Track 15 · Applied Statistics for Professionals · Lesson 3 of 12

Returns, risk and volatility

Describe the distribution of investment returns, measure and annualise volatility, and estimate a historical Value-at-Risk while stating what it cannot tell you.

35 minAdvanced8 graded exercisesMembers

A risk committee reviewing three funds asks “how risky is each one, and how bad can a normal bad day get?” The standard answers — volatility and Value-at-Risk — are easy to compute and easy to misread: returns have fat tails, quiet and turbulent periods cluster, and a VaR figure is not a worst case.

Analysts work with returns, not prices. A simple daily return is the percentage change in price from one day's close to the next. Returns are comparable across assets of very different prices, and their spread is what we call risk.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Compute simple returns and compound them over several days
Measure volatility and annualise it with √250, stating the independence assumption
Estimate historical and normal one-day VaR and explain why they differ
Recognise fat tails and volatility clustering, and report VaR as a threshold, not a worst case

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Formula sheet

Simple returns and compounding
$r_t = \frac{P_t - P_{t-1}}{P_{t-1}} \times 100\%, \qquad 1 + R_{\text{multi-day}} = \prod_{t} \left(1 + \frac{r_t}{100}\right)$
Square-root-of-time annualisation (convention used in this course)
$\sigma_{\text{annual}} \approx \sigma_{\text{daily}} \times \sqrt{250}$
Historical and normal one-day VaR, reported as a positive loss
$\text{VaR}_{95\%}^{\text{hist}} = -Q_{0.05}(r), \qquad \text{VaR}_{95\%}^{\text{normal}} = -(\bar{r} + z_{0.05}\, s) = -(\bar{r} - 1.645\, s)$

Reference

Weighted and geometric means
A weighted mean gives some values more influence than others; a geometric mean averages multiplicative change such as growth rates.
Coefficient of variation
The coefficient of variation is the standard deviation divided by the mean — spread relative to size, without units.
Skewness and excess kurtosis
Skewness measures asymmetry and excess kurtosis measures tail heaviness relative to the normal distribution (normal = 0 for both).

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.