Present value: what a future dollar is worth now

Why it matters

The U.S. Treasury borrows for two years by issuing a two-year note, and Kitalpha records the market’s yield for that maturity on 2-Year Treasury Par Yield as 4.67% as of 2026-09-15. Now suppose someone holds a promise of US$1,000 to be paid in exactly two years. Is that promise worth US$1,000 today? Most readers sense that the answer is “a little less”, but by how much, and why that amount rather than another? Working it out is the whole of this lesson, and it is the step that makes sums due on different dates comparable at all.

The concept

Present value is the amount today that stands in for a stated sum due on a stated future date. Discounting is the arithmetic that finds it: moving a future sum back to the present at a rate of interest, which in this use is called the discount rate. The idea rests on the previous lesson. Money in hand can be lent, and at a positive yield it grows, so an amount due later is matched by a smaller amount now, namely the one that grows to it by the due date. A dollar due in a year is worth less than a dollar today, and the discount rate says how much less.

The arithmetic is compounding run backwards. Under compounding, a starting amount multiplied by (1 + y), with y the annual rate as a decimal, gives the balance a year later; multiplied by (1 + y) squared, the balance two years later. Discounting divides instead. Take 5% and one year: US$1,000 due in a year is worth 1,000 ÷ 1.05, or US$952.38, today, because US$952.38 lent at 5% grows to US$1,000 in a year. For two years the divisor is 1.05 squared, or 1.1025, and the present value is US$907.03. The divisor is called the discount factor, and one more factor of (1 + y) is applied for every additional year of waiting.

Three wrong turns are common. The first is to skip the step altogether: to treat US$1,000 due in two years as US$1,000. The second is to subtract the interest instead of dividing: US$1,000 less 5% of US$1,000 for each of two years is US$900, which is too low, because the interest that closes the gap accrues on the smaller present amount, not on the full future sum. The third is simple-interest discounting: dividing by 1 + 0.10 rather than by 1.05 squared, which gives US$909.09 rather than US$907.03. The gap between those last two is small over two years and widens with every added year, exactly as the compounding gap did in the previous lesson.

Why does this matter beyond the arithmetic? Because amounts on different dates cannot be added or compared as they stand. US$1,000 today plus US$1,000 due in two years is not US$2,000 of value today; it is US$1,000 plus the present value of the later payment. Present value places every sum on one date, today, so that sums due at different times can be totalled, ranked or set against a price. This is the same idea a Treasury yield encodes: the yield for a given maturity is the rate at which the market discounts a promise from the Treasury of that length.

Which rate to use? For a sum due from the Treasury in two years, the natural discount rate is the two-year par yield: the annual yield, recorded daily, at which the market lends to the Treasury for that long. The chart below discounts US$1,000 at today’s 2-year yield for every horizon from zero to ten years. Look for the solid line falling away from the dashed line of the undiscounted amount: the gap at each year is the discount for that much waiting, and it keeps widening.

Worked example

Take the 2-year Treasury yield from the opening, 4.67% as of 2026-09-15, as the discount rate for US$1,000 due in two years. Every figure below comes from that record, and the steps are shown in full.

Read the steps in order. The quoted percentage becomes a decimal by dividing by 100. One plus that decimal, squared, is the factor that two years of compounding would multiply a balance by; here it is the divisor. US$1,000 divided by it is the present value: the amount that, lent today at the record’s yield with the interest left in place, grows to exactly US$1,000 on the due date. The next line shows the undiscounted US$1,000 for contrast, and the last line is the difference between them, the discount for two years of waiting. Notice that the discount is smaller than two years’ interest on US$1,000 itself, because the interest that closes the gap accrues on the smaller present amount, not on the full future sum. That is why subtracting the interest from US$1,000 gives the wrong answer.

Faded example

Now the second record: the 5-year Treasury yield, quoted at 4.83% as of 2026-09-15, as the discount rate for US$1,000 due in five years. The first two steps are done for you below. Complete the last one yourself, then reveal the answer and compare.

Retrieval check

Mark your confidence before each answer. Every option carries an explanation, so read the feedback for the options you rejected as well as the one you chose.

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