Question

There are always two zero coupon bonds with face value $1000 available: one with a maturity of 1 year, one with 3 years. Suppose the yield curve is currently flat at 5%. What investment strategy would you choose to generate $1000 in two years from now?

(Please write quick explanation for each step of the process, thank you so much!)

Answer 1

Let P_i and D_i denote respectively the price and duration of the Zero coupon bond (ZCB) with maturity of i year.

Let A be the amount required in two years. Hence, A = 1,000

D₁ = 1 ; D₃ = 3

P₁ = 1000 / (1 + y)¹ = 1,000 / (1 + 5%) = $ 952.38

P₃ = 1,000 / (1 + y)³ = 1,000 / (1 + 5%)³ = $  863.84

PV (A) = A / (1 + y)² = 1,000 / (1 + 5%)² = $ 907.03

Let the investment strategy is to have N₁ and N₃ number of the ZCBs to generate A in two years time.

The present value should match hence,

N₁ x P₁ + N₃ x P₃ = PV (A)

Hence, N₁ x 952.38 + N₃ x 863.84 = 907.03

Hence, 952.38N₁ + 863.84N₃ = 907.03 ------Equation (1)

Hence, in order to match the duration:

Weighted average of the duration of the portfolio = Duration of A

Hence, (N₁ x P₁ x D₁ + N₃ x P₃ x D₃) / (N₁ x P₁ + N₃ x P₃) = Duration of A

Or, (N₁ x P₁ x D₁ + N₃ x P₃ x D₃) = (N₁ x P₁ + N₃ x P₃) x Duration of A = PV (A) x Duration of A

Hence, N₁ x 952.38 x 1 + N₃ x 863.84 x 3 = 2 x 907.03

Or, 952.38N₁ + 2,591.51N₃ = 1,814.06 ------------------ Equation (2)

Equation (2) - Equation (1) gives:

(2,591.51 - 863.84)N₃ = (1,814.06 - 907.03)

Hence, N₃ =   0.5250

Hence, N₁ = (907.03 - 863.84N₃) / 952.38 = 0.4762

Hence, the investment strategy should be:

• Buy today, 0.4762 no. of ZCB maturing in year 1 and hold them till maturity
• Buy today, 0.5250 no. of ZCB maturing in year 3
• Use the maturity proceeds from ZCB maturing in year 1, to buy ZCB maturing in year 3, at the end of year 1
• Sell all the ZCB maturing in year 3, at the end of year 2