Deriving the Compound Future-Value Formula

Why Derive the Compound Future-Value Formula?

When money earns compound interest, each period's interest becomes part of the balance that can earn interest in later periods. The future value is therefore not found by adding the same dollar amount of interest each year. It is found by applying the periodic growth factor repeatedly. Deriving the formula makes clear what the exponent counts, which rate belongs in the formula, and how the calculation changes when deposits are made on different dates.

Here, $PV$ is the amount invested at the valuation date, $FV$ is its value after the stated number of periods, and $i$ is the effective interest rate for one such period. For example, if periods are years, $i$ must be an effective annual rate. The formula below concerns one initial balance with no additional deposits or withdrawals during the holding period.

Definition: Under compound interest at a constant effective rate $i$ per period, a balance at the beginning of a period is multiplied by $1+i$ to obtain the balance at the end of that period. The compound future value after $n$ whole periods is the resulting balance, where $n$ is a nonnegative integer.

Deriving the Formula From Periodic Growth

Let $B_k$ denote the balance after $k$ periods. At the start, before any period has passed, $B_0=PV$. During each period, the interest earned is $i$ times the balance at that period's start. The ending balance is therefore the starting balance plus that interest:

B_{k+1}=B_k+iB_k=(1+i)B_k.

Applying this recurrence successively gives $B_1=PV(1+i)$, $B_2=PV(1+i)(1+i)$, and then one further factor of $1+i$ for each additional period. The exponent is the number of times the one-period growth factor is applied.

Formula: If a single amount $PV$ earns a constant effective rate $i$ per period for $n$ whole periods, with no intervening cash flows, its future value is

$$ FV=PV(1+i)^n. $$

The formula includes the zero-period case. If $n=0$, no period has elapsed, so the balance remains $PV$. Since $(1+i)^0=1$, the formula gives $FV=PV$, as it should. For positive whole-number $n$, the formula follows by induction: after $k$ periods, suppose $B_k=PV(1+i)^k$. One more period gives

B_{k+1}=(1+i)B_k =(1+i)PV(1+i)^k =PV(1+i)^{k+1}.

Starting with $B_0=PV=PV(1+i)^0$, this step establishes the formula for every nonnegative whole-number period count. The derivation also explains why the exponent is not a dollar amount or an interest rate: it counts the number of applications of the periodic growth factor.

Worked Examples: Applying the Repeated Growth Factor

Worked Example: One Balance Over Four Years

A hypothetical reserve account has an initial balance of $1,200. It earns an effective annual interest rate of 5%, compounded annually, for four years. There are no deposits or withdrawals. Find the balance at the end of year four, first by showing each period and then using the formula. Name the method and identify a likely mistake.

Method. The period is one year, so $i=0.05$, and the number of periods is $n=4$. Each year's opening balance is multiplied by $1.05$.

Year 1: $1{,}200.00(1.05)=1{,}260.00$.
Year 2: $1{,}260.00(1.05)=1{,}323.00$.
Year 3: $1{,}323.00(1.05)=1{,}389.15$.
Year 4: $1{,}389.15(1.05)=1{,}458.6075$.

Using the formula gives the same result:

FV=1{,}200(1.05)^4 =1{,}200(1.21550625) =1{,}458.6075.

Rounding the final amount to the nearest cent gives $1,458.61. The check is that the formula result agrees with applying the growth factor four times; it is also greater than the initial balance because the rate is positive. A common mistake is to use $n=5$ because the timeline displays five dates, from time 0 through time 4. There are only four elapsed annual periods, so the exponent is 4, not 5.

Worked Example: Distinguishing the Rate Per Period

A hypothetical firm places $2,500 in an account that earns an effective annual rate of 4%, compounded annually. The balance remains invested for three years. Calculate the future value and show why applying simple interest would give a different answer.

Method. Since the rate is effective per year and each period is one year, use $i=0.04$ and $n=3$. The interest earned in each year stays in the account and earns interest in later years.

Year 1: $2{,}500.00(1.04)=2{,}600.00$.
Year 2: $2{,}600.00(1.04)=2{,}704.00$.
Year 3: $2{,}704.00(1.04)=2{,}812.16$.

The formula confirms the period-by-period calculation:

FV=2{,}500(1.04)^3 =2{,}500(1.124864) =2{,}812.16.

The final value is $2,812.16. As a check, the first year's interest is $100.00, but later interest is calculated on balances that include prior interest. The simple-interest calculation $2{,}500[1+3(0.04)]=2{,}800.00$ omits that compounding and is not the answer under the stated model. The formula applies because the given 4% is the effective rate for each annual period; a rate quoted for some other period cannot be inserted without first matching the rate period to the number of periods.

