Chapter 6 Efficient Diversification

E(r p ) = W 1 r 1 + W 2 r 2 W 1 = W 2 = = Two-Security Portfolio Return E(r p ) = 0.6(9.28%) + 0.4(11.97%) = 10.36% Easy Wi = % of total money invested in security i % 11.97% r1r1 r2r2 6-2

Portfolio Variance and Standard Deviation: Hard! 6-3 Consider something simple first instead  is always in the range __________ inclusive. Consider 1, 0, -1 benchmarks, ranges in between Which value is ideal for diversification? (use logic, or math formula of portfolio variance in your book) Again Chapter 11 in FINC301

Summary: Portfolio Risk/Return Two Security Portfolio Amount of risk reduction depends critically on _________________________. Adding securities with correlations _____ will result in risk reduction. If risk is reduced by more than expected return, what happens to the return per unit of risk (the Sharpe ratio)? correlations or covariances < 1 6-4

Extending Concepts to All Securities Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result. The set of portfolios that provide the optimal trade-offs are described as the efficient frontier. The efficient frontier portfolios are dominant or the best diversified possible combinations. All investors should want a portfolio on the efficient frontier. … Until we add the riskless asset 6-5

6.3The Optimal Risky Portfolio With A Risk-Free Asset 6.4 Efficient Diversification With Many Risky Assets 6-6

Including Riskless Investments The optimal combination becomes linear A single combination of risky and riskless assets will dominate 6-7

Dominant CAL with a Risk-Free Investment (F) CAL(P) = Capital Market Line or CML dominates other lines because it has the the largest slope Slope = (E(r p ) - rf) /  p (CML maximizes the slope or the return per unit of risk or it equivalently maximizes the Sharpe ratio) Regardless of risk preferences some combinations of P & F dominate 6-8

Practical Implications The analyst or planner should identify what they believe will be the best performing well diversified portfolio, call it P. P may include funds, stocks, bonds, international and other alternative investments. This portfolio will serve as the starting point for all their clients. The planner will then change the asset allocation between the risky portfolio and “near cash” investments according to risk tolerance of client. The risky portfolio P may have to be adjusted for individual clients for tax and liquidity concerns if relevant and for the client’s opinions. o o o o 6-9

6.5 A Single Index Model: CAPM 6-10 Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic. That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors Tremendous computational advantage makes it practical!

Sharpe Ratios and alphas – When ranking portfolios and security performance we must consider both return & risk “Well performing” diversified portfolios provide high Sharpe ratios: Sharpe = (r p – r f ) /  p You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify 6-11

Sharpe Ratios and alphas “Well performing” individual stocks held in diversified portfolios can be evaluated by the stock’s alpha in relation to the stock’s unsystematic risk. Seeking Positive Alphas 6-12

6.6 Risk of Long-Term Investments Are Stock Returns Less Risky in the Long Run? 6-13