Continuous compounding · RiskAtoms Academy

Know these first

L1.3.02 Simple and compound interest
L1.3.03 Present value and discounting

Math Atoms

∑ The number e and continuous growth

By the end you will be able to

1Convert between discrete rates (compounded m times a year) and continuous rates.
2Explain continuous compounding as the limit of compounding more and more often, and where e comes from.
3Grow and discount money with continuous rates, using e^(r_c T) and e^(−r_c T).
4Explain why continuous rates and log returns are convenient: they add up over time.

Why it matters

In 1683 the mathematician Jacob Bernoulli asked a banker's question. If a bank paid 100% a year, and added the interest more and more often, how rich could you get? The answer is not “infinitely rich”. It is 2.71828…, a number now called e.

Core explanation

Compounding without limit

In L1.3.02 we saw that adding interest more often helps, but each step helps less: 6% monthly beats 6% quarterly by only a little. Push that idea to its end. Add interest every day, every hour, every second, every instant. That is continuous compounding.

Bernoulli's version makes the pattern easy to see. Start with 1 at 100% a year:

Interest added	m	1 grows to
Annually	1	2.000000
Semi-annually	2	2.250000
Quarterly	4	2.441406
Monthly	12	2.613035
Weekly	52	2.692597
Daily	365	2.714567
Hourly	8,760	2.718127
Every minute	525,600	2.718279

The growth keeps rising but slows to a crawl, closing in on a ceiling of 2.718281…. That ceiling is the number e. At any rate r, compounding without limit for one year multiplies money by $e^{r}$ . At 5%, that is 1.051271, an effective annual rate of 5.127%.

Why would anyone use it? No bank account literally adds interest every instant. But continuous rates are the standard language of derivatives pricing and risk models, for two reasons. The formulas are cleaner: growth is a single $e^{rT}$ , with no m. And continuous rates add up. Grow at 3% for a year and then 4% for a year, and the total is exactly $e^{0.03 + 0.04} = e^{0.07}$ . Discrete rates do not add like that.

The same idea applies to investment returns. The continuously compounded return, or log return, of a price moving from 100 to 110 is ln(1.10) = 9.53%. If the price then falls to 99, the log return is -10.54%. Add them and you get -1.01%, exactly the log return from 100 to 99. The simple returns, +10% and −10%, add to zero, which is wrong: the investor lost 1%.

Because each convention gives a different number for the same growth, you will often need to convert. Two rates are equivalent if they turn 1 into the same amount over a year.

The formal version

Two new tools appear here: the number $e$ and its partner, the natural logarithm ln. The Math Atom on e and continuous growth covers both. We write $r_c$ for a continuous rate and $r_m$ for a rate compounded $m$ times a year.

Why continuous rates add. One line of exponent rules:

e^{r_1} \times e^{r_2} = e^{r_1 + r_2}

So growing at $r_1$ then at $r_2$ is the same as growing once at their sum. For log returns, $\ln(P_2/P_0) = \ln(P_1/P_0) + \ln(P_2/P_1)$ .

The formulas, for reference

\mathrm{FV}_T = \mathrm{PV}\,e^{r_c T} \qquad \mathrm{PV} = \mathrm{FV}_T\,e^{-r_c T}

r_c = m \ln\Big(1 + \frac{r_m}{m}\Big) \qquad r_m = m\big(e^{r_c/m} - 1\big)

With $m = 1$ : $r_c = \ln(1 + r_1)$ and $r_1 = e^{r_c} - 1$ .

Try it

Start with Bernoulli's 100% and slide the frequency from annual to every minute: the dot climbs towards the dashed ceiling at e. Then set the rate to 5% and read the table: the same rate in every convention.

Try itThe compounding race

Open the rate converter →

Nominal annual rate100%

Interest addedAnnually (m = 1)

1 grows in a year to

2.000000

Limit e^1.002.718282

Still to gain0.718282

Growth of 1 in a year, by frequency (1 + r/m)^m e^r, the ceiling

Horizontal axis: m, the number of times interest is added a year.

100% annually is the same as…

Compounded annually	100.000%
Compounded semi-annually	82.843%
Compounded quarterly	75.683%
Compounded monthly	71.356%
Continuous	69.315%

Continuous rate: 69.315%. Check it with the exponents and logarithms calculator.

Worked example

A bond rate into a model rate

A risk analyst in Singapore has a US Treasury yield of 5% a year, compounded semi-annually (the US Treasury convention). Her pricing model needs continuous rates. Try each step before revealing it.

Converting and using a continuous rate

Step 1 of 5

1
Set up the conversion
$r_c = 2 \ln\big(1 + 0.05/2\big) = 2 \ln(1.025)$

Regulatory lens · looking ahead

No rules apply yet. Market risk models, such as value at risk in later levels, are usually built on log returns, and derivative pricing (Black–Scholes) uses continuous rates throughout.

Sparks

SparkA number found by asking about interest

The number e was not found by geometers. It turned up in Jacob Bernoulli's 1683 study of compound interest, as the limit of (1 + 1/m)^m. Leonhard Euler later gave it the letter e and showed how often it appears across mathematics.

SparkBlack–Scholes discounts with e

The Black–Scholes option formula, published in 1973, values the strike price paid at expiry as

K e^{-rT}

: the continuous discount factor from this lesson. When you meet it later in the course, that part will already be familiar.

Key takeaways

Continuous compounding is the limit of adding interest more and more often; 1 at 100% for a year becomes e ≈ 2.71828.
Grow with FV = PV e^(r_c T) and discount with PV = FV e^(−r_c T).
Convert with r_c = m ln(1 + r_m/m) and back with r_m = m(e^(r_c/m) − 1).
A continuous rate is always a little lower than the discrete rate that gives the same growth.
Continuous rates and log returns add up over time, which is why risk and pricing models use them.

Reaction

Check what stuck

Question 1 · objective 3

£1,000 grows at a continuously compounded rate of 6% a year for 5 years. What is it worth at the end, in £?

Question 2 · objective 1

What continuously compounded rate is equivalent to 8% a year compounded annually? Answer in %, to three decimals.

Question 3 · objective 1

A continuously compounded rate is 5%. What is the equivalent annually compounded rate, in %, to three decimals?

Question 4 · objective 2

Which rate grows money fastest over a year?

A6% continuously compoundedB6.1% compounded semi-annuallyC6.15% compounded annually

Question 5 · objective 3

What is €500 due in 2 years worth today at a continuously compounded rate of 3%, in €?

Glossary

Continuous compounding: Interest added at every instant, the limit of compounding more and more often. Growth over T years is e^(r_c T).
e: The number 2.71828…, the limit of (1 + 1/m)^m as m grows without end.
Natural logarithm (ln): The natural logarithm: the power you must raise e to, to get a number. It undoes e^x.
Continuous rate (r_c): An annual rate compounded continuously.
Equivalent rates: Two rates are equivalent if they grow money by the same amount over the same time.
Log return: The continuously compounded return over a period: ln(end value ÷ start value). Log returns over successive periods add up.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives (Pearson), chapter 4, section on continuous compounding. Intermediate.
Eli Maor, e: The Story of a Number (Princeton University Press, 1994). Beginner, historical.

Origins

Hull, Options, Futures, and Other Derivatives (Pearson).
Maor, e: The Story of a Number (Princeton University Press, 1994).
Black and Scholes, “The pricing of options and corporate liabilities”, Journal of Political Economy 81(3), 1973.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.

Previous · L1.3.04

Annuities and perpetuities

Next · L1.3.06

Day counts and rate conventions