Question

2. An investor plans to retire in 10 years. As part of the retirement portfolio, the investor buys a newly issued, 12-year, 8% annual coupon payment bond. The bond is purchased at par value, so its yield-to-maturity is 8.00% stated as an effective annual rate.

a. Calculate the approximate Macaulay duration for the bond, using a 1 bp (0.01%) increase and decrease in the yield-to-maturity and calculating the new prices per 100 of par value to six decimal places.

b. Calculate the duration gap at the time of purchase. (Hint: An investor plans to retire in 10 years. So, this investor’s investment horizon is 10 years.)

c. Does this bond at purchase entail the risk of higher or lower interest rates?

d. A bond is currently trading for 98.722 per 100 of par value. If the bond’s yield-to-maturity (YTM) rises by 10 basis points, the bond’s full price is expected to fall to 98.669. If the bond’s YTM decreases by 10 basis points, the bond’s full price is expected to increase to 98.782. What is the bond’s approximate convexity?

Answer 1

Part A

The approximate duration of the bond is 7.5361

let pv at 0 equals to 100 and considering the same change at pv+ and pv-

Pv+ = 8(1.0801)*1 +...+108( 1.0801)12 = 99.924678

Pv-= 8(1.0801)*1+ .....+108(1.0801)12 = 100.075400

so duration for approximately is calculated as The macular duration basically is calculated by multiplying the time period by the periodic coupon payment and dividing the same by the resulting values plus 1 periodic yield to time to maturity

i.e

(100.075400-99.924678)/2*0.0001*100 = 7.5361

Approximately macuralry duration is 7.5361*1.08= 8.1390

Part b

Duration gap can be found out by subtracting the 10 years i.e 8.1390-10 = -1.8610

at the time of purchase, the duration gap is negative for this bond

Part c

So as the negative duration gap basically means the risk of low interest rates however, to be intact reinvesting bonds less than the coupon rate is higher when compared to selling the same at a price above with constant yields.

Part D

formuale for approximate convexity is = [pv_+pv++-(2*pv0)]/ change in yield * Pv0

i.e pv_ = price after ytm decreased

pv+ = price after ytm increased

pv0 = actual price

so,

Approximate Convexity = [98.782+98.669-(2*98.722)]/0.0012*98.722 = 70.906