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 8.59 %​, ​13-year bond​ that's priced at $ 1,092.14 to yield 7.47 %.

2. A 7.784 %​, ​15-year bond​ that's priced at $ 1023.51 to yield 7.52 %.

3. A​ 20-year stripped Treasury​ (zero coupon)​ that's priced at $ 197.76 to yield 8.27 %.

4. A​ 24-year, 7.49 % bond​ that's priced at $ 965.54 to yield 7.81 %. 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 $ 380,000 each into bonds 1 and 3 and $ 120,000 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? 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.

Bond​ 1: 13​ years, 8.59 %​, priced to yield 7.47 %.

The duration of this bond is ____ years.  ​

The modified duration of this bond is ____ years.  

Bond​ 2: 15​ years, 7.784 % priced to yield 7.52 %.

The duration of this bond is ____ years. 

The modified duration of this bond is ____ years.  

Bond​ 3: 20​ years, zero​ coupon, priced to yield 8.27 %.

The duration of this bond is ____ years.  

The modified duration of this bond is ____ years.  

Bond​ 4: 24​ years, 7.49 %​, priced to yield 7.81 %.

The duration of this bond is ____ years.  ​

The modified duration of this bond is _____ years.  ​

b. Find the duration of the whole bond portfolio if Elliot puts $ 250,000 into each of the 4 U.S. Treasury bonds.

The duration of this portfolio is ____ years.  

c. Find the duration of the portfolio if Elliot puts $ 380,000 each into bonds 1 and 3 and $ 120,000 each into bonds 2 and 4.

The duration of this portfolio is ____ years.  ​

d. Which portfolio - the portfolio in part b or the portfolio in part c - should Elliot select if he thinks rates are about to head up and he wants to avoid as much price volatility as​ possible?  ​(Choose the best answer​ below.)

A. 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.

B. He should invest in either portfolio. The portfolio in part c has a higher duration than the portfolio part b. If rates are about to​ rise, then it is equally safe to invest in either​ portfolio, because both portfolios would be equally price volatile.

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

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

Please explain how to calculate in excel....

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 $.

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	8.590%	7.47%	2	1	8.30	8.00
2	1/1/2019	1/1/2034	7.784%	7.52%	2	1	9.17	8.84
3	1/1/2019	1/1/2039	0.000%	8.27%	2	1	20.00	19.21
4	1/1/2019	1/1/2043	7.490%	7.81%	2	1	11.27	10.85

Bond​ 1: 13​ years, 8.59 %​, priced to yield 7.47 %.

The duration of this bond is 8.30 years.  ​

The modified duration of this bond is 8.00 years.  

Bond​ 2: 15​ years, 7.784 % priced to yield 7.52 %.

The duration of this bond is 9.17 years. 

The modified duration of this bond is 8.84 years.  

Bond​ 3: 20​ years, zero​ coupon, priced to yield 8.27 %.

The duration of this bond is 20 years.  

The modified duration of this bond is 19.21 years.  

Bond​ 4: 24​ years, 7.49 %​, priced to yield 7.81 %.

The duration of this bond is 11.27 years.  ​

The modified duration of this bond is 10.85 years.  ​

b. Find the duration of the whole bond portfolio if Elliot puts $ 250,000 into each of the 4 U.S. Treasury bonds.

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 + 9.17 + 20.00 + 11.27) = 12.19 years

c. Find the duration of the portfolio if Elliot puts $ 380,000 each into bonds 1 and 3 and $ 120,000 each into bonds 2 and 4.

W1 = W3 = 380,000 / 1,000,000 = 0.38

W2 = W4 = 120,000 / 1,000,000 = 0.12

The duration of this portfolio is = 0.38 x 8.30 + 0.12 x 9.17 + 0.38 x 20.00 + 0.12 x 11.27 = 13.21 years

d. Which portfolio - the portfolio in part b or the portfolio in part c - should Elliot select if he thinks rates are about to head up and he wants to avoid as much price volatility as​ possible?  ​(Choose the best answer​ below.)

The correct answer is the first option i.e. option A. 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.