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1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc Author: Nick Bagley Objective To understand the theory of personal portfolio selection in theory and in practice

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2 Chapter 12 Contents 12.1 The process of personal portfolio selection12.1 The process of personal portfolio selection 12.2 The trade-off between expected return and risk12.2 The trade-off between expected return and risk 12.3 Efficient diversification with many risky assets12.3 Efficient diversification with many risky assets

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3 Objectives To understand the process of personal portfolio selection in theory and practiceTo understand the process of personal portfolio selection in theory and practice

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4 Introduction How should you invest your wealth optimally?How should you invest your wealth optimally? –Portfolio selection Your wealth portfolio containsYour wealth portfolio contains –Stock, bonds, shares of unincorporated businesses, houses, pension benefits, insurance policies, and all liabilities

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5 Portfolio Selection Strategy There are general principles to guide you, but the implementation will depend such factors as your (and your spouse’s)There are general principles to guide you, but the implementation will depend such factors as your (and your spouse’s) –age, existing wealth, existing and target level of education, health, future earnings potential, consumption preferences, risk preferences, life goals, your children’s educational needs, obligations to older family members, and a host of other factors

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The Process of Personal Portfolio Selection Portfolio selectionPortfolio selection –the study of how people should invest their wealth –process of trading off risk & expected return to find the best portfolio of assets & liabilities Narrower dfn: consider only securitiesNarrower dfn: consider only securities Wider dfn: house purchase, insurance, debtWider dfn: house purchase, insurance, debt Broad dfn: human capital, educationBroad dfn: human capital, education

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The Life Cycle The risk exposure you should accept depends upon your ageThe risk exposure you should accept depends upon your age Consider two investments (rho=0.2)Consider two investments (rho=0.2) –Security 1 has a volatility of 20% and an expected return of 12% –Security 2 has a volatility of 8% and an expected return of 5%

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8 Price Trajectories The following graph show the the price of the two securities generated by a bivariate normal distribution for returnsThe following graph show the the price of the two securities generated by a bivariate normal distribution for returns –The more risky security may be thought of as a share of common stock or a stock mutual fund –The less risky security may be thought of as a bond or a bond mutual fund

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10 Interpretation of the Graph The graph is plotted on a log scale in so that you can see the important featuresThe graph is plotted on a log scale in so that you can see the important features The magenta bond trajectory is clearly less risky than the navy-blue stock trajectoryThe magenta bond trajectory is clearly less risky than the navy-blue stock trajectory The expected prices of the bond and the stock are straight lines on a log scaleThe expected prices of the bond and the stock are straight lines on a log scale

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11 Interpretation of the Graph Recall the log scale: the volatility increases with the length of the investmentRecall the log scale: the volatility increases with the length of the investment You begin to form the conjecture that the chances of the stock price being less than the bond price is higher in earlier yearsYou begin to form the conjecture that the chances of the stock price being less than the bond price is higher in earlier years

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12 Generating More Trajectories This was just one of an infinite number of trajectories generated by the same 2 means, 2 volatilities, and the correlationThis was just one of an infinite number of trajectories generated by the same 2 means, 2 volatilities, and the correlation –I have not cheated you, this was indeed the first trajectory generated by the statistics –the following trajectories are not reordered nor edited Instructor: On slower computers there may be a delayInstructor: On slower computers there may be a delay

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15 …and Lots More!

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16 From Conjecture to Hypothesis You are probably ready to make the hypothesis thatYou are probably ready to make the hypothesis that –the probability of the high-risk, high-return security will out-perform the low-risk, low- return increases with time

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17 But: I promised to be perfectly frank and honest (pfah) with you about the ordering of the simulated trajectoriesI promised to be perfectly frank and honest (pfah) with you about the ordering of the simulated trajectories The next trajectory truly was the next trajectory in the sequence, honest!The next trajectory truly was the next trajectory in the sequence, honest!

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19 Implication for Investors –If you are older, the average remaining life of the investment is relatively short, and there is a larger probability that an investment in the risky security will result in a loss –This is not serious if you have substantial assets, in which case you can afford to take the risk, and enjoy higher expected returns

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20 Implication for Investors –If you are younger, the average remaining life of retirement investment is longer, and there is only a small probability that an investment in the risky security will be less than the “safer” one –Investing in the less risky security will almost always result in a significantly smaller retirement income

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21 Implication for Investors –Relatively early during a typical life cycle, there may be a need to liquidate some invested funds, perhaps for a house deposit, a child’s education, or an uninsured medical emergency –In the case where liquidating an investment early may damage long-term goals, some precautionary funds should be kept in lower- risk securities

