Compounding and Discounting

If you value bonds or interest rate derivatives, you need to put rates on the same compounding convention before you discount future cash flows.

Simple interest and compound interest

Consider a hypothetical example in which 100 yen is deposited for three years at 2%. Simple interest is calculated on the principal only,¹ so each year's interest stays at 2 yen, which is 2% of 100 yen. After three years, you have 100 yen plus 2 yen of interest three times, or 106 yen.

Under compound interest, interest is added to the principal, and the next year's interest is charged on that total². In the second year, the 102 yen you have after one year earns 2.04 yen (2% of 102 yen) and grows to 104.04 yen. In the third year, 2% of 104.04 yen is added, giving 106.12 yen³.

Because interest also earns interest under compounding, at the same rate the gap from simple interest widens as the period gets longer¹. For a depositor, this means receiving more. For a borrower, the amount to repay grows by compounding in the same way².

Differences by compounding frequency

With semiannual compounding at 2%, interest of 1% is added every six months. The 100 yen becomes 101 yen after six months, and after one year 1% of 101 yen is added, giving 102.01 yen. In general, to compound several times a year, you divide the annual rate by the number of periods in a year and add that rate once per period⁴.

The result is 0.01 yen more than the 102 yen from annual compounding, because the 1 yen of interest added after six months itself earns interest over the second half of the year. So even at the same 2%, semiannual compounding makes the amount grow by 2.01% over one year. As you compound more often, the amount after one year rises slightly, as the following table shows⁵.

However many times you compound, the amount after one year does not grow without limit; it approaches a fixed value. Letting the number of periods grow without limit is called continuous compounding, and at 2% it gives about 102.020 yen after one year⁶.

Deep diveFormulas for the future amount by compounding frequency

With an annual rate $r$ compounded $m$ times a year, an amount $PV$ today grows to the following amount $FV$ after $n$ years. This is because interest at the rate $r/m$ is added each period, and this is repeated $m \times n$ times.

$$FV = PV \times \left(1 + \frac{r}{m}\right)^{m n}$$

As $m$ grows without limit, $(1 + r/m)^{m}$ approaches Euler's number $e$ (about 2.71828) raised to the power $r$. So with continuous compounding, the amount after $n$ years is:

$$FV = PV \times e^{r n}$$

Discounting reverses compounding

With semiannual compounding at 2%, dividing the 102.01 yen you have after one year by 1.01 twice brings you back to 100 yen today. Compounding multiplies an amount today by a growth factor to find a future amount; discounting divides a future amount by the same factor to find the amount today⁷.

The value today of money to be received or paid in the future (a cash flow) is called its present value. At 2% with annual compounding, the present value of 100 yen received in 10 years is 82.03 yen: 100 yen divided by 1.02 ten times. The present value of 1 yen in the future is called the discount factor, and in this example it is 0.8203. How to use discount factors to find present values is covered in Present Value and Discount Factors.

Deep diveFormulas for the discount factor by compounding frequency

Solving the future amount formula for the amount today gives the discount factor $DF$ for $n$ years with compounding $m$ times a year:⁷

$$DF = \frac{1}{\left(1 + \frac{r}{m}\right)^{m n}}$$

With continuous compounding, the discount factor is:

$$DF = e^{-r n}$$

Worked example: frequency differences over 10 years

At 2% over 10 years, the table compares, for each convention, what 100 yen today grows to in 10 years and the present value of 100 yen received in 10 years. Under simple interest, the present value is 100 yen divided by the 10-year growth factor of 1.2 (100 ÷ 1.2).

The more often interest is compounded, the larger the amount after 10 years, and the smaller the discount factor and present value⁸. Between annual and continuous compounding, the difference is 0.24 yen in the amount after 10 years and 0.16 yen in present value.

When the frequency gap widens

When the rate is positive, the gap between the future amounts under annual and continuous compounding widens as the rate rises and as the period lengthens. At 5%, the amount after 10 years is 162.89 yen with annual compounding and 164.87 yen with continuous compounding, a gap of 1.98 yen. That is more than eight times the gap of 0.24 yen at 2%. If the rate stays at 2% and the period is extended to 20 years, the amounts are 148.59 yen with annual compounding and 149.18 yen with continuous compounding, a gap of 0.59 yen.

