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
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