The number e and continuous growth · RiskAtoms Academy

By the end you will be able to

1Explain e as the limit of (1 + 1/n)ⁿ as growth is split into ever more pieces.
2Use eˣ and ln as inverses that undo each other.
3Calculate continuous growth with e^(rt), and compare it with compounding in steps.
4Convert between a continuous rate and a yearly rate.

Why you need it

A bank pays 100% interest a year. Paid once, £1 becomes £2. Paid monthly, a little more. Paid every second, do you get rich without limit? No: you can never pass 2.71828. That number is e.

L1.3.05 Continuous compounding
Continuous compounding, e^(rt), and converting between continuous and yearly rates.

You will also meet e and ln in log returns and in many pricing models later on. M-A04 on logarithms comes first.

The idea

Splitting growth into smaller and smaller pieces

In 1683 the Swiss mathematician Jacob Bernoulli asked the bank question above. Take 100% growth over a year. Split it into 2 pieces of 50%, and £1 becomes 1.5 × 1.5 = £2.25. Split it into 12 pieces, and you multiply by (1 + 1/12) twelve times. More pieces always give a little more, but the extra shrinks each time:

Interest added	Pieces n	(1 + 1/n)ⁿ
Once a year	1	2.000000
Twice a year	2	2.250000
Quarterly	4	2.441406
Monthly	12	2.613035
Daily	365	2.714567
Hourly	8,760	2.718127
Every second	31,536,000	2.718282

The values creep up towards a ceiling they never pass. That ceiling is called e, about 2.71828. Like π, its decimals go on for ever. The letter was made popular by Leonhard Euler in the 18th century.

Think firstBefore you read on

With 10% growth a year instead of 100%, what would compounding every second come close to?
Is e a variable, like x, that changes from problem to problem?

The rules

e as a limit. New notation: $\lim$ , the limit.

eˣ and ln. A power with base e, and the log that undoes it.

Continuous growth. What happens to compounding when the pieces become infinitely small.

Converting rates. This extends what you already know, so here it is in a few lines. A continuous rate $r$ and a yearly rate $R$ are equivalent when one year of growth is the same:

e^{r} = 1 + R

Subtract 1 from both sides to get $R$ , or take ln of both sides to get $r$ :

R = e^{r} - 1 \qquad\qquad r = \ln(1 + R)

And doubling time is exact with continuous growth: $e^{rt} = 2$ gives $t = \ln 2 / r$ . At 7%, that is 9.90 years.

The rules on one card

e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828 \qquad e^x = \lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n

\ln(e^x) = x \qquad e^{\ln y} = y \qquad A = A_0\,e^{rt} \qquad R = e^r - 1 \qquad r = \ln(1 + R)

Try it

Drag the slider to split the year into more pieces. Watch the dot climb towards the dashed line, fast at first and then ever more slowly. Count how many decimal places match e.

Try itChasing e

Pieces per year, n1

Growth over the year, r

(1 + 1/1)¹ =

2.00000000

e2.71828183

Still to go0.71828183

Decimal places that match0

(1 + r/n)ⁿthe limit, eʳ

Across: the number of pieces, each mark ten times the last. Up: what 1 grows to in a year.

Worked examples

1. 1,000 grows at 5% a year for 3 years

Step 1 of 3

1
Continuous growth
$1{,}000 \times e^{0.05 \times 3} = 1{,}000 \times e^{0.15} = 1{,}000 \times 1.161834 = 1{,}161.83$

2. A continuous rate of 4%, as a yearly rate

Step 1 of 2

1
One year of continuous growth
$e^{0.04} = 1.040811$

3. A yearly rate of 6%, as a continuous rate

Step 1 of 2

1
Take ln of the growth factor
$r = \ln(1.06) = 0.058269$ , so 5.8269%, compounded continuously.

Common mistakes

Each of these is tempting. Decide what is wrong before you reveal the fix.

“Compound infinitely often and the money grows without limit.”
“e^(a + b) = eᵃ + eᵇ.”
“5% for 3 years: e^(5 × 3).”
“A 6% continuous rate and a 6% yearly rate are the same.”

Reaction

Check what stuck

Question 1 · objective 1

Split 100% growth into 4 equal pieces. What is (1 + 1/4)⁴, to four decimal places?

Question 2 · objective 4

A rate of 10% a year is compounded continuously. What is the equivalent yearly rate, in %, to two decimals?

%

Question 3 · objective 2

What is ln(e⁵)?

A5Be⁵C5eDln 5

Question 4 · objective 3

2,000 grows continuously at 3% a year for 10 years. What is it worth at the end, to two decimals?

Question 5 · objective 4

What continuous rate grows money as fast as 8% a year compounded yearly? Answer in %, two decimals.

%

Glossary

e: A constant, about 2.71828, that (1 + 1/n)ⁿ gets closer and closer to as n grows. The base of natural growth.
Limit: The value an expression gets as close as you like to, as something (here n) grows without end.
eˣ (exp): e raised to the power x. Often written exp(x) on calculators and in spreadsheets.
Natural log (ln): The natural log: the power you raise e to, to get a number. It undoes eˣ.
Continuous growth: Growth added at every instant rather than in steps: A = A₀ e^(rt).
Equivalent rates: Two rates, quoted differently, that grow money by the same amount over a year.