Question

Today is 1 July 2020. Joan has a portfolio which consists of two different types of financial instruments (henceforth referred to as instrument A and instrument B). Joan purchased all instruments on 1 July 2013 to create this portfolio and this portfolio is composed of 36 units of instrument A and 34 units of instrument B.

Instrument A is a zero-coupon bond with a face value of 100. This bond matures at par. The maturity date is 1 January 2030. Instrument B is a Treasury bond with a coupon rate of j2 = 4.53% p.a. and face value of 100. This bond matures at par. The maturity date is 1 January 2023.

(a) Calculate the current price of instrument A per $100 face value. Round your answer to four decimal places. Assume the yield rate is j2 =2.57% p.a.

(b) Calculate the current price of instrument B per $100 face value. Round your answer to four decimal places. Assume the yield rate is j2 = 2.57% p.a. and Joan has just received the coupon payment.

(c) What is the duration of instrument B? Express your answer in terms of years and round your answer to three decimal places. Assume the yield rate is j2 = 2.57% p.a.

(d) Based on the price in part a and part b, and the duration value in part c, calculate the current duration of Joan’s portfolio. Express your answer in terms of years and round your answer to two decimal places.

Answer 1

a). Price of instrument A = par value/(1+ r/2)^(n*2) where par value = 100; r (yield rate) = 2.57% p.a.; n (time to maturity in years) = 16.5

= 100/(1+2.57%/2)^(16.5*2) = 65.6161

b). Price of instrument B: FV (par value) = 100; PMT (semi-annual coupon) = annual coupon rate*par value/2 = 4.53%*100/2 = 2.265; rate (semi-annual rate) = yield rate/2 = 2.57%/2 = 1.285%; N (number of coupon payments remaining) = 19, solve for PV.

Current price = 116.4282

c). Duration calculation (using DURATION() formula in excel):

Settlement date - July 1, 2013; Maturity date - January 1, 2023; rate (coupon rate) = 4.53%; yield = 2.57%; frequency = 2; redemption (value paid out at maturity) = 100; basis = 0 (or default), solve for DURATION.

Duration = 7.970 years

d). Duration of a zero-coupon bond = time to maturity, so duration of Instrument A = 16.5 years.

Portfolio duration = sum of weighted durations

Weights calculation:

Total portfolio value (TV) = sum of (number of bonds*current price per bond)

= (36*65.6161)+(34*116.4282) = 6,320.74

Weight of Instrument A = (36*65.6161)/6,320.74 = 37.37%

Weight of Instrument B = 100%-weight of Instrument A = 100%-37.37% = 62.63%

Portfolio duration = (37.37%*16.5)+(62.63%*7.970) =11.16 years