Question

1. Say you are an insurance company manager who issued a zero-coupon bond that promises to pay $13,400.96 six years from today. The present value (PV) of the obligation today is $10,000.

1. (2) What is the YTM of this obligation?
2. (8) You wish to fund the obligation using the following 2 bonds:
   1. Bond A:

   Maturity: 13 years

Coupon rate: 5.29% (Annually)

Price:   $1200

2. Bond B:

Maturity: 2 years

Coupon rate: 0.5% (Annually)

Price: $800

                       

Find the optimal weights for an immunization strategy (A portfolio that is not exposed to interest rate risk). How many bonds A and bonds B should you buy? (No need to show me how the income grows over the years)

Answer 1

Part (a)

YTM of the obligation = (FV / PV)^{(1 / n)} - 1 = (13,400.96 / 10,000)^(1/6) - 1 = 5.00%

Part (b)

As a first step we need to find the modified duration of each of the two bonds and the liability.

Modified duration of the liability = Duration / (1 + YTM) = 6 / (1 + 5%) =  5.7143

Please see the table below for calculation of the modified duration of the bonds. We will have to first calculate the YTM of each of the bonds and making use of YTM, we will calculate the modified duration. I am resorting to excel. Yellow colored cells contain the relevant values. Adjacent cells in blue contain the excel formula used to get the relevant values.

Let w be the weight of the bond A and hence 1 - w will be the weight of bond B in the immunizing portfolio.

Modified duration of the portfolio = Modified duration of liability

Hence, w x 9.7132 + (1 - w) x 1.7745 =  5.7143

Hence, w = 0.496278597 = 49.63%

Hence, weight of Bond A in the immunizing portfolio = w = 49.63%

and that of Bond B = 1 - w = 1 - 49.63% = 50.37%

Also, if N_A and N_B be the number of bonds A and B we buy respectively then,

w = weight of Bond A = N_A x P_A / total value of portfolio = N_A x 1,200 / 10,000 = 49.63%

Hence, N_A =  4.1357

Similarly, 1 - w = 50.37% = N_B x P_B / total value = N_B x 800 / 10,000

Hence, N_B = 50.37% x 10,000 / 800 =  6.2965