Question

You are managing a portfolio of $1 million. Your target duration is 3 years, and you can choose from two bonds: a zero-coupon bond with time to maturity of 5 years, and a bond with an annual coupon rate of 8% and time to maturity of 2 years, both with yield to maturity of 5%. Assume both bonds have a face value of $1000. (a) how much of each bond will you hold in your portfolio (b) how will these fractions change next year if target duration is now 2 years and the interest rates do not change

Answer 1

a) Target duration = 3 years

duration for zero coupon bond = time to maturity = 5 years

Duration for bond with annual coupon of 8%

cash flow in year 1, CF1 = coupon rate*face value = 0.08*1000 = 80

Cash flow in year 2 , CF2 = coupon + face value = 80+1000 = 1080

YTM, (r) = 5% = 0.05

Present value(PV) of CF1, (PV1) = CF1/(1+r) = 80/(1.05) = 76.19047619

PV of CF2 , (PV2)= CF2/(1+r)² = 1080/(1.05)² = 979.5918367

Total of Present value = PV1 + PV2 =  76.19047619 + 979.5918367 = 1055.782313

Weight of CF1 = PV of CF1/ total of PV = 76.19047619/1055.782313 = 0.072164948

weight of CF2 =  PV of CF2/ total of PV = 979.5918367/1055.782313 = 0.927835052

duration = (1*weight of CF1) +( 2*weight of CF2) = 0.072164948 + 1.855670103 = 1.927835052 = 1.9278 years

let w = proportion of zero coupon bond in portfolio

1-w = proportion of annual coupon bond in portfolio

Target duration = (w*duration of zero coupon bond) + ((1-w)*duration of annual coupon bond)

3 = (w*5) + ((1-w)*1.9278

3 = 5w + 1.9278 - 1.9278w

3 = 1.9278 + 3.0722w

w = (3-1.9278)/3.0722 = 0.349000716

(1-w ) = (1-0.349000716) = 0.650999283

weight of zero coupon bond in portfolio = 0.349000716 or 34.9000716% = 34.90% or 35%

weight of annual coupon bond in portfolio = 0.650999283 or 65.0999283% = 65.10% or 65%

b)

duration of zero coupon bond after 1 year = 4 years

duration of annual coupon bond = 1 year

Target duration = 2

2 = (w*4)+[(1-w)*1] = 4w + 1 - w= 3w + 1

2 = 3w + 1

3w = 1

w = 1/3 = 0.33333

weight of zero coupon bond in portfolio after 1 year = 0.3333 = 33.33%

weight of annual coupon bond in portfolio after 1 year = 1-w = 1-0.3333 = 0.6667 = 66.67%