BFF5520 Chap.2 Risk and Portfolio Theory
Risk and Portfolio Theory
Portfolio theory measures expected return as a weighted average but treats risk through variances and covariances. Diversification is valuable when assets do not move perfectly together. The feasible opportunity set shows every attainable combination, while the efficient frontier retains portfolios that cannot be improved in expected return without accepting more risk.
Adding a risk-free asset creates capital allocation lines, and tangency selects the risky portfolio with the greatest Sharpe ratio. Risk aversion then determines how much an investor lends to or borrows from the risk-free position.
What this chapter covers
- 01
State-contingent expected return
- 02
Variance and standard deviation
- 03
Covariance and correlation
- 04
Two-asset portfolio risk
- 05
Diversification and common exposure
- 06
Efficient and dominated portfolios
- 07
Capital allocation line
- 08
Tangency portfolio and Sharpe ratio
Worked example · free
Scale a risky fund with a safe asset
- +1Find the risky premium: 12% - 3% = 9%.
- +2Expected return is 3% + 0.70(9%) = 9.3%.
- +1Standard deviation is 0.70(15%) = 10.5% because the safe asset carries no volatility.
- +1Sharpe ratio is (9.3% - 3%)/10.5% = 0.60, matching the risky fund's 9%/15%.
- +1Interpret the remaining 30% as lending that scales down both excess return and risk along the same line.
Key terms
- Expected return
- Expected return is the probability-weighted average of possible returns or the weighted average of component expected returns in a portfolio.
- Covariance
- Covariance measures whether two asset returns tend to deviate from their means in the same or opposite directions.
- Correlation
- Correlation rescales covariance between minus one and plus one so co-movement can be compared across pairs.
- Efficient frontier
- The efficient frontier contains portfolios offering the highest expected return for each risk level or lowest risk for each expected return.
- Capital allocation line
- A capital allocation line contains combinations of a selected risky portfolio and a risk-free asset.
- Sharpe ratio
- The Sharpe ratio is expected excess return divided by standard deviation and equals the slope of a capital allocation line.
Risk and Portfolio Theory FAQ
Why is portfolio risk not a weighted average of asset risk?
Portfolio variance contains covariance terms as well as weighted standalone variances. Those cross terms capture how holdings move together. Low or negative co-movement can make total risk much lower than a simple average of individual standard deviations.
What makes a portfolio efficient?
An efficient portfolio is not dominated by another feasible portfolio. No alternative offers more expected return at the same risk or less risk at the same expected return. The lower branch of the risky opportunity set is dominated.
How does correlation affect diversification?
Lower correlation reduces the covariance contribution to portfolio variance when weights and individual volatilities stay fixed. Expected return does not change because correlation is absent from its weighted-average formula. Diversification improves without changing the component return forecasts.
What does a risky weight above one mean?
A risky weight above one means the investor borrows at the risk-free rate and invests more than initial wealth in the risky portfolio. Expected excess return and volatility are both magnified, so losses are leveraged as well as gains.
Why can investors share one tangency portfolio?
Under the model's common beliefs and borrowing assumptions, tangency identifies the risky portfolio with the greatest Sharpe ratio. Risk tolerance then changes the mix between that portfolio and the risk-free asset rather than changing the risky portfolio itself.
Exam move
Master three layers in order. First compute expected return, variance, covariance and correlation with consistent periods. Second explain diversification through co-movement rather than the number of holdings. Third distinguish the efficient frontier of risky assets from the capital allocation line created with the risk-free asset.
Recompute one allocation with a weight below one, one at one and one above one, and say what lending or borrowing means economically. On every covariance question, predict the direction before calculating: lower correlation should lower portfolio risk while leaving expected return unchanged.
Construct a small state table with probabilities and two asset returns, calculate each expected return, and keep the deviations visible until covariance is complete. This exposes whether a negative product belongs in the sum and prevents correlation from being confused with covariance. Repeat the table after changing only one probability so you can explain why every moment must be recomputed.
Sketch feasible portfolios before searching for a minimum: mark dominated points, the minimum-variance point, the efficient branch and the tangency point as distinct objects. Next create three complete portfolios using the same risky fund, including a leveraged position, and reconcile their weights to one. Audit answers with boundary cases.
A risky weight of zero should reproduce the risk-free return and zero volatility; a risky weight of one should reproduce the fund; correlation of one should remove the curved diversification benefit. End with an economic sentence about consumption of risk capacity, because a correct standard deviation without an allocation interpretation is incomplete.
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