Investments
Risk and return
Expected return is the probability-weighted average outcome; risk is the dispersion around it.
Definition
Risk and return are two sides of one decision. Expected return is the probability-weighted average of possible returns. Risk is usually measured as the standard deviation of those returns: the wider the spread, the greater the uncertainty.
Because risk can be diversified, only the systematic part is rewarded with a premium. Risk-adjusted measures such as the Sharpe ratio compare excess return per unit of risk.
Intuition
You are not paid for taking risk. You are paid for taking risk you cannot remove by diversifying.
When to use it
- Comparing investments with different risk profiles.
- Building the cost of equity and portfolio decisions.
When not to use it
- When returns are highly non-normal and standard deviation understates tail risk.
Formula
Expected return
E(R) = Σ (pi × Ri)
Formula
Variance
Var = Σ pi × (Ri − E(R))²
Formula
Sharpe ratio
Sharpe = (E(R) − Rf) ÷ SD
Excess return earned per unit of total risk.
Worked example
A two-state payoff
A 50% chance of +20% and a 50% chance of 0%. The risk-free rate is 4%.
- 1
Expected return
0.5 × 20% + 0.5 × 0% = 10%.
- 2
Variance
0.5 × 10%² + 0.5 × (−10%)² = 0.01.
- 3
Standard deviation
Square root of 0.01 = 10%.
- 4
Sharpe
(10% − 4%) ÷ 10% = 0.6.
Expected return 10%, volatility 10%, Sharpe 0.6. It earns 0.6 units of excess return for each unit of risk.
Common mistakes
- Comparing raw returns without adjusting for risk.
- Confusing volatility (total risk) with beta (systematic risk).
- Using historical volatility as a guarantee of future volatility.
How it connects
relates to
Useful to study together, in either order.
is an advanced form of
The source extends the target once the basics are secure.
affects
Changing the source moves the target.