Asset Allocation

Optimal Risky Portfolios- Asset AllocationsBKM Ch 7Asset AllocationIdeafrom bank account to diversified portfolioprinciples are the same for any number of stocksDiscussionA. bonds and stocksB. bills, bonds and stocksC. any number of risky assets1/16/2010Bahattin Buyuksahin, JHU , Investment2Diversification and Portfolio RiskMarket riskSystematic or nondiversifiableFirm-specific riskDiversifiable or nonsystematic1/16/2010Bahattin Buyuksahin, JHU , Investment3Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

1/16/2010Bahattin Buyuksahin, JHU , Investment4Figure 7.2 Portfolio Diversification

1/16/2010Bahattin Buyuksahin, JHU , Investment5Covariance and CorrelationPortfolio risk depends on the correlation between the returns of the assets in the portfolioCovariance and the correlation coefficient provide a measure of the way returns two assets vary1/16/2010Bahattin Buyuksahin, JHU , Investment6Two-Security Portfolio: Return

1/16/2010Bahattin Buyuksahin, JHU , Investment7 = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security ETwo-Security Portfolio: Risk

1/16/2010Bahattin Buyuksahin, JHU , Investment8Two-Security Portfolio: Risk ContinuedAnother way to express variance of the portfolio:

1/16/2010Bahattin Buyuksahin, JHU , Investment9D,E = Correlation coefficient of returns

Cov(rD,rE) = DEDED = Standard deviation of returns for Security DE = Standard deviation of returns for Security ECovariance1/16/2010Bahattin Buyuksahin, JHU , Investment10Range of values for 1,2+ 1.0 > r >-1.0If r = 1.0, the securities would be perfectly positively correlatedIf r = - 1.0, the securities would be perfectly negatively correlatedCorrelation Coefficients: Possible Values1/16/2010Bahattin Buyuksahin, JHU , Investment11Table 7.1 Descriptive Statistics for Two Mutual Funds

1/16/2010Bahattin Buyuksahin, JHU , Investment12 2p = w1212+ w2212+ 2w1w2 Cov(r1,r2)+ w3232 Cov(r1,r3)+ 2w1w3 Cov(r2,r3)+ 2w2w3Three-Security Portfolio

1/16/2010Bahattin Buyuksahin, JHU , Investment13Asset Allocation Portfolio of 2 risky assets (contd)examplesBKM7 Tables 7.1 & 7.3BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off)portfolio opportunity set (BKM7 Fig. 7.5)minimum variance portfoliochoose wD such that portfolio variance is lowestoptimization problemminimum variance portfolio has less risk than either component (i.e., asset)1/16/2010Bahattin Buyuksahin, JHU , Investment14Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

1/16/2010Bahattin Buyuksahin, JHU , Investment15Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

1/16/2010Bahattin Buyuksahin, JHU , Investment16Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

1/16/2010Bahattin Buyuksahin, JHU , Investment17Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

1/16/2010Bahattin Buyuksahin, JHU , Investment18Minimum Variance Portfolio as Depicted in Figure 7.4Standard deviation is smaller than that of either of the individual component assetsFigure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

1/16/2010Bahattin Buyuksahin, JHU , Investment19Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

1/16/2010Bahattin Buyuksahin, JHU , Investment20The relationship depends on the correlation coefficient-1.0 < < +1.0The smaller the correlation, the greater the risk reduction potentialIf r = +1.0, no risk reduction is possibleCorrelation Effects1/16/2010Bahattin Buyuksahin, JHU , Investment21Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

1/16/2010Bahattin Buyuksahin, JHU , Investment22The Sharpe RatioMaximize the slope of the CAL for any possible portfolio, pThe objective function is the slope:

1/16/2010Bahattin Buyuksahin, JHU , Investment23Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

1/16/2010Bahattin Buyuksahin, JHU , Investment24Figure 7.8 Determination of the Optimal Overall Portfolio

1/16/2010Bahattin Buyuksahin, JHU , Investment25Asset AllocationFinding the optimal risky portfolio: II. FormallyIntuitivelyBKM7 Figs. 7.6 and 7.7improve the reward-to-variability ratiooptimal risky portfolio tangency point (Fig. 7.8)Formally:

1/16/2010Bahattin Buyuksahin, JHU , Investment26Asset Allocation 18formally (continued)

