Lecture 5: Portfolio Diversification and Supporting Financial InstitutionsPortfolio DiversificationEqually-Weighted Portfolio When Asset Returns are Independent & Same VarianceInvestment Companies as Providers of DiversificationDoubts about DiversificationEquity Premium PuzzleDominance of Stocks over Fixed Incomes?Survey of Institutional Investors, Shiller, 1993But is Equity Premium Robust?International EvidenceOptimal Portfolio Diversification in General CasePortfolio Variance, Three AssetsEfficient Portfolio Frontier with Two AssetsEfficient Portfolio FrontierRisk-Return Trade-Off and Holding PeriodsSlide 16Mutual Fund TheoremCapital Asset Pricing Model (CAPM)BetaDo Mutual Funds Hold Optimal Portfolio?Index FundsSlide 22Alfred Cowles 1891-1984Survey of Individual Investors 1999Persistence of Mutual Fund PerformanceSlide 26Slide 27Growth of Mutual FundsLecture 5: Portfolio Diversification and Supporting Financial InstitutionsPortfolio Diversification•All that should matter to an investor is the performance of the entire portfolio.•Mean and variance of portfolio matter•Law of large numbers means that spreading over many independent assets reduces risk, has no effect on expected return.Equally-Weighted Portfolio When Asset Returns are Independent & Same Variance•Same dollar value in each asset•Rebalancing each period•Portfolio expected return equals average of asset expected returns•Portfolio standard deviation equals asset standard deviation divided by •Square root rulenInvestment Companies as Providers of Diversification•Investment trusts (before 1940s)•Mutual funds (especially index funds)•Closed end investment companies•Unit investment trustsAll these institutions can enable small investors to overcome transactions cost and lumpiness problems in achieving diversified portfoliosDoubts about Diversification•Complete diversification would imply holding much in fixed incomes, real estate, etc. But hasn’t stock market outperformed these?Equity Premium Puzzle•Geometric average real stock market return 1871-1997: 7.0% (Siegel Table 1-1).•Geometric average real fixed-income return 1871-1997: 1.7% (Siegel Table 1-2)•Equity premium = 7.0%-1.7%=5.3%•Puzzle: Why has equity premium been so high?Dominance of Stocks over Fixed Incomes?•No thirty-year period since 1831-1861 when the return on either long-term or short-term bonds exceeded that on equities. (Siegel p. 15)Survey of Institutional Investors, Shiller, 1993“There is no thirty-year period since 1860 in which US government bonds have outperformed stocks.” Have you heard roughly this claim (even if details, such as the use of 30 years) are different?1. Yes, often 52%2. Yes, once or twice 22%3. No 26%But is Equity Premium Robust?•Geometric average US real stock market return 1802-1997: 7.0% (Siegel Table 1-1)•Geometric average real fixed income return 1802-1870: 5.1% (Siegel, Table 1-2)•Equity premium = 7.0% - 5.1% = 1.9%•Equity premium was much smaller then.International Evidence•Median real stock market appreciation rate for 39 countries 1926-96: 0.8% per year.•Real stock market appreciation rate for US 1926-96: 4.3% per year.(Philippe Jorion and William Goetzmann, Journal of Finance 54:953-80, 1999.)So, US equity premium may reflect a selection bias.Optimal Portfolio Diversification in General Case•Drop assumption of equal weighting, independence and equal variance•Put xi dollars in ith asset, I=1,..,n, where the xi sum to $1.•Portfolio expected value = •Portfolio variance (two assets) =)(E1iniireturnx),cov()1(2)var()1()var(2111221121returnreturnxxreturnxreturnx Portfolio Variance, Three Assets•Portfolio variance = )1where(),cov(2),cov(2),cov(2)var()var()var(31323231312121323222121iixreturnreturnxxreturnreturnxxreturnreturnxxreturnxreturnxreturnxEfficient Portfolio Frontier with Two Assets•Frontier expresses portfolio standard deviation in terms of portfolio expected return r rather than in terms of x1.• 2121rrrrx12221122222112122122)())((2)()(rrrrrrrrrrrrrrEfficient Portfolio FrontierRisk-Return Trade-Off and Holding PeriodsMutual Fund Theorem•All investors, regardless of risk preferences, will hold a combination of the riskless asset and the tangency portfolio of all risky assets.•Therefore, only one asset need be made available to investors: a mutual fund that holds the tangency portfolio.Capital Asset Pricing Model (CAPM)•CAPM Asserts that all investors hold their optimal portfolio•Consequence of the mutual fund theorem: all investors hold the same portfolio of risky assets, the tangency portfolio•Therefore the CAPM says that the tangency portfolio equals the market portfolioBeta•The CAPM implies that the expected return on the ith asset is determined from its beta.•Beta (i) is the regression slope coefficient when the return on the ith asset is regressed on the return on the market.•Fundamental equation of the CAPM:)(fmifirrrr Do Mutual Funds Hold Optimal Portfolio?•Completely diversified funds do not exist•Mutual funds are classified into equity funds, fixed-income funds, etc•Mutual funds are further classified into styles: growth, income, blue-chip, etc.•Index funds account invest in stock price indexes, such as S&P 500Index Funds•$350 billion, or 8% of stock market invested in equity index funds in 2000.•Much of this is in specialty index funds, such as Internet funds.•Some other broadly diversified funds, however, may substitute for market index funds.Alfred Cowles 1891-1984•Yale ’13•Investment advisor, NYC•Founded Econometric Society, Cowles Foundation•Econometrica 1933 “Can Stock Market Forecasters Forecast?”Survey of Individual Investors 1999“Trying to time the market, to get out before it goes down and in before it goes up, is:1. A smart thing to do; I can reasonably expect to be a success at it. 11%2. Not a smart thing to do; I can’t reasonably expect to be a success at it. 83%3. No opinion 5%Persistence of Mutual Fund Performance•Mutual funds that performed well last year tend to a little better than average this year. Effect is weak.William Goetzmann and Roger Ibbotson, Journal of Finance, 54:953-80, 1999.Survey of Individual Investors 1999“Trying to pick individual stocks, for example, if and when Ford Motor stock will go up, or IBM stock will go up, is:1. A smart thing to do; I can