Why Equal Nominal Totals Can Have Different Values

Why Equal Totals May Not Mean Equal Values

Two payment schedules can add to the same dollar amount and still represent different values today. The reason is timing: a dollar received sooner can be invested for longer, while a later dollar has less time to earn interest. To compare streams, place every payment on a common valuation date by discounting it at the rate appropriate to its timing.

This comparison depends on the cash-flow dates and the discount rate, not just the sum of the amounts. The effective annual rate $i$ is the interest earned over one year, with interest compounded annually. In the examples below, payments occur at year-end, the rate is constant, and the valuation date is time 0. These assumptions make each discount period one year.

Definition: For a stream with cash flow $C_t$ at the end of year $t$, its undiscounted total is $\sum_{t=1}^{n} C_t$. Its present value at time 0, using a constant effective annual rate $i$, is the sum of the cash flows discounted to time 0. Equal undiscounted totals do not, by themselves, imply equal present values.

Here, $C_t$ is the amount paid at the end of year $t$. It can be a receipt or a payment; when comparing schedules with positive cash flows, a larger present value means a more valuable stream to the recipient. In general, cash flows with different signs must be discounted with their signs intact. A conclusion about which schedule is better cannot be based on payment timing alone if some amounts are outflows.

Valuing a Dated Stream

With annual end-of-year payments and a constant effective annual rate, one dollar due at the end of year $t$ has present value $v^t$, where $v=1/(1+i)$. Multiplying by the payment amount gives its present value; adding the results for all dates gives the value of the stream.

\operatorname{PV}_0=\sum_{t=1}^{n} C_t v^t =\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}, \qquad v=\frac{1}{1+i}.

The exponent is the number of annual periods between the valuation date and the payment date. For example, a payment at the end of year 3 is discounted by $(1+i)^3$, not by $1+i$. If payments occur on dates that are not annual year-ends, the time measurement and compounding convention must match the stated rate. The formula above is specifically for annual periods under annual effective compounding.

Record each amount and date.
Write the cash flow at each time $t$, preserving whether it is a receipt or a payment.

Identify the valuation date and rate.
Confirm the rate is effective annual, and count the annual periods from each payment date back to the valuation date.

Discount and add.
Calculate $C_t/(1+i)^t$ separately for each payment, then sum those present values.

Compare like with like.
Compare present values at the same date and check separately whether the undiscounted totals are equal.

Why Earlier Payments Have Greater Present Value

Suppose a positive payment $C$ is moved from the end of year $t+1$ to the end of year $t$. For a positive effective annual rate, $v<1$. The change in present value is the present value at the earlier date minus the present value at the later date:

Cv^t-Cv^{t+1} =Cv^t(1-v) =Cv^t\frac{i}{1+i}>0 \qquad (C>0,\ i>0).

Thus, moving a positive payment one year earlier increases its present value when the annual effective rate is positive. This does not change the payment’s nominal amount or the stream’s undiscounted total. It changes only how long the amount is discounted. If $i=0$, then $v=1$ and timing has no effect on present value. If $-1<i<0$, then $v>1$, and a later positive payment has greater present value under this formula. The usual “earlier is more valuable” conclusion therefore relies on a positive rate.

This result applies directly when one stream can be obtained from another simply by moving positive payments earlier or later. When the streams have different patterns of early and late payments, there may be no general ranking: the discount rate can affect not only the size of the difference, but also which stream has the larger present value.

Worked Example: One Equal Payment at Two Different Dates

A hypothetical agreement offers either \$1,000 at the end of year 1 or \$1,000 at the end of year 3. Find the present value of each alternative and the difference at an effective annual rate of 10%. The undiscounted total is \$1,000 under either alternative.

Method. Discount each payment to time 0 using its own date. This avoids the mistake of applying only one year of discounting to both alternatives.

For the year-1 payment:

\operatorname{PV}_1 =\frac{\$1{,}000}{1.10} =\$909.090909\ldots =\$909.09.

For the year-3 payment:

\operatorname{PV}_3 =\frac{\$1{,}000}{(1.10)^3} =\frac{\$1{,}000}{1.331} =\$751.3148009\ldots =\$751.31.

Using the unrounded values, the difference is $\$909.090909\ldots-\$751.3148009\ldots=\$157.7761082\ldots$, or \$157.78 to the nearest cent. Both alternatives have the same nominal total, but the year-1 payment has the higher present value. The calculation confirms the one-year timing result: with a positive rate, the earlier positive payment is more valuable at time 0.

Equal Totals With Payments Around a Middle Date

A useful comparison is a stream split evenly between two dates and a second stream paid entirely at the midpoint. These streams have the same total, but the split stream can have a higher present value when the rate is positive. The calculation below shows why; it does not imply that every pair of schedules has the same ranking.

Worked Example: A Split Payment Versus a Middle-Date Payment

Stream A pays \$5,000 at the end of year 1 and \$5,000 at the end of year 3. Stream B pays \$10,000 at the end of year 2. Both have an undiscounted total of \$10,000. Compare their present values at an effective annual rate of 10%.

Method. Discount each payment to time 0, then compare the sums. The common mistake is to treat equal nominal totals as equal present values without accounting for their dates.

For Stream A, the year-1 present value is:

\frac{\$5{,}000}{1.10} =\$4{,}545.454545\ldots

Its year-3 present value is:

\frac{\$5{,}000}{1.331} =\$3{,}756.574003\ldots

Adding the unrounded amounts gives $\operatorname{PV}_A=\$8{,}302.028548\ldots=\$8{,}302.03$.

