The Future Value Formula

The basic future value formula

The future value formula calculates how much an investment will be worth at a future date given a specific growth rate:

$FV = PV \times \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• FV = Future value of the investment
• PV = Present value (initial investment amount)
• r = Annual interest rate (as a decimal)
• n = Number of times interest is compounded per year
• t = Number of years the money is invested
• $\left(1 + \frac{r}{n}\right)^{nt}$ = The growth factor

Future value of a series (annuity)

When you make regular contributions in addition to a lump sum, you need the future value of a series formula.

Deposits at end of period (ordinary annuity)

$FV = PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$

Deposits at beginning of period (annuity due)

$FV = PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \times \left(1 + \frac{r}{n}\right)$

Where:

• PMT = The regular payment/deposit amount
• The annuity due formula simply multiplies by $\left(1 + \frac{r}{n}\right)$ to account for one extra period of interest on each deposit

Combined formula (lump sum + contributions)

$FV = PV \times \left(1 + \frac{r}{n}\right)^{nt} + PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$

Example 1: Lump sum only

You invest $5,000 at 6% annual interest, compounded monthly, for 5 years.

$FV = \$5,000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 5}$ $FV = \$5,000 \times (1.005)^{60}$ $FV = \$5,000 \times 1.34885$ $FV = \$6,744.25$

• Initial investment: $5,000.00
• Interest earned: $1,744.25
• Future value: $6,744.25

Example 2: Monthly contributions

You start with $10,000 and deposit$200 per month at the end of each month. The interest rate is 7% compounded monthly for 15 years.

Step 1: Future value of the initial $10,000

$FV_{pv} = \$10,000 \times \left(1 + \frac{0.07}{12}\right)^{12 \times 15}$ $FV_{pv} = \$10,000 \times (1.00583)^{180}$ $FV_{pv} = \$10,000 \times 2.8489$ $FV_{pv} = \$28,489.00$

Step 2: Future value of $200 monthly contributions

$FV_{pmt} = \$200 \times \frac{(1.00583)^{180} - 1}{0.00583}$ $FV_{pmt} = \$200 \times \frac{2.8489 - 1}{0.00583}$ $FV_{pmt} = \$200 \times 317.14$ $FV_{pmt} = \$63,428.00$

Step 3: Total future value

$\text{Total FV} = \$28,489.00 + \$63,428.00 = \$91,917.00$

• Total deposits: $10,000 + ($200 × 180) = $46,000
• Total interest earned: $91,917.00 -$46,000 = $45,917.00

Example 3: Annuity due vs ordinary annuity

You deposit $500 per month at 5% interest for 20 years.

Depositing at the beginning of each period earns you an extra $854 over 20 years — just by shifting your deposit timing.

Compounding frequency comparison

$10,000 invested at 5% for 10 years:

The difference between annual and daily compounding is about **$198** over 10 years on a$10,000 investment.

The Rule of 72

A quick way to estimate how long it takes for an investment to double:

$\text{Doubling Time} \approx \frac{72}{\text{Interest Rate}}$

This is a useful mental shortcut, though it's less accurate at very high or very low rates.

Present value vs future value

The present value formula is simply the future value formula rearranged to solve for the present:

$PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}$

This tells you how much you need to invest today to reach a future target.

Example: How much do you need to invest today to have $50,000 in 10 years at 6% compounded monthly?

$PV = \frac{\$50,000}{\left(1 + \frac{0.06}{12}\right)^{120}}$ $PV = \frac{\$50,000}{1.8194}$ $PV = \$27,472.00$

You would need to invest **$27,472 today** to have$50,000 in 10 years at 6%.