# Portfolio Diversification

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Portfolio DiversificationChapters 7 and 8 Investments (BKM)

Systematic and specific riskWhat would be the source of risk of ADIB? 1. Systematic risk: general economy conditions (business cycle, inflation, interest rates, and exchange rates) 2. Specific risk: firm-specific risk What is the risk that we can reduce?y

Portfolio Risk as a Function of the Number of Stocks in the Portfolio

Diversiable vs. nondiversiable riskWe cannot eliminate the risk that comes from common sources y Risk cannot be reduced to zero by diversifying our portfolio y The remaining component is: market risk, or systematic risk, or nondiversifiable risk y The risk that can be eliminatated by diversification is unique risk, or firmspecific risk, or nonsystematic risk, or diversiable risky

Portfolio DiversificationNYSE stocks Equally-weighted portfolios randomly selected

The power of diversification is limited by systematic risk

TwoTwo-Security Portfolio: ReturnW1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = return on Security 1 r2 = return on Security 2 E(): expected return rp = W1r1 + W2r2 E(rp) = W1E(r1 ) + W2E(r2 )n

w !1i i !1

TwoTwo-Security Portfolio: RiskWp2 = w12W12 + w22W22 + 2W1W2 Cov(r1r2) W12 = Variance of Security 1 W22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Risk and returnThe expected return of the portfolio is a weighted average of the expected returns of the assets that form the portfolio. The weight is the proportion invested in each asset y The variance of the portfolio is not a weighted average of the individual asset variances y The variance is reduced if the covariance term is negativey

CovarianceCov(r1r2) = V1,2W1W2 V1,2 = Correlation coefficient of returnsW1 = Standard deviation of returns for Security 1 W2 = Standard deviation of returns for Security 2

ExerciseCalculate the expected return and the variance of the portfolio that consists of 40% of debt and the remaining in equity

Correlation Coefficients: Possible ValuesRange of values for V1,2 + 1.0 > V > -1.0If V= 1.0, the securities would be perfectly positively correlated If V= - 1.0, the securities would be perfectly negatively correlated If V then the variance is reduced

CorrelationIt is always better to add to your portfolios assets with lower or, even better, negative correlation with your existing positions y Portfolios of less than perfectly correlated assets always offer better risk-return opportunities than the individual component securities on their own y The lower the correlation between the assets, the greater the gain of diversificationy

Portfolio variance

ThreeThree-Security Portfoliorp = W1r1 + W2r2 + W3r3 E(rp) = W1E(r1) + W2E(r2) + W3E(r3 ) W2p=

W12W12

+

W22W12

+ W32W32

+ 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)

Correlation and variance

Portfolio Expected Return as a Function of Investment Proportions

Portfolio Standard Deviation as a Function of Investment Proportions

Portfolio risk and returnW1 and W2 can be 1 (short sell) y Portfolio standard deviation decreases and then increases y Where is the minimum-variance portfolio? y How much is the variance of the minimum-variance portfolio? Compare it the variance of the two assetsy

Portfolio Expected Return as a function of Standard Deviation

Portfolio opportunity set: possible combinations of the two assets

The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

Optimal risky portfolioWe should find the weights that give the highest slope of the Capital Allocation Line (CAL) y The objective function is the slope (Sharpe ratio: reward-to-volatility) of the CAL:yS E (rp

!p

p

) rp 1

f

W ) ! W

E (r

E ( r1 ) W W1

2

E ( r2 ) ! 1

Given that

W

2

Optimal risky portfolioy

In the case of two risky assets, the weights of the optimal risky portfolio are:[ E(r1 ) rf ]W 2 [E(r2 ) rf ]COV(r1 , r2 ) [ E(r1 ) rf ]W 2 [ E(r1 ) rf E(r2 ) rf ]COV(r1 , r2 ) W1 ! 1 W22 2

W1 !

Optimal complete portfolioThe optimal complete portfolio is formed once the optimal risky portfolio is set y The optimal complete portfolio consists of the optimal risky portfolio and the Tbills y Given the risk aversion A, the proportion invested in the risky portfolio isy

E ( rp ) rf y! AW 2 p

Determination of the Optimal Overall Portfolio

Steps to form the optimal complete portfolio1.

2. 3.

y

Specify the return characteristics of all securities (expected returns, variances, covariance) Establish the risky portfolio, P (characteristics of P) Allocate funds between risky and the risk-free asset (calculate the proportion invested in each asset) Do exercise p 222 BKM (concept check 3)

Portfolio selection model: MarkowitzGeneralize the portfolio construction model to many risky securities and a riskfree asset y First step: determine the minimumvariance frontier: the minimum variance portfolio for any targeted expected returny

The Minimum-Variance Frontier of Risky Assets Minimum-

MinimumMinimum-variance portfolioThe bottom part of the efficient frontier is inefficient. Why? y Portfolios with the same risk have different expected returns y Second step: introduce the risk-free asset and search for CAL with the highest reward-to-volatility ratio y Find the tangent CAL to the efficient frontiery

Optimal complete portfolioy

Last step: choose between the optimal risky portfolio and the risk-free asset

Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

Single Factor Modelri = E(Ri) + iF + e i = index of a securities particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor

Single Index Model: Security Market Line (SML)(ri - rf) =Risk Prem Ei

E i + i(rm - rf) + eiMarket Risk Prem or Index Risk Prem

= the stocks expected return if the (rm - rf) = 0 markets excess return is zero i(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements

Risk Premium FormatLet: Ri = (ri - rf) Rm = (rm - rf) Risk premium format

Ri = Ei + i(Rm) + ei

Components of RiskMarket or systematic risk: risk related to the macro economic factor or market index. y Unsystematic or firm specific risk: risk not related to the macro factor or market index. y Total risk = Systematic + Unsystematicy

Measuring Components of RiskWi2 = Fi2 Wm2 + W2(ei)where;

Wi2 = total variance Fi2 Wm2 = systematic variance W2(ei) = unsystematic variance

Examining Percentage of VarianceTotal Risk = Systematic Risk + Unsystematic Risk y Systematic Risk/Total Risk = V2 y i2 W m2 / W2 = V2 y Covariance =product of betas*market index risky

( ri , r j ) ! F i F j Wy

2

M

Correlation=product of correlations with the market indexVij

! V

iM

* V

jM

Portfolio construction and the singlesingle-index modelOnce we have estimated the SML for all assets, the securities that will be chosen are those that have the highest Alphas ( ) y Positive-Alpha securities are underpriced: long position y Negative-Alpha securities are overpriced: short positiony

Efficient Frontiers with the Index Model and FullFullCovariance Matrix