Question

a. A 10-year 5% coupon bond has a yield of 8% and a duration of 7.85 years. If the bond yield increases by 60 basis points, what is the percentage change in the bond price?

b. Alpha Insurance Company is obligated to make payments of $2 million, $3 million, and $4 million at the end of the next three years, respectively. The market interest rate is 8% per annum.

i. Determine the duration of the company’s payment obligations.

ii. Suppose the company’s payment obligations are fully funded and immunized using both 6-month zero coupon bonds and perpetuities. Determine how much of each of these bonds the company will hold in the portfolio.

I would like to know the answer of question bi and bii.

Answer 1

a. Modified Duration helps to provide estimate the % change of bond price given to change its yield to maturity.

% price change of bond= - Annual modified duration X change in the annual yield to maturity

                                  = - 7.85 X 0.0060= -0.047100 or 4.71 % change

So, If the yield increases by 60 basis points or increase by 0.6% (100 basis points=1%), the estimated decrease in the price of the bond is 4.71%.

b. First we have to calculate the maccaulay duration and then with that we are able to calculate the modified Duration.

i. Period    CashFlow         PV                 Weights                      Period X Weight

    1              2 million             1.852 million             0.244                          0.244

    2              3 million             2.572 miillion            0.338                          0.677

    3              4 million             3.175 million             0.418                          1.253

                                   Total 7.599 million 1.000     2.174

The first two columns of show the number of periods to the receipt of the cash flow. The third column is the present value of the cash flow. For example, the final payment is 4 million and its present value is 3.175 million.

4/ (1.08)^3= 3.175

The sum of the present values is the full Payment. The fourth column is the weight, the share of total market value corresponding to each cash flow. The final payment of 4 million is 41.8% of the total market value.

3.175/ 7.599= 0.418

The sum of the weights is 1.00000. The fifth column is the number of periods to the receipt of the cash flow (the first column) multiplied by the weight (the fourth column). The sum of that column is 2.174.

It is the Macaulay duration statistic divided by one plus the yield per period.
ModDur = MacDur/ 1+yield

           = 2.174/ 1.08

           = 2.013 duration

ii. A perpetuity is a bond that does not mature whereas zero coupon bond is the bond that are given in discount and redeemed at par. There is no principal to redeem in the case of perpetual bond. Perpetual bond is basically the callable bond.