Question

The following table shows average bond yields across bonds (with semi-annual coupons) according to their ratings.

Rating	Bond-equivalent yield
AAA	4.4%
AA	4.6%
A	4.9%
BBB	5.4%
BB	5.9%
B	6.5%

a)   A company is considering issuing 20-year bonds with face value of $1,000 and semi-annual coupons. If the bonds are rated A and the company would like to sell the bonds at par, what would the semi-annual coupon be?

b)   What is the duration of the bonds (assuming the coupons from part a)?

c)   What is the modified duration? And based on the modified duration, how much is the bond price expected to change for a 0.5% change in the interest rate?

d)   Suppose that the Federal Reserve decided to reduce its federal funds target rate by 1% immediately after the bond issue, causing the average yield on long-term A-rated bonds to decrease by 0.5%. What is the new bond value and duration? And how does the change in value compare to the answer in part c)?

e)   Suppose instead that poor firm performance announced immediately after the bond issue triggered a downgrading of the bonds to BBB. What is the new bond value and duration? And how does the change in value compare to the answer in part c)?

f)   Based on your calculations of durations above, what is the relation between duration and yield to maturity?

Answer 1

MV: Price of bond

FV: face value of bond = 100

t: time to maturity in years = 20

n compounding frequency (semi-annuall) = 2

(a)

y: yield of bond Rated A = 4.9% = 0.049

(a) C: coupon rate (= yield when bond is issued at par) = 4.9%

Coupon = 4.9%*100 = 4.9 (4.9/2 = 2.45 will be paid every 6 months from date of issue)

(b)D: Duration

y+ = 4.9% + 0.05% = 4.95%

P+ price of bond when (yield=4.95% but coupon = 4.9% using above) = 99.370

y- = 4.9% - 0.05% = 4.85%

P- price of bond when (yield=4.85% but coupon = 4.9% using above) = 100.636

P: price of bond (when bond is issued at par its price is equal to face value = 100) = 100

(b)Duration using above equation = 12.66 years

Modified duration =

(c)Modified duration = 12.66/(1+4.9%/2) = 12.357years

If yield decreases by 0.5%, then price of bond = 100.636 (calculated above)