Question

Today is 15 November 2018. To construct a portfolio, Jake just purchased two different Treasury bonds, henceforth referred to as Bond A and Bond B. Assume the yield rate for these financial instruments was j₂ = 3.15% p.a.

• Bond A has a coupon rate of j₂ = 3.05% p.a. and a face value of $100. The maturity date of this bond is 15 May 2020.

• Bond B has a coupon rate of j₂ = 2.95% p.a. and a face value of $100. The maturity date of this bond is 15 January 2022.

a. Calculate the duration and the modified duration of Treasury bond A. Give your answer in terms of years, rounded to three decimal places.

b. Calculate the price Jake paid for bond A (rounded to three decimal places).

c. Calculate Jake’s purchase price of bond B using the RBA approach (rounded to three decimal places).

d. Jake purchased 101 units of bond A and 199 units of bond B to establish his portfolio. Based on your results from part a, b and c, calculate th duration of Jake’s portfolio, given that bond B has duration of 3.016 years. Give your answer in terms of years, rounded to two decimal places.

Answer 1

a). Duration of Bond A = 1.478 years

Modifide duration of Bond A = 1.455 years

b). Price of Bond A = 99.85

Calculated using Excel formulas:

c). Price of Bond B (using RBA approach):

Price of bond is calculated as

where i = annual yield/200 = 3.15%/200 = 0.0001575

v = 1/(1+i) = 1/(1+0.0001575) = 0.9998

n (number of half years from the next coupon date to maturity) = 6

a angle n = (1- v^n)/i = (1-0.9998^6)/0.0001575 = 5.9967

f (number of days from the settlement date to the next coupon date 15 Jan 2019) = 61

d (number of days from the last coupon date 15 Jul 2018 to the next coupon date 15 Jan 2019) = 184

g = annual coupon/2 = 2.95%/2 = 0.01475

So, P = (0.0008^(61/184))*[(0.01475*(1+5.9987) + (100*0.9998^6)] = 100.004

d). Amount invested in Bond A = number of bonds bought*price per bond = 101*99.85 = 10,085.31

Amount invested in Bond B = 199*100.004 = 19,900.70

Weight of Bond A = 10,085.31/(10,085.31+19,900.70) = 0.336

Weight of Bond B = 1-0.336 = 0.664

Portfolio duration = sum of weighted duration

= (0.336*1.478) + (0.664*3.016) = 2.50 years