Future value and compounding

Why it matters

Today the U.S. Treasury quotes a yield on its 13-week bill, and Kitalpha records it on 13-Week Treasury Bill (coupon-equivalent) as a figure of 4.07% as of 2026-09-15. Suppose US$1,000 were placed at that rate and the interest were left in place year after year. Would the balance after two years be the starting amount plus twice the yearly interest, or something more? The difference between those two answers is the whole of this lesson, and it decides how every longer-dated figure on the site is read.

The concept

Interest is the price of time: the amount a borrower pays for the use of money over a period. A yield is that price expressed as a percentage per year, which is the convention every Treasury record on Kitalpha follows. Simple interest applies the rate to the original amount every period. If US$1,000 is lent at 5% under simple interest, each year adds the same US$50, and after two years the balance is US$1,100. The growth is a straight line.

Compounding changes one thing: at the end of each period the interest earned is added to the balance, and the next period’s interest is calculated on that larger balance. The rate does not change, but the base it is applied to does. In the same example, year one adds US$50 as before. Year two applies 5% to US$1,050, adding US$52.50 rather than US$50. The extra US$2.50 is interest earned on the first year’s interest. After two years the balance is US$1,102.50, and every later year adds a little more than the year before.

The arithmetic is compact. Write the yearly rate as a decimal, so 5% becomes 0.05. One year of growth multiplies the balance by 1.05. Two years multiplies by 1.05 twice, which is 1.05 squared, or about 1.1025. In general, a starting amount grows to the starting amount multiplied by (1 + y) raised to the power of the number of years, where y is the decimal rate. The exponent is what distinguishes compounding from a straight line: simple interest multiplies y by the number of years, compounding multiplies (1 + y) by itself that many times.

Two features follow, and both matter when reading data. First, the gap between compound and simple growth is not a fixed amount; it widens every year, because each year’s interest is earned on a base that has itself grown. Over short horizons the two lines are hard to tell apart, which is why the difference is easy to underestimate. Over long horizons the divergence dominates. Second, the compounding frequency matters as well as the rate. Quoted yields are annual, but a bill that matures in thirteen weeks is repaid long before a year has passed, so a quoted annual figure describes a rate, not the growth actually earned over the bill’s life. A later lesson takes up how quoted and effective rates differ; here the point is simply that the rate is per year and the calculation must respect the period.

The one visual for this lesson plots a US$1,000 balance for ten years at today’s quoted bill yield under both rules. Look for where the two lines separate: they begin together, part slowly, and the gap between them grows without limit.

Worked example

Take the record from the opening: the 13-week bill on 13-Week Treasury Bill (coupon-equivalent), quoted at 4.07% as of 2026-09-15. To see compounding, treat that annual rate as if it were earned for two full years on US$1,000, with the interest left in place. Every number 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. Year one multiplies US$1,000 by one plus that decimal. Year two multiplies the year-one balance, not the original US$1,000, by the same factor again. The final line shows the simple-interest balance for the same two years, which adds the yearly interest twice to the original amount; the difference between the two balances is the interest earned on year-one interest. That difference is small after two years and is the seed of everything the chart above shows over ten.

Faded example

Now a second record: the 26-week bill on 26-Week Treasury Bill (coupon-equivalent), quoted at 4.21%. The first two steps are done for you below. Complete the last one yourself, then reveal the answer and compare.

Retrieval check

Before each answer, mark how confident you are. Every option carries an explanation, so read the feedback for the options you rejected as well as the one you chose.

Your summary