Why the Time Horizon Matters
A single amount does not have one value independent of its date. If money earns interest at a positive rate, leaving a deposit invested for more periods produces a larger future amount. In the same model, a fixed payment due farther in the future has a smaller present value: it must be discounted for more periods to express it in today’s dollars.
The comparison in this tutorial holds the payment amount and interest rate fixed and changes only the number of years. This isolates the effect of time. It is useful in corporate finance when comparing values at different dates, and it is aligned with the SOA VEE Accounting and Finance syllabus.
These formulas describe a single payment, not a sequence of deposits or withdrawals. They assume the annual effective rate remains unchanged throughout the horizon and that the amount earns interest each year. There are no fees, taxes, or other cash flows in the model. The positive-rate condition matters: the direction of the comparisons below follows from \(i>0\).
Future Value Increases With the Horizon
For an initial deposit \(C\), each additional year multiplies its current accumulated value by \(1+i\). Since \(i>0\), this factor exceeds 1. The amount therefore grows from one year to the next, and a longer investment horizon produces a larger future value for the same deposit.
The increase between two horizons is not a separate deposit. It is the difference between the accumulated values at those dates. A longer horizon includes the interest accumulated during the additional years, including interest earned on earlier interest under annual compounding.
Worked Example: Extending a Deposit from Two Years to Six
A hypothetical business deposits $2,400 today in an account earning an effective annual rate of 5%. Find the future value after 2 years and after 6 years, then calculate the increase attributable to the additional 4 years. Assume the rate is constant and interest is compounded annually.
Method. Use \(\operatorname{FV}_n=C(1+i)^n\) at each horizon. The initial amount is $2,400 and \(i=0.05\).
Two years. The accumulation factor is \(1.05^2=1.1025\). Thus the future value is \(2{,}400(1.1025)=\$2{,}646.00\).
Six years. The accumulation factor is \(1.05^6=1.340095640625\). Thus the future value is \(2{,}400(1.340095640625)=\$3{,}216.2295375\).
Increase over the additional four years. Subtract the unrounded values: \(3{,}216.2295375-2{,}646.00=\$570.2295375\), which rounds to $570.23. The six-year value is greater, as expected at a positive rate. Dollar amounts are rounded to the nearest cent only at the final step.
Present Value Decreases With the Horizon
Now consider a fixed amount due at a future date. To express that amount at time 0, divide by the accumulation factor for the number of years until payment. Each additional year adds another discounting period, so the present value falls when the rate is positive.
For example, the present value of a $1,000 payment due in one year is greater than the present value of the same $1,000 payment due in three years, at the same positive rate. This does not mean the stated payment itself has changed. It means the amount needed today to be financially equivalent to that payment has changed because the dates differ.
Worked Example: Comparing Two Due Dates for the Same Payment
A hypothetical supplier is owed $5,000, payable either in 2 years or in 5 years. Find the present value of each alternative at an effective annual rate of 8%, and find how much lower the present value is when payment is delayed to year 5. Assume annual compounding and no other cash flows.
Method. Discount the same future payment to time 0 using \(\operatorname{PV}_n=C/(1+i)^n\), with \(C=\$5{,}000\) and \(i=0.08\).
Payment in 2 years. The discount denominator is \(1.08^2=1.1664\). Therefore, \(\operatorname{PV}_2=5{,}000/1.1664=\$4{,}286.6941015\ldots\), or $4,286.69 to the nearest cent.
Payment in 5 years. The discount denominator is \(1.08^5=1.4693280768\). Therefore, \(\operatorname{PV}_5=5{,}000/1.4693280768=\$3{,}402.915986\ldots\), or $3,402.92 to the nearest cent.
Decrease caused by the later date. Using the unrounded values, the decrease is \(4{,}286.6941015\ldots-3{,}402.915986\ldots=\$883.7781155\ldots\), which rounds to $883.78. The year-5 payment has a lower present value, as expected. All dollar answers are rounded to the nearest cent only at the final step.
Why the Direction of Change Must Hold
The relationship can be justified directly from the formulas. Compare values at two horizons \(n\) and \(n+k\), where \(k\) is a positive whole number. For a future value, factor the shorter-horizon amount out of the longer-horizon formula. The additional accumulation factor is greater than 1. For present value, the additional factor is its reciprocal, which is less than 1.
\(\operatorname{FV}_n=C(1+i)^n\) and \(\operatorname{FV}_{n+k}=C(1+i)^{n+k}\).