1/16/2010Bahattin Buyuksahin, JHU , Investment27Asset Allocation 19Example (BKM7 Fig. 7.8)1. plot D, E, riskless2. compute optimal risky portfolio weightswD = Num/Den = 0.4; wE = 1- wD = 0.63. given investor risk aversion (A=4), compute w*

bottom line: 25.61% in bills; 29.76% in bonds (0.7439x0.4); rest in stocks

1/16/2010Bahattin Buyuksahin, JHU , Investment28Figure 7.9 The Proportions of the Optimal Overall Portfolio

1/16/2010Bahattin Buyuksahin, JHU , Investment29Markowitz Portfolio Selection ModelSecurity SelectionFirst step is to determine the risk-return opportunities availableAll portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations1/16/2010Bahattin Buyuksahin, JHU , Investment30Markowitz Portfolio Selection ModelCombining many risky assets & T-Billsbasic idea remains unchanged1. specify risk-return characteristics of securitiesfind the efficient frontier (Markowitz)2. find the optimal risk portfoliomaximize reward-to-variability ratio3. combine optimal risk portfolio & riskless assetcapital allocation1/16/2010Bahattin Buyuksahin, JHU , Investment31finding the efficient frontierdefinition set of portfolios with highest return for given risk minimum-variance frontiertake as given the risk-return characteristics of securitiesestimated from historical data or forecastsn securities -> n return + n(n-1) var. & cov.use an optimization programto compute the efficient frontier (Markowitz)subject to same constraintsMarkowitz Portfolio Selection Model1/16/2010Bahattin Buyuksahin, JHU , Investment32Finding the efficient frontier (contd)optimization constraints portfolio weights sum up to 1no short sales, dividend yield, asset restrictions, Individual assets vs. frontier portfoliosBKM7 Fig. 7.10short sales -> not on the efficient frontierno short sales -> may be on the frontier example: highest return assetMarkowitz Portfolio Selection Model1/16/2010Bahattin Buyuksahin, JHU , Investment33Figure 7.10 The Minimum-Variance Frontier of Risky Assets

1/16/2010Bahattin Buyuksahin, JHU , Investment34Markowitz Portfolio Selection Model ContinuedWe now search for the CAL with the highest reward-to-variability ratio1/16/2010Bahattin Buyuksahin, JHU , Investment35Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

1/16/2010Bahattin Buyuksahin, JHU , Investment36Markowitz Portfolio Selection Model ContinuedNow the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

1/16/2010Bahattin Buyuksahin, JHU , Investment37Figure 7.12 The Efficient Portfolio Set

1/16/2010Bahattin Buyuksahin, JHU , Investment38Capital Allocation and the Separation PropertyThe separation property tells us that the portfolio choice problem may be separated into two independent tasksDetermination of the optimal risky portfolio is purely technicalAllocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference1/16/2010Bahattin Buyuksahin, JHU , Investment39Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

1/16/2010Bahattin Buyuksahin, JHU , Investment40The Power of DiversificationRemember:

If we define the average variance and average covariance of the securities as:

We can then express portfolio variance as:

1/16/2010Bahattin Buyuksahin, JHU , Investment41Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

1/16/2010Bahattin Buyuksahin, JHU , Investment42Risk Pooling, Risk Sharing and Risk in the Long RunConsider the following:1 p = .999p = .001Loss: payout = $100,000No Loss: payout = 01/16/2010Bahattin Buyuksahin, JHU , Investment43Risk Pooling and the Insurance PrincipleConsider the variance of the portfolio:

It seems that selling more policies causes risk to fallFlaw is similar to the idea that long-term stock investment is less risky

1/16/2010Bahattin Buyuksahin, JHU , Investment44Risk Pooling and the Insurance Principle ContinuedWhen we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:

1/16/2010Bahattin Buyuksahin, JHU , Investment45Risk SharingWhat does explain the insurance business?Risk sharing or the distribution of a fixed amount of risk among many investors1/16/2010Bahattin Buyuksahin, JHU , Investment46An Asset Allocation Problem

Bahattin Buyuksahin, JHU , Investment1/16/201047An Asset Allocation Problem 2Perfect hedges (portfolio of 2 risky assets)

perfectly positively correlated risky assetsrequires short salesperfectly negatively correlated risky assetsBahattin Buyuksahin, JHU , Investment