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Time Horizons –Planning horizon The total length of time for which one plansThe total length of time for which one plans –Decision horizon The length of time between decisions to revise a portfolioThe length of time between decisions to revise a portfolio –Trading horizon The shortest possible time interval over which investors may revise their portfoliosThe shortest possible time interval over which investors may revise their portfolios

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Risk Tolerance Your tolerance for bearing risk is a major determinant of portfolio choicesYour tolerance for bearing risk is a major determinant of portfolio choices –It is the mirror image of risk aversion –Whatever its cause, we do not distinguish between capacity to bear risk and attitude towards risk

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Role of Professional Asset Managers Most people have neither the time nor the skill necessary to optimize a portfolio for risk and returnMost people have neither the time nor the skill necessary to optimize a portfolio for risk and return –Professional fund managers provide this service as individually designed solutions to the precise needs of a customer ($$$$)individually designed solutions to the precise needs of a customer ($$$$) a set of financial products which may be used together to satisfy most customer goals ($$)a set of financial products which may be used together to satisfy most customer goals ($$)

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Trade-Off between Expected Return and Risk Assume a world with a single risky asset and a single riskless assetAssume a world with a single risky asset and a single riskless asset The risky asset is, in the real world, a portfolio of risky assetsThe risky asset is, in the real world, a portfolio of risky assets The risk-free asset is a default-free bond with the same maturity as the investor’s decision (or possibly the trading) horizonThe risk-free asset is a default-free bond with the same maturity as the investor’s decision (or possibly the trading) horizon

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26 Trade-Off between Expected Return and Risk The assumption of a risky and riskless security simplifies the analysisThe assumption of a risky and riskless security simplifies the analysis

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27 Combining the Riskless Asset and a Single Risky Asset Assume that you invest w proportion of your wealth in a risky security and (1-w) proportion of your wealth in a riskless securityAssume that you invest w proportion of your wealth in a risky security and (1-w) proportion of your wealth in a riskless security Let r s and r f be the returns on the risky security and the riskless security, respectively.Let r s and r f be the returns on the risky security and the riskless security, respectively.

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28 Combining the Riskless Asset and a Single Risky Asset Your statistics background tells you how to determine the expected return and volatility of any two-security portfolioYour statistics background tells you how to determine the expected return and volatility of any two-security portfolio Form a new random variable, the return of the portfolio, r P, from the two security return variables, r s and r fForm a new random variable, the return of the portfolio, r P, from the two security return variables, r s and r f r P = w*r s + (1-w) r f r P = w*r s + (1-w) r f

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29 Combining the Riskless Asset and a Single Risky Asset –The expected return of the portfolio is the weighted average of the expected returns on the securities: E(r P )= w*E(r s ) + (1-w)*r f E(r P )= w*E(r s ) + (1-w)*r f or, or, E(r P )= r f + w*(E(r s ) - r f ) E(r P )= r f + w*(E(r s ) - r f )

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30 Combining the Riskless Asset and a Single Risky Asset –The volatility of the portfolio is not quite as simple: Variance = P = (w * s ) 2 + Variance = P = (w * s ) 2 + 2w*(1-w) s * f + ((1-w)* f ) 2 2w*(1-w) s * f + ((1-w)* f ) 2 Std. dev. = P = ((w * s ) 2 + 2w*(1-w) s * f + ((1-w)* f ) 2 ) 1/2 Std. dev. = P = ((w * s ) 2 + 2w*(1-w) s * f + ((1-w)* f ) 2 ) 1/2

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31 Combining the Riskless Asset and a Single Risky Asset –We know something special about the portfolio, namely that riskless security has zero standard deviation or f = 0, and P becomes: P = ((w * s ) 2 + 2w*(1-w) s *0 + ((1-w)* 0) 2 ) 1/2 P = |w| * s and w = P s

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32 Combining the Riskless Asset and a Single Risky Asset Case 1: w > 0 Case 1: w > 0 Substituting w = P s from the previous slide into the last equation on slide 29 we get: Substituting w = P s from the previous slide into the last equation on slide 29 we get: E(r P )= r f + [(E(r s ) - r f )/ s ] P E(r P )= r f + [(E(r s ) - r f )/ s ] P

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33 Combining the Riskless Asset and a Single Risky Asset Case 2: w < 0 Case 2: w < 0 Substituting w = - P s from slide 31 into the last equation on slide 29 we get: Substituting w = - P s from slide 31 into the last equation on slide 29 we get: E(r P )= r f - [(E(r s ) - r f )/ s ] P E(r P )= r f - [(E(r s ) - r f )/ s ] P

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34 Illustration Consider the set of all portfolios that may be formed by investing (long and or short) inConsider the set of all portfolios that may be formed by investing (long and or short) in –a risky security with a volatility of 20% and an expected return of 15% –a riskless security with a volatility of 0% and a known return of 5%

