Deriving Present–Future Value Equivalence

Why Present and Future Values Must Agree

A single cash amount can be expressed at different points in time. For example, a deposit made today may grow into a larger amount after several years, while a payment due in the future can be translated into an equivalent amount today. These are not competing ways to value the cash flow: under the same interest assumptions, they must give consistent answers.

This consistency matters in corporate finance because cash-flow alternatives are often stated at different dates. A business comparing a payment now with a payment later needs a common valuation date. The same idea is useful when checking a calculation: if a present value is accumulated to the future and then discounted back, it should return to the original amount, subject only to rounding.

Definition: Under annual compound interest at an effective annual rate $i$, the present value of a single payment is its equivalent amount at the valuation date, and its future value is the equivalent amount at a later date. If $C_0$ is the amount at time 0 and $C_n$ is the amount at the end of $n$ years, the two amounts are equivalent when the same interest model converts one into the other.

In this tutorial, assume a single payment, no intervening deposits or withdrawals, a constant effective annual interest rate, and annual compounding for an integer number $n$ of years. The rate must satisfy $i>-1$, so the annual accumulation factor $1+i$ is positive. The examples use positive rates. If rates change over time, compounding is not annual, or additional cash flows occur between the dates, the simple single-rate formula below must be adapted.

The Present–Future Value Relationship

For a constant effective annual rate $i$, each year multiplies an amount by $1+i$. Repeating that annual accumulation for $n$ years gives the future value of an amount at time 0. The earlier material on calculating present value by discounting gives the corresponding operation in reverse.

C_n=C_0(1+i)^n \qquad\text{and}\qquad C_0=\frac{C_n}{(1+i)^n}.

The first expression accumulates an amount forward from time 0 to time $n$. The second discounts an amount back from time $n$ to time 0. The central result is that using either expression on the same payment gives a consistent pair of values.

Theorem: Under the stated constant-rate, annual-compounding model, accumulating the present value of a single payment to time $n$ gives that payment’s future value. Conversely, discounting that future value to time 0 returns the original present value: $$ \left(\frac{C_n}{(1+i)^n}\right)(1+i)^n=C_n, \qquad \frac{C_0(1+i)^n}{(1+i)^n}=C_0. $$

Why the Equivalence Holds

The relationship follows directly from what it means to accumulate or discount under compound interest. Starting with $C_0$, one year of interest produces $C_0(1+i)$. A second year applies the same annual factor to that new balance, producing $C_0(1+i)^2$. Continuing through $n$ years gives $C_0(1+i)^n$. If this result is the future payment $C_n$, solve the equation for $C_0$ by dividing both sides by $(1+i)^n$, which is positive under the model’s condition $i>-1$. Thus $C_0=C_n/(1+i)^n$. Substituting that expression back into the accumulation formula gives $C_n$ exactly.

C_n=C_0(1+i)^n \quad\Longrightarrow\quad \frac{C_n}{(1+i)^n}=C_0 \quad\Longrightarrow\quad \left(\frac{C_n}{(1+i)^n}\right)(1+i)^n=C_n.

This justification also explains why the equivalence is a useful check rather than an additional assumption: both calculations use the same model and the same time interval. Rounding a reported result can create a tiny apparent difference, so keep full precision during the calculation and round only the final money amount.

Worked Example: Checking a Reserve Deposit at Two Dates

A hypothetical firm deposits $7,000 now into an account earning an effective annual rate of 8%, compounded annually. Find the value after three years, then discount that amount to verify the original deposit. Report dollar amounts to the nearest cent.

Method. First accumulate the deposit for three annual periods. Then use the same rate and duration to discount the unrounded future value. This checks that the two date-based values agree.

The annual accumulation factor is $1+0.08=1.08$. Over three years, the factor is $1.08^3=1.08\times1.08\times1.08=1.1664\times1.08=1.259712$. Therefore,

C_3=\$7{,}000(1.259712)=\$8{,}817.984.

The future value, rounded to cents, is $8,817.98. To check the equivalence, retain the unrounded amount in the discounting step:

C_0=\frac{\$8{,}817.984}{1.259712}=\$7{,}000.

The discounted amount equals the original deposit. The check works because both calculations use the same 8% annual rate and the same three-year interval. The common mistake this avoids is discounting a rounded future value and treating a tiny rounding difference as a failure of equivalence.

Worked Example: Verifying a Two-Year Accumulation

A hypothetical project account receives $12,000 today and earns 4% effective annual interest, compounded annually. Calculate the account value at the end of year 2, then verify that discounting the future amount returns $12,000. Round final money amounts to cents.

Method. Calculate the two-year accumulation factor, multiply the present amount by it, and discount the unrounded result using that same factor.

