Portfolio optimization with warren and bill
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The document discusses portfolio optimization and the efficient frontier. It explains that the efficient frontier shows the set of optimal portfolios that maximize return for a given level of risk. To find the efficient frontier, the risk and return of all potential investments must be analyzed to select a portfolio with minimum risk for its expected return level. Proper diversification across low-correlated assets is important to reduce non-systematic risk and move a portfolio toward the efficient frontier.
Portfolio optimization with warren and bill
- 1. Stodder, Efficient FrontierPortfolio Optimization MethodologyHow to Find that Efficient Frontier!Theoretical & “Real Life” Examples
- 4. Security Market Line EquationRequired Return=Risk Free + Risk Premiumon Stock iRate on Stock iRequired Return=Risk Free + βi(Market Risk)on Stock iRate PremiumRi= Rrf+ βi(Rm - Rrf)
- 5. Beta of the Market = 1βi= (Ri– Rrf)/(Rm - Rrf) So if Ri = Rm, βi = βm thenβm = (Rm– Rrf)/(Rm - Rrf) = 1
- 7. How do We Find the Efficient Frontier?Basic Strategy:Find the Standard Deviation(σi) and Mean Return(μi) of every stock Stock i.For any given rate of return, find the minimal standard deviation portfolio that can achieve that return.
- 8. At what point do we have least risk?
- 10. Use More Than One Basket for Your Eggs The Axiom The Concept of Risk Aversion Revisited Preliminary Steps in Forming a Portfolio The Reduced Security Universe Security Statistics Interpreting the Statistics The Role of Uncorrelated Securities The Variance of a Linear Combination Diversification and Utility The Concept of Dominance
- 11. The Efficient Frontier Optimum Diversification of Risky Assets The Minimum Variance Portfolio The Effect of a Risk free Rate The Efficient Frontier with Borrowing Different Borrowing and Lending Rates Naive Diversification The Single Index Model
- 12. Don’t put all your eggs in one basket.
- 13. Failure to diversify may violate the terms of a fiduciary trust.
- 14. Risk aversion seems to be an instinctive traitin human beings.Preliminary Steps in Forming a PortfolioIdentify a collection of eligible investments known as the security universe.Compute statistics for the chosen securities. e.g. mean of return variance / standard deviation of return matrix of correlation coefficients
- 15. Preliminary Steps in Forming a PortfolioInsert Figure 16-1 here.
- 16. Preliminary Steps in Forming a PortfolioInsert Figure 16-2 here.
- 17. Preliminary Steps in Forming a Portfolio Interpret the statistics. 1. Do the values seem reasonable?2. Is any unusual price behavior expected to recur?3. Are any of the results unsustainable?4. Low correlations: Fact or fantasy?
- 18. wherexi = the proportion invested in security iThe Role of Uncorrelated SecuritiesThe expected return of a portfolio is a weighted average of the component expected returns.The Role of Uncorrelated SecuritiesInsert Table 16-5 here.
- 19. two-securityportfolio risk= riskA + riskB+ interactive riskThe Role of Uncorrelated SecuritiesThe total risk of a portfolio comes from the variance of the components and from the relationships among the components. betterperformanceexpected returnriskThe Role of Uncorrelated SecuritiesInvestors get added utility from greater return. They get disutility from greater risk.
- 20. The point of diversification is to achieve a given level of expected return while bearing the least possible risk.The Role of Uncorrelated SecuritiesA portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk.Efficient frontierimpossibleportfoliosexpected returndominatedportfoliosrisk (standard deviation of returns)The Efficient Frontier : Optimum Diversification of Risky Assets The efficient frontier contains portfolios that are not dominated.
- 21. single securitywith the highestexpected returnexpected returnminimum varianceportfoliorisk (standard deviation of returns)The Efficient Frontier : The Minimum Variance PortfolioThe right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio.The Efficient Frontier : The Minimum Variance PortfolioInsert Figure 16-6 here.
- 22. Efficient frontier:Rf to M to CimpossibleportfoliosCexpected returnMdominatedportfoliosRfrisk (standard deviation of returns)The Efficient Frontier : The Effect of a Riskfree RateWhen a riskfree investment complements the set of risky securities, the shape of the efficient frontier changes markedly.ED
- 23. The Efficient Frontier : The Effect of a Riskfree RateIn capital market theory, point M is called the market portfolio.The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line.Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the riskfree rate is often called a lending portfolio.
- 24. Efficient frontier:the ray from Rf through Mimpossibleportfoliosborrowingexpected returnMlendingdominatedportfoliosRfrisk (standard deviation of returns)The Efficient Frontier with BorrowingBuying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio.Efficient frontier : RL to M, the curve to N, then the ray from Nimpossibleportfoliosexpected returnNMRBdominatedportfoliosRLrisk (standard deviation of returns)The Efficient Frontier : Different Borrowing and Lending RatesMost of us cannot borrow and lend at the same interest rate. As portfolio size increases,total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest.total riskNondiversifiable risknumber of securitiesThe Efficient Frontier : Naive DiversificationNaive diversification is the random selection of portfolio components without conducting any serious security analysis.2040
- 25. The Efficient Frontier : Naive DiversificationThe remaining risk, when no further diversification occurs, is pure market risk. Market risk is also called systematic risk and is measured by beta.A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and vice versa.
- 26. The Efficient Frontier : The Single Index ModelA pairwise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to a benchmark measure.The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market.
- 27. The Efficient Frontier : The Single Index ModelBeta is the statistic relating an individual security’s returns to those of the market index.The Efficient Frontier : The Single Index ModelThe relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta.The Efficient Frontier : The Single Index ModelInsert Figure 16-11 here.
- 28. The Efficient Frontier : The Single Index ModelInsert Figure 16-12 here.
- 29. Use More Than One Basket for Your Eggs
- 30. The Axiom
- 31. The Concept of Risk Aversion Revisited
- 32. Preliminary Steps in Forming a Portfolio
- 33. The Reduced Security Universe
- 35. Interpreting the Statistics
- 36. The Role of Uncorrelated Securities
- 37. The Variance of a Linear Combination
- 38. Diversification and Utility
- 39. The Concept of DominanceWhat did we learn? The Efficient Frontier
- 40. Optimum Diversification of Risky Assets
- 41. The Minimum Variance Portfolio
- 42. The Effect of a Riskfree Rate
- 43. The Efficient Frontier with Borrowing
- 44. Different Borrowing and Lending Rates
- 46. The Single Index ModelAppendix: Arbitrage Pricing Theory Theory presumes that market return is determined by a number of distinct, unidentifiable macroeconomic factors
- 47. Four factors that make the market move:
- 48. The economy
- 49. Fed policy
- 50. Valuation
- 51. Investor sentimentAppendix: Arbitrage Pricing Theory
- 52. Appendix: Arbitrage Pricing Theory