Returns & Volatility

Measure what you expect to earn and how much it swings

Finance FundamentalsRisk & ReturnFree preview

⏱️ About 15 min

A stock might return +30% in a boom, +10% in a normal year, or -20% in a recession. What return should you expect, and how risky is that?

💡

The big idea: The expected return is the probability-weighted average of the possible outcomes; volatility - the standard deviation of those outcomes - measures how far returns typically stray from that average. Together they summarise reward and risk in two numbers.

🎯 By the end, you'll be able to

Compute an expected return as a probability-weighted average.
Compute variance and standard deviation (volatility) of returns.
Explain why two assets with the same mean can differ sharply in risk.

📎 Helpful to know first

Comfort with percentages and simple averages.

Return is what you earn; risk is how much it varies

A future return is uncertain: it depends on which state of the world arrives. The expected return summarises the outcomes into one number by weighting each possible return by its probability. But the average alone hides the risk - a steady 10% and a coin-flip between +40% and -20% can share the same mean.

\[ E[R] = \sum_{i} p_{i} \, R_{i} \]

Measuring the spread

$R_{i}$ is the return in state $i$ and $p_{i}$ its probability. To measure risk we look at how far each outcome sits from the mean. Variance is the probability-weighted average of those squared gaps; its square root, the standard deviation, is the volatility, back in the same units as returns.

\[ \sigma^{2} = \sum_{i} p_{i} \,(R_{i} - E[R])^{2}, \qquad \sigma = \sqrt{\sigma^{2}} \]

⚠️ Why square the gaps

Squaring stops positive and negative deviations from cancelling and punishes large misses more than small ones. Taking the square root at the end returns the measure to percentage points, so a volatility of 17% is directly comparable to a return of 10%.

🎮 Expected Return & Volatility Explorer LIVE

Predict first: Predict first: if you raise the boom return but lower the recession return by the same amount, does the volatility rise or fall?

Scenario returns weighted by probability; the widget reports the expected return and the volatility (standard deviation).

📝 Worked example: A stock returns +30% in a boom (probability 0.3), +10% in a normal year (0.5), and -20% in a recession (0.2). Find its expected return and volatility.

1. Expected return: $0.3(30) + 0.5(10) + 0.2(-20) = 9 + 5 - 4 = 10\%$.
2. Deviations from the mean: $+20$, $0$, $-30$ percentage points.
3. Weighted squared deviations: $0.3(20)^{2} + 0.5(0)^{2} + 0.2(-30)^{2} = 120 + 0 + 180 = 300$.
4. Variance $= 300$ (percent-squared).
5. Volatility $= \sqrt{300} = 17.32\%$.

✓ Expected return = 10%; volatility = 17.32%

✏️ Practice: A stock returns +25% with probability 0.6 and -15% with probability 0.4. What are its expected return and volatility?

💡 Hint

Mean $= 0.6(25) + 0.4(-15)$; variance $= 0.6(25-9)^{2} + 0.4(-15-9)^{2}$.

Answer

Expected return = 9%; volatility = 19.60%.

Check your understanding

1. The expected return of a risky asset is:

Expected return weights each possible outcome by its probability and sums them - a probability-weighted average.

2. Two assets have the same expected return but different volatilities. The riskier one has the:

Volatility is the standard deviation of returns; a larger standard deviation means outcomes stray further from the mean, which is more risk.

✅ Key takeaways

Expected return is the probability-weighted average of the outcomes.
Variance $= \sum p_{i}(R_{i}-E[R])^{2}$; volatility is its square root.
Two assets can share a mean yet differ greatly in risk.

➡️ One asset has a return and a volatility; the next lesson shows why holding two assets together can be less risky than either alone.

Want to test yourself on this? Try the Finance Fundamentals Aptitude test →