Calculating Contributions for an Annuity with Monthly Payments

Summary

The document shows how to calculate a fixed monthly contribution that grows to a target future value. It converts the investment horizon from years to months and converts the stated annual growth rate into a periodic monthly rate. With no initial lump sum, the future value of the regular contributions is set equal to the target, and the expression is rearranged to solve for the monthly payment.

An example applies the method to a retirement corpus goal, monthly deposits, and annual contribution growth. However, the question’s stated return assumption and the answer’s conversion appear inconsistent: the answer converts a 5% annual rate to a monthly rate, although the question gives a 3% return. The displayed contribution is therefore tied to the answer’s assumptions and should not be treated as valid for the stated 3% return without recalculation. The example also uses a simplified periodic compounding setup and does not discuss payment timing, taxes, fees, or uncertainty in realized returns.

Key ideas

• Convert the investment horizon into the same periods used for contributions.
• Convert an annual rate to a monthly rate before applying a monthly contribution formula.
• Set the accumulated value of contributions equal to the target and rearrange to solve for the payment.
• The example uses a rate inconsistent with the return stated in the question, so its payment figure needs recalculation.
• The calculation simplifies compounding and does not account for costs or uncertain returns.

Tags

# Is there a formula for present value of a growing annuity with yearly payment growth and monthly payments?


# Is there a formula for present value of a growing annuity with yearly payment growth and monthly payments?












I have seen formulas that have cracked the future value of growing annuity where there are monthly payments and yearly growth rates.

But given a future value, is it possible to derive the present value of annuity that is monthly in nature and grows on a yearly basis.

E.g. I want a retirement corpus of say $1,000,000 in 2040. What should be my monthly savings over the next 24 years such that monthly savings amount grows by 5% every year. The rate or return can be assumed to be 3%

## Answer by CCL (score 1)

https://quant.stackexchange.com/a/28064

A simple query on google could have given you the answer...

Let's define

- lumpsum q

- periodic contribution a

- y periods

- a periodic rate i

$$q*(1+i)^y + a( ((1+i)^y-1) / i ) - a = f$$

Suppose we do not want an initial investment $q=0$.

2040 - 2016 = 24 years. As you want to know the monthly contribution, everything needs to be converted to months. Thus 24 years equals $y=288$ months.

The 5% yearly return needs to be converted to a monthly return as well. $1,05^{1/12} - 1 = i=0.4\% $.

Finally let's put a monthly contribution of $a=\$1857.66$.

Putting that in the formula gives us +/- $1,000,000.

By rearranging the formula, we isolate the contribution amount a

$$a = (i * (f -q*(i+1)^y))/((i+1)^y-i-1)$$

which gave us a monthly contribution of $1857.66

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