Every investment decision trades off return — what you hope to earn — against risk — the chance that the outcome differs from what you expect, including the chance of losing money. Investors generally demand a higher expected return for taking more risk. Measuring both, rather than looking at returns alone, is the foundation of portfolio management and of models such as modern portfolio theory and the CAPM.
What you will learn
- Calculate expected return from scenarios and from historical data
- Calculate variance and standard deviation as measures of risk
- Distinguish systematic and unsystematic risk and other types of risk
- Use risk-adjusted measures such as the Sharpe ratio
Measuring return
- Holding period return = (ending value − beginning value + income) ÷ beginning value.
- Expected return — the probability-weighted average of possible outcomes.
- Historical average return — the average of past returns, used as one input to expectations; compound growth over time is measured by CAGR.
Expected return and standard deviation
E(R) = Σ pᵢ × Rᵢ
Variance σ² = Σ pᵢ × (Rᵢ − E(R))², and Standard deviation σ = √σ²
where pᵢ is the probability of scenario i and Rᵢ the return in that scenario.
Worked example: two investments
Worked example: expected return and risk from scenarios
An analyst considers a share and a bond over the next year (illustrative figures):
| Scenario | Probability | Share return | Bond return |
|---|---|---|---|
| Economic boom | 30% | 25% | 8% |
| Normal | 50% | 10% | 7% |
| Recession | 20% | −15% | 5% |
Share:
- E(R) = 0.30 × 25 + 0.50 × 10 + 0.20 × (−15) = 7.5 + 5.0 − 3.0 = 9.5%.
- Variance = 0.30 × (25 − 9.5)² + 0.50 × (10 − 9.5)² + 0.20 × (−15 − 9.5)² = 72.08 + 0.13 + 120.05 = 192.25.
- Standard deviation = √192.25 = 13.87%.
Bond: E(R) = 6.9%; standard deviation = 1.04%.
The share offers 2.6 percentage points more expected return, but with far greater uncertainty, including a possible 15% loss.
Historical risk
With historical data, the standard deviation is calculated from the sample of past returns. For annual returns of 12%, −8%, 20%, 5% and 16%, the average is 9.0% and the sample standard deviation is 11.0% (dividing the sum of squared deviations by n − 1 = 4). If returns were roughly normally distributed, about two-thirds of years would fall within one standard deviation of the average (−2% to 20%) — though real returns often show more extreme outcomes than a normal distribution suggests.
The risk–return trade-off
Across asset classes, higher average returns have historically come with higher volatility: cash and short-term government bills at the low end, longer-term bonds in the middle and equities at the higher end. The extra return investors expect for bearing risk is the risk premium:
Risk premium = Expected return − Risk-free rate
This relationship is a tendency over long periods, not a guarantee in any particular year — high-risk investments can and do produce large losses.
Types of risk
| Risk | Description |
|---|---|
| Market (systematic) risk | Affects the whole market: economic cycles, interest rates, inflation — cannot be diversified away |
| Specific (unsystematic) risk | Affects one company or industry: management, competition, lawsuits — can be reduced by diversification |
| Interest rate risk | Bond prices fall when interest rates rise — see bond pricing |
| Inflation risk | Returns may not keep up with rising prices |
| Credit risk | A borrower may not pay interest or principal |
| Liquidity risk | An asset cannot be sold quickly at a fair price |
| Currency risk | Foreign investments change value when exchange rates move |
The split between systematic and unsystematic risk is central: because diversification removes specific risk at little cost, markets reward investors mainly for bearing systematic risk — the idea behind diversification and the CAPM’s beta.
Risk-adjusted returns
Comparing returns without considering risk can mislead. The Sharpe ratio measures excess return per unit of total risk:
Sharpe ratio
Sharpe ratio = (Portfolio return − Risk-free rate) ÷ Standard deviation
Using the scenario example with a risk-free rate of 4%: share (9.5 − 4) ÷ 13.87 = 0.40; bond (6.9 − 4) ÷ 1.04 = 2.78. The bond provides much more excess return per unit of risk on these assumptions, although an investor seeking higher absolute returns might still hold shares — preferably as part of a diversified portfolio, which can improve the risk–return trade-off.
Other measures include the Treynor ratio (excess return per unit of beta), maximum drawdown (the largest peak-to-trough fall) and value at risk.
Risk in personal decisions
Standard deviation is a statistical measure; for an individual investor, the practical risk is not having the money needed when it is needed. That depends on the time horizon: an investor needing cash in one year faces a much greater chance of a loss from shares than one investing for twenty years. Matching investments to goals is therefore as important as measuring volatility — see introduction to investment.
Common mistakes
- Comparing investments by return alone.
- Assuming that past volatility fully describes future risk.
- Ignoring risks that standard deviation misses, such as liquidity and credit risk.
- Believing high-risk investments guarantee high returns over any period.
- Holding concentrated positions and bearing unrewarded specific risk.
Key terms
- Expected return — the probability-weighted average of possible returns.
- Standard deviation — a measure of how widely returns vary around their average.
- Risk premium — the expected return above the risk-free rate.
- Systematic risk — market-wide risk that cannot be diversified away.
- Sharpe ratio — excess return divided by standard deviation.
Frequently asked questions
Is a higher standard deviation always bad?
It means more uncertainty, both up and down. It is bad if it exceeds what your goals and time horizon can bear.
What is the risk-free rate?
Usually the yield on short-term government securities, such as treasury bills, in the investor’s currency.
Can I eliminate investment risk?
You can reduce specific risk by diversifying, but market risk remains. Only accepting lower returns removes most risk.
Key takeaways
- Expected return is a probability-weighted average; standard deviation measures how far outcomes may vary.
- Higher expected returns generally require bearing more risk, but not every risk is rewarded.
- Diversification removes specific risk; systematic risk is what investors are paid for.
- Use risk-adjusted measures such as the Sharpe ratio and match risk to your time horizon.
Explore this in ARBIMETRA LAB
Apply these ideas to your own data. Files are analysed in your browser and are not uploaded.
- Correlation Analysis: Pearson, Spearman and Kendall correlation.
- GARCH(1,1): Conditional volatility estimation.
- Portfolio Return & Risk: Return, variance, covariance by weight.
- Sharpe Ratio: Excess return per unit of volatility.