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 2011 to create this portfolio and this portfolio is composed of 27 units of instrument A and 22 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 = 2.40% 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 =4.23% 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 = 4.23% 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 = 4.23% 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

Part (a): Instrument A:

Present value of Zero coupon bond= F/(1+r)^n

Where

F= Redemption value (given as $100),

r= Yield rate per period= 4.23%/2

n= Time to maturity= 9 years*2= 18

Plugging the values,

Price per $100 face value= 100/(1+4.23%/2)^18 = $68.6101

Part (b): Instrument B:

Price per $ 100 face value= $95.7015

Part (c):

Duration of instrument B= 2.428 years.

Calculations given below.

Duration of portfolio= Weighted average of duration of individual bonds

Given,

Number of instrument A= 27 and Number of instrument B=22

Price as above: A= $45.1437 and B= $95.7035

Value of A= 27*68.6101 = 1852.4735

Value of B= 22*95.7035 =2105.4770

Weight of A= 1852.4735/(1852.4735+2105.4770) = 0.46803858

Weight of B= 1- 0.46803858 = 0.53196142

Duration of Zero coupon bond is equal to term to maturity= 9 years

Therefore,

Duration of portfolio= 0.46803858*9 + 0.53196142*2.428

=4.212347 + 1.291602 = 5.50 years.