FINS5513 Chap.3 Markowitz Portfolio Optimisation
Markowitz Portfolio Optimisation
Week 3 converts return, variance and covariance inputs into an opportunity set. Diversification comes from combining assets whose returns are less than perfectly positively correlated. The minimum-variance frontier contains the smallest-risk portfolio available for each target expected return. Its leftmost point is the global minimum-variance portfolio.
The lower branch is dominated, while the upper branch is the efficient frontier. Introducing a risk-free asset creates straight capital allocation lines because the risk-free variance and covariance are zero. The line through the risky portfolio with the greatest slope identifies the tangency portfolio P star, which maximises the Sharpe ratio. That portfolio is not the same object as the global minimum-variance portfolio.
Portfolio construction chooses P star from market inputs; investor capital allocation then chooses the fraction y invested in P star from the risk premium, its variance and the investor's risk aversion. A weight below one means lending part of wealth at the risk-free rate, while a weight above one represents borrowing under the course assumptions.
What this chapter covers
- 01
Diversification benefit as correlation falls below plus one
- 02
Minimum-variance frontier for target expected returns
- 03
Global minimum-variance portfolio and the dominated lower branch
- 04
Efficient frontier as the upper portion of the risky opportunity set
- 05
Risk-free asset, complete portfolios and the capital allocation line
- 06
Tangency portfolio P star and the maximum Sharpe ratio
- 07
Separation of risky-fund construction from investor allocation
- 08
Optimal risky weight from risk premium, variance and risk aversion
Worked example · free
Investor allocation after the tangency fund is known
- +1The risky premium is 12% minus 3%, which equals 9%.
- +1The optimal risky weight is 0.09 divided by 5 times 0.15 squared, which equals 0.80.
- +1The risk-free weight is 0.20 and complete expected return is 3% plus 0.80 times 9%, which equals 10.20%.
- +1Complete standard deviation is 0.80 times 15%, which equals 12%. The investor lends rather than borrows.
Key terms
- Diversification Benefit
- The reduction in portfolio risk produced by imperfect correlation among component returns.
- Minimum-Variance Frontier
- Portfolios offering the lowest available variance for each target expected return.
- Global Minimum-Variance Portfolio
- The risky portfolio with the smallest standard deviation in the feasible set.
- Efficient Frontier
- The non-dominated upper branch of the minimum-variance frontier.
- Capital Allocation Line
- The risk-return line formed by combining a risk-free asset with one risky portfolio.
- Tangency Portfolio
- The risky portfolio whose capital allocation line has the maximum Sharpe-ratio slope.
Markowitz Portfolio Optimisation FAQ
What does lower correlation change?
Lower correlation bends the opportunity set left by reducing portfolio variance. It does not change the weighted expected return when asset returns and weights are held fixed.
Is the minimum-variance portfolio always optimal?
No. The global minimum-variance portfolio minimises risk, while the tangency portfolio maximises the Sharpe ratio. An investor's complete portfolio then combines the tangency portfolio with borrowing or lending.
What does a risky weight above one mean?
It means the investor borrows at the assumed risk-free rate and invests more than total wealth in the tangency portfolio. The interpretation relies on the course's frictionless borrowing assumption.
Why is the capital allocation line straight?
Expected return and standard deviation both scale linearly with the risky weight when the other asset is risk free. Eliminating the weight gives a line with Sharpe-ratio slope.
How does risk aversion affect portfolio choice?
Higher risk aversion reduces the optimal weight in the tangency portfolio because the investor applies a larger penalty to variance. It does not change the tangency portfolio under separation.
Exam move
Draw the sequence as four boxes: risky inputs, frontier, tangency portfolio and investor allocation. Label the global minimum-variance portfolio at the leftmost point and shade the lower branch as dominated. Add the risk-free intercept and draw the steepest feasible capital allocation line to locate P star. Then solve y from the risky premium, variance and risk aversion.
Practise verbal distinctions because the diagrams use similar points: GMVP minimises standard deviation, P star maximises Sharpe, and the complete portfolio combines P star with the risk-free asset. Use direction checks: lower correlation moves feasible portfolios left, higher risk aversion lowers y, and y greater than one implies a negative risk-free weight.
State the borrowing and homogeneous-input assumptions when interpreting leverage or separation.
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