Portfolio Optimization (Markowitz)


Mean-Variance Portfolio Optimization


Diversification is an investment strategy which reduces portfolio risk without necessarily reducing portfolio return.

  • It works because the expected return on a portfolio is the weighted-average of the expected returns of the assets in the portfolio, but the standard deviation of the portfolio is less than the weighted average of the individual standard deviations of the assets in the portfolio.

Diversification: The Math

Say we have two risky assets, A and B, in our portfolio. The expected return on our portfolio is:

`E(r_p) = w_AE(r_A) + w_BE(r_B)`

where `w`  denotes the weight of the asset in our portfolio. We see that the expected return on a portfolio is the weighted-average of the expected returns of the individual assets in the portfolio. However the variance of the portfolio is:

`\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\sigma_A\sigma_B\rho_{A,B}`

Importantly, the portfolio variance is a function of the correlation coefficient between the assets in the portfolio, but the expected return is not.

Now, assume that `\rho_{A,B} = 1`, then:

`\sigma_p^2 = (w_A\sigma_A + w_B\sigma_B)^2 \Rightarrow \sigma_p = w_A\sigma_A + w_B\sigma_B`

and the risk on a portfolio is the weighted-average of the risk of the individual assets in the portfolio.

  • However, in practice `\rho_{A,B} < 1` and so risk on a portfolio will be less than the weighted-average of the risk of the individual assets in the portfolio. This is the benefit of diversification. The ability to reduce risk (risk is decreasing with correlation) without necessarily reducing the expected return. The expected return is not a function of asset correlations.

Diversification: An Economic Argument

In economic terms, we can think of the risk on an assets as being decomposable into market and firm-specific induced risks.

  • Market risks (such as the risk of a sudden increase in interest rates) are common across all assets, and are not diversifiable.
  • Examples of firm-specific risks are a fire at a Ford factory or a drop in Facebook users.

As you add assets to your portfolio the good news from one company text to offset the bad news from another. Adding enough assets, eventually all the firm-specific risk is offset and you hold only market risk.

We are diversifying our portfolio if, given the portfolio has a set size, we split this amount across more and more assets.

  • Note, diversification doesn’t mean you add more money to your portfolio and invest it in a new asset.

Portfolio Frontier with Two Risky Assets and Varying Correlation

The following widget shows the efficient frontier for a portfolio of two risky assets. The first risky asset (call it ‘Debt’) has a 9% expected return and 11% standard deviation. The second portfolio (call it ‘Equity’) has a 12% expected return and a 20% standard deviation. You are free to change the correlation coefficient between Debt and Equity returns, and see the resulting effect on the efficient frontier.

What you should note, is that as you lower the correlation coefficient, you can receive the same expected return for less risk. That is, investors benefit form the lower correlation. If the correlation coefficient is -1, then you can construct a risk-free portfolio. See below for the calculation.

Correlations and Diversification

So we can see, holding expected returns constant, as we lower the correlation, we lower the portfolio risk without lowering the portfolio return.

  • Often portfolio managers will look for assets with low or negative correlations with the other assets in their portfolio as a way of limiting risk.

  • You can use the following app to investigate the correlation between assets.

Optimal Diversification and Markowitz

In 1952 Harry Markowitz published Portfolio Selection, which introduced the idea of diversifying optimally.

  • Fixing the portfolio expected return we find the weights on each asset in the portfolio such that risk (portfolio variance) is minimized.

  • Note, here we assume either the investor ignores portfolio skewness and kurtosis in their utility function, or returns are distributed according to an elliptical distribution (such as the normal distribution).

  • Doing this over all portfolio expected returns gives us a set of portfolios which have minimum risk. This portfolio is known as the mean-variance fontier.

Efficient Frontier

As you can see in the previous app the mean-variance frontier forms one side of a hyperbola.

  • The section of the frontier from the minimum variance portfolio upwards is known as the efficient frontier—investors would hold one of these portfolios.

  • The lower half offers the same portfolio variance offered on the efficient frontier, however with a lower expected return.

Optimal Portfolios

If we add a risk free asset we can find the unique optimal risky portfolio by choosing the risky portfolio which maximizes the slope of the line starting at the risk free rate and passing one of the set of risky portfolios (capital allocation line).

  • Of great importance is that the optimal portfolio is not a function of investor risk preferences. In fact, two investors who consider the same set of stocks, and have the same estimates of the stock return parameters (expected returns and variance/covariances) will have the same optimal portfolio.

  • The Capital Asset Pricing Model builds on Markowitz by assuming all investors hold the market portfolio.

The app on the next slide allows you to enter 5 stocks, and see the mean-variance frontier, and the optimal portfolio with the capital allocation line passing through it. We allow short selling stocks in the app, which benefits the frontier. By better we mean the frontier portfolios have higher return per unit risk.

Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

Learn more about how to contribute here.