89

P12 = $30 + $1,000/(1+0.04)2 = $981.14

P13 = $1,000

Bond Price Behavior as Maturity approaches

Bond Price

$1,169.36

$1,148.77 Miller Bond

$1,0000 (Face Value)

Periods until Maturity

$864.10 Modigliani Bond

$847.53

All else held equal, the premium over par value for a premium bond declines as maturity approaches, and the discount from par value for a discount bond declines as maturity approaches. This is called “pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity lengths.

Also, notice that the price of each bond when no time is left to maturity is the par value, even though the purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the clean price of the bond.

6.8 Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial YTM on both bonds is the coupon rate, 7 percent. If the YTM suddenly rises to 9 percent:

PLaurel = $35 + $1,000/(1+0.045)4 = $964.12

PHardy =$35 + $1,000/(1+0.045)30 = $837.11

The percentage change in price is calculated as:

Percentage change in price = (New price – Original price) / Original price

PLaurel% = ($964.12 – $1,000) / $1,000 = – 0.0359, or –3.59%

PHardy% = ($837.11 – $1,000) / $1,000 = – 0.1629, or –16.29%

If the YTM suddenly falls to 5 percent:

PLaurel = $35 + $1,000/(1+0.025)4 = $1,037.62

PHardy = $35 + $1,000/(1+0.025)30 = $1,209.30

PLaurel% = ($1,037.62 – $1,000) / $1,000 = +0.0376, or 3.76%

PHardy% = ($1,209.30 – $1,000) / $1,000 = +0.2093, or 20.93%

Bond Price Behavior vs Yield to Maturity

Bond Price

$1,209.3

$1,037.62

$1,000

$964.12 Laurel Bond $837.11

Hardy Bond

5% 7% 9% Yield to Maturity

All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in interest rates. Notice also that for the same interest rate change, the gain from a decline in interest rates is larger than the loss from the same magnitude change. For a plain vanilla bond, this is always true.

6.9 Initially, at YTM of 10 percent, the prices of the two bonds are:

PFaulk = $30 + $1,000/(1+0.05)24 = $724.03

PGonas = $70 + $1,000/(1+0.05)24 = $1,275.97

If the YTM rises from 10 percent to 12 percent:

PFaulk = $30 + $1,000/(1+0.06)24 = $623.49

PGonas = $70 + $1,000/(1+0.06)24 = $1,125.50

The percentage change in price is calculated as:

Percentage change in price = (New price – Original price) / Original price

PFaulk% = ($623.49 – $724.03) / $724.03 = – 0.1389, or –13.89%

PGonas% = ($1,125.50 – $1,275.97) / $1,275.97 = – 0.1179, or –11.79%

If the YTM declines from 10 percent to 8 percent:

PFaulk = $30 + $1,000/(1+0.04)24 = $847.53

PGonas = $70 + $1,000/(1+0.04)24= $1,457.41

PFaulk% = ($847.53 – $724.03) / $724.03 = + 0.1706, or 17.06%

PGonas% = ($1,457.41 – $1,275.97) / $1,275.97 = + 0.1422, or 14.22%

All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates.

6.10 The bond price equation for this bond is:

P0 = $1,050 = $31 + $1,000/(1+r)18

Using a spreadsheet, financial calculator, or trial and error we find:

r = 2.744%

This is the semiannual interest rate, so the YTM is:

YTM = 2 2.744% = 5.49%

The current yield is:

Current yield = Annual coupon payment / Price = $62 / $1,050 = .0590 or 5.90%

The effective annual yield is the same as the EAR, so using the EAR equation from the previous chapter:

Effective annual yield = (1 + 0.02744)2 – 1 = 0.0556, or 5.56%

6.11 The company should set the coupon rate on its new bonds equal to the required return. The required return can be observed in the market by finding the YTM on outstanding bonds of the company. So, the YTM on the bonds currently sold in the market is:

P = $1,063 = $35 + $1,000/(1+r)40

Using a spreadsheet, financial calculator, or trial and error we find:

r = 3.218%

This is the semiannual interest rate, so the YTM is:

YTM = 2 3.218% = 6.44%

6.12 To find the number of years to maturity for the bond, we need to find the price of the bond. Since we already have the coupon rate, we can use the bond price equation, and solve for the number of years to maturity. We are given the current yield of the bond, so we can calculate the price as:

Current yield = 0.0842 = $90/PV

PV = $90/0.0842 = $1,068.88

Now that we have the price of the bond, the bond price equation is:

PV = $1,068.88 = $90{[(1 – (1/1.0781 t)]/0.0781} + $1,000/(1.0781)t

We can solve this equation for t as follows:

1,068.88 (1.0781)t = 1,152.37 (1.0781)t – 1,152.37 + 1,000

152.37 = 83.49(1.0781)t

1.8251 = 1.0781t

t = log 1.8251 / log 1.0781 = 8.0004 8 years

The bond has 8 years to maturity.

6.13 The bond has 9 years to maturity, so the bond price equation is:

P = $1,053.12 = $36.20 + $1,000/(1+r)18

Using a spreadsheet, financial calculator, or trial and error we find:

r = 3.226%

This is the semiannual interest rate, so the YTM is:

YTM = 2 3.226% = 6.45%

The current yield is the annual coupon payment divided by the bond price, so:

Current yield = $72.40 / $1,053.12 = 0.0687, or 6.87%

6.14 We found the maturity of a bond in Problem 6.12. However, in this case, the maturity is indeterminate. A bond selling at par can have any length of maturity. In other words, when we solve the bond pricing equation as we did in Problem 6.12, the number of periods can be any positive number.

6.15 To find the capital gains yield and the current yield, we need to find the price of the bond. The current price of Bond P and the price of Bond P in one year is:

P: P0 = $90 + $1,000/(1+0.07)10 = $1,140.47

P1 = $90 + $1,000/(1+0.07)9 = $1,130.30

Current yield = $90 / $1,140.47 = 0.0789, or 7.89%

The capital gains yield is:

Capital gains yield = (New price – Original price) / Original price

Capital gains yield = ($1,130.30 – $1,140.47) / $1,140.47 = – 0.0089, or –0.89%

The current price of Bond D and the price of Bond D in one year is:

D: P0 = $50 + $1,000/(1+0.07)10 = $859.53

P1 = $50 + $1,000/(1+0.07)9 = $869.70

Current yield = $50 / $859.53 = 0.0582 or 5.82%

Capital gains yield = ($869.70 – $859.53) / $859.53 = 0.0118, or 1.18%

All else held constant, premium bonds pay a high current income while having price depreciation as maturity nears; discount bonds pay a lower current income but have price appreciation as maturity nears. For either bond, the total return is still 7%, but this return is distributed differently between current income and capital gains.

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