Portfolio DiversificationPortfolio DiversificationChapters 7 and 8Investments (BKM)

“Don’t put all of your eggs in one basket”

Systematic and specific Systematic and specific riskriskWhat 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?

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

Diversiable vs. Diversiable vs. nondiversiable risknondiversiable riskWe cannot eliminate the risk that

comes from common sourcesRisk cannot be reduced to zero by

diversifying our portfolioThe remaining component is:

market risk, or systematic risk, or nondiversifiable risk

The risk that can be eliminatated by diversification is unique risk, or firm-specific risk, or nonsystematic risk, or diversiable risk

Portfolio DiversificationPortfolio DiversificationNYSE stocksEqually-weighted portfolios randomly selected

The power of diversification is limited by systematic risk

W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2r1 = return on Security 1r2 = return on Security 2E(): expected return

rp = W1r1 + W2r2

E(rp) = W1E(r1 ) + W2E(r2 )1

n

1iiw

Two-Security Portfolio: Two-Security Portfolio: ReturnReturn

p2 = w1

212 + w2

222 + 2W1W2 Cov(r1r2)

12 = Variance of Security 1

22 = Variance of Security 2

Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Two-Security Portfolio: Two-Security Portfolio: RiskRisk

Risk and returnRisk 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

The variance of the portfolio is not a weighted average of the individual asset variances

The variance is reduced if the covariance term is negative

1,2 = Correlation coefficient of returns

Cov(r1r2) = 1,212

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

CovarianceCovariance

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

Range of values for 1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

If then the variance is reduced

Correlation Coefficients: Correlation Coefficients: Possible ValuesPossible Values

Correlation Correlation It is always better to add to your

portfolios assets with lower or, even better, negative correlation with your existing positions

Portfolios of less than perfectly correlated assets always offer better risk-return opportunities than the individual component securities on their own

The lower the correlation between the assets, the greater the gain of diversification

Portfolio variancePortfolio variance

2p = W1

212 + W2

212

+ 2W1W2

rp = W1r1 + W2r2 + W3r3

Cov(r1r2)

+ W323

2

Cov(r1r3)+ 2W1W3

Cov(r2r3)+ 2W2W3

Three-Security PortfolioThree-Security Portfolio

E(rp) = W1E(r1) + W2E(r2) + W3E(r3 )

Correlation and varianceCorrelation and variance

Portfolio Expected Return as a Function Portfolio Expected Return as a Function of Investment Proportionsof Investment Proportions

Portfolio Standard Deviation as a Portfolio Standard Deviation as a Function of Investment ProportionsFunction of Investment Proportions

Portfolio risk and returnPortfolio risk and returnW1 and W2 can be <0 or >1 (short

sell)Portfolio standard deviation

decreases and then increasesWhere is the minimum-variance

portfolio?How much is the variance of the

minimum-variance portfolio? Compare it the variance of the two assets

Portfolio Expected Return as a function Portfolio Expected Return as a function of Standard Deviation of Standard Deviation

Portfolio opportunity set: possible combinations of the two assets

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

Optimal risky portfolioOptimal risky portfolioWe should find the weights that

give the highest slope of the Capital Allocation Line (CAL)

The objective function is the slope (Sharpe ratio: reward-to-volatility) of the CAL:

1W WGiven that

)()()(

)(

21

2211

rEWrEWrE

rrES

p

p

fpp

Optimal risky portfolioOptimal risky portfolioIn the case of two risky assets,

the weights of the optimal risky portfolio are:

21

2122

21

2122

11

1

),(])()([])([

),(])([])([

1

2

WW

rrCOVrrErrErrE

rrCOVrrErrEW

fff

ff

Optimal complete Optimal complete portfolioportfolioThe optimal complete portfolio is

formed once the optimal risky portfolio is set

The optimal complete portfolio consists of the optimal risky portfolio and the T-bills

Given the risk aversion A, the proportion invested in the risky portfolio is

pA

rfrpEy

2

)(

Determination of the Optimal Overall Determination of the Optimal Overall PortfolioPortfolio

Steps to form the optimal Steps to form the optimal complete portfoliocomplete portfolio1. Specify the return characteristics

of all securities (expected returns, variances, covariance)

2. Establish the risky portfolio, P (characteristics of P)

3. 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: Portfolio selection model: MarkowitzMarkowitzGeneralize the portfolio

construction model to many risky securities and a risk-free asset

First step: determine the minimum-variance frontier: the minimum variance portfolio for any targeted expected return

The Minimum-Variance Frontier of Risky The Minimum-Variance Frontier of Risky AssetsAssets

Minimum-variance Minimum-variance portfolioportfolioThe bottom part of the efficient

frontier is inefficient. Why?Portfolios with the same risk have

different expected returnsSecond step: introduce the risk-

free asset and search for CAL with the highest reward-to-volatility ratio

Find the tangent CAL to the efficient frontier

Optimal complete Optimal complete portfolioportfolioLast step: choose between the

optimal risky portfolio and the risk-free asset

Risk Reduction of Equally Weighted Portfolios Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universesin Correlated and Uncorrelated Universes

ri = E(Ri) + ßiF + eßi = index of a securities’ particular

return to the factorF= 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 Factor ModelSingle Factor Model

(ri - rf) = i + ßi(rm - rf) + ei

Risk Prem Market Risk Prem or Index Risk Prem

i= the stock’s expected return if the market’s excess return is zero

ßi(rm - rf) = the component of return due to

movements in the market index

(rm - rf) = 0

ei = firm specific component, not due to market

movements

Single Index Model: Security Single Index Model: Security Market Line (SML)Market Line (SML)

Let: Ri = (ri - rf)

Rm = (rm - rf)

Risk premiumformat

Ri = i + ßi(Rm) + ei

Risk Premium FormatRisk Premium Format

Market or systematic risk: risk related to the macro economic factor or market index.

Unsystematic or firm specific risk: risk not related to the macro factor or market index.

Total risk = Systematic + Unsystematic

Components of RiskComponents of Risk

i2 = i

2 m2 + 2(ei)

where;

i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

Measuring Components of Measuring Components of RiskRisk

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = 2

ßi2

m2 / 2 = 2

Covariance =product of betas*market index risk

Correlation=product of correlations with the market index

Examining Percentage of Examining Percentage of VarianceVariance

Mjiji rrCOV 2),(

jMiMij *

Portfolio construction and Portfolio construction and the single-index modelthe single-index modelOnce we have estimated the SML

for all assets, the securities that will be chosen are those that have the highest Alphas (α)

Positive-Alpha securities are underpriced: long position

Negative-Alpha securities are overpriced: short position

Efficient Frontiers with the Index Model Efficient Frontiers with the Index Model and Full-Covariance Matrixand Full-Covariance Matrix