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 8.58 %?, ?13-year bond? that's priced at $ 1, 092.20 to yield 7.46 %.

2. A 7.782 %?, ?15-year bond? that's priced at $ 1017.02 to yield 7.59 %.

3. A? 20-year stripped Treasury? (zero coupon)? that's priced at $ 198.14 to yield 8.26 %.

4. A? 24-year, 7.44 % bond? that's priced at $ 960.16 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 $ 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?

Answer 1

Therefore the duration of the bond will be 11.95612.83714058 years
Solution)
1) Assumed FV of each Bond to be $ 1000

Duration = ? PV(C_t) * t

                              P_b

              

Bond 1: 85.8825(11.7091) *13

                              1092.2

               = 8.317011 years

Modified duration =       Duration in years

                                             1+ Yield-to-maturity

= 8.317011/(1+0.0746)

                                       =  7.739634
Bond 2: Duration = ? PV(C_t) * t

                                             P_b

= 77.8(8.77792) * 15 /1017.02 = 8.7317336 years

Modified duration = Duration in years

                              1+ Yield-to-maturity

= 8.731736/ (1+0.0759)

                                             = 8.115748%

Bond 3: for zero coupon rate bonds the duration remains the same. Hence duration is 20 years

               Modified duration =        Duration in years

                                                            1+ Yield-to-maturity

= 20/(1+0.0826)

                                                            = 18.474044 years

Bond 4: Duration = ? PV(C_t) * t

                                             P_b

= 74.4(10.697741)*24/960.16 = 10.776178 years

Modified duration =        Duration in years

                                             1+ Yield-to-maturity

= 10.776178/ (1+0.0781)

                                                            = 9.995527 years

2) When Elliot invests $250,000 in each of the four bonds, the weighted average duration of the portfolio is:

Bond Particulars	Amount Invested	Weight	Bond Duration	W. Bond Duration
Bond 1 13 years, 8.58%	$250,000	0.25	8.317011	2.07925275
Bond 2 15 years, 7.7825%	$250,000	0.25	8.7317336	2.1829334
Bond 3 20 years, 0%	$250,000	0.25	20	5
Bond 4 24 years, 7.44%	$250,000	0.25	10.77617792	2.694044479
TOTAL	$1,000,000	1	47.82492252	11.95623063

Therefore the duration of the bond will be 11.956 years

3) When  Elliot puts $ 350,000 each into bonds 1 and 3 and $ 150,000 each into bonds 2 and 4, , the weighted average duration of the portfolio is:

Bond Particulars	Amount Invested	Weight	Bond Duration	W. Bond Duration
Bond 1 13 years, 8.58%	$350,000	0.35	8.317011	2.91095385
Bond 2 15 years, 7.7825%	$150,000	0.15	8.7317336	1.30976004
Bond 3 20 years, 0%	$350,000	0.35	20	7
Bond 4 24 years, 7.44%	$150,000	0.15	10.77617792	1.616426687
TOTAL	$1,000,000	1	47.82492252	12.83714058

Therefore the duration of the bond will be 12.83714058 years

4) Because of the weighting scheme in duration regardless of the coupon rate and the maturity of different bonds, a bond with longer duration is more volatile that one with shorter duration. This is because duration is directly related to price volatility. Hence it would be safer to invest in portfolio b as it has a shorter duration.