Question

1.

a.Suppose you will go to graduate school for 3 years beginning in year 5. Tuition is $34,657 per year, due at the end of each school year. What is the Macaulay duration (in years) of your grad school tuitions? Assume a flat yield curve of 0.05.

b.Suppose in the question above, the tuition obligations have a Macaulay duration of 4.54 in years, and that you wish to immunize against the tuition payments by buying a single issue of a zero coupon bond. What maturity zero coupon bond should you buy?

c.Suppose in question a, the tuition obligations have a Macaulay duration of 6.86 in years and a present value of 44,225. In order to immunize against the tuition payments by investing in some combination of two bonds with duration 3.48 and 9.91, what is the dollar amount that you should invest in the bond with duration 9.91?

Answer 1

Part (a):

Macaulay Duration of grad school fee= 6.9675 years

Calculation as below:

Part (b):

In order to immunize the future payment obligation, duration of the investment should be equal to that of the future obligation. Duration of a zero coupon bond is equal to its term to maturity. It is given that duration of the obligation is 4.54

Therefore, maturity of the zero coupon bond bought shall be 4.54 years.

Part (c ):

Duration of portfolio is the weighted average of duration of individual bonds.

Given, duration of bond 1= 3.48 years and bond 2= 9.91 years

Portfolio duration needed= 6.86 years

Let the proportion of bond 1 be x

Then, 3.48x + 9.91(1-x) = 6.86

3.48x – 9.91x= 6.86-9.91

-6.43x = -3.05

Therefore, proportion of bond 1= 3.05/6.43= 0.474339

Also given, present value of investment= 44,225

Therefore, amount to be invested in bond 2 with duration of 9.91 years= 44225*(1-0.474339)

= $23,247.36