The present value of a general annuity is the amount of money you would need today to make a series of regular payments in the future, considering that the interest is compounded at a different frequency than the payments are made.

In general annuities, you need to adjust the interest rate to match the payment period before using the formula.

Formula for Present Value of a General Annuity

To calculate the present value of a general annuity, use this formula:

Where:

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

For general annuities, the interest rate $i$ is adjusted to match the payment period. To find $i$, use:

Where:

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

Example

You will pay $200 every quarter for 2 years, and the annual interest rate is 8%, compounded monthly. How much money is this series of payments worth today?

Step 1: Identify the Values

• $R=200$,
• $j=8\%=0.08$ (annual interest rate),
• $m=12$ (the interest is compounded monthly),
• $f=3$ (there are 3 months in a quarter),
• $n=2\times 4=8$ (2 years with 4 quarters per year).

Step 2: Find the Effective Interest Rate ($i$)

The interest rate per quarter is found using the formula

Substituting in values,

So the effective interest rate per quarter is approximately $2.01$.

Step 3: Use the Present Value Formula

Now substitute the values into the formula for the present value of an annuity

Substituting

• $R=200$,
• $i=0.0201$,
• $n=8$,

Final Answer

The present value of the general annuity is approximately $1464.47.