\(\operatorname{FV}_{n+k}/\operatorname{FV}_n=(1+i)^k>1\), because \(i>0\) and \(k>0\). Thus the longer-horizon future value is larger.
\(\operatorname{PV}_{n+k}/\operatorname{PV}_n=(1+i)^{-k}<1\). Thus the same payment due later has a lower present value.
This comparison also shows the role of the assumptions. The amount \(C\) and rate \(i\) are held fixed, and the rate is positive. If the payment amounts differ, a longer horizon alone does not determine which present value is larger; both amount and timing matter. If rates vary over time, the constant-rate formulas used here do not describe the whole accumulation or discounting process.
Reading the Comparison Correctly
A useful habit is to identify both the amount and the valuation date before calculating. A deposit available today and a payment due in the future are not interchangeable descriptions. For a deposit, move forward to find its future value. For a future payment, move backward to find its present value. The horizon is the number of annual periods between the payment date and the valuation date.
The calculation of a difference deserves particular care. Calculate each value at full precision, subtract those unrounded values, and round the final result. Subtracting displayed amounts that have already been rounded can occasionally change the answer by a cent. The worked example with the $5,000 payment illustrates why the difference is calculated from the unrounded present values.
Worked Example: Tracking Both Values Across Several Horizons
A hypothetical reserve of $3,000 can remain invested at a constant effective annual rate of 4%. Compare its future value after 3 years and 8 years. Find the increase over the additional 5 years, and explain why the result is consistent with the model. Assume annual compounding, with no deposits or withdrawals.
Method. Apply the single-deposit future-value formula at each horizon, keeping the starting amount and rate fixed.
Three years. \(1.04^3=1.124864\). The future value is \(3{,}000(1.124864)=\$3{,}374.592\), or $3,374.59.
Eight years. \(1.04^8=1.3685690506\ldots\). The future value is \(3{,}000(1.3685690506\ldots)=\$4{,}105.7071518\ldots\), or $4,105.71.
Increase. Subtracting before rounding gives \(4{,}105.7071518\ldots-3{,}374.592=\$731.1151518\ldots\), or $731.12 to the nearest cent. The increase is positive because the additional five years multiply the three-year value by \(1.04^5>1\). The calculation checks against the positive-rate rule.
Common Mistake: Reversing the Effect of Time
A common error is to say that a later payment has a higher present value simply because it is received later. That mixes up the payment date with its value at time 0. The future amount may be unchanged, but discounting for more years reduces what that amount is worth today at a positive rate.
For instance, for a fixed $5,000 due in 2 years or 5 years at 8%, an incorrect comparison might multiply by \(1.08^n\) to obtain present values. That would produce \(5{,}000(1.08)^2=\$5{,}832.00\) and \(5{,}000(1.08)^5=\$7{,}346.64\). These are accumulated amounts, not present values. The correction is to divide by the accumulation factors: the present values are approximately $4,286.69 and $3,402.92, respectively, so the later payment has the lower present value.
A second trap is rounding each intermediate value and then using those rounded values to calculate a difference. Keep full precision during the calculations and round only the requested final result. State the direction of the result as a check: with a positive rate and the same payment amount, extending the horizon raises future value and lowers present value.
Check Your Understanding
For each problem, assume the stated effective annual rate is constant, compounding is annual, and no fees or other cash flows apply. Round dollar amounts to the nearest cent at the final step.
- A hypothetical account receives a $1,000 deposit today and earns 6% annually. Find its future values after 2 years and 5 years, then find the increase. (Answer: $1,123.60 after 2 years; $1,338.23 after 5 years; increase $214.63.)
- A $3,600 payment is due in either 1 year or 4 years. At 5%, calculate each present value and the decrease when the payment date moves to year 4. Calculate the decrease before rounding. (Answer: $3,428.57; $2,961.73; decrease $466.84.)
- A hypothetical fund invests $2,500 at 4%. Compare its future values after 3 years and 7 years, and calculate the increase using unrounded values. (Answer: $2,812.16 after 3 years; $3,289.83 after 7 years; increase $477.67.)
- A $10,000 payment is due in 1 year or 3 years. At 10%, find both present values and how much lower the present value is for the later due date. (Answer: $9,090.91; $7,513.15; decrease $1,577.76.)
- A hypothetical deposit of $1,500 earns 5% annually. Find its value after 2 years and after 5 years, and state how much the additional 3 years increase the future value. (Answer: $1,653.75; $1,914.42; increase $260.67.)