Presentation 2: Objectives  To introduce you with the Principles of Investment Strategy and Modern Portfolio Theory  Topics to be covered  Indifference curve and utility function  Diversification  Mean and Variance of a portfolio  Efficient Frontier  Effect of Correlation  Skills for the above 1

Asset Allocation  Revisit  The process of selecting assets from a variety of different asset classes, designed to balance investors’ expected returns with their tolerance for risk.  A fundamental approach designed to reduce risk, preserve profits and improve total returns

Rational Investor Attitude to Risk  Blue: E, Red: B

Rational Investor Attitude to Risk  Risk Loving - high return for high risk  Risk neutral - indifferent to risk  Risk averse - low risk, low return  Attitudes to risk reflected by shape of utility (indifference) curves

Indifference Curve  Indifference curve captures the various risk- return scenario providing the same utility level for investors.  The curve (Utility Function) of an individual investor allows us to measure the subjective value the individual would place on investment.

Indifference Curve

 Indifference curve and risk aversion A B C

 Utility Function Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s 2 = variance of returns Indifference Curve

 Utility Function  An asset with expected rate of return 10% and standard deviation 28%.  What is the utility level of this specific asset to an risk Conservative(4) investors.  How about an risk Aggressive (2) investor? Indifference Curve

 Select the portfolio where the highest attainable indifference curve is tangential (just touching) to the efficient frontier. Indifference Curve

 Which is most accurate about indifference curve? A. For a risk averse person the indifference curve is flatter B. Investors expected utility may be different along the indifference curve C. Indifference Curve do not intersect Indifference Curve

Asset Allocation  Diversification  Investment Strategy  Having eggs in many basket or having eggs in the right baskets

Diversification 13 Source: Sardonic Salad

Diversification  Diversification can help reduce risk only when you combine the right assets.  Right assets whose value movements are not in a perfect synchrony.  Risk averse investor will diversify to at least some extent  more risk-averse investors diversifying more completely than less risk-averse investors.

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Mean, Variance and Covariance of a Portfolio  If we have a portfolio of assets we could compute the expected return and the variance of the whole portfolio.  Suppose assets 1,2, …., n are held in the proportions X 1, X 2, …, X n then the expected return on a portfolio is given by:-  where X 1 + X 2 + … + X n = 1.  Note that JP Morgan uses this as an approximation (although the returns are calculated using log returns as opposed to percentage returns). 23

Mean, Variance and Covariance of a Portfolio  When computing the variance we also have to concern ourselves with how the asset returns vary together - the covariance.  If returns tend to move in opposite directions then this reduces the overall variability of the portfolio.  But if returns tend to move in the same direction then the variability of the portfolio is increased.  In the analysis that follows  12 refers to the covariance between assets 1 and 2. 24

Mean, Variance and Covariance of a Portfolio  Variance of Portfolio with 2 assets:  Can we derive the above? 25

Mean, Variance and Covariance of a Portfolio  Applying the derivation explained for 2 assets above now prove the following for 3 assets case: 26

Mean, Variance and Covariance of a Portfolio  A measure of the association between two assets which is always in the range +1 to -1 is the correlation coefficient. This is defined as:- 27 The covariance and correlation coefficient always have the same sign

Mean, Variance, Covariance and Correlation of a Portfolio  Thus for a two variable portfolio:- 28

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The opportunity set under risk: Efficient Portfolios  How can we use the previous analysis to construct portfolios? 31

The opportunity set under risk: Efficient Portfolios  Suppose we consider a two asset (1 and 2) portfolio with proportion X 1 invested in asset 1 and X 2 (1- X 2 ) invested in asset 2. 32

The opportunity set under risk: Efficient Portfolios  The expected return of such a portfolio is:-  The variance is then:- 33

The opportunity set under risk: Efficient Portfolios  Let us focus on the correlation coefficient  12.  If the correlation coefficient is negative between 1 and 2 then the risk of a single asset (1 or 2) portfolio will be reduced by combining them together.  Since  12 < 0 then the returns on 1 and 2 tend to move in opposite directions and will partially offset each other.  When the return on one is high the return on the other is low (and vice versa) and as a result portfolio returns are less variable. 37

The opportunity set under risk: Efficient Portfolios  If  12 is +1 then the returns on assets will always tend to move in the same direction and there is no risk reduction. 38

Negative correlation  When correlation between two assets is “-1” you have the ultimate in diversification benefits and a risk free portfolio.  The graph on the next slide shows such an outcome. Perfect negative correlation gives a mean combined return for two securities over time equal to the mean for each of them, so the returns for the portfolio show no variability. 39

40  Any returns above and below the mean for each of the assets are completely offset by the return for the other asset, so there is no variability in total returns, that is, no risk, for the portfolio. Negative correlation

Perfect Positive Correlation (  = +1) 41

Perfect Positive Correlation (  = -1) 42

No Correlation (  = 0) 43

Actual Correlation (r ) 44

45 Single Index Model  “The mean variance approach” to portfolio analysis involves estimating the mean and variance of alternative portfolios and then selecting the portfolio that offers the best mean-variance combination.  What do we need to do this ?  Mean and variance of each asset.  Correlation between each asset.  Mean and variance of differing combinations of assets.  This is fine when the portfolio consists of only two assets. What if there are 20 ?

46  If there are 20 assets we need the same info as for 2:-  Mean and variance of each asset  40 parameters  Correlation between each asset  each one of the assets can be correlated with each of the 19 other assets = 19 x 20 = 380.  But correlation matrices are symmetric = 190  So for each combination we need 230 parameters, or:-  2N + N(N-1)/2 parameters for an N asset portfolio.  E.g. N = 200, parameters = (199)/2 = 20,300 parameters. Single Index Model

47 Thank you

48 Variance (X) = Cov (X, X) Variance (Rp) = Cov (Rp, Rp) When R 1, R 2 are the returns from X1 and X2 respectively Variance = Cov (x 1 R 1 + X 2 R 2, X 1 R 1 + X 2 R 2 ) Applying Cov(A+B,C+D) = Cov(A,C)+Cov(A,D)+Cov(B,C)+Cov(B,D) Variance (Rp) = X 2 1 Var(R 1 ) +X 2 2 Var(R 2 )+ 2X 1,X 2 Cov(R 1,R 2 ) Replacing Covariance by Correlation: X 2 1 Var (R 1 ) + +X 2 2 Square Var (R 2 )+ 2X 1,X 2 Corr(R 1 R 2 ) SD(R 1 ) SD(R 2 ) Note to Formula Derivation