Question

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?

Answer 1

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:

V_t = 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 = V_t / (1 + R)^T = $ 67,543 / (1 + 3.05%)¹² = $  47,098