Question

Invest an amount P=2 100 000 between 2 bonds A, B – to be immunized against interest rate risk:

A: FV= 1000, c=0%, y=5%, n=2

B: FV= 1000, c=0%, y=5%, n=10

The change in yield was 1%.

Investment horizon: IH=4

​a) Construct a portfolio – how much do you invest into A and B?

​b) Calculate the future value V to IH for changed yield?

​c) Calculate the yield to IH after the change of the yield?

Answer 1

As given in the question both bonds are zero coupon bonds and hence the maturity of the bonds is equal to the duration of the bonds. It is given that the Investment Horizon (IH)= 4 and hence we need to construct a portfolio with duration equal to 4.

(a) PV of Bond A = 1000/(1+5%)^2 = 907.03

PV of Bond B = 1000/(1+5%)^10 = 613.91

If the number of bonds are x and y respectively for Bond A & Bond B,

Total Amount: 907.03x + 613.91y = 2100000 .....Eqn (1)

Duration: (2*907.03)x + (10*613.91)y = 4*2100000 ......Eqn (2)

Solving (1) & (2), x= 1736 , y = 855

Therefore, Invest 1574604 in Bond A & 524893 in Bond B

(b) Change in Yield = 1%

FV (Bond A) at IH = 4 years = (1736*1000)*(1+6%)^2 = 1950570

FV (Bond B) at IH = 4 years = (855*1000)/(1+6%)^6 = 602741.30

Therefore, FV of portfolio at IH = 4 years = 1950570 + 602741.30 = 2553311.30

On repeating the above exercise with decrease in yield, i.e, @ 4%, the FV of portfolio would remain the same implying that the interest rate risk has been hedged

(c) To calculate yield of above portfolio at IH = 4 years,

Y = C+ ((FV-PV)/n)/((FV+PV)/2)

Solving the equation with C=0, n=4, FV = 2553311.30, PV = 2100000

Therefore, on computing Y = 4.87%