Portfolio Construction 01/26/09

2 Portfolio Construction Where does portfolio construction fit in the portfolio management process? What are the foundations of Markowitz’s Mean- Variance Approach (Modern Portfolio Theory)? Two- asset to multiple asset portfolios. How do we construct optimal portfolios using Mean Variance Optimization? Microsoft Excel Solver.

3 Portfolio Construction How do we incorporate IPS requirements to determine asset class weights? What are the assumptions and limitations of the mean-variance approach? How do we reconcile portfolio construction in practice with Markowitz’s theory?

4 Portfolio Construction within the larger context of asset allocation IPS provides us with the risk tolerance and return expected by the client Capital Market Expectations provide us with an understanding of what the returns for each asset class will be

5 Portfolio Construction within the larger context of asset allocation C1: Capital Market Conditions I1: Investor’s Assets, Risk Attitudes C2: Prediction Procedure C3: Expected Ret, Risks, Correlations I2: Investor’s Risk Tolerance Function I3: Investor’s Risk Tolerance M1: Optimizer M2: Investor’s Asset Mix M3: Returns

6 Portfolio Construction within the larger context of asset allocation Optimization, in general, is constructing the best portfolio for the client based on the client characteristics and CMEs. When all the steps are performed with careful analysis, the process may be called integrated asset allocation.

7 Mean Variance Optimization The Mean-Variance Approach, developed by Markowitz in the 1950s, still serves as the foundation for quantitative approaches to strategic asset allocation. Mean Variance Optimization (MVO) identifies the portfolios that provide the greatest return for a given level of risk OR that provide the least risk for a given return.

8 Mean Variance Optimization TO develop an understanding of MVO, we will derive the relationship between risk and return of a portfolio by looking at a series of three portfolios: One risky asset and one risk-free asset Two risky assets Two risky assets and one risk-free asset We will then generalize our findings to portfolios of a larger number of assets.

9 MVO: One risky and one risk-free asset For a portfolio of two assets, one risky (r) and one risk-free (f), the expected portfolio return is defined as: Since, by definition, the risk-free asset has zero volatility (standard deviation), the portfolio standard deviation is:

10 MVO: One risky and one risk-free asset With the portfolio return and standard deviation equations, we can derive the Capital Allocation Line (CAL): Notice that the slope of this line represent the Sharpe ratio for asset r. It represents the reward-to-risk ratio for asset r.

11 MVO: One risky and one risk-free asset With one risky and one risk-free asset, an investor can select a portfolio along this CAL based on his risk / return preference.

12 MVO: Two risky assets With two risky assets (1 and 2), as long as the correlation between the two assets is less than 1, creating a portfolio with the two assets will allow the investor to obtain a greater reward-to-risk ratio than either of the two assets provide.

13 MVO: Two risky assets Portfolio expected return and standard deviation can be calculated as follows:

14 MVO: Two risky assets Remember that the correlation coefficient can be calculated as: Where and n = number of historical returns used in the calculations.

15 MVO: Two risky assets These values (as well as asset returns and standard deviations) can be easily calculated on a financial calculator or Excel.

16 MVO: Two risky assets By altering weights in the two assets, we can construct a minimum-variance frontier (MVF). The turning point on this MVF represents the global minimum variance (GMV) portfolio. This portfolio has the smallest variance (risk) of all possible combinations of the two assets. The upper half of the graph represents the efficient frontier.

17 MVO: Two risky assets The weights for the GMV portfolio is determined by the following equations:

18 MVO: Two risky and one risk-free asset We know that with one risky asset and the risk-free asset, the portfolio possibilities lie on the CAL. With two risky assets, the portfolio possibilities lie on the MVF. Since the slope of the CAL represents the reward-to-risk ratio, an investor will always want to choose the CAL with the greatest slope.

19 MVO: Two risky and one risk-free asset The optimal risky portfolio is where a CAL is tangent to the efficient frontier. This portfolio provides the best reward-to-risk ratio for the investor. The tangency portfolio risky asset weights can be calculated as:

20 MVO: All risky assets (market) and one risk-free asset We can generalize our previous results by considering all risky assets and one risk-free asset. The tangency (optimal risky) portfolio is the market portfolio. All investors will hold a combination of the risk-free asset and this market portfolio. In this context, the CAL is referred to as the Capital Market Line (CML).

21 Investor Risk Tolerance and CML To attain a higher expected return than is available at the market portfolio (in exchange for accepting higher risk), an investor can borrow at the risk free- rate. Other minimum variance portfolios (on the efficient frontier) are not considered.

22 Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier RFR M CML Borrowing Lending

23 Assumptions / Limitations of Markowitz Portfolio Theory Investors take a single-period perspective in determining their asset allocation. Drawback: Investors seldom have a single-period perspective. In a multiple-period horizon, even Treasury bills exhibit variability in returns Possible Solutions: Include the “risk-free asset” as a risky asset class. If investors have a liquidity need, construct an efficient frontier and asset allocation on the funds remaining after the liquidity need is satisfied.

24 Assumptions / Limitations of Markowitz Portfolio Theory Investors base decisions solely on expected return and risk. These expectations are derived from historical returns. Drawback: Optimal asset allocations are highly sensitive to small changes in the inputs, especially expected returns. Portfolios may not be well diversified. Potential solutions: Conduct sensitivity tests to understand the effect on asset allocation to changes in expected returns.

25 Assumptions / Limitations of Markowitz Portfolio Theory Investors can borrow and lend at the risk- free rate. Drawback: Borrowing rates are always higher than lending rates. Certain investors are restricted from purchasing securities on margin. Potential solutions: Differential borrowing and lending rates can be easily incorporated into MVO analysis. However, leverage may be practically irrelevant for many investors (liquidity, regulatory restrictions).

26 Practical Application of MVO MVO can be used to determine optimal portfolio weights with a certain subset of all investable assets. An efficient frontier can be constructed with inputs (expected return, standard deviation and correlations) for the selected assets.

27 Practical Application of MVO MVO can be either unconstrained, in which case we do not place any constraints on the asset weights, or it can be constrained.

28 Practical Application of MVO Unconstrained Optimization The simplest optimization places no constraints on asset-class weights except that they add up to 1. With unconstrained optimization, the asset weights of any minimum variance portfolio is a linear combination of any other two minimum variance portfolios.

29 Practical Application of MVO Constrained Optimization The more useful optimization for strategic asset allocation is constrained optimization. The main constraint is usually a restriction on short sales.

30 Practical Application of MVO Constrained Optimization We can determine asset weights using the corner portfolio theorem. This theorem states that the asset weights of any minimum variance portfolio is a linear combination of any two adjacent corner portfolios. Corner portfolios define a segment of the efficient frontier.

31 Practical Application of MVO Excel Solver is a powerful tool that can be used to determine optimal portfolio weights for a set of assets. To use the tool, we need expected returns and standard deviations for our assets as well as a set of constraints that are appropriate for the portfolio.

32 Readings RB 7 RB 8 (pgs ) RM 3 (5, – 6.1.4)