1/16/201048An Asset Allocation Problem 3

Bahattin Buyuksahin, JHU , Investment1/16/201049CHAPTER 8Index ModelsFactor ModelBahattin Buyuksahin, JHU , InvestmentIdeathe same factor(s) drive all security returnsImplementation (simplify the estimation problem)do not look for equilibrium relationshipbetween a securitys expected returnand risk or expected market returnslook for a statistical relationshipbetween realized stock returnand realized market return 1/16/201051Factor Model 2Bahattin Buyuksahin, JHU , InvestmentFormallystock return = expected stock return + unexpected impact of common (market) factors+ unexpected impact of firm-specific factors

1/16/201052Index ModelBahattin Buyuksahin, JHU , InvestmentFactor modelproblemwhat is the factor? Index Modelsolution market portfolio proxyS&P 500, Value Line Index, etc.

1/16/201053Reduces the number of inputs for diversificationEasier for security analysts to specializeAdvantages of the Single Index Model1/16/2010Bahattin Buyuksahin, JHU , Investment54 i = index of a securities particular return to the factor m = Unanticipated movement related to security returns ei = Assumption: a broad market index like the S&P 500 is the common factor.Single Factor Model

1/16/2010Bahattin Buyuksahin, JHU , Investment55Single-Index ModelRegression Equation:

Expected return-beta relationship:

1/16/2010Bahattin Buyuksahin, JHU , Investment56Single-Index Model ContinuedRisk and covariance:Total risk = Systematic risk + Firm-specific risk:Covariance = product of betas x market index risk:

Correlation = product of correlations with the market index

1/16/2010Bahattin Buyuksahin, JHU , Investment57Index Model and DiversificationPortfolios variance:

Variance of the equally weighted portfolio of firm-specific components:

When n gets large, becomes negligible

1/16/2010Bahattin Buyuksahin, JHU , Investment58Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient p in the Single-Factor Economy

1/16/2010Bahattin Buyuksahin, JHU , Investment59Figure 8.2 Excess Returns on HP and S&P 500 April 2001 March 2006

1/16/2010Bahattin Buyuksahin, JHU , Investment60Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

1/16/2010Bahattin Buyuksahin, JHU , Investment61Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

1/16/2010Bahattin Buyuksahin, JHU , Investment62Figure 8.4 Excess Returns on Portfolio Assets

1/16/2010Bahattin Buyuksahin, JHU , Investment63Alpha and Security AnalysisMacroeconomic analysis is used to estimate the risk premium and risk of the market indexStatistical analysis is used to estimate the beta coefficients of all securities and their residual variances, 2 ( e i )Developed from security analysis1/16/2010Bahattin Buyuksahin, JHU , Investment64Alpha and Security Analysis ContinuedThe market-driven expected return is conditional on information common to all securitiesSecurity-specific expected return forecasts are derived from various security-valuation models The alpha value distills the incremental risk premium attributable to private informationHelps determine whether security is a good or bad buy

1/16/2010Bahattin Buyuksahin, JHU , Investment65Single-Index Model Input ListRisk premium on the S&P 500 portfolioEstimate of the SD of the S&P 500 portfolion sets of estimates ofBeta coefficientStock residual variancesAlpha values1/16/2010Bahattin Buyuksahin, JHU , Investment66Optimal Risky Portfolio of the Single-Index ModelMaximize the Sharpe ratioExpected return, SD, and Sharpe ratio:

1/16/2010Bahattin Buyuksahin, JHU , Investment67Optimal Risky Portfolio of the Single-Index Model ContinuedCombination of:Active portfolio denoted by AMarket-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by MModification of active portfolio position:

When

1/16/2010Bahattin Buyuksahin, JHU , Investment68The Information RatioThe Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

1/16/2010Bahattin Buyuksahin, JHU , Investment69Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

1/16/2010Bahattin Buyuksahin, JHU , Investment70Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

