Consider the following series of payments which start at time t = 0:5, 7, 9, 11... What is the value of this series of payments at time t = 6? Effective annual interest rate is 8% p.a.
Question
Consider the following series of payments which start at time t = 0:5, 7, 9, 11... What is the value of this series of payments at time t = 6? Effective annual interest rate is 8% p.a.
Solution
This is an arithmetic series with an initial term of 5 and a common difference of 2. The payments are made at the beginning of each period, so this is an annuity due.
The formula for the present value of an annuity due is:
PV = P * (1 - (1 + r)^-n) / r * (1 + r)
where: PV = present value of the annuity P = initial payment per period r = interest rate per period n = number of periods
However, in this case, we want to find the value of the series at time t = 6, not the present value. This means we need to calculate the future value of the annuity due at time t = 6.
The formula for the future value of an annuity due is:
FV = P * ((1 + r)^n - 1) / r * (1 + r)
But since the payments are increasing by 2 each period, we need to calculate the future value of each payment separately and then sum them up.
The future value of each payment at time t = 6 is:
FV = 5 * (1 + 0.08)^6 + 7 * (1 + 0.08)^5 + 9 * (1 + 0.08)^4 + 11 * (1 + 0.08
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