ECON90034 Chap.4 Mean-Variance Portfolio Theory
Mean-Variance Portfolio Theory
Portfolio theory describes every investment by its expected return and standard deviation. With two risky assets, the expected return of a mix is a weighted average, but its risk depends on the correlation between the assets, which decides whether the set of possible portfolios is a straight line, two segments or a curve. You find the minimum variance portfolio and the efficient frontier above it.
Adding a risk-free asset creates the capital market line, which touches the frontier at the market portfolio, so every investor holds the same risky mix and chooses only how much to lend or borrow. The review lecture points to the tutorial and Assignment 1 for the expected question style, with two investors placed on one line.
The chapter's example uses two independent assets and a risk-free rate, finds the minimum variance and market portfolios, and places one cautious and one bold investor on the line with dollar amounts.
What this chapter covers
- 01
Expected return and variance of a two-asset portfolio
- 02
Covariance and correlation, and the cases of plus one, minus one and zero
- 03
The minimum variance portfolio and the efficient frontier
- 04
Indifference curves of risk-averse, risk-neutral and risk-loving investors
- 05
The risk-free asset and the capital market line
- 06
The market portfolio and the market price of risk
- 07
Placing lenders and borrowers on the line with dollar amounts
Worked example · free
One portfolio, three correlations
- 1Expected return does not depend on correlation: 0.5 × 9 + 0.5 × 5 = 7%.
- 1Correlation +1: σ = 0.5 × 6 + 0.5 × 3 = 4.5%.
- 1Correlation 0: σ2 = 0.25 × 36 + 0.25 × 9 = 11.25, so σ = 3.354%.
- 1Correlation −1: σ = |0.5 × 6 − 0.5 × 3| = 1.5%.
Key terms
- Minimum variance portfolio
- The mix of the risky assets with the lowest variance; the efficient frontier starts there and runs upwards.
- Efficient frontier
- The upper part of the set of risky portfolios, where no other portfolio gives a higher expected return for the same standard deviation.
- Market portfolio
- The risky portfolio where the capital market line touches the efficient frontier; in equilibrium its weights match each asset's share of total market value.
- Market price of risk
- The gradient of the capital market line, the extra expected return the market pays for each extra point of standard deviation.
- Diversification
- Combining assets whose returns do not move perfectly together, so that the portfolio's standard deviation is below the weighted average of the assets' own.
Mean-Variance Portfolio Theory FAQ
Why is the lower part of the frontier never chosen?
Each portfolio on the lower branch has a partner directly above it with the same standard deviation and a higher expected return, so any investor who prefers more return at the same risk picks the upper one.
How do two investors on the same capital market line differ?
Both hold the market portfolio in the same proportions. The more cautious one also lends at the risk-free rate and sits left of the market portfolio; the bolder one borrows and invests more than their wealth in it.
What do indifference curves look like in a mean and standard deviation diagram?
Upward-sloping for a risk-averse investor, who needs more return to accept more risk; flat for a risk-neutral investor, who cares only about expected return; and downward-sloping for a risk lover, who enjoys risk.
Can a portfolio of two risky assets have zero risk?
Only with correlation of exactly minus one. Then a particular mix, with weights inversely proportional to the standard deviations, cancels all variation and sits on the vertical axis.
What is the market price of risk?
It is the gradient of the capital market line: the market portfolio's expected return minus the risk-free rate, divided by the market portfolio's standard deviation. It tells you how much extra expected return each extra point of risk earns along the line.
How do I find the market portfolio's weights?
Write the expected return of the risky mix as a function of the weight in the first asset, set it equal to the market portfolio's expected return given in the question, and solve. The second weight is one minus the first.
Exam move
Practise the full chain on fresh numbers until it is automatic: write return and variance as functions of one weight, differentiate for the minimum variance portfolio, recover market weights from the given market return, write the capital market line, then place each investor with a share in the market and dollar amounts.
Sketch the frontier, the line and both investors' points every time, and keep returns in percent throughout. Before each exam-style question, write the three formulas you will need, portfolio return, portfolio variance and the capital market line, so that every later step only substitutes numbers.
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