Annuity Formula

Calculating an Annuity-Due

1) Solving the Total Amount

You make monthly payments of $100 in a 9% annuity-due.
What is the total amount after 7 years?

The payment per period is $100
the total number of periods 'n' is:
12 periods per year for 7 years, equals 12*7 = 84
the interest rate we must use is
.09 ÷ 12 = .0075

Putting these numbers into the formula:
Total = 100 * [((1.0075)^85 -1)/.0075] -100
Total = 100 * [(1.88725097807133 -1)/.0075] -100
Total = 100 * (.88725097807133 / .0075) -100
Total = 11,830.0130409511 -100
Total = 11,730.01

2) Solving the Periodic Payment

(Using the data from question 1)
A monthly 9% annuity-due is worth $11,730.01 after 7 years?
What was the periodic payment?

((1+r)⁽ⁿ⁺¹⁾ -1) ÷ r equals
((1+.0075)⁽⁸⁵⁾ -1) ÷ .0075 =
(1.88725097807133 -1) ÷ .0075 =
0.88725097807133 ÷ .0075 =
118.30013040951 minus one equals
117.30013040951
Periodoc Payment = 11,730.01 ÷ 117.30013040951
Periodoc Payment = 100.00

3) Solving for Years

(Using the data from question 1)
A 9% annuity-due with $100 monthly investments is now worth $11,730.01.
How many years did this take?

Log [rate * (total/periodic amount) + (1+rate)]
Log [.0075 * (11,730.01/100) + 1.0075] =
Log (1.88725075) =
0.275829606633691

Log (1+rate)
Log (1.0075) =
0.00324505481314708

0.275829606633691 ÷ 0.00324505481314708 =
84.9999838265255 = 85 (rounded)
85 -1= 84
84/12 = 7 years

4) Solving for Rate
The annuity-due formula cannot be solved for rate.
The annuity-due calculator solves for rate by using a trial and error process.

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