Question

Peter Piper is a​ 35-year-old bank executive who has just inherited a large sum of money. Having spent several years in the​ bank's investments​ department, he's well aware of the concept of duration and decides to apply it to his bond portfolio. In​ particular, Elliot intends to use $ 1 million of his inheritance to purchase 4 U.S. Treasury​ bonds:

1. An 8.55 %​, ​13-year bond​ that's priced at $ 1087.98 to yield 7.48 %.

2. A 7.801 %​, ​15-year bond​ that's priced at $ 1020.50 to yield 7.57 %.

3. A​ 20-year stripped Treasury​ (zero coupon)​ that's priced at $ 200.05 to yield 8.21 %.

4. A​ 24-year, 7.42 % bond​ that's priced at $ 950.76 to yield 7.88 %.

Note that these bonds are semiannual compounding bonds.

a. Find the duration and the modified duration of each bond.

b. Find the duration of the whole bond portfolio if Elliot puts $ 250000 into each of the 4 U.S. Treasury bonds.

c. Find the duration of the portfolio if Elliot puts $ 330000 each into bonds 1 and 3 and $ 170 comma 000 each into bonds 2 and 4.

d. Which portfolio b or c should Elliot select if he thinks rates are about to head up and he wants to avoid as much price volatility as​ possible? Explain. From which portfolio does he stand to make more in annual interest​ income? Which portfolio would you​ recommend, and​ why?

Answer 1

I have used the excel functions DURATION and MDURATION to solve this question. Please see the table below. Please be guided by the second row to understand the mathematics. The cells highlighted in yellow contain your answer. Figures in parenthesis, if any, mean negative values. All financials are in $.

Part (a)

Bond	Settlement	Maturity	Coupon	Yield	Frequency	Basis	Duration	Modified Duration
	A	B	C	D	E	F	'=DURATION(A,B,C,D,E,F)	'=MDURATION(A,B,C,D,E,F)
1	1/1/2019	1/1/2032	8.550%	7.48%	2	1	8.30	8.00
2	1/1/2019	1/1/2034	7.801%	7.57%	2	1	9.16	8.82
3	1/1/2019	1/1/2039	0.000%	8.21%	2	1	20.00	19.21
4	1/1/2019	1/1/2043	7.420%	7.88%	2	1	11.25	10.82

Part (b)

Weight of each of the bond = w = 250,000 / 1,000,000 = 1/4 = 0.25

The duration of this portfolio = weighted average duration = 0.25 x (8.30 + 9.16 + 20.00 + 11.25) = 12.18 years

Part (c)

W1 = W3 = 330,000 / 1,000,000 = 0.33

W2 = W4 = 170,000 / 1,000,000 = 0.17

The duration of this portfolio is = 0.33 x 8.30 + 0.17 x 9.16 + 0.33 x 20.00 + 0.17 x 11.25 = 12.81 years

Part (d)

He should invest in the portfolio in part b. The portfolio in part c has a higher duration than the portfolio in part b. If rates are about to​ rise, then it is safer to invest in the portfolio in part b​, because it would be less price volatile than the other portfolio.