Question

# A friend wants to borrow money from you. He states that he will pay you \$3,800...

A friend wants to borrow money from you. He states that he will pay you \$3,800 every 6 months for 13 years with the first payment exactly 6 years and six months from today. The interest rate is 6.1 percent compounded semiannually. What is the value of the payments today?

Sum of present value of N numbers of annuity A, at any time time t, when first annuity starts at t + 1 is given by:

Vt = A / R x [1 - (1 + R)-N] where R = interest rate per period

Here, payment frequency is semi annual. Hence, period is half year.

A = \$ 3,800

R = semi annual interest rate = 6.1% / 2 = 3.05%

N = number of equally spaced annuities = 2 x 13 = 26

First annuity is paid out at the end of period t + 1 = 6.5 years away from now = 2 x 6.5 = 13 half years = 13 periods from now. Hence, t + 1 = 13, hence t = 12

Hence, V12 = A / R x [1 - (1 + R)-N] = \$ 3,800 / 3.05% x [1 - (1 + 3.05%)-26] = \$  67,543

Since we want the value today, we need to discount it further for 12 periods, hence

Value today = V0 = Vt / (1 + R)T = \$ 67,543 / (1 + 3.05%)12 = \$  47,098