Question

1. Suppose you will go to graduate school for 4 years beginning in year 4. Tuition is $25,942 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 0.06. Assume annual compounding.

2. Suppose in the question above, the tuition obligations have a Macaulay duration of 3.93 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? Assume annual compounding.

3. Suppose in question 1, the tuition obligations have a Macaulay duration of 6.75 in years and a present value of 59,578. In order to immunize against the tuition payments by investing in some combination of two bonds with duration 2.99 and 8.49, what is the dollar amount that you should invest in the bond with duration 8.49? Assume annual compounding.

Answer 1

(1)

Year	PVF at 6%	Cash Flow	PV of Cash flow	Present value of a Cash flows * Year of Reciept
A	B	C	B*C	C*A
4	0.7921	$        25,942	$ 20,548.49	$    82,193.98
5	0.7473	$        25,942	$ 19,385.37	$    96,926.86
6	0.7050	$        25,942	$ 18,288.09	$ 109,728.52
7	0.6651	$        25,942	$ 17,252.91	$ 120,770.38
-	-	Sum	$ 75,474.86	$ 409,619.74
Duration =	75,474.86 / 409,619.74	5.43 Years

(2) As there are no intermediate payments for a Zero Coupon Bond, Duration for a Zero Coupon Bond wil be same as Maturity period.

Given, The tuition obligations have a Macaulay duration of 3.93 in years.

To immunize against the tuition payments by buying a single issue of a zero coupon bond.

The maturity zero coupon bond should be equal to Duration of Payments required to be Immunised.

Hence, To immunize against the tuition payments by buying a single issue of a zero coupon bond, The Maturity of the Zerocoupon Bonds should be equal to Duration of Tution payments = 3.93 Years.

(3) Given, tuition obligations have a Macaulay duration of 6.75

Present value of Tution Payments = $59,578

In Order to Immunise the Tution fee payments against Interest Rate Risk, The Investment should be done in such a way that the Weighted Average Duration of the Portfolio should equal to Duration of the Payments.

Bond	Duration	PV of Investment
A	2.99	$ 59,578 - x
B	8.49	x
	PV of Investment	$        59,578

Let 'x' be the amount of investment made in the Bond having Duration 8.49.  

{[($ 59,578 - x)*2.99] + 8.49 x} / $59,578 = 6.75

$ 178,138.22 +(8.49 - 2.99) x = $402,151.50

(8.49 - 2.99) x = $402,151.50 - $ 178,138.22

x = $224,013.28 / 5.5

x = $40,729.69