Question

I seek the formula for manual calculation of the payment of an ordinary annuity when the interest is not compounded at the same frequency as the deposits (e.g. compound daily with monthly payments). This is an annuity that I buy now, the present value is large, I receive monthly payments, the future value is zero. TI-83 can do this, but I need the formula for manual calculation, and I haven't yet found it anywhere. The only formula that I have includes only future value. Don't I need a formula that includes both present value and future value? This is the formula that I have:

A & = & \mbox{ accumulated balance after t years} \\
& = & FV \\
& = & \mbox{ future value of the annuity} \\
PMT & = & \mbox{ periodic payment} \\
& = & \mbox{ need to find this} \\
r & = & \mbox{ APR written as a decimal} \\
m & = & \mbox{ compounding frequency} \\
i & = & \mbox{ periodic interest rate} \\
& = & \frac { r } { m } \\
n & = & \mbox{ number of payments per year} \\
t & = & \mbox{ number of years} \\
A & = & FV \\
& = & PMT
\frac {
\left [ \left ( 1 + i \right )^{\frac { m } { n }} \right ]^{(nt)} - 1}
{ ( 1 + i )^{\frac { m } { n }} - 1 } \\
& = & PMT
\frac {
\left [
\left ( 1 + \frac { r } { m } \right )^{\frac { m } { n }} \right ]^{(nt)} - 1}
{ ( 1 + \frac { r } { m } )^{\frac { m } { n }} - 1 }

Here is the specific question: Suppose that you have just received an inheritance of \$300,000. You use the money to set up an annuity to fund a stream of monthly payments to yourself over the next 20 years; at the end of the 20-year period, the account will be completely depleted. Assuming that you can earn 6\% interest compounded daily on this investment, what will be your monthly payments?
Here is the TI-83 solution:

N & = & 240 \quad \mbox{12 payments per year for 20 years } \\
I\% & = & 6 \quad \mbox{Given interest rate} \\
PV & = & 300,000 \quad \mbox{The given present value} \\
PMT & = & ?? \quad \mbox{Again, solving for the payment} \\
FV & = & 0 \quad \mbox{At the end, the account is depleted} \\
P/Y & = & 12 \quad \mbox{payments are on a monthly basis} \\
C/Y & = & 12 \quad \mbox{compounding is on a monthly basis} \\
PMT & : & END

Solving for the unknown, we get

PMT = -2149.29

In other words, this investment will give us monthly payments of \$2149.29 over the next 20 years.

Thank you.

Answer 1

The formula that can be used to calculate the payment can be derived with the use of following formula for calculating Present Value given as below:

Present Value = Payment*[(1-(1+Rate of Interest)^(-Period))/Rate of Interest]

Rearranging formula, we get,

Payment = Present Value*(Rate of Interest)/(1-(1+Rate of Interest)^(-Period))

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Payment with Daily Compounding (as mentioned in the Question)

Here, compounding is daily, so we will have to calculate EAR (effective annual rate) first as follows:

EAR =(1+Rate of Interest/365)^(365) - 1 = (1+ 6%/365)^(365) - 1 = 6.18%

Now, we can calculate payment with the use of formula provided above:

Payment = 300,000*(6.18%/12)/(1-(1+6.18%/12)^(-240)) = $2,180.56

_____

Payment with Normal Interest Rate Compounding (as calculated in the Question)

Here, compounding is taken annually (which is not correct as per the details provided in the question).

We can calculate payment with the use of formula provided above:

Payment = 300,000*(6%/12)/(1-(1+6%/12)^(-240)) = $2,149.29 (same as calculated with the use of Financial Calculator)

_____

Notes:

The formula for calculating the payment would remain the same. Only the interest rate will change depending on the type of compounding.