Question

Consider a bond portfolio, which consists of 1,000 units each of three bonds:

• Bond A with annual coupons, a coupon rate of 7%, maturity of 6 years, and YTM of 6.5%.

• Bond B, a perpetuity with coupon rate of 7.5%, and YTM of 6.0%. •

Bond C, a zero coupon bond with YTM of 6.9% and maturity of 5 year.

a. Calculate the total current market value of the portfolio.

b. Calculate the modified duration of the portfolio

c. What would be the new market value of the portfolio if interest rates were to decrease by 50 basis points across board?(use modified duration approach)

d. Suggest a strategy the portfolio manager can use to mitigate the portfolio’s exposure to the yield increase.

Answer 1

Bond A: annual coupon 7%, Maturity Value (assumed) : $1000 and Term : 6 years and YTM = 6.5%. The current price will be the present value of future cash flows discounted at 6.5%;

Price = 70/(1+6.5%) + 70/(1+6.5%)² + ... + (1000 + 70)/(1+6.5%)⁶ = 1024.21

Bond B : Perpetuity coupon 7.5% and YTM 6%. Assuming face value of $ 1000, then the current price is simply the annual coupon divided by the YTM.

Price = 75/6% = 1250

Bond C : 5 year zero coupon bond YTM 6.9%. At face value (assumed) of $1000, the current price is:

Price = 1000/(1+6.9%)⁵ = 716.33

Hence, the total market value of the portfolio : 1000 * 1024.21 + 1000 * 1250 + 1000 * 716.33 = $2,990,540.00

Modified duration = Macaulay Duration / {1+ yield/n} ; where n is the frequency of coupon payment in a year. Macaulay duration is simply the PV of cash flows multiplied by the respective time period number and divided by sum of present value of all cash flows.

Bond A Duration and Modified Duration:

The modified duration will be = 5.11/ (1+6.5%/1) = 4.80

Bond B: Perpetuity duration = (1+YTM) / YTM and Modified Duration = Duration / (1+YTM) ; hence Modified Duration of Perpetuity = 1/YTM = 1/6% = 16.67

Bond C : Modified duration will same as time to maturity i.e 5

Portfolio Modified Duration is the weighted average of respective bond durations.

Portfolio Duration : (1024210/2990540) * 4.8 + (1250000/2990540) * 16.67 + (716330/2990540) * 5 = 9.81

Now if the yields decrease by 50 bps, the portfolio market value will increase by 9.81*0.5 = 4.90%

Hence new market value will be : 2990540 * (1+4.90%) = $ 3,137,216.79

The portfolio manager can mitigate the risk by immunising the bond portfolio which is align the portfolio duration to investors' investment horizon.