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36 Sub-Optimal Investments Investments on the higher part of the line (i.e., case 1 on slide 32) are always preferred to investments on the lower part of the line (i.e., case 2 on slide 33) so for our current purposes we may ignore the lower line. That is, we will not sell the risky asset short and invest the proceeds in the riskless securityInvestments on the higher part of the line (i.e., case 1 on slide 32) are always preferred to investments on the lower part of the line (i.e., case 2 on slide 33) so for our current purposes we may ignore the lower line. That is, we will not sell the risky asset short and invest the proceeds in the riskless security

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37 Observations –An investor with a low risk tolerance may invest in a portfolio containing a small % of risky securities, and a correspondingly higher % of riskless securities –An investor with a high tolerance for risk may sell risk-free securities he does not own (also called short selling), and invest the proceeding in the risky investment –They both use the same two securities

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38 Achieving a Target Expected Return (1) Suppose you want an expected return of 20% on your portfolio. What should be the allocation between the risky security and the riskless security?Suppose you want an expected return of 20% on your portfolio. What should be the allocation between the risky security and the riskless security? Assume E(r s ) = 15%, s = 20%, and r f = 5%Assume E(r s ) = 15%, s = 20%, and r f = 5% Compute w and 1-wCompute w and 1-w

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39 Achieving a Target Expected Return (1) Rearranging the last equation on slide 29 we get: w = (E(r P ) - r f )/(E(r s ) - r f )Rearranging the last equation on slide 29 we get: w = (E(r P ) - r f )/(E(r s ) - r f ) w = ( )/( ) = 150% w = ( )/( ) = 150% 1-w = 100% - 150% = -50%1-w = 100% - 150% = -50%

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40 Achieving a Target Expected Return (1) Assume that you manage a $50,000,000 portfolioAssume that you manage a $50,000,000 portfolio A w of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the differenceA w of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the difference Borrowing at the risk-free rate is requiredBorrowing at the risk-free rate is required

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41 Achieving a Target Expected Return (1) How risky is this strategy?How risky is this strategy? P = |w| * s = 1.5 * 0.20 = 0.30 The portfolio has a volatility of 30%The portfolio has a volatility of 30%

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42 Important Observation It doesn’t require much skill to leverage a portfolio; stockbrokers will let most investors trade “on margin”It doesn’t require much skill to leverage a portfolio; stockbrokers will let most investors trade “on margin” When evaluating an investment’s performance, you must examine both the risk and the expected returnWhen evaluating an investment’s performance, you must examine both the risk and the expected return

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43 Returning to the Example You can leverage the funds expected returns up or downYou can leverage the funds expected returns up or down If you want an expected returns of 10%, or, 20%, 30%, 40%, 50%, 60%… you can have it (under the condition you can continue to borrow at the risk-free rate)If you want an expected returns of 10%, or, 20%, 30%, 40%, 50%, 60%… you can have it (under the condition you can continue to borrow at the risk-free rate)

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44 Portfolio Efficiency An efficient portfolio is defined as the portfolio that offers the investor the highest possible expected rate of return at a specific riskAn efficient portfolio is defined as the portfolio that offers the investor the highest possible expected rate of return at a specific risk We now investigate more than one risky asset in a portfolioWe now investigate more than one risky asset in a portfolio

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Efficient Diversification with Many Assets We have consideredWe have considered –Investments with a single risky, and a single riskless, security –Investments where each security shares the same underlying return statistics We will now investigate investments with more than one (heterogeneous) stockWe will now investigate investments with more than one (heterogeneous) stock

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46 Portfolio of Two Risky Assets Recall from statistics, that two random variables, such as two security returns, may be combined to form a new random variableRecall from statistics, that two random variables, such as two security returns, may be combined to form a new random variable A reasonable assumption for returns on different securities is the linear model:A reasonable assumption for returns on different securities is the linear model:

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47 Equations for Two Shares The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be trueThe sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be true The expected return on the portfolio is the sum of its weighted expectationsThe expected return on the portfolio is the sum of its weighted expectations

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48 Equations for Two Shares Ideally, we would like to have a similar result for riskIdeally, we would like to have a similar result for risk –Later we discover a measure of risk with this property, but for standard deviation:

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49 Mnemonic There is a mnemonic that will help you remember the volatility equations for two or more securitiesThere is a mnemonic that will help you remember the volatility equations for two or more securities To obtain the formula, move through each cell in the table, multiplying it by the row heading by the column heading, and summingTo obtain the formula, move through each cell in the table, multiplying it by the row heading by the column heading, and summing

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50 Variance with 2 Securities

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51 Variance with 3 Securities

