Question

Once upon a time, the treasurer of “Mighty Corporation” (MCO) decided to issue a bond (hereafter: The bond) The bond would have a 20-year life and promised that the holder of the bond would receive:

• 20 annual payments of $80,
• Along with the 20^th payment, MCO promised to return the principal of $1,000. In other words, the final payments received at maturity would be ($1000+$80) = $1080.

Question #3:

The insurance company wants to use The Bond as part of a portfolio designed to “immunize” an obligation that it foresees will come due in 15 years. (Hint: Example 11.2 in the book is very useful here).

• To create a bond portfolio with an overall duration of 15 years, the insurance company decides to buy zero coupon bonds with a maturity of 20 years (and with a YTM of 7%) to combine with The Bond . To bring the combination of The Bond plus The Zero to a combined portfolio duration of 15 years, what would be the correct mix of The Bond relative to the 20-year Zero?
  • A) percentage amount of The Bond in the portfolio
  • B) percentage amount of The Zero in the portfolio
  • C) Can you suggest a simpler investment that would give the insurance company a bond investment with a 15 year duration?

Answer 1

Years	Cash flows	Present Value at 7%	Years*Cash flows
1	80	75	75
2	80	70	140
3	80	65	196
4	80	61	244
5	80	57	285
6	80	53	320
7	80	50	349
8	80	47	372
9	80	44	392
10	80	41	407
11	80	38	418
12	80	36	426
13	80	33	432
14	80	31	434
15	80	29	435
16	80	27	434
17	80	25	431
18	80	24	426
19	80	22	420
20	1,080	279	5,582
210	2,600	1,106	12,217

Duration of 20 years MCO bond = 12,217/1,106= 11.05 years

Duration of 20 years zero coupon bond = 20 years

Target duration of insurance company = 15 years

Assuming we invest x% in zero coupon bond and (1-x)% in the 8% coupon bearing bond,

Thus, 20x+11.05(1-x)=15

On solving the above equation we get,

X=0.44 or 44% and so 1-x = 0.56 or 56%

Ans A. Proportion to be invested in The Bond = 56%.

And B. Proportion to be invested in zero coupon bond = 44%.