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**Efficient Diversification**

CHAPTER 6

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**Diversification and Portfolio Risk**

Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio? Uncertainty at the market level Uncertainty at the firm level Market risk Systematic or Nondiversifiable Firm-specific risk Diversifiable or nonsystematic If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk If it is diversified, the total risk has only market risk

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**Diversification and Portfolio Risk**

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**Diversification and Portfolio Risk**

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**Figure 6.1 Portfolio Risk as a Function of the Number of Stocks**

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**Covariance and Correlation**

Why the std (total risk) decreases when more stocks are added to the portfolio? The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond state Prob. stock (%) Bond (%) Portfolio Recession Normal Boom

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**Covariance and Correlation**

What is the E(rs) and σs? What is the E(rb) and σb? What is the E(rp) and σp? E(r) σ Bond Stock Portfolio

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**Covariance and Correlation**

When combining the stocks into the portfolio, you get the average return but the std is less than the average of the std of the 2 stocks in the portfolio Why? The risk of a portfolio also depends on the correlation between 2 stocks How to measure the correlation between the 2 stocks Covariance and correlation

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**Covariance and Correlation**

Prob rs E(rs) rb E(rb) P(rs- E(rs))(rb- E(rb)) Cov (rs, rb) = -114 The covariance tells the direction of the relationship between the 2 assets, but it does not tell the whether the relationship is weak or strong Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99

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**Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of**

returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2

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**Correlation Coefficients: Possible Values**

Range of values for r 1,2 -1.0 < r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated If ρ = 0, no correlation

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**Two Asset Portfolio St Dev – Stock and Bond**

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**Numerical Example: Bond and Stock**

Returns E(Bond) = 6% E(Stock) = 10% Standard Deviation Bond = 12% Stock = 25% Correlation Coefficient (Bonds and Stock) = 0

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**Numerical Example: Bond and Stