Question

A pension fund has an average Macaulay duration of its liabilities equal to 15 years. The fund is looking at 3-year maturity zero coupon bonds and 3% yield perpetuities to immunize its interest rate risk. How much of its portfolio should it allocate to the zero-coupon bonds to immunize if there are no other assets funding the plan?

Answer 1

Answer,

We have a liability of pension fund within 15 years but we couldn't find bond that paying the desired interest rate with a 15-year duration. So we should invest in this portfolio of zero-coupon bond and perpetuity bond which ultimately gives a combined duration of 15 years.

Step 1 -DURATION OF ZCB & PERPETUITY BOND

Duration of ZCB=Term of bond=3

Duration of perpetual bond=(1+Y) / Y Y=Yield

=(1+0.03 )/ 0.03 =34.33

STEP 2 - PROPORTION OF ALLOCATION

BOND	PROPORTION	DURATION	WEIGHTED DURATION
ZCB	P	3	3P
PERPETUITY BOND	1-P	34.33	34.33-34.33P

Weighted duration of bond is 15 years,so,

3P+34.33-34.33P=15

Rearrange the equation,

19.33=31.33P

P=0.62

1-P=0.38

Conclusion:62% of portfolio should allocate to ZCB and 38% to perpetuity bond.