As the number of assets in the portfolio increases, note how the number of covariance terms in the expansion increases as the square of the number of variance terms

Standard deviation No. of shares in portfolio Diversifiable / idiosyncratic risk Systematic risk As we add additional assets, we can lower overall risk. Lowest achievable risk is termed “systematic”, “non-diversifiable” or “market” risk Lowest risk with n assets

Percentage of risk on an individual security that can be eliminated by holding a random portfolio of stocks US73 UK65 FR67 DE56 IT60 BE80 CH56 NE 76 International89 Source: Elton et al. Modern Portfolio Theory

Add assets…especially with low correlations Even without low correlations, you lower variance as long as not perfectly correlated Low, zero, or (best) negative correlations help lower variance best An individual asset’s total variance doesn’t much affect the risk of a well-diversified portfolio

Building the efficient frontier: combining two assets in different proportions 1, 0 0, 1 0.5, , 0.25 Standard deviation Mean return

Risk and return reduced through diversification  =  = +1  = - 1  = +0.5  = 0 Mean return Standard deviation

Efficient frontier of risky assets μpμp pp x x x x x x x x x x x x x x x x B A C

Capital Market Line and market portfolio (M)  Capital Market Line =Tangent from risk-free rate to efficient frontier rfrf A B M  μ m - r f mm μ μmμm

So far we said nothing about preferences!

Individual preferences μpμp pp A B  Z ER p I2I2 I1I1 I 2 > I 1 Y Mean return Standard deviation

Capital Market Line and market portfolio (M)  r A B M  μ m - r mm μ μmμm IAIA Investor A reaches most preferred M-V combination by holding some of the risk-free asset and the rest in the market portfolio M giving position A

Capital Market Line and market portfolio (M)  r A B M  μ m - r mm μ μmμm IBIB B is less risk averse than A. Chooses a point that requires borrowing some money and investing everything in the market portfolio

Some lessons from our toy exercise for daily returns It’s laborious to compute the efficient set Curvature is not that great except for negatively correlated assets We “know” that these means and covariances are going to be bad estimates of next weeks process…so how stable do we think asset returns are generally…. …is it just a question of longer samples or do covariances etc change over time?

Issues in using covariance matrix for portfolio decisions Expected returns are very volatile – past not a good guide Covariances also volatile, but less so If we try to estimate covariances from past data –(i) we need a lot of them (almost n 2 /2 for n assets) –(ii) lots of noise in the estimation But a simplifying model seems to fit well:

What is β? Could get it from past historic patterns (though experience shows these are not stable and tend to revert to mean… …adjustments possible (Blume, Vasicek) Could project it from asset characteristics (e.g. if no market history) Dividend payout rate, asset growth, leverage, liquidity, size (total assets), earnings variability

Why use single index model? (Instead of projecting full matrix of covariances) 1.Less information requirements 2.It fits better!