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52 Note: The correlation of a with b is equal to the correlation of b with aThe correlation of a with b is equal to the correlation of b with a For every element in the upper triangle, there is an element in the lower triangleFor every element in the upper triangle, there is an element in the lower triangle – so compute each upper triangle element once, and just double it This generalizes in the expected mannerThis generalizes in the expected manner

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53 Correlated Common Stocks The next slide shows statistics of two common stock with these statistics:The next slide shows statistics of two common stock with these statistics: –mean return 1 = 0.15 –mean return 2 = 0.10 –standard deviation 1 = 0.20 –standard deviation 2 = 0.25 –correlation of returns = 0.90 –initial price 1 = $57.25 –initial price 2 = $72.625

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55 Correlation The two shares are highly correlatedThe two shares are highly correlated –They track each other closely, but even adjusting for the different average returns, they have some individual behavior

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57 Observation Shorting the high-risk, low-return stock, and re-investing in the low-risk, high- return stock, creates efficient portfoliosShorting the high-risk, low-return stock, and re-investing in the low-risk, high- return stock, creates efficient portfolios –Shorting high-risk by 80% of the net wealth crates a portfolio with a volatility of 20% and a return of 19% (c.f. 15% on security 1) –Shorting by 180% gives a volatility of 25%, and a return of 24% (c.f. 10% on security 2)

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58 Observation In order to generate a portfolio that generates the same risk, but with a higher returnIn order to generate a portfolio that generates the same risk, but with a higher return –Compute the weights of the minimum portfolio, W 1 (min-var), W 2 (min-var) (Formulae given later)(Formulae given later) –Use the relationship W i (sub-opt) +W i (opt) = 2 * W i (min-var)W i (sub-opt) +W i (opt) = 2 * W i (min-var)

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59 Formulae for Minimum Variance Portfolio

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60 Formulae for Tangent Portfolio

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61 Example: What’s the Best Return given a 10% SD?

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62 Selecting the Preferred Portfolio The procedure is as followsThe procedure is as follows –Find the portfolio weights of the tangent portfolio of the line (CML) through (0, rf) –Determine the standard deviation and expectation of this point –Construct the equation of the CML –Apply investment criterion

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63 Achieving the Target Expected Return (2): Weights Assume that the investment criterion is to generate a 30% returnAssume that the investment criterion is to generate a 30% return This is the weight of the risky portfolio on the CMLThis is the weight of the risky portfolio on the CML

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64 Achieving the Target Expected Return (2):Volatility Now determine the volatility associated with this portfolioNow determine the volatility associated with this portfolio This is the volatility of the portfolio we seekThis is the volatility of the portfolio we seek

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65 Achieving the Target Expected Return (2): Portfolio Weights

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66 Investment Strategies We have examined two strategies in detail whenWe have examined two strategies in detail when –the volatility is specified –the return is specified Additionally, one of the graphs indicated an approach to take when presented with investor’s risk/return preferencesAdditionally, one of the graphs indicated an approach to take when presented with investor’s risk/return preferences

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67 Portfolio of Many Risky Assets In order to solve problems with more than two securities requires tools such as quadratic programmingIn order to solve problems with more than two securities requires tools such as quadratic programming The “Solve” function in Excel may be used to obtain solutions, but it is generally better to use a software package such as the one that came with this bookThe “Solve” function in Excel may be used to obtain solutions, but it is generally better to use a software package such as the one that came with this book

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68 Chapter Assumptions The theory underlying this chapter is essentially just probability theory, but there are financial assumptionsThe theory underlying this chapter is essentially just probability theory, but there are financial assumptions

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69 –We do not have to assume that the generating process of returns is normal, but we do assume that the process has a mean and a variance. This is may not be the case in real life

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70 –We assumed that the process was generated without inter-temporal correlations. Some investors believe that there is valuable information in old price data that has not been incorporated into the current price--this runs counter to many rigorous empirical studies.

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71 –There are no “hidden variables” that explain some of the noise

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72 –Investors make decisions based on mean- variances alone statistics such as skewness & kurtosis have been ignoredstatistics such as skewness & kurtosis have been ignored

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73 We have made the assumption the we can lend at the risk-free rate, and that we can “short” common stock aggressivelyWe have made the assumption the we can lend at the risk-free rate, and that we can “short” common stock aggressively

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74 Summary There is no single investment strategy that is suitable for all investors; nor for a single investor for his whole lifeThere is no single investment strategy that is suitable for all investors; nor for a single investor for his whole life Time makes risky investments more attractive than safer investmentsTime makes risky investments more attractive than safer investments In practice, diversification has somewhat limited power to reduce riskIn practice, diversification has somewhat limited power to reduce risk

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