Question

You will be paying $11,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 9%.

a. What is the present value and duration of your obligation?

b. What maturity zero-coupon bond would immunize your obligation?

c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 10%. What happens to your net position, that is an increase or decrease in value, to the difference between the value of the bond and that of your tuition obligation?

d. What if rates fall immediately to 8%? What happens to your net position, that is an increase or decrease in value, and by how much?

Round duration to 4 decimal places and all other answers to 2 decimal places.

Answer 1

a. What is the present value and duration of your obligation?

PV = C1 x (1 + y)^-1 + C2 x (1 + y)^-2 = 11,000 x (1 + 9%)^-1 + 11,000 x (1 + 9%)^-2 =  19,350.22

Duration = [1 x C1 x (1 + y)^-1 + 2 x C2 x (1 + y)^-2 ] / PV

= [1 x 11,000 x (1 + 9%)^-1 + 2 x 11,000 x (1 + 9%)^-2 ] / 19,350.22

=  1.4785 years

b. What maturity zero-coupon bond would immunize your obligation?

The maturity should be same as = duration =  1.4785 = 1.48 years.

And the face value of ZCB = FV of the ZCB = PV x (1 + y)ⁿ =  19,350.22 x (1 + 9%)^1.4785 = 21,979.61

c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 10%. What happens to your net position, that is an increase or decrease in value, to the difference between the value of the bond and that of your tuition obligation?

y = 10%

Value of obligations = C1 x (1 + y)^-1 + C2 x (1 + y)^-2 = 11,000 x (1 + 10%)^-1 + 11,000 x (1 + 10%)^-2 = 19,090.91

Value of ZCB = FV / (1 + y)ⁿ = 21,979.61 / (1 + 10%)^1.4785 = 19,090.71

Net position = Asset - liability = 19,090.71 - 19,090.91 = -0.21

i.e. declines (decreases in value) by 0.21

d. What if rates fall immediately to 8%? What happens to your net position, that is an increase or decrease in value, and by how much?

Value of obligations = C1 x (1 + y)^-1 + C2 x (1 + y)^-2 = 11,000 x (1 + 8%)^-1 + 11,000 x (1 + 8%)^-2 =  19,615.91

Value of ZCB = FV / (1 + y)ⁿ = 21,979.61 / (1 + 8%)^1.4785 = 19,615.70

Net position = Asset - liability =  19,615.91 - 19,615.70 = 0.21

i.e. increases in value by 0.21