In simple annuities, why is the amount of payment for the first deposit R(1+i)^n-1?
Shouldn't it be n?
In simple annuities, where the interest is not compounded, the amount of payment for the first deposit is not simply n, but rather R(1+i)^n-1. Let me break it down for you:
The formula for the future value of a simple annuity is given by:
FV = R * ((1 + i)^n - 1) / i
Where:
FV = Future value of the annuity
R = Regular payment amount
i = Interest rate per period
n = Number of periods
To understand why it's R(1+i)^n-1 and not just n, we need to consider the formula derivation. The term (1+i)^n - 1 represents the accumulation of the growth of the annuity over n periods, while the division by i is used to adjust for the compounding effect.
Now, let's see how this formula relates to the amount of the first payment:
When you make the first deposit, it has not had a chance to accrue any interest yet. So, the only factor affecting the value of the first payment is the number of compounding periods, which is n.
However, notice that the formula for the annuity's future value includes a (1+i)^n term, which accounts for the interest growth over the n periods. Therefore, to determine the contribution of the first payment to the future value of the annuity, we need to subtract the accumulated interest from the total future value.
By rearranging the formula, we can isolate R:
R = FV * i / ((1 + i)^n - 1)
Now, when you substitute FV by its equivalent form (R * ((1 + i)^n - 1) / i) in the above equation, you will get:
R = R * ((1 + i)^n - 1) / i * i / ((1 + i)^n - 1)
The (1 + i)^n - 1 terms cancel out, resulting in:
R = R
This shows that the analysis is consistent, and R is indeed the correct value for the first deposit in a simple annuity.