To find the future value of a general annuity, you calculate how much money you will have in the future when regular payments are made, but the interest is compounded on a different schedule than the payments. The future value accounts for both the payments and the interest earned.

Formula for Future Value of a General Annuity

The formula is:

Where:

• $F$ = future value of the annuity
• $R$ = regular payment amount
• $i$ = effective interest rate per payment period
• $n$ = total number of payments

Since interest is compounded differently than the payment schedule, you need to calculate the effective interest rate per payment period.

The effective interest rate $i$ is calculated as

where

• $j$ = annual nominal interest rate,
• $m$ = number of compounding periods per year,
• $f$ = number of compounding periods in one payment interval.

Example

You deposit $200 every 6 months for 3 years into an account that earns 6% annual interest, compounded quarterly. How much money will you have at the end of 3 years?

Step 1: Identify the values

• $R=200$,
• $j=6\%=0.06$ (annual nominal interest rate),
• $m=4$ (there are 4 quarters per year),
• $f=2$ (There are 2 quarters per every 6-month payment period),
• $n=3\times 2=6$ (2 payments per year over 3 years).

Step 2: Find the effective interest rate ($i$)

The effective interest rate ($i$) is given by the formula

Substitute the values,

The effective semi-annual interest rate is $i=0.030225$, or $3.0225$

Step 3: Use the future value formula

Now substitute into the future value formula:

Final Answer

The future value of the annuity is approximately $1,294.41.

Key Points:

1. Adjust the interest rate: Use the effective interest rate per payment period.
2. Plug into the formula: Use the annuity formula to calculate the future value.