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.

1. Calculate the total current market value of the portfolio.
2. Calculate the modified duration of the portfolio
3. What would be the new market value of the portfolio if interest rates were to increase by 60 basis points across board? (use modified duration approach)
4. Suggest a portfolio adjustment the manager can use to mitigate the portfolio’s exposure to the yield increase.

Answer 1

Assume face value of each bond to be $ 1,000

Market value of bond A = PV (Rate, period, PMT, FV) = PV (6.5%, 6, -7% x 1000, -1000) = $  1,024.2051

Market value of bond B = Coupon / yield = 7.5% x 1000 / 6% = $ 1,250.0000

Market value of Bond C = Face value / (1 + yield)ⁿ = 1,000 / (1 + 6.9%)⁵ = $ 716.3272523

Part (a)

Portfolio comprises of 1,000 units of each of the three bonds.

hence, market value of the portfolio = 1000 x ( 1,024.2051 + 1,250.00 + 716.3272523) = $  2,990,532.32

Part (b)

For Bond A:

Year	Cash flows	PV of Ct	t x Pvt
t	Ct	PVt = Ct / (1 + 6.5%)^t
1	70	65.73	65.73
2	70	61.72	123.43
3	70	57.95	173.85
4	70	54.41	217.65
5	70	51.09	255.46
6	1070	733.31	4,399.85
Total		1,024.21	5,235.96

Duration = 5,235.96 / 1,024.21 = 5.11 years

Modified duration = Duration / (1 + yield) = 5.11 / (1 + 6.5%) =  4.80

Modified duration of Bond B = Modified duration of a perpetuity = 1 / yield = 1 /  6% =  16.67

Modified duration of a zero coupon bond = Years to maturity / (1 + yield) = 5 / (1 + 6.%) = 4.68

Hence, modified duration of the portfolio = 9.73 as shown below.

Bond	Market Value	Number	Value in portfolio	Proportion	Modified duration	Proportion x Modified duration
A	1,024.21	1,000	1,024,205	34.25%	4.80	1.64
B	1,250.00	1,000	1,250,000	41.80%	16.67	6.97
C	716.33	1,000	716,327	23.95%	4.68	1.12
Total			2,990,532			9.73

Part (c)

60 basis points = 60 / 100 = 0.6%

%age change in value = -%age change in interest rate x Modified duration = - 0.6% x 9.73 = -5.84%

Hence, the new market value of the portfolio if interest rates were to increase by 60 basis points across board? (use modified duration approach) = 2,990,532 x (1 - 5.84%) = $  2,815,931

Part (d)

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

• Portfolio manager can undertake a duration hedging. He can introduce new position in the portfolio such that modified duration of the portfolio is zero.
  • Portfolio manager can short bonds in the portfolio
  • Portfolio manager can undertake convexity hedging