Question

You currently have pension fund assets of $15 million in a bond portfolio with a Macaulay

duration of 10.

Your liabilities are $2.7 million per year starting 30 years from today (i.e. at time 30) and

lasting for 30 years (i.e. the last payment is at time 60).

The yield curve is flat with spot rates constant at 4% for all maturities.

(a) Compute the present value of your liabilities. Do you have enough assets to cover those

liabilities?

(b) Compute the Macaulay duration and the modified duration of your liabilities.

(c) What is the change in the funding status (assets less liabilities) of the fund if interest

rates rise by 0.10%?

(d) You are worried that a drop in interest rates will increase the value of your liabilities.

To hedge against this exposure, you would like to invest your cash in a portfolio of

bonds such that any change in the value of your liabilities is exactly compensated by

a change in the value of your assets.

The market has a 1-year zero-coupon bond and a 100-year zero-coupon bond. How

would you allocate your assets to ensure that your pension fund is hedged against an

interest rate change?

Answer 1

A)

Value of liability at the beginning of 30th year to next 30 years = Annuity liability * 1-(1+r)^-n / r => 2.7 * (1 - (1.04)^-30 / 0.04) => 2.7 * 17.2920 => 46.68

Value of liability today = 46.68 / 1.04^30 =>14.39.

The present value of liability is 14.39 million, whereas the assets remaining in the portfolio is 15 million, so it is sufficient enough to cover liabilities.

B) Computation of Macaulay duration

29 Yearly iability (B) ( AB) 8.1 Present value Total C= No of today years (N) (C/1.04^N) 31 0.800442696 5.4 32 1.539312876 33 2.220162802 10.8 34 2.846362567 13.5 35 3.421108855 16.21 36 3.947433294 18.9 37 4.428210426 21.6 38 4.866165303| 24.3 39 5.263880737| 27 40 5.623804206 29.7 41 5.948254449 32.4 42 6.239427743 35.1 43 6.499403899 37.8 44 6.730151967 40.5 45 6.93353568 43.2 46 7.111318646 45.9 47 7.26516929 48.6 48 7.396665567 51.3 49 7.507299453 54 50 7.598481228 56.7 51 7.671543547 59.4 52 7.727745332 62.1 53 7.768275465 64.8 54 7.794256319 67.5 55 7.806747114 70.2 56 7.806747114 72.9 57 7.795198672 75.6 58 7.772990129 59 7.740958576 811 60 7.699892483 181.7709464 78.3 2.7 Total

Macaulay Duration = 181.77 / 14.39 => 12.63

Modified duration = Macaulay Duration / 1+YTM => 12.63 / 1.04 => 12.14

3.Value of liability if the interest rates raise by 0.10%

= Annuity liability * 1-(1+r)^-n / r => 2.7 * (1 - (1.041)^-30 / 0.041) => 2.7 * 17.0839=> 46.12

Value of liability today = 46.12/ 1.041^30 =>13.8155.

The entity is not required to fund additionally due to change in interest rate as there were sufficient funds

4.

When the interest rates drop, it will result in increase in liabilities.

The investor has yearly liabilities to be paid, so this can be hedged by purchasing one year zero coupon bonds and sell one year futures of 100 year zero coupon bond year after year.

The investor has to purchase the bond by discounting at the expected rate of return and sells futures contract at the same discount rate.

For example, a liability is due for $909 in an year with interest 1,000 (909 *1.1) then it can be hedged by purchasing a one year bond having coupon rate of 10% by purchasing today at $909 (1,000/1.1), and selling 100 year zero coupon bond one year futures at a same discount rate.

If during the year discount rate in the market interest raises to 11% the value of bond will become 901(1,000 / 1.11).

The liability to be paid will be $1,009 (909 *1.11) instead of 1,000, this is solved as follows

The investor can realize $1,000 from one year bond and in the market the investor can sell the futures at 1,000 and earn profit because the value of the 100 year zero coupon bond would have decreased due to increase in interest rate (i.e., the value of 100 year zero coupon bond after an year when interest rates raise to 11% is (100 * (1-(1.11)^-99) / 0.11 + 1000 / 1.11^99 => 909) so the investor can realize 91 as profit), this profit can be used to settle increase in interest rate.

When the interest rate decrease the investor will realize $1,000 on maturity of one year bond and pay off its liabilities at a lower value due to reduced interest rate and using such profit to close off futures position loss.