Question

Assume that a pension fund is fully funded and that the manager plans to invest all money in a single-maturity zero to immunize the fund. The yield curve is flat at 6% and the pension fund has the following obligations:

Time in years.    0 1 2 3 4 5 6 7 8 9 10

Obligation (in $MM) $5 7 10 12 15 18 20 25 30 35

c. How much needs to be invested in the zero coupon bond to immunize the fund?

d. What is the Macaulay duration of the fund’s obligations?

e. What maturity zero coupon bond should be purchased to immunize the fund?

f. Instead of investing in a single zero coupon bond, say you have access to a 2 year zero coupon bond and a 10 year zero coupon bond. What percentage of your investment should be invested in the 2 year and what percentage in the 10 year?

Answer 1

c) Amount to be invested

=present value of obligations

=5/1.06+7/1.06^2+10/1.06^3+12/1.06^4+15/1.06^5+18/1.06^6+20/1.06^7+25/1.06^8+30/1.06^9+35/1.06^10

=119.0337

So, Amount invested in Zero coupon bonds to immunize= $113.0337 MM

d) Macaulay Duration of Fund's obligation

=(5/1.06*1+7/1.06^2*2+10/1.06^3*3+12/1.06^4*4+15/1.06^5*5+18/1.06^6*6+20/1.06^7*7+25/1.06^8*8+30/1.06^9*9+35/1.06^10*10)/119.0337

= 6.6066 years

e) Maturity of the zero coupon Bond should be the same as the Macaulay Duration of obligations

So, Amount should be invested in a zero coupon bond of 6.6066 years maturity

f) As the duration of a portfolio is the weighted average duration of individual component securities

Let w be the weight of 2 year ZCB and (1-w) be the weight of 10 year ZCB

Then

w*2+(1-w)*10 = 6.6066

=>w = 0.42417545

So, One should invest 42.417545% of investment in 2 year Zero Coupon Bond

and 57.582455% of investment in 10 year Zero Coupon Bond