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Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory Aims • Explain the principles of portfolio diversification • Demonstrate the construction of the efficient frontier • Show the trade-off between risk and return • Derive the Capital Market Line (CML) • Show the calculation of the optimal portfolio choice based on the mean and variance of portfolio returns. Overview • Investors choose a set of risky assets (stocks) plus a risk-free asset. • The risk-free asset is a term deposit or government Treasury bill. • Investors can borrow or lend as much as they like at the risk-free rate of interest. • Investors like return but dislike risk (risk averse). Preferences of Expected return and risk • We have seen how expected return is defined in Lecture 2. • The investor faces a number of stocks with different expected returns and differ from each other in terms of risk. • The expected return on the portfolio is the weighted mean return of all stocks. First moment. • Risk is measured in terms of the variance of returns or standard deviation. Second moment. • Investor preferences are in terms of the first and second moments of the distribution of returns. Investor Utility function U U E ( R p ), p U U U1 0; U2 0 E ( R p ) p dU dU dU U1 U2 0 dE ( R p ) d p dU dU dE ( R p ) U1 / 0 d p dE ( R p ) d p U2 Preference Function E(Rp) Expected return U2 U0 σp Risk Expected return n R p i Ri i 1 R p 1R1 2 R2 E R p 1E ( R1 ) 2 E ( R2 ) Risk E R p E(Rp ) E 1 R1 E ( R1 ) 2 R 2 E ( R2 ) 2 2 2 p 12 12 2 2 212 E R1 E ( R1 ) R2 E ( R2 ) 2 2 E R1 E ( R1 ) R2 E ( R2 ) CovR1 , R 2 CovR1 , R2 1, 2 Var( R1 )Var( R2 ) p 12 12 2 2 212 1, 2 1 2 2 2 2 Return and risk • How do return and risk vary relative to each other as the investor alters the proportion of each of the assets in the portfolio? • Assume that returns, risk and the covariance are fixed and simply vary the weights in the portfolio. • Let E(R1)=8.75% and E(R2)=21.25 • Let w1=0.75 and w2=0.25 • E(Rp)=.75x8.75+.25x21.25=11.88 • σ1=10.83, σ2=19.80, ρ1,2=-.9549 Portfolio Risk • σ2p=(0.75)2x(10.83)2+(0.25)2x(19.80)2+2x(0 .75)x(0.25)x(-0.95)x(10.83)x(19.80) • =13.7 • σp=√13.7=3.7 • Calculate risk and return for different weights Portfolio risk and return Equity 1 Equity 2 E(Rp) Risk State w1 w2 1 1 0 8.75% 10.83% 2 0.75 0.25 11.88% 3.70% 3 0.5 0.5 15% 5% 4 0 1 21.25 19.8% Locus of risk-return points Expected return (0,1) (.5,.5) (.75,.25) (1,0) Risk=standard deviation Risk – return locus • Can see that the locus of risk and returns vary according to the proportions of the equity held in the portfolio. • The proportion (0.75,0.25) is the lowest risk point with highest return. • The other points are either higher risk and higher return or low return and high risk. • The locus of points vary with the correlation coefficient and is called the efficient frontier Choice of weights • How does the portfolio manager choose the weights? • That will depend on preferences of the investor. • What happens if the number of assets grows to a large number. • If n is the number of assets then will need n(n-1)/2 covariances - becomes intractable • A short-cut is the Single Index Model (SIM) where each asset return is assumed to vary only with the return of the whole market (FTSE100, DJ, etc). • For ‘n’ assets the efficient frontier defines a ‘bundle’ of risky assets. ‘n’ asset case E R p i E Ri n i 1 n n i j i j ij 2 p i 1 j 1 How is the efficient frontier derived? • The shape of the efficient frontier will depend on the correlation between the asset returns of the two assets. • If the correlation is ρ = +1 then the portfolio risk is the weighted average of the risk of the portfolio components. • If the correlation is ρ = -1 then the portfolio risk can be diversified away to zero • When ρ < +1 then not all the total risk of each investment is non-diversifiable. Some of it can be diversified away Correlation of +1 p 2 12 (1 ) 2 2 2 1, 2 (1 ) 1 2 2 1, 2 1 p 2 12 (1 ) 2 2 2 (1 ) 1 2 2 ( 1 (1 ) 2 ) 2 1 (1 ) 2 Correlation of -1 p 2 12 (1 ) 2 2 2 (1 ) 1 2 2 p 1 (1 ) 2 2 min risk p 1 (1 ) 2 