Why the Interest Rate Changes a Single-Sum Value
A single cash flow can be described at different points in time. A deposit today grows to a future value, while a payment due in the future can be expressed as a present value. The interest rate affects these two calculations in opposite directions. For a fixed term, a higher rate increases the future value of a fixed deposit but decreases the present value of a fixed future payment.
The distinction between what is held fixed matters. In a future-value comparison, the starting deposit is fixed. In a present-value comparison, the future payment is fixed. If both the rate and the cash-flow amount are allowed to change, the direction of the result may not be determined by the rate alone.
A rate comparison should use the same time unit as the compounding assumption. Here, \(i\) is an effective annual rate and \(n\) is measured in years. The rate is applied once for each year. Rates such as 4% and 7% are entered as \(0.04\) and \(0.07\), respectively. The amount of interest earned on a deposit is its future value minus its original deposit; it is not the future value itself.
Why the Two Directions Are Opposite
The accumulation factor for \(n\) years is \((1+i)^n\). When \(i\) increases, this factor increases, so multiplying a fixed positive deposit by it produces a larger future value. For a fixed future payment, present value instead divides by that same accumulation factor. A larger positive denominator produces a smaller present value.
The comparison can be stated precisely. Let \(i_2>i_1>-1\) be two rates and keep \(n\) fixed. Then \(1+i_2>1+i_1>0\). Raising both positive quantities to the same positive integer power preserves the inequality:
Multiplying this inequality by the fixed positive deposit \(C_0\) gives a higher future value at \(i_2\). Taking reciprocals reverses the inequality, and multiplying by the fixed positive future payment \(F_n\) gives a lower present value at \(i_2\). This is the reason for the opposite directions; it follows directly from accumulation and discounting, not from a special rule for particular rates.
For \(n=0\), there is no period over which to accumulate or discount, so the rate has no effect. A zero cash flow also has value zero at every rate. The direction rules above therefore require a positive cash flow and a positive fixed term, as assumed. They hold for rates below zero as well, provided \(i>-1\); the annual accumulation factor remains positive and increases with the rate.
Comparing Future Values
To compare the future values of one deposit under two rates, keep the deposit and term unchanged, calculate each accumulation factor, and multiply by the deposit. A higher rate means more interest is credited in each year, and the interest credited in earlier years also earns interest in later years. This compounding is why the rate's effect is not generally just the initial deposit multiplied by the rate difference.
Worked Example: Comparing Two Deposit Rates
A hypothetical reserve account receives a deposit of $8,000 today. It earns interest compounded annually for 3 years. Compare its future value at effective annual rates of 4% and 8%, and find the difference in future values.
Method. Use \(\operatorname{FV}_n=C_0(1+i)^n\) for each rate, holding the $8,000 deposit and the 3-year term fixed.
At 4%, the accumulation factor is \(1.04^3=1.04\times1.04\times1.04=1.0816\times1.04=1.124864\). Thus the unrounded future value is \($8,000\times1.124864=$8,998.912\), or $8,998.91 to the nearest cent.
At 8%, the accumulation factor is \(1.08^3=1.08\times1.08\times1.08=1.1664\times1.08=1.259712\). Thus the unrounded future value is \($8,000\times1.259712=$10,077.696\), or $10,077.70 to the nearest cent.
The difference, calculated before rounding, is \($10,077.696-$8,998.912=$1,078.784\), or $1,078.78 to the nearest cent. As a check, the interest earned is $8,998.912 − $8,000 = $998.912 at 4% and $10,077.696 − $8,000 = $2,077.696 at 8%; the difference in interest earned is also $1,078.784. The higher rate gives the higher future value, as expected. This method avoids treating the rate difference as if it were a one-time addition to the deposit.
Comparing Present Values
For a fixed future payment, use the discount factor or divide by the accumulation factor. Keep both the payment amount and its due date fixed when comparing rates. A higher rate gives each dollar due in the future a smaller value today because a smaller deposit would grow to that same payment at the higher rate.
Worked Example: Valuing a Fixed Future Payment
A hypothetical contract requires a payment of $12,100 exactly 2 years from today. Find its present value at annual effective rates of 5% and 10%, and compare the results. Assume annual compounding and no other cash flows.
Method. Since the future payment is fixed, use \(\operatorname{PV}_0=F_n/(1+i)^n\) at each rate.
At 5%, the 2-year accumulation factor is \(1.05^2=1.1025\). The present value is \($12,100/1.1025=$10,975.056689\ldots\), or $10,975.06 to the nearest cent.
At 10%, the accumulation factor is \(1.10^2=1.21\). The present value is \($12,100/1.21=$10,000.00\).
