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Approximating Roots, Logs and Powers

Screens at research and trading firms often expect numerical answers without a calculator: an annualised volatility, a compounded growth factor, a probability that involves a large power, or a square root that falls out of a variance. Candidates who try to compute these exactly run out of time. Candidates who carry a few constants and expansions get within a percent in seconds, and say so. This lesson is that kit.

Constants worth memorising

Quantity Value Where it appears
ln⁡2\ln 2 0.693 Doubling and half-lives
ln⁡10\ln 10 2.303 Converting between log bases
log⁡102\log_{10} 2 0.301 Orders of magnitude, 210≈1032^{10} \approx 10^3
log⁡103\log_{10} 3 0.477 Orders of magnitude
252\sqrt{252} 15.87 Annualising daily volatility
1/2π1/\sqrt{2\pi} 0.399 The peak of the standard normal density
ee 2.718 Continuous compounding

Expansions for small x

For small xx:

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