Powers, Roots & Logarithms

Easy2–3 hrsPowersRootsLogarithmsRule of 72

Overview

Powers and logarithms appear constantly in trading — from compound returns to options (log-normal model) to vol scaling. The two key mental shortcuts: Rule of 72 for doubling times, and the √252 ≈ 16 scaling factor for converting daily vol to annual. These come up on every trading floor.

How to Recognize

→Compound interest, option pricing, doubling times
→'Approximately how long to double at X% per year?'
→Risk questions with exponential decay/growth

Step-by-Step Approach

1.Rule of 72: years to double ≈ 72/rate%. At 8%: 72/8 = 9 years
2.ln(1+x) ≈ x for small x (within 1%). So 5% annual growth: ln(1.05)≈0.05
3.Memorize: √2≈1.414, √3≈1.732, √5≈2.236, √10≈3.162
4.Powers of 2: 2¹⁰≈1000, 2²⁰≈10⁶, 2³⁰≈10⁹

Key Formulas

01Rule of 72:

T_{double} \approx 72/r\%

\ln(1+x) \approx x

for

x

small;

e^x \approx 1+x

for small

x

e \approx 2.718

e^{0.5} \approx 1.65

e^{0.1} \approx 1.105

04Annualizing:

\sigma_{annual} = \sigma_{daily} \times \sqrt{252} \approx \sigma_{daily} \times 16

Worked Examples

0Quick Example

Daily vol = 1%. Annual vol ≈ 1% × √252 ≈ 1% × 16 = 16%.

Problem

A stock has daily vol of 1.5%. What is the approximate annual volatility?

Solution

Annual vol = daily vol × √(trading days). Trading days ≈ 252.

√252 ≈ 15.87 ≈ 16. (Memorize this!)

Annual vol ≈ 1.5% × 16 = 24%.

Answer

Annual vol ≈ 24%. Quick formula: daily vol × 16 = annual vol.

Common Mistakes

!
Annualizing vol by multiplying by 252 instead of √252. Variance scales linearly, volatility scales with √time.
!
Rule of 72 is approximate — use for quick estimates, not precise calculations.

Practice Problems

Click "Show Answer" to reveal

Annualize a monthly vol of 3%.

Hint: Monthly to annual: × √12 ≈ × 3.46.

At 10% annual return, how long to 10× your money?

Hint: Use ln(10)/ln(1.1) or note: double every 7.2 years × 3.32 doublings.

Approximate √0.5.

Hint: √(1/2) = 1/√2 ≈ 1/1.414.

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