The efficient frontier

1. The efficient frontier

Let's visualize what you have been doing

2. Changing target return

in a plot showing on the x-axis the portfolio volatility and on the y-axis the expected return.

3. Changing target return

By fixing the return target, you searched along the horizontal dashed line for the portfolio that has the minimum variance.

4. Changing target return

We can, of course, do this for other

5. Changing target return

return targets.

6. Changing target return

If we take a higher return target, the corresponding optimal portfolio

7. Changing target return

will have a higher volatility. If we take a lower return target, the optimal portfolio will have a lower volatility.

8. Changing target return

This corresponds to the classical risk/reward trade-off in finance: if you desire a higher expected return, you must accept taking greater risk. If we do the optimization for all possible return targets,

9. The efficient frontier

we obtain the so-called efficient frontier: it is the curve connecting the expected return/volatility couples of the mean-variance efficient portfolios. There are no portfolios possible above the frontier, and all portfolios below the frontier are dominated by the portfolios on the frontier: for the same level of volatility, they offer the highest possible expected return.

10. Minimum variance portfolio

Note that the efficient frontier

11. Minimum variance portfolio

starts at an expected return, which is higher than the risk-free rate. This is required since investing in a risky portfolio makes only sense if the increase in expected return compared to the risk-free rate is sufficiently high compared to the risk taken. The portfolio at which the efficient frontier starts is called the minimum variance portfolio.

12. Minimum variance portfolio

It is the portfolio that solves the problem of minimizing the variance, without any constraint on expected returns. All other portfolios on the efficient frontier have a higher volatility and a higher expected return.

13. Maximum Sharpe ratio portfolio

For each portfolio on the efficient frontier, we can evaluate the risk-return trade-off by computing the portfolio’s Sharpe ratio. It is the portfolio expected return in excess of the risk-free rate, divided by the portfolio volatility.

14. Maximum Sharpe ratio portfolio

Graphically, it corresponds to the slope of the line connecting the risk-free asset

15. Maximum Sharpe ratio portfolio

and the risky portfolio If we draw this line for

16. Maximum Sharpe ratio portfolio

each of the efficient portfolios, we obtain

17. Maximum Sharpe ratio portfolio

as a special case the portfolio on the frontier for which the line is tangent to the efficient frontier. This is called the tangency portfolio. Note that it is impossible to find a portfolio with a higher Sharpe ratio. This tangency portfolio is thus the maximum Sharpe ratio portfolio. It offers the highest possible reward in terms of excess return per unit of portfolio volatility.

18. Time for practice

In the exercises, you will learn how to construct the efficient frontier

19. Time for practice

by running a

20. Time for practice

for-loop

21. Time for practice

over the set of

22. Time for practice

possible return targets.

23. Let's practice!

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