The factor for two years is $1.04^2=1.04\times1.04=1.0816$. Thus the value at the end of year 2 is

C_2=\$12{,}000(1.0816)=\$12{,}979.20.

Now discount the future amount to time 0:

C_0=\frac{\$12{,}979.20}{1.0816}=\$12{,}000.

The calculation checks exactly: $12,979.20 at the end of year 2 is equivalent to $12,000 today under the stated model. A common mistake is to use the one-year factor $1.04$ when the payment is two years away. The two periods require $1.04^2$, not $1.04$.

Comparing Payments at a Common Date

The equivalence gives two valid approaches when comparing payments at different dates: discount every amount to the present, or accumulate every amount to a common future date. Do not compare an amount at time 0 directly with an amount at time $n$; their dollar figures refer to different dates.

Identify each payment date.
Mark when each amount is received or paid, and choose a common date for the comparison.

Apply the matching operation.
Move an amount forward by accumulating it, or backward by discounting it, using the stated rate and number of periods.

Compare equivalent amounts.
Once all amounts are expressed at the same date, compare them and state which alternative has the larger value at that date.

Worked Example: Comparing a Payment Now With One Later

A hypothetical supplier offers a company a choice between paying $6,100 today or paying $6,000 exactly two years from today. For this comparison, the company uses an effective annual rate of 4%, compounded annually. Determine which option has the lower cost, using both time 0 and the end of year 2 as comparison dates. Report final money amounts to cents.

Method. First discount the later payment to time 0. Then independently accumulate the payment due today to the end of year 2. Both methods must indicate the same preference.

The two-year factor is $1.04^2=1.0816$. Discounting the $6,000 due in two years gives

\frac{\$6{,}000}{1.0816} =\$5{,}547.337278\ldots \approx \$5{,}547.34.

At time 0, the later payment has a value of $5,547.34, less than the $6,100 payment due today. Therefore, the two-year payment is the lower-cost option on a present-value basis. The difference in present values is $\$6{,}100-\$5{,}547.337278\ldots=\$552.662722\ldots$, or $552.66 to cents.

For the independent check, accumulate $6,100 to the end of year 2:

\$6{,}100(1.0816)=\$6{,}597.76.

At year 2, paying today is equivalent to a cost of $6,597.76, while paying later costs $6,000. The later payment is lower by $\$6{,}597.76-\$6{,}000=\$597.76$ at that date, confirming the same preference. The two difference amounts are stated at different dates, so they need not be numerically equal. The common trap is comparing $6,100 directly with $6,000 without adjusting either amount for timing.

Common Mistake: Accumulating When You Should Discount

Suppose a $6,000 payment is due two years from now at 4% effective annual interest. A wrong calculation for its present value would be

\$6{,}000(1.0816)=\$6{,}489.60.

That calculation accumulates the $6,000 forward; it does not find the value today of a payment already stated at a future date. To move the payment back to time 0, divide by the two-year factor:

\frac{\$6{,}000}{1.0816} =\$5{,}547.337278\ldots \approx \$5{,}547.34.

A reliable check is to take the calculated present value and accumulate it to the stated payment date. Using the unrounded value gives $\$5{,}547.337278\ldots\times1.0816=\$6{,}000$.

Key takeaway: For a single payment under a constant effective annual rate and annual compounding, the present and future values are two date-specific expressions of the same cash flow. Accumulating one to the other date and discounting back must reproduce the original amount.

Check Your Understanding

For each problem, use the stated annual compound-interest model and compare values only after placing them at a common date.

A hypothetical fund invests $4,000 today at an effective annual rate of 3%, compounded annually for two years. Find its year-2 value and verify it by discounting. (Answer: $4,243.60 at year 2; discounting gives $4,000.00.)
A single payment of $10,000 today earns 6% effective annual interest, compounded annually for three years. Find its future value and state the present value obtained by discounting that future value. (Answer: $11,910.16 at year 3; present value $10,000.00.)
A contract requires a single $7,500 payment in two years. At an effective annual rate of 5%, compounded annually, find its present value to cents. (Answer: $6,802.72.)
A supplier offers either $5,200 paid today or $6,000 paid in three years. Use an effective annual rate of 5%, compounded annually. Which payment has the lower cost at time 0, and by how much? (Answer: The $6,000 payment in three years; its present value is $5,183.03, which is $16.97 less than $5,200.)
A company can pay $5,500 today or $6,000 in two years. At an effective annual rate of 4%, compounded annually, compare both costs at the end of year 2. (Answer: Paying today accumulates to $5,948.80, so it is lower by $51.20 than the $6,000 payment due at year 2.)

What You'll Learn