Question

c. What is the present value (i.e., the value at t = 0, where t counts years) of the following stream of payments if the effective annual interest rate is .03 (i.e., 3% per annum): The first payment of $100 is made in 10 years (at t = 10). There are 10 more payments after that first payment that occur every 1.5 years. Each of these subsequent payments are 5% larger than the previous payment.

Answer 1

Effective annual interest rate = 3%

Effective interest rate for 1.5 years = (1+Effective annual interest rate)^1.5 - 1 = (1+3%)^1.5 - 1 = 4.533583%

First payment = $100

growth rate of payments = 5%

Second payment = 100*(1+5%) = $105

The present value of the final 10 payments at year t=10 can be calculated using the PV for growing annuity formula

Where P = Second payment = $105

r = Effective interest rate for 1.5 years = 4.533583%

g = growth rate of payments = 5%

n = number of payments = 10

PV = -22512.049757 * -0.045525477 = 1024.871809

Total PV at 10 years = First payment + The present value of the final 10 payments

= 100 + 1024.871809 = 1124.871809

Present value now = Total PV at 10 years / (1+Effective annual interest rate)¹⁰

= 1124.871809 / (1+3%)¹⁰

= 1124.871809 / 1.343916

= 837.0102681

Present value now = $837.01