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 $ 1million of his inheritance to purchase 4 U.S. Treasury​ bonds:

1. An 8.59 %​, ​13-year bond​ that's priced at $ 1,093.02to yield 7.46 %.

2. A 7.777 %​, ​15-year bond​ that's priced at $ 1021.98 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, 7.47 % bond​ that's priced at $ 955.08 to yield 7.89 %.

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 $ 330,000 each into bonds 1 and 3 and $ 170,000 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

Duration and modified duration of each bond can be easily calculated using excel formulas. Excel formulas require settlement date and maturity date. the purchase date of the bond is settlement date and maturity date is the date on which bond will be retired. both of these dates have not been given in the question.

So let's assume all 4 bonds are purchased today. Bond 1's settlement date will be 8/23/2019 and maturity date will be 8/23/2032 after 13-years. same way for all other bonds.

a. Calculation of duration and modified duration

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*8.30 + 0.25*9.17 + 0.25*20 + 0.25*11.23 = 2.075 + 2.2925 + 5 + 2.8075 = 12.175

c. in this option investment amount in each bond is not equal and is different. so we need to calculate new weight of each bond in the portfolio as per their investment amount. Bond 1 and 3 have same amount of investment and so are bond 2 and 4.

weight of bond 1 and 3 = $330,000/$1,000,000 = 0.33 each

Weight of bond 2 and 4 = $170,000/$1,000,000 = 0.17 each

Portfolio duration = 0.33*8.30 + 0.17*9.17 + 0.33*20 + 0.17*11.23 = 2.739 + 1.5589 + 6.6 + 1.9091 = 12.81

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 than portfolio c. the main reason for lower duration of portfolio b is that bond 3 has the highest duration among all bonds and portfolio b has lower weight in bond 3 compared to portfolio c.

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.