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 8.68%​,​13-year bond​ that's priced at $1,099.55 to yield 7.47%.

2. A 7.852%​,​15-year bond​ that's priced at $ 1028.65$to yield 7.53 %.

3. A​ 20-year stripped Treasury​ (zero coupon)​ that's priced at $198.52 to yield 8.25%.

4. A​ 24-year, 7.43%bond​ that's priced at $955.96 to yield 7.84 %.

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 $250,000 into each of the 4 U.S. Treasury bonds.

c. Find the duration of the portfolio if Elliot puts $350,000 each into bonds 1 and 3 and $150,000

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?

a. The duration and modified duration can be calculated using a​ spreadsheet, such as Excel. It gives the precise duration measure because it avoids the​ rounding-off errors, which are inevitable with manual calculations.

Answer 1

The formulas of duration and modified duration in excel require settlement and maturity date. Settlement date is the date at which bond is purchased and maturity date is the date at which bond will be retired.

For settlement and maturity date, let's assume all 4 bonds are purchased today and will mature at the end of their life.

for Bond 1, settlement date is today's date i.e. 8/23/2019 on which bond is purchased. bond's life is 13 years. so, after 13 years it will mature on maturity date of 8/23/2032.

a. Calculation of duration and modified duration using excel

b. Total bond portfolio value is $1,000,000 and $250,000 invested in each of the 4 bonds. So weight of each bond in the portfolio is $250,000/$1,000,000 = 0.25.

Portfolio duration = weight of the bond*duration of the bond

Portfolio duration = 0.25*8.28 + 0.25*9.15 + 0.25*20 + 0.25*11.27 = 2.07 + 2.2875 + 5 + 2.8175 = 12.175

c. If different amounts will be used to purchase each bond then weight of each bond in the portfolio value be different. so we need to calculate the new weights of each bond first.

weight of bond 1 and 3 = $350,000/$1,000,000 = 0.35 each

Weight of bond 2 and 4 = $150,000/$1,000,000 = 0.15 each

Portfolio duration = 0.35*8.28 + 0.15*9.15 + 0.35*20 + 0.15*11.27 = 2.898 + 1.3725 + 7 + 1.6905 = 12.96

d. Elliot should select portfolio b if he thinks rates are about to head up and he wants to avoid as much price volatility as​ possible. Portfolio b has lower duration than portfolio c. Interest rate and bond prices have inverse relationship. if interest rate rises then bonds which have longer duration will fall in value more than bonds which have shorter duration and vice versa.

So, portfolio b which has shorter duration than portfolio c will fall in value less.

From portfolio c he stands to make more in annual interest​ income because it has higher weight in bond 1 which pays the highest coupon interest.

Coupon rates are fixed. so portfolio will get same coupon interest irrespective of higher interest rates in future.

hence portfolio b is recommended because it will fall in value less when interest rates rise.