The present value at 5% exceeds the present value at 10% by \($10,975.056689\ldots-$10,000=$975.056689\ldots\), or $975.06 to the nearest cent. Check the calculations by accumulating each unrounded present value for 2 years: \($10,975.056689\ldots\times1.1025=$12,100\), and \($10,000\times1.21=$12,100\). Both reproduce the stipulated payment. The fixed future payment therefore has a lower present value at the higher rate.
Size of the Effect
The direction of the change is consistent, but its size depends on the rate difference, the starting or future amount, and the accumulation factor over the fixed term. The percentage change in future value is not usually equal to the percentage-point change in the interest rate. For example, the rate increase from 2% to 7% is 5 percentage points, but the future value changes according to the ratio of the two accumulation factors, not by a simple 5% adjustment.
Worked Example: Measuring the Relative Change
A hypothetical fund receives $6,000 today and earns annual compound interest for 4 years. Compare its future values at 2% and 7%, then calculate the percentage increase from the lower-rate future value. Round money to the nearest cent and the percentage to two decimal places.
Method. Calculate both accumulated values without intermediate rounding, subtract to obtain the dollar increase, and divide that increase by the lower-rate value for the percentage increase.
At 2%, \(1.02^4=1.08243216\), so the future value is \($6,000\times1.08243216=$6,494.59296\), or $6,494.59.
At 7%, \(1.07^4=1.31079601\), so the future value is \($6,000\times1.31079601=$7,864.77606\), or $7,864.78.
The unrounded dollar increase is \($7,864.77606-$6,494.59296=$1,370.18310\), or $1,370.18. The percentage increase, using the lower-rate future value as the base, is \($1,370.18310/$6,494.59296\times100=21.097\ldots\%\), or 21.10% to two decimal places.
As a check on the percentage, the ratio of the two future values is also the ratio of their accumulation factors: \(1.31079601/1.08243216=1.21097\ldots\). This is approximately 1.211 times the lower-rate value, consistent with an increase of about 21.10%. The percentage is not the 5-percentage-point rate difference: each year's accumulation is compounded.
Common Mistake: Accumulating Instead of Discounting
A frequent error is to multiply a future payment by \((1+i)^n\) when asked for its value today. That calculation moves a value forward in time; it does not bring a future payment back to the present. It can even produce a present value greater than the future payment at a positive rate, which should prompt a check of the direction.
For example, suppose $10,000 is due in 2 years at 8% effective annual interest. The incorrect calculation is \($10,000\times1.08^2=$10,000\times1.1664=$11,664\). This accumulates the payment and answers a different question. The correct present value is \($10,000/1.1664=$8,573.388\ldots\), or $8,573.39 to the nearest cent. Accumulating that present value checks the result: \($8,573.388\ldots\times1.1664=$10,000\). The payment is fixed at $10,000 in the future, and the amount needed today is smaller at this positive rate.
Why Rate Comparisons Matter
In corporate finance, a single amount may be discussed either as money invested now or as money payable later. The rate comparison helps keep those perspectives distinct: a higher return raises the ending balance of a fixed investment, while a higher discount rate lowers the amount assigned today to a fixed future obligation. The calculations are also useful as a direction check before using a calculator or spreadsheet.
When interpreting a comparison, verify three items before calculating:
- Which amount is fixed? Use the same initial deposit for a future-value comparison, or the same future payment for a present-value comparison.
- What is the valuation date? Identify whether the requested value is at the start or the end of the term.
- Are the rate and term units consistent? An effective annual rate with annual compounding pairs with a term measured in years.
Finally, round only after calculating with the unrounded accumulation factor. Rounding each year's factor or an intermediate present value can create a small discrepancy, especially when values are compared or checked by reversing the calculation.
Check Your Understanding
For each problem, assume annual compounding, no fees or other cash flows, and round dollar amounts to the nearest cent.
- A hypothetical account receives $2,500 today for 3 years. Find its future value at 2% and at 5%. (Answer: $2,653.02 at 2%; $2,894.06 at 5%.)
- A payment of $7,000 is due in 2 years. Find its present value at 4% and 6%. (Answer: $6,471.89 at 4%; $6,229.98 at 6%.)
- A $9,000 deposit earns interest for 3 years. Compare the future values at 3% and 8%, and give the dollar difference. (Answer: $9,834.54 at 3%; $11,337.41 at 8%; difference $1,502.87.)
- A hypothetical obligation of $15,000 is due in 1 year. Compare its present value at 0% and 5%, and state the direction of change as the rate rises. (Answer: $15,000.00 at 0%; $14,285.71 at 5%; present value decreases.)
- A $5,000 deposit is invested for 3 years at either 1% or 4%. Find both future values and the percentage increase from the lower-rate value. (Answer: $5,151.51 at 1%; $5,624.32 at 4%; increase $472.82, or 9.18%.)