At 5% over 10 years, the discount factor is 0.6139 with annual compounding, 0.6103 semiannually, 0.6084 quarterly, 0.6072 monthly, and 0.6065 with continuous compounding.

Converting rates with different conventions to compare them

The same "2%" gives different future amounts and discount factors under different compounding conventions. So rates quoted under different conventions can be compared once they are converted to the same convention. A semiannually compounded rate of 2% equals 2.01% with annual compounding and about 1.99% with continuous compounding⁹.

A 2006 paper in the US Federal Reserve Board's Finance and Economics Discussion Series says that continuous compounding may be the simplest way to express yields. It also explains that a widely used convention expresses yields with semiannual compounding (coupon-equivalent yields)⁹. In the data published with the paper, yields on bonds that pay no interest before maturity (zero-coupon bonds) are continuously compounded, and yields on bonds that pay interest are semiannually compounded¹⁰. Even a single source can list rates under different conventions side by side.

Deep diveConverting between semiannual and continuous compounding

A semiannually compounded rate $y_{ce}$ and the continuously compounded rate $y$ that gives the same discount factor are related as follows⁹. $\ln x$ is the function that gives the power to which $e$ must be raised to get $x$¹¹.

$$y = 2 \ln\left(1 + \frac{y_{ce}}{2}\right)$$

If $y_{ce} = 2\%$, then $y = 2 \ln(1.01)$, or about 1.990%.

Assumptions and limits

The worked examples use simple period lengths such as one year and six months, and apply the same rate to every period. In actual transactions, the length of each interest period is converted from days into years. How that is done is covered in Day Count Conventions.

When rates differ by period, the relationship between the rate from today to a future date (the spot rate) and the rate from one future date to another (the forward rate) is not covered on this page.

Notes

1. Japan Securities Dealers Association, "単利" (Simple interest), in "投資の時間 金融・証券用語集" (Time for investing: glossary of financial and securities terms), the "意味" (Meaning) and "解説" (Explanation) sections (in Japanese). ↩ ↩²

2. Japan Financial Literacy and Education Corporation, "複利：ふくり" (Compound interest), in "用語・金融商品解説" (Glossary of terms and financial products) (in Japanese). ↩ ↩²

3. The amount after three years is 106.1208 yen, rounded to two decimal places. ↩

4. Akira Yamazaki, "キャッシュフローと現在価値" (Cash flows and present value), lecture slides for "ファイナンス論I" (Finance Theory I), Faculty of Business Administration, Hosei University, the "付利期間" (Compounding period) section, slides 20–27 (in Japanese). ↩

5. Because the differences between frequencies are small, this table alone is rounded to three decimal places. Before rounding, the amounts are 102.01505 yen (quarterly), 102.01844 yen (monthly), and 102.02013 yen (continuous). ↩

6. Akira Yamazaki, "キャッシュフローと現在価値" (Cash flows and present value), "微小付利期間の将来価値" (Future value with very short compounding periods), "ネイピア数と連続複利" (Euler's number and continuous compounding), and "連続時間の将来価値と現在価値" (Future value and present value in continuous time), slides 31–33 (in Japanese). ↩

7. The same slides, "現在価値（２）" (Present value (2)), slide 12, and "n 年満期、1/m 年複利の割引因子" (Discount factor for n-year maturity with compounding every 1/m of a year), slide 37. The slides call the discount factor "割引因子". ↩ ↩²

8. Amounts and present values are rounded to two decimal places of a yen, and discount factors to four decimal places. Present values are calculated by multiplying the unrounded discount factor by 100 and then rounding. The values for annual, semiannual, quarterly, monthly, and continuous compounding match the discount factors and present values in the simulator (2%, 10 years). ↩

9. Refet S. Gürkaynak, Brian Sack, and Jonathan H. Wright, "The U.S. Treasury Yield Curve: 1961 to the Present" (Federal Reserve Board, Finance and Economics Discussion Series 2006-28, 2006), section 2.1, pages 2–3, equations (2)–(4). ↩ ↩² ↩³

10. The same paper, page 24. ↩

11. The same paper, page 2, equations (1) and (2). The paper gives one equation that finds the continuously compounded yield from the discount factor using ln, and the reverse equation. ↩