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

Preference Function E(Rp) Expected return σp Risk U0 U2

Expected return

Risk

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 8.75% 10.83% 2 0.75 0.25 11.88% 3.70% 3 0.5 15% 5% 4 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

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

Correlation of -1

Check

Correlation < +1

Efficient frontier E(Rp) Ρ = -1 -1 < Ρ < +1 Ρ = +1 σp

The general case – applied to two assets

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

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 10% 0% 2 0.5 16.25% 12.44% 3 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 σp E(Rp) -0.5 borrowing + 1.5 in risky bundle 0.5 lending + 0.5 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 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-diversifiable risk Number of securities 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.