Module 5 — MPT framework, assumptions, investor risk attitudes, expected return & risk calculations, efficient frontier, optimization
Diversification — "don't put all your eggs in one basket" — was known long before 1952, but investors had no way to quantify its benefits. In 1952, Harry Markowitz published "Portfolio Selection" in The Journal of Finance (he won the Nobel Prize in Economics in 1990).
MPT provides a framework for constructing and selecting portfolios based on (a) expected performance of investments and (b) the risk appetite of the investor. Markowitz introduced covariance/correlation to quantify diversification and mathematically demonstrated that variance of returns is a meaningful measure of portfolio risk.
| Investor Type | Behaviour | Certainty Equivalent Rate (CER) |
|---|---|---|
| Risk Averse | Rejects a "fair game" (zero risk-premium prospect); invests in risk-free opportunities or those with positive expected risk premium. Greater the risk, greater the demanded premium. Makes a downward adjustment to utility for risk. | Highly risk-averse investor may set CER below the risk-free rate and reject the risky portfolio; less risk-averse investor sets a higher CER and may accept it. |
| Risk Neutral | Evaluates opportunities solely on expected return, with no regard to risk; the amount of risk is irrelevant; provides no penalty for risk. | CER = Expected rate of return on the risky portfolio. |
| Risk Seeking | Willing to engage in a "fair game"; makes an upward adjustment to utility (opposite of risk-averse investor). | Assigns CER above the certainty-equivalent a risk-averse investor would assign. |
| State | Probability | Stock A | Stock B |
|---|---|---|---|
| I — Boom | 0.3 | 15% | 25% |
| II — Normal | 0.5 | 10% | 20% |
| III — Recession | 0.2 | 2% | 1% |
RA = 0.3(15%) + 0.5(10%) + 0.2(2%) = 9.9%
RB = 0.3(25%) + 0.5(20%) + 0.2(1%) = 17.7%
Risk = variability in return. Standard deviation (square root of variance) is the most well-known risk measure — larger SD = greater dispersion = greater risk.
| Constituent | Weight (% of portfolio) | Expected Return | Weighted Return |
|---|---|---|---|
| A | 0.2 | 0.09 | 0.018 |
| B | 0.1 | 0.12 | 0.012 |
| C | 0.3 | 0.15 | 0.045 |
| D | 0.4 | 0.18 | 0.072 |
Expected Return of Portfolio = 0.018 + 0.012 + 0.045 + 0.072 = 0.147 (14.7%)
The first term is the weighted variance of investment 1, the second the weighted variance of investment 2, and the third the weighted covariance between 1 and 2 (covariance = correlation × SD1 × SD2).
| Correlation coefficient (A,B) | 0.5 |
| Expected return on A | 0.15 |
| Std. deviation of A | 0.05 |
| Expected return on B | 0.15 |
| Std. deviation of B | 0.05 |
| Weight of A & B | 0.50 each |
E(σ²port) = (0.50² × 0.05²) + (0.50² × 0.05²) + (2 × 0.50 × 0.50 × 0.5 × 0.05 × 0.05)
E(σ²port) = 0.001875 ⇒ E(σport) = √0.001875 = 0.0433 (4.33%)
A 3-security portfolio variance has six terms: 3 weighted variances + 3 weighted covariances (I&II, II&III, I&III).
Number of covariance terms = (50² − 50) / 2 = (2500 − 50)/2 = 1225 (in addition to the 50 individual variances).
If two securities are perfectly (positively) correlated, the risk-return opportunity set is a straight line connecting them — both portfolio return and SD are linear (weighted-average) combinations. There is no diversification benefit when correlation = +1.
As correlation falls below +1, the combination curve bows inward (to the left) — offering risk reduction through diversification, with the bowing becoming most pronounced as correlation approaches −1.
The Efficient Frontier is the set of optimal portfolios offering the highest expected return for a given level of risk, or the lowest risk for a given level of expected return. Plotting all feasible combinations of securities produces an "umbrella"-shaped curve — this curve is the Efficient Frontier.
An optimum portfolio combines investments with desirable risk-return characteristics for a given set of constraints. To use the MPT framework, the portfolio manager must estimate:
The accuracy of portfolio allocation outputs depends entirely on the accuracy of statistical inputs (returns, risk, correlations). The number of correlation estimates required grows rapidly — for a 50-security portfolio there are 1225 correlation estimates (using (n²−n)/2). The potential error from these estimations is called estimation risk.
For larger portfolios, a variance-covariance matrix must be calculated — typically using spreadsheets or financial modelling tools.
1. Stock A has expected returns of 15% (Boom, prob 0.3), 10% (Normal, prob 0.5) and 2% (Recession, prob 0.2). What is its expected rate of return?
2. In Markowitz's portfolio theory, what statistical measure did he use to quantify the benefit of diversification between two assets?
3. A risk-neutral investor's Certainty Equivalent Rate (CER) for a risky portfolio is:
4. For a portfolio of two securities with weight 0.5 each, both having standard deviation of 0.05 and correlation coefficient of 0.5, the portfolio variance E(σ²port) is approximately:
5. The Efficient Frontier represents: