In plain English
In finance the raw data is almost never the price — it is the return, the percentage change. Returns are comparable across assets and across time in a way prices are not. Once you have returns, the average tells you reward, the standard deviation tells you risk, and dividing one by the other tells you whether the reward was worth it.
The advanced view
Prices are non-stationary; simple returns are approximately stationary and log returns are additive over time, which is why continuously compounded returns r = ln(P₁/P₀) are the modelling standard. Under i.i.d. returns, variance scales linearly in time and volatility with the square root of time. Empirical returns violate i.i.d.: they exhibit excess kurtosis, volatility clustering and negative skew, so normal-based risk measures understate tail loss.
Three return definitions, and interviewers do notice the difference. The simple return is (P₁ − P₀ + income)/P₀. The arithmetic mean return averages those period returns and answers "what do I expect next period?". The geometric mean compounds them and answers "what did I actually earn over the whole period?". The geometric mean is always the lower of the two, and the gap widens with volatility — which is precisely why a volatile fund can post a positive average return and still leave you poorer.
Log returns exist to make time arithmetic easy. Because ln(P₂/P₀) = ln(P₂/P₁) + ln(P₁/P₀), log returns add up across periods while simple returns must be multiplied. For small moves the two are nearly identical (a 2% simple return is a 1.98% log return); for large moves they diverge badly, and only the log version keeps a price from going negative in a simulation.
Volatility is quoted annually, so you have to scale. Under independence, variances add across periods, which means volatility grows with the square root of time: multiply a daily standard deviation by √252, a monthly one by √12. This single move — the square-root-of-time rule — turns up in option pricing, value-at-risk and every risk report you will ever read.
Then the honest caveat. Real return distributions have fatter tails and more negative skew than the normal distribution, and their volatility clusters: calm weeks follow calm weeks, and crashes arrive in clusters. So a 95% value-at-risk computed from a normal assumption is a floor, not a bound, and it tells you nothing about how bad the bad day is. Expected shortfall — the average loss given that you breached VaR — answers that, and is the reason regulators moved toward it.
Return and risk formulas
Simple return R = (P₁ − P₀ + D) / P₀ Log return r = ln(P₁ / P₀) ≈ R for small R Geometric mean = [(1+R₁)(1+R₂)…(1+Rₙ)]^(1/n) − 1 Annualised vol σ_ann = σ_period × √(periods per year) daily × √252, weekly × √52, monthly × √12 Sharpe ratio = (R_p − R_f) / σ_p VaR (normal, 95%) = μ − 1.645σ (99%: μ − 2.326σ) Expected shortfall = average loss beyond the VaR threshold
Essential vocabulary
- Arithmetic vs. geometric mean
- Expected next-period return vs. realised compound return. Geometric ≤ arithmetic, and the gap ≈ σ²/2.
- Volatility
- Standard deviation of returns, annualised by convention. The market's shorthand for risk.
- Volatility clustering
- High-volatility periods follow high-volatility periods. The reason GARCH models exist.
- Fat tails (excess kurtosis)
- Extreme returns occur far more often than a normal distribution implies. Kurtosis above 3 is the marker.
- Value-at-risk (VaR)
- A loss threshold breached with a given probability over a given horizon. Says nothing about severity beyond it.
- Expected shortfall (CVaR)
- Average loss conditional on exceeding VaR. Coherent, and what VaR should have been.
- Sharpe ratio
- Excess return per unit of volatility. The standard reward-for-risk comparison across strategies.
Common pitfalls
- ×Averaging returns arithmetically and calling it performance. Volatility means the compounded outcome is lower.
- ×Annualising volatility by multiplying by 252 instead of √252. Off by a factor of about 16.
- ×Treating VaR as a worst case. It is the best of the bad days, not the worst.
- ×Using log returns to describe portfolio performance to a client. Report simple returns; model in logs.
- ×Assuming normality for anything tail-related, then being surprised by a 5-sigma day twice a decade.
- ×Comparing Sharpe ratios computed over different horizons or with different risk-free rates.
Worked example — from daily data to a risk number
Step 1 of 10
- 1Price moves from 100 to 102 in one day
Finance connection
This phase is the plumbing under portfolio theory, option pricing and risk management. σ from here is the σ in Black–Scholes, the σ_p in the Sharpe ratio, and the input to every capital-at-risk conversation.
Deeper
Deeper: arithmetic vs. geometric, and why volatility drags
Arithmetic mean return overstates what an investor actually earns when returns are volatile. Geometric (compound) return ≈ arithmetic − σ²/2. Up 50% then down 50% leaves you at 0.75 — an arithmetic mean of 0 and a geometric mean of −13.4%.
This volatility drag is why risk reduction adds return, not just comfort, and why leveraged products decay in choppy markets. When quoting performance, state which mean you used and over what compounding frequency.
Must know cold
- ✓Geometric ≈ arithmetic − σ²/2.
- ✓Annualised volatility = daily σ × √252.
- ✓Sharpe = (return − rf) ÷ σ; it scales with √t too.
- ✓Log returns add across time; simple returns add across assets.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
Daily volatility 1.1%. Annualise it, and compute the drag on a 10% arithmetic mean return.
Exercise 2
Fund A returns 12% with σ 20%; Fund B 8% with σ 10%. rf is 2%. Which is better, and what does leverage do?