Portfolio Risk and Return: Part I

Every allocation decision trades expected return against risk, and the exam expects you to quantify that trade-off. This module reviews how the major asset classes have behaved, how risk aversion is captured in a utility function, how to measure return, risk and co-movement from historical data, why combining imperfectly correlated assets lowers risk, and how an investor arrives at an optimal portfolio on the efficient frontier and the capital allocation line.

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What you will learn

1. Describe the historical risk and return characteristics of the major asset classes and distinguish nominal returns, real returns and risk premiums.

Historically, the ranking of average return and of volatility is the same: bills, government bonds, corporate bonds, large-cap equities, small-cap equities.
Risk premium = expected return minus the risk-free rate; it is earned on average, not every year.
Real return = (1 + nominal return) / (1 + inflation) - 1; subtracting inflation is only an approximation.
Risky asset returns tend to be negatively skewed with fat tails, so standard deviation understates crash risk.
Liquidity, fees and taxes change the return an investor actually keeps.

2. Explain risk aversion, calculate an investor's utility for a risky portfolio, and interpret indifference curves.

Risk averse: same expected return, lower risk preferred; A > 0.
Risk neutral: A = 0, indifferent to risk; risk seeking: A < 0.
U = E(r) - 0.5 x A x variance, with returns and standard deviation in decimals.
Higher A means steeper indifference curves and a preference for less risky portfolios.
Investors prefer indifference curves further to the upper left (north-west).

3. Calculate and interpret the mean, variance, standard deviation, covariance and correlation of asset returns from historical data.

Sample variance and covariance divide by n - 1; the population formula divides by n.
Standard deviation is in the same units as returns and measures total risk.
Covariance = average cross-product of deviations; its sign shows direction of co-movement.
Correlation = covariance / (product of standard deviations), bounded by -1 and +1.
Historical estimates vary with the sample period; correlations are especially unstable.

4. Calculate the expected return and standard deviation of a portfolio and explain how combining assets that are less than perfectly correlated reduces portfolio risk.

Portfolio expected return is the weighted average of asset expected returns.
Portfolio standard deviation is below the weighted average of standard deviations whenever correlation < +1.
At correlation +1 there is no diversification benefit; at -1 risk can be eliminated.
With many equally weighted assets, portfolio variance falls towards the average covariance, not to zero.
Correlations among risky assets often rise in crises, shrinking diversification benefits.

5. Describe and interpret the minimum-variance frontier, the efficient frontier and the global minimum-variance portfolio of risky assets.

The minimum-variance frontier holds, for each expected return, the portfolio with the lowest variance.
The global minimum-variance portfolio is the left-most point of the frontier: lowest possible risk.
Only the part of the frontier above the GMV is efficient; portfolios below it are dominated.
Adding assets with low correlation shifts the efficient frontier up and to the left.
An investor's optimal risky portfolio lies where their indifference curve is tangent to the efficient frontier (before introducing a risk-free asset).

6. Explain how an investor selects an optimal portfolio by combining a risk-free asset with risky assets along a capital allocation line, given the investor's risk aversion.

Combining the risk-free asset with a risky portfolio produces a straight line, the CAL.
CAL slope = (E(RP) - Rf) / σP, the Sharpe ratio of the risky portfolio.
The optimal risky portfolio produces the steepest CAL, tangent to the efficient frontier.
Optimal weight in the risky portfolio y* = (E(RP) - Rf) / (A σP²).
Risk aversion determines the mix of risk-free and risky assets, not which risky portfolio is held.

Module at a glance

Higher-returning asset classes have historically been more volatile; risk premiums are long-run averages that vary widely.
Real return = (1 + nominal) / (1 + inflation) - 1.
Utility U = E(r) - 0.5Aσ²; higher A means steeper indifference curves and less risky choices.
Correlation = covariance / (σ1σ2); historical estimates depend on the sample period.
Portfolio risk falls below the weighted average of asset risks whenever correlation is below +1.
Only the frontier above the global minimum-variance portfolio is efficient.
The CAL's slope is the Sharpe ratio; the optimal risky weight is y* = (E(RP) - Rf) / (Aσ²).

This free module has 6 lessons, 18 practice questions with full explanations.

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