Separate Deposits Have Separate Growth Periods

The single-sum formula applies to one amount invested at one date. If more than one deposit is made, apply the formula separately to each deposit, using the number of periods from that deposit's date to the valuation date. Then add the resulting values. This follows from the balance recurrence: each deposit contributes its own compounded amount to the total balance.

Choose a valuation date.
State the date at which the total accumulated value is required.

Count periods for each deposit.
Count from the deposit date to the valuation date, not from the account's opening date unless those dates coincide.

Compound each amount separately.
For a deposit $C_j$ invested for $n_j$ periods, its value at the valuation date is $C_j(1+i)^{n_j}$, when the same effective rate $i$ applies throughout.

Add the accumulated amounts.
The total is the sum of the future values of all deposits, with no deposit counted more than once.

Worked Example: Deposits Made on Different Dates

A hypothetical project fund receives $500 immediately, $300 at the end of year 1, and $200 at the end of year 2. The fund earns an effective annual rate of 6%, compounded annually, and there are no withdrawals. Find the total fund value at the end of year 4. Explain the timing of each deposit.

Method. Use the end of year 4 as the valuation date. The immediate $500 deposit earns interest for four full years. The $300 deposit made at the end of year 1 earns interest for three full years. The $200 deposit made at the end of year 2 earns interest for two full years.

The three accumulated amounts are:

\begin{aligned} 500(1.06)^4 &=500(1.26247696)=631.23848,\\ 300(1.06)^3 &=300(1.191016)=357.30480,\\ 200(1.06)^2 &=200(1.1236)=224.72000. \end{aligned}

Adding them gives

$$ 631.23848+357.30480+224.72000=1{,}213.26328. $$

Rounded to the nearest cent, the fund value is $1,213.26. A check by year-end balances gives the same amount: at year 1 the balance is $500(1.06)+300=830$; at year 2 it is $830(1.06)+200=1{,}079.80$; at year 3 it is $1{,}079.80(1.06)=1{,}144.588$; and at year 4 it is $1{,}144.588(1.06)=1{,}213.26328$. A common error is to assign every deposit four periods of growth. That would incorrectly give the later deposits interest for periods before they were made.

Conditions, Interpretation, and Common Traps

The equation $FV=PV(1+i)^n$ is exact under the stated compound-interest model: the same effective rate applies in each period, the periods match the rate, and no additional cash flows occur for the single amount. The period count $n$ is a nonnegative integer in this tutorial. If rates change across periods, use the actual growth factor for each period rather than one constant factor:

FV=PV(1+i_1)(1+i_2)\cdots(1+i_n),

where $i_k$ is the effective rate for period $k$. For example, a balance earning 4% for one year and 6% for each of the following two years grows by $1.04(1.06)^2$, not by one constant rate unless an equivalent rate is specifically calculated. The constant-rate formula also does not by itself account for deposits or withdrawals made during the holding period; such cash flows must be accumulated from their own dates.

Several exam traps can be prevented by identifying the time unit before substituting values:

Counting dates instead of elapsed periods. A timeline with times 0, 1, 2, and 3 has three periods from time 0 to time 3, not four.
Using the wrong rate period. An annual effective rate belongs with annual periods. The exponent and rate must refer to the same period length.
Applying a single-sum formula to multiple deposits as if they were all made at time 0. Each deposit has its own investment duration.
Rounding too early. Keep sufficient precision in intermediate balances and round the final dollar amount to the nearest cent when requested.

In corporate finance, the formula is a basic way to project a known amount forward under a stated interest assumption. Its practical usefulness depends on keeping the model's assumptions visible: the calculated value is a mathematical accumulation under the specified rate and timing, not a guarantee that an actual account or investment will earn that rate.

Key takeaway: Compound growth applies the factor $1+i$ once per period. Begin with $PV$, count the elapsed periods, and use $FV=PV(1+i)^n$. For deposits made on different dates, compound each deposit for its own number of periods and then add the resulting values.

Check Your Understanding

For each problem, use the stated timing and effective rate. Round dollar answers to the nearest cent.

A hypothetical reserve starts with $900 and earns an effective annual rate of 3%, compounded annually, for two years with no cash flows. What is its value at the end of year 2? (Answer: $954.81)
A $1,600 deposit earns an effective annual rate of 7% for three years, compounded annually. Find the future value and state how many times the annual growth factor is applied. (Answer: $1,960.07; 3 times)
A fund receives $400 immediately and another $250 at the end of year 1. It earns an effective annual rate of 5%, compounded annually. Find the total at the end of year 3. (Answer: $738.68)
A deposit of $750 earns 0% effective interest per year for six years, with no deposits or withdrawals. Find its value and explain what the zero rate does to the growth factor. (Answer: $750.00; the growth factor is 1 each year)
A $1,000 balance earns an effective annual rate of 4% during year 1 and 6% during each of years 2 and 3. Find its value at the end of year 3 without treating the rate as constant. (Answer: $1,168.54)

What You'll Learn