In: Statistics and Probability
Solution:
(a)
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:
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)
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) duration of the company’s payment obligations:
The duration of a recurring payment (or cash received) is nothing but the weighted average of the time of payment, which can be calculated as follows:
1 | 2 | 3 | 4 | ||
YEAR | Payment Obligation | PV:Nominal Value/(1+annual rate)^year | Share of cash flow in PV (col-3/sum(col-3)) | Weighted time of payment col-4*col-1 | |
1 | 2000000 | 1851851.852 | 0.243690456 | 0.243690456 | |
2 | 3000000 | 2572016.461 | 0.338458967 | 0.676917934 | |
3 | 4000000 | 3175328.964 | 0.417850577 | 1.25355173 | |
Sum | 7599197.277 | 2.17416012 |
So the duration would be 2.174 years
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) how much of each of these bonds the company will hold in the portfolio:
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.
a) six months zero coupon bonds:
The primary point to understand here is that the immunization simply means that the bonds in which the company would invest to meet its future obligations can be safely assumed to have a non-fluctuating 8% interest rate. With that in mind, we can draw a cash flow structure which pays a semi annual interest rate of 4% (8%/2). The cash flow table would look like this:
1 | 2 | 3 | 4 | ||
Payment Obligation | Nominal Value of Bond with Reinvested Int | Less the Payment Obligation (col-3-col-4) | |||
0.5 | - | XXX*(1.04) | XXX*(1.04)-(col-2) | ||
1 | 20,00,000 | (XXX*(1.04)-(col-2))*1.04 | |||
1.5 | - | ||||
2 | 30,00,000 | ||||
2.5 | - | ||||
3 | 40,00,000 | ||||
Initial bond portfolio Value | XXX |
Populating the rest of the values, and solving for XXX we get the initial bond portfolio value.
1 | 2 | 3 | 4 |
Payment Obligation | Nominal Value of Bond with Reinvested Int | Less the Payment Obligation (col-3-col-4) | |
0.5 | - | 78,77,774 | 78,77,774 |
1 | 20,00,000 | 81,92,885 | 61,92,885 |
1.5 | - | 64,40,601 | 64,40,601 |
2 | 30,00,000 | 66,98,225 | 36,98,225 |
2.5 | - | 38,46,154 | 38,46,154 |
3 | 40,00,000 | 40,00,000 | - |
Initial bond portfolio Value | 75,74,783.10 |
Please note that this result can be calculated very easily using the goal seek function in EXCEL.
Ans. 75,74,783.10
b) Perpetuity:
The same obligation can be fulfilled by a perpetuity paying cash flows, the table for it can be drawn in the similar way.
Payment Obligation | Nominal Value of perpetuity payments+ Reinvested Int |
Less the Payment Obligation (col-3-col-4) | |
0.5 | - | 14,44,980 | 14,44,980 |
1 | 20,00,000 | 29,47,759 | 9,47,759 |
1.5 | - | 24,30,650 | 24,30,650 |
2 | 30,00,000 | 39,72,856 | 9,72,856 |
2.5 | - | 24,56,750 | 24,56,750 |
3 | 40,00,000 | 40,00,000 | - |
Corpus for perpetuity | 3,61,24,500.89 |
Ans: 3,61,24,500.89