The Return and Risk of a Financial Portfolio

Investors hold portfolios, not single assets, and a portfolio's risk depends as much on how its holdings move together as on how volatile each one is. This module shows how to compute a portfolio's expected return and risk, why diversification works, how the minimum-variance and efficient frontiers are formed, and how risk aversion, the risk-free asset and the market portfolio lead to the capital allocation line and the capital market line.

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What you will learn

1. Calculate and interpret the expected return, variance and standard deviation of a portfolio, and explain how covariance and correlation determine the benefits of diversification.

Portfolio expected return = weighted average of the assets' expected returns.
Two-asset variance = w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 rho s1 s2.
If rho < +1, portfolio standard deviation is below the weighted average of the standard deviations.
With many assets, portfolio variance approaches the average covariance; specific risk diversifies away, systematic risk does not.
Correlations tend to rise in crises, reducing diversification benefits.

2. Describe and calculate the minimum-variance portfolio of two assets and explain the minimum-variance frontier and the efficient frontier.

Combinations of imperfectly correlated assets can have less risk than either asset alone.
Two-asset minimum-variance weight: w1 = (s2^2 - Cov) / (s1^2 + s2^2 - 2Cov).
Minimum-variance frontier: the lowest-risk portfolio for each expected return.
Efficient frontier: the part of the frontier above the global minimum-variance portfolio.
Optimizer results are very sensitive to estimation errors in the inputs.

3. Explain how risk aversion and utility determine an investor's optimal portfolio, and calculate the optimal allocation between a risky portfolio and the risk-free asset along the capital allocation line.

Utility U = E(R) - 0.5 A sigma^2, with returns and variance in decimals.
A > 0 risk-averse, A = 0 risk-neutral, A < 0 risk-seeking; higher A means steeper indifference curves.
The CAL combines the risk-free asset with a risky portfolio; its slope is the Sharpe ratio.
The optimal risky portfolio is the tangency portfolio with the highest Sharpe ratio.
Optimal weight in the risky portfolio y* = [E(R_P) - R_f] / (A sigma_P^2); y* > 1 means borrowing.

4. Explain how the capital allocation line extends to the market portfolio and the capital market line, and calculate the expected return and risk of portfolios on the capital market line.

With homogeneous expectations, the common tangency portfolio is the market portfolio.
CML: E(R_P) = R_f + [(E(R_M) - R_f) / sigma_M] sigma_P; its slope is the market Sharpe ratio.
Two-fund separation: all investors hold the risk-free asset and the market portfolio in different proportions.
Lending portfolios lie left of M; borrowing (leveraged) portfolios lie right of M.
A higher borrowing rate kinks the CML; the CML underpins passive market-cap index investing.

Module at a glance

Portfolio expected return is a weighted average; portfolio standard deviation is below the weighted average unless rho = +1.
Two-asset variance: w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 rho s1 s2.
Minimum-variance weight: w1 = (s2^2 - Cov) / (s1^2 + s2^2 - 2Cov).
The efficient frontier is the part of the minimum-variance frontier above the global minimum-variance portfolio.
Utility U = E(R) - 0.5 A sigma^2; optimal risky share y* = [E(R_P) - R_f] / (A sigma_P^2).
The CAL's slope is the Sharpe ratio; the best risky portfolio is the tangency portfolio.
CML: E(R_P) = R_f + [(E(R_M) - R_f) / sigma_M] sigma_P; all investors mix the market portfolio with lending or borrowing.

This free module has 4 lessons, 12 practice questions with full explanations.

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