0 1 2 2 0 2 1 2 Check 2 2 p 1 1 2 1 2 1 2 1 2 1 2 p 0 1 2 1 2 Correlation < +1 p 1 (1 ) 2 Efficient frontier E(Rp) Ρ = -1 -1 < Ρ < +1 Ρ = +1 σp The general case – applied to two assets p 2 12 (1 ) 2 2 2 1, 2 (1 ) 1 2 2 p 2 12 (1 ) 2 2 2 1, 2 (1 ) 1 2 2 2 d p 2 2 12 2(1 ) 2 2 1, 2 1 2 4 1, 2 1 2 0 2 d 12 (1 ) 2 1, 2 1 2 2 1, 2 1 2 0 2 ( 12 2 ) 2 2 1, 2 1 2 1, 2 1 2 2 2 ( 12 2 2 1, 2 1 2 ) 2 1, 2 1 2 2 2 2 ( 2 1, 2 1 ) 2 [ 1 2 2 1, 2 1 2 ] 2 Efficient Frontier E(Rp) X Y σp Risk-free asset • Lets introduce a risk-free asset that pays a rate of interest Rf. • The rate Rf is known with certainty and has zero variance and therefore no covariance with the portfolio. • Such a rate could be a short-term government bill or commercial bank deposit. One bundle of risky assets • Take one bundle of risky assets and allow the investor to lend or borrow at the safe rate of interest. The investor can; • Invest all his wealth in the risky bundle and undertake no lending or borrowing. • Invest less than his total wealth in the single risky bundle and the rest in the risk-free asset. • Invest more than his total wealth in the risky bundle by borrowing at the risk-free rate and hold a levered portfolio. • These choices are shown by the transformation line that relates the return on the portfolio with one risk-free asset and risk. Transformation line E R p R f (1 ) RN p 2 2 (1 ) 2 N (1 ) f N fn 2 f 2 (1 ) 2 p 2 2 N p (1 ) N Linear Opportunity set • Let the risk-free rate Rf = 10% and the return on the bundle of assets RN = 22.5%. • The standard deviation of the returns on the bundle σN = 24.87%. • The weights on the risky bundle and the risk-free asset can be varied to produce a range of new portfolio returns. Portfolio Risk and Return State T-bill Equity E(Rp) σp (1-φ) φ 1 1 0 10% 0% 2 0.5 0.5 16.25% 12.44% 3 0 1 22.5% 24.87% 4 -0.5 1.5 28.75% 37.31% Transformation line • The transformation line describes the linear risk- return relationship for any portfolio consisting of a combination of investment in one safe asset and one ‘bundle’ of risky assets. • At every point on a given transformation line the investor holds the risky assets in the same fixed proportions of the risky portfolio ωi. Transformation line E(Rp) -0.5 borrowing + 1.5 0.5 lending + 0.5 in risky bundle in risky bundle No lending all investment in bundle Rf All lending σp A riskless asset and a risky portfolio • An investor faces many bundles of risky assets (eg from the London Stock Exchange). • The efficient frontier defines the boundary of efficient portfolios. • The single risky asset is replaced by a risky portfolio. • We can find a dominant portfolio with the riskless asset that will be superior to all other combinations. Combining risk-free and risky portfolios E(Rp) C B A Rf σp Borrowing and Lending • The investor can lend or borrow at the risk- free rate of interest rate. • The risk-free rate of interest Rf represents the rate on Treasury Bills or some other risk-free asset. • The efficiency boundary is redefined to include borrowing. Borrowing and lending frontier C E(Rp) B Rf A σp Combined borrowing and lending at different rates of interest • The investor can borrow at the rate of interest Rb • Lend at the rate of interest Rf • The borrowing rate is greater than the risk- free rate. Rb > Rf • Preferences determine the proportions of lending or borrowing, Combining borrowing and lending D E(Rp) Q C B Rb P Rf A σp Separation Principle • Investor makes 2 separate decisions • Given knowledge of expected returns, variances and covariances the investor determines the efficient frontier. The point M is located with reference to Rf. • The investor determines the combination of the risky portfolio and the safe asset (lending) or a leveraged portfolio (borrowing). Market portfolio and risk reduction Portfolio risk Diversifiable risk Non- Number of diversifiable securities risk 20 Summary • We have examine the theory of portfolio diversification • We have seen how the efficient frontier is constructed. • We have seen that portfolio diversification reduces risk to the non-diversifiable component.