Question

My pension plan will pay me $10,500 once a year for a 10-year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.

a. What is the duration of its obligation to me? The current interest rate is 5.0% per year. (Do not round intermediate calculations. Round your answer to 4 decimal places.)

Duration=9.0990

b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

                                               Investment

5yr zero coupon bond=--------?

20yr zero coupon bond=--------?

c. What will be the face value of the holdings in each zero? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

                                           Face Value

5yr zero coupon bond=--------?

20yr zero coupon bond=--------?

Answer 1

a) Duration of an annuity starting immediately is given by

Σ(1 + C./(1 + r)/P

where Ci is the cash flow in year i

r is the interest rate and P is the present value (at the end of 4 years) of the normal annuity starting after 4 years (1st payment at the end of 5th year)

Here, P = 10500/1.05 + 10500/1.05^2 + ... + 1500/1.05^10

=10500/0.05* (1-1/1.05^10) = $81078.22

So, Duration = (10500*1/1.05 + 10500*2/1.05^2+ .... + 10500*10/1.05^10) / 81078.22

= 413424.72/81078.22 = 5.099 years

Now, as the Annuity started after 4 years

Duration of the deferred annuity = 4+5.099 =  9.099 years

b) Duration of Liability = Duration of Asset for immunisation

So, if x is the weight of 5 year zero bonds and (1-x) is the weight of 20 year zero bonds

then, Duration of portfolio = weighted average duration of individual assets

and since Duration of zero coupon bonds is the same as maturity

Duration of portfolio = x*5+ (1-x) *20 = duration of liability  = 9.099

=> 15*x = 10.901

=> x= 0.7267 i.e. 72.67% in 5 year zero bonds and 27.33 % in 20 year zero bonds

Value of liability today = 81078.22/1.05^4 = $66703.25

So, amount invested in 5 year zero coupon bond = 0.7267*66703.25 = $48475

amount invested in 20 year zero coupon bond = 0.2733*66703.25 = $18228

c) Face value is the future value of these bonds

So,

face Value of 5 year zero coupon bond = 48475*1.05^5 = $61868

face Value of 20 year zero coupon bond = $18228*1.05^20 =$48364