1/16/2010Bahattin Buyuksahin, JHU , Investment71Index Model: Industry PracticesBahattin Buyuksahin, JHU , InvestmentBeta booksMerrill Lynchmonthly, S&P 500Value Lineweekly, NYSEetc.Idearegression analysis 1/16/201072Index Model: Industry Practices 2Bahattin Buyuksahin, JHU , InvestmentExample (Merrill Lynch differences, Table 8.3)total (not excess) returnsslopes are identicalsmallnesspercentage price changesdividends?S&P 500adjusted betabeta = (2/3) estimated beta + (1/3) . 1sampling errors, convergence of new firmsexploiting alphas (Treynor-Black)1/16/201073Table 8.3 Merrill Lynch, Pierce, Fenner & Smith, Inc.: Market Sensitivity Statistics

1/16/2010Bahattin Buyuksahin, JHU , Investment74Table 8.4 Industry Betas and Adjustment Factors

1/16/2010Bahattin Buyuksahin, JHU , Investment75Using Index ModelsBahattin Buyuksahin, JHU , Investment

1/16/201076Using Index Models 2Bahattin Buyuksahin, JHU , Investment

1/16/201077Using Index Models 3Bahattin Buyuksahin, JHU , Investment

1/16/201078Using Index Models 4Bahattin Buyuksahin, JHU , Investment

1/16/201079Question:Suppose that there are many stocks in the market and that the characteristics of Stocks A and B are given as follows:

Stock

Expected Return

Standard Deviation

-------------------------------------------------------------------------------------------

A

10%

5%

B

15%

10%

-------------------------------------------------------------------------------------------

Correlation = -1-------------------------------------------------------------------------------------------Suppose that it is possible to borrow at the risk-free rate, Rf.

What must be the value of the risk-free rate?

Answer:The trick is to construct a risk-free portfolio from Stocks A and B that rules out arbitrage.

Since Stocks A and B are perfectly negatively correlated, a risk-free portfolio can be constructed and its rate of return in equilibrium will be the risk-free rate.

To find the proportions of this portfolio (wA invested in Stock A and wB = 1 - wA in Stock B), set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to:

(p = ( wA(A - wB(B(=>0 = ( 5wA - 10 (1-wA) (=>wA =0.6667

The expected rate of return on this risk-free portfolio is:

E(R) = 0.6667 x 10% + (0.3333 x 15%) ( 11.67%.

Question:Consider the two (excess return) index-model regression results for Stocks A and B:

RA = 1% + 1.2RM

R-square (A) = 0.576

Residual standard deviation-N (A)= 10.3%

RB = -2% + 0.8RM

R-square (B) = 0.436

Residual standard deviation-N (B) = 9.1%

(a) Which stock has more firm-specific risk?

(b) Which stock has greater market risk?(c) For which stock does the market explain a greater fraction of return variability?(d) Which stock had an average return in excess of that predicted by the CAPM?Answer:

(a) Firm-specific risk is measured by the residual standard deviation.

Thus, Stock A has more firm-specific risk, 10.3 > 9.1.

(b) Market risk is measured by beta, the slope coefficient of the regression.

Stock A has a larger beta coefficient, 1.2 > 0.8.

(c) R-square measures the fraction of the total variation in asset returns explained by the market return.

Stock As R-square is larger than Stocks Bs R-square, that is, 0.576 > 0.436.

(d) The average rate of return in excess of that predicted by the CAPM is measured by alpha, the intercept of the security characteristic line (SCL).

Alpha (A) = 1% is larger than alpha (B) = -2%.

Question:Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively.

The beta of stock A is 0.8, while that of stock B is 1.5.

The T-bill rate is currently 6%. The expected rate of return on the S&P 500 index is 12%.

The standard deviation of stock A is 10% annually, while that of stock B is 11%.

(a) If you currently hold a well-diversified portfolio, would you choose to add either of these stocks to your holdings?

(b) If instead you could invest only in bills plus only one of these stocks, which stock would you choose? Explain your answer using either a graph or a quantitative measure of the attractiveness of the stocks.Answer:

(a) The alpha of Stock A is:

(A = rA [rf + (A (rM rf)]

=> (A = 11 [6 + 0.8(12 6)] = 0.2%.

For Stock B,

(B = rB [rf + (B (rM rf)]

=> (B = 14 [6 + 1.5(12 6)] = -1%.

Thus, Stock A would be a good addition. Taking a short position in B may be desirable.

(b) The reward to variability ratio of the stocks is:

SA = (11 6)/10 = 0.50

SB = (14 6)/11 = 0.73

Stock B is superior when only one can be held.