1. A SSET A LLOCATION Portfolio Management Ali Nejadmalayeri

2. Asset Allocation Strategic Asset Allocation –Set weights for general asset classes to meet return and risk objectives Tactical Asset Allocation –Short-term changes to SAA to take advantage of expected relative performance of different asset classes

3. Strategic Asset Allocation Set target and permissible target ranges for asset class weights Meet return and risk objectives –Specifies the desired “systematic” risk Evidence suggest that large fraction of total return variation is due to asset allocation

4. SAA Approaches Asset Only –Black-Litterman model Start with global value-weighted index Deviate from those weights to reflect investor’s view of expected returns and variations Asset/Liability Approaches –Cash flow matching Inflows and outflows are matched –Immunization Weighted averages durations are matched

5. SAA & IPS SAA & Return Objectives –Find weights that achieves desired return SAA & Risk Objective –Since investors are risk-averse, or U P = E(R P ) – 0.005  λ A  σ P then, shortfall risk has to be managed Sharpe ratio, SFRatio (Roy’s Ratio), etc.

6. What is an Asset Class? Asset Class should be: 1.Assets in the class should relatively homogenous 2.Asset classes should be mutually exclusive 3.Asset classes should be diversifying 4.Asset classes, as a group, should account for the preponderance of world wealth 5.Asset classes should have the capacity to absorb majority of investor’s portfolio without damaging liquidity

7. When to Add a New Asset Class? Beyond the obvious, the following should hold:

8. Finding Optimal Portfolio Unconstrained MVF –Asset weights of any MVF is a linear combination of asset weights in two other MVFs Sign-Constrained MVF –Find adjacent “Corner Portfolios” –Asset weights of any MVF are positive linear combination of the corresponding weights in the adjacent corner portfolios

9. Frontiers Return Risk  = 1.0  0 <  < 1.0  < 0 Efficient Frontiers Minimum Variance Frontiers Corner Portfolios Global Minimum Variance Portfolio

10. How to Optimize? An Algorithmic Approach to Finding Corner Portfolios

11. Portfolio Construction Given a set of selected Securities Finding Appropriate Asset Weights Optimizing the Portfolio: Highest Return for a Given Level of Risk

12. Optimal Portfolio Define the Risk Level Given the set of assets, Find the Bundle that Maximizes the Portfolio Return (Markowitz Optimization) –Define Measures of Return and Risk –Account for the Covariation of Asset Returns –Maximize Portfolio Return, or Minimize Portfolio Risk

13. Risk and Return In finance, we ALWAYS perceive everything in a forward looking way so: –Return and Risk are Expected Measures Q: How Does One Make Up Expectation about Future Return and Risk? A: Either History tells, or a Model Defines

14. How Construct EF? With Historical Information: –1 st, find asset returns from prices –2 nd, find return on an equally weighted portfolio –3 rd, find the average and std. dev. of returns for the portfolio –4 th, use SOLVER to determine that given a level of return, what are the variance minimizing weights

15. Historical Measures: Return Ordinary we know of transaction prices, so: –If P beg and P end are price of an asset at the beginning and end of an unit period of time, say one month, and CF is the additional cash flow payment to holders of the asset at the end of the period, then:

16. Expected Return by History Let’s assume for T period we know that returns are given: R 1, …, R T, then Expected Return, E(R), is:

17. Risk by History Ordinary we measure risk with variance, Var(R). Let’s assume for T period we know that returns are given: R 1, …, R T, then Risk (variance), Var(R), is:

18. How Construct EF? With Non-Historical Expectations: –1 st, use the correlation (variance-covariance) structure, find average and std. dev. of returns for the portfolio –2 nd, use SOLVER to determine that given a level of return, what are the variance minimizing weights

19. Covariation by History Ordinary we measure covariation with covariance, Cov(R) and correlation, Corr(R). Let’s assume for T period we know that returns for two assets are given: asset X; R X 1, …, R X T, and asset Y; R Y 1, …, R Y T then Covariance, Cov(R), is: the Correlation, Corr(R), is:

20. Portfolio Variance Say we have N assets with N expected returns of E (R 1 ), …, E (R N ), N variances of Var (R 1 ), …, Var (R N ), and N  N pairs of correlations,  1,1, …,  i,j,…,  N,N. Then the variance of portfolio with weights of w 1, …, w N is given:

21. Implementation: 1 st, Set-up the Problem

22. Implementation: 2 nd, Simplify Correlations

23. Implementation: 3 rd, Weights  Stdevs

24. Implementation: 4 th, Weights  Stdev’s  Corr.’s

25. Implementation: Last, Sum All Elements