Question

Sam Strother and Shawna Tibbs are vice presidents of Mutual of Seattle Group Health Cooperative and codirectors of the organization's pension fund management division. The unions that represent the GHC hospital staff have requested an investment seminar so that they better understand the decisions being made on behalf of their members. Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions. a. What is the value of a ten-year, $1,000 par value bond with a 10 percent annual coupon if its required rate of return is 10 percent? b. What would be the value of the bond described in question a. if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing investors to require a 13 percent return? Would we now have a discount or a premium bond? c. What would be the value of the bond described in question a. if, just after it had been issued, the expected inflation rate fell by 3 percentage points, causing investors to require a 7 percent return? Would we now have a discount or a premium bond? d. What would happen to the value of the ten-year bond over time if the required rate of return remained at 13 percent, remained at 7 percent, or remained at 10 percent? Graph your results using the table below: Value of Bond in Given Year: N 7% 10% 13% 0 1 2 3 4 5 6 7 8 9 10 e. What is the yield to maturity on a ten-year, 9 percent annual coupon, $1,000 par value bond that sells for $887.00? f. What are the total return, the current yield, and the capital gains yield for the bond in question e.? (Assume the bond is held to maturity and the company does not default on the bond.)

Answer 1

Price of a bond is present value of all cashflows associated with the bond - namely coupons and maturity value, discounted at the required rate of return or current market interest rate.

Mathematically, price of a bond is shown as:

where P is price of a bond, C is the annual coupon, i is the required rate of return or YTM, M is the maturity value, n is the number of years to maturity.

Basic principle of bond pricing says -

YTM > coupon rate - price of bond would be lower than par. (at Discount)

YTM = coupon rate - price of bond would be equal to par.

YTM < coupon rate - price of bond would be higher than par. (at Premium)

Now let us calculate what is asked in different questions:

a. Required rate of return = 10%

Here, YTM = 10%, Coupon rate = 10%. Since YTM = Coupon Rate

Bond would trade at par, i.e., $1,000. (You can try putting values in formula mentioned above, I will go by concept and apply formula in next parts)

b. Required rate of Return = 13%

YTM = 13%, n = 10 years, C = $100 {= $1000 * 10%}, M = $1,000

Substituting values in formula:

P = $837.21

Since the current price is less than the par value, bond is trading at discount.

c. Required rate of Return = 7%

YTM = 7%, n = 10 years, C = $100 {= $1000 * 10%}, M = $1,000

Substituting values in formula:

P = $1,210.71

Since the current price is higher than the par value, bond is trading at premium.

d. For bond with YTM = 10%, the price of the bond would remain at $1000 if the YTM stayed at 10%.

For bond with YTM = 13% and which was trading at discount, the price would increase with maturity, such that at maturity, price of bond would become $1000

For bond with YTM = 7% and which was trading at premium, the price would decrease with maturity, such that at maturity, price of bond would become $1000

Using Excel here, to calculate the value of bond at different years for 3 YTM and graphs them

e. YTM can be approximated through manual calculation. Exact YTM value requires excel use.

C = $90, F = $1000, P = $887, n = 10

Approx YTM = 101.3/943.5

Approx YTM = 10.74%

By Excel calculation:

Hence, YTM = 19.91% (if you can't use Excel, use 10.74% as answer)

f. Current Yield = Annual Coupon/Current price = $90/$887 = 10.15%

Capital gains (over the life of bond) = ($1000 - $887)/$887 = 12.74%

Total return of Bond = 10.15% + 12.74% = 22.89%