Question

Elliot Karlin 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 8.69 %​, ​13-year bond​ that's priced at $ 1 comma 100.37 to yield 7.47 %.
2. A 7.821 %​, ​15-year bond​ that's priced at $ 1024.08 to yield 7.55 %.
3. A​ 20-year stripped Treasury​ (zero coupon)​ that's priced at $ 198.52 to yield 8.25 %.
4. A​ 24-year, 7.46 % bond​ that's priced at $ 957.11 to yield 7.86 %.
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 $ 250 comma 000 into each of the 4 U.S. Treasury bonds.
c. Find the duration of the portfolio if Elliot puts $ 380 comma 000 each into bonds 1 and 3 and $ 120 comma 000 each into bonds 2 and 4.
d. Which portfoliolong dashb or clong dashshould 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

Part (a)

I have used the excel functions Duration and Mduration. Inputs are settlement date which i have arbitrarily assumed to be 01/01/2019 for each of them. This doesn't impact any of the answers. Maturity date has been assumed to be the same date i.e. (01/01) but as many years later as is the maturity of the bond. Type is 2 which reflects semi annual coupon payment.

Bond	Settlement	Maturity	Coupon	Yield	Duration	Modified Duration
	A	B	C	D	E = DURATION (A, B, C, D, 2)	F = MDURATION (A, B, C, D, 2)
13 years bond	1/1/2019	1/1/2032	8.69%	7.47%	8.2791	7.9810
15 year bond	1/1/2019	1/1/2034	7.82%	7.55%	9.1561	8.8230
20-year Zero coupon bond	1/1/2019	1/1/2039	0%	8.25%	20.0000	19.2077
24 year bond	1/1/2019	1/1/2043	7.46%	7.86%	11.2516	10.8261

Part (b)

Equal weights across all four. Hence, duration = average duration = (8.2791 + 9.1561 + 20.0000 + 11.2516) / 4 = 12.1717

Part (c)

Duration = Weighted average duration = 0.38 x 8.2791 + 0.12 x 9.1561 + 0.38 x 20 + 0.12 x 11.2516 = 13.1950

Part (d)

Duration is a measure of volatility or interest rate risk. A lower duration assures a relatively lower decline in price when interest rate rises. Hence, Elliot should select the first portfolio where an amount of $ 250,000 is invested across each of the four bonds. This portfolio has lower duration and hence should have lower reduction in value when interest rate heads up.

Lower the proportion of zero coupon bond in the portfolio, higher will be the interest income. Hence, the first portfolio where an amount of $ 250,000 is invested across each of the four bonds, will give the higher interest income.

I would have advised the first portfolio where an amount of $ 250,000 is invested across each of the four bonds. This portfolio has lower duration as well as higher interest income.