For Stream B:

\operatorname{PV}_B =\frac{\$10{,}000}{(1.10)^2} =\frac{\$10{,}000}{1.21} =\$8{,}264.462810\ldots =\$8{,}264.46.

The difference, calculated from unrounded values, is $\$8{,}302.028548\ldots-\$8{,}264.462810\ldots=\$37.565738\ldots$, or \$37.57 to the nearest cent. Stream A has the higher present value. Its first payment arrives before year 2, while its second arrives after year 2; the earlier payment’s gain in present value is larger than the later payment’s loss in this particular comparison.

The Rate Can Change the Size or Direction of a Comparison

For any one stream, a higher positive discount rate lowers the present value of a positive payment made in the future. It does not follow that the difference between two streams must change by the same amount or even keep the same sign. The timing patterns matter. A stream with payments spread far apart can behave differently from one concentrated in earlier years, and their relative values can change as the rate changes.

Worked Example: How a Higher Rate Changes a Comparison

Stream A pays \$1,000 at the end of year 1 and \$1,000 at the end of year 5. Stream B pays \$2,000 at the end of year 3. Each stream has an undiscounted total of \$2,000. Compare their present values at effective annual rates of 5% and 10%.

Method. Value both streams at each rate using the same year-end dates. This comparison shows how a rate change can alter the size of the difference, even when the payment schedules and nominal totals stay fixed.

At 5%, Stream A has present value:

\begin{aligned} \operatorname{PV}_A &=\frac{\$1{,}000}{1.05}+\frac{\$1{,}000}{(1.05)^5}\\ &=\$952.380952\ldots+\$783.526166\ldots\\ &=\$1{,}735.907119\ldots =\$1{,}735.91. \end{aligned}

At the same rate, Stream B has present value:

\operatorname{PV}_B =\frac{\$2{,}000}{(1.05)^3} =\frac{\$2{,}000}{1.157625} =\$1{,}727.675197\ldots =\$1{,}727.68.

Stream A is higher by $\$1{,}735.907119\ldots-\$1{,}727.675197\ldots=\$8.231922\ldots$, or \$8.23.

At 10%, Stream A is:

\operatorname{PV}_A =\frac{\$1{,}000}{1.10}+\frac{\$1{,}000}{(1.10)^5} =\$909.090909\ldots+\$620.921323\ldots =\$1{,}530.012232\ldots =\$1{,}530.01.

Stream B is:

\operatorname{PV}_B =\frac{\$2{,}000}{1.331} =\$1{,}502.629602\ldots =\$1{,}502.63.

At 10%, Stream A is higher by $\$1{,}530.012232\ldots-\$1{,}502.629602\ldots=\$27.382630\ldots$, or \$27.38. The ranking is unchanged in this example, but the difference grows from \$8.23 at 5% to \$27.38 at 10%.

A ranking can also reverse when the timing patterns differ. For example, suppose Stream A pays \$1,000 at the end of year 1 and \$1,000 at the end of year 6, while Stream B pays \$1,000 at the end of year 2 and \$1,000 at the end of year 3. At 5%, the present values are approximately \$1,698.60 for A and \$1,770.87 for B, so B is higher by \$72.27. At 100%, the values are \$515.63 for A and \$375.00 for B, so A is higher by \$140.63. The rate affects each dated payment by a different discount factor, which can change the ordering of two complete streams.

Common Mistake: Comparing Only the Nominal Totals

A common incorrect calculation is: “Stream A totals \$10,000 and Stream B totals \$10,000, so they have equal value.” The conclusion does not follow unless the valuation rate is zero or the timing patterns happen to produce equal present values. For instance, in the split-payment example, adding \$5,000 and \$5,000 gives the same nominal total as \$10,000, but discounting at 10% gives \$8,302.03 for Stream A and \$8,264.46 for Stream B.

The correction is to keep the nominal-total comparison and the present-value comparison separate. First sum the amounts without discounting if the question asks for the nominal total. Then discount every amount to the stated common date if it asks for value. Do not round each discounted payment prematurely; retain sufficient precision through the sum and round the final dollar result to cents.

Key takeaway: Equal undiscounted totals describe equal sums of cash amounts, not equal values at a common date. Under a positive effective annual rate, each positive payment’s present value depends on its date. Compare streams by discounting each cash flow at its own time, and do not assume the ranking of different timing patterns is independent of the rate.

Check Your Understanding

For each question, use the stated annual effective rate and value all payments at time 0. Round dollar answers to the nearest cent.

A hypothetical contract pays \$2,400 at the end of year 1 or \$2,400 at the end of year 2. At 8%, which date gives the higher present value, and by how much? (Answer: The year-1 payment, by \$164.61.)
Stream A pays \$3,000 at the end of year 1 and \$3,000 at the end of year 3. Stream B pays \$6,000 at the end of year 2. At 10%, which stream has the higher present value, and by how much? (Answer: Stream A, by \$22.54.)
Stream A pays \$6,000 at the end of year 1 and \$6,000 at the end of year 3; Stream B pays \$12,000 at the end of year 2. At 10%, compare their present values. (Answer: Stream A, by \$45.08.)
Stream A pays \$1,000 at the end of year 1 and \$1,000 at the end of year 6; Stream B pays \$1,000 at the end of year 2 and \$1,000 at the end of year 3. Which stream has the higher present value at 5%, and which at 100%? (Answer: Stream B at 5%, by \$72.27; Stream A at 100%, by \$140.63.)
Two streams each pay a total of \$9,000, but on different dates. At an effective annual rate of 0%, do they necessarily have equal present values under the year-end convention? Explain briefly. (Answer: Yes. At 0%, every discount factor is 1, so each present value equals its \$9,000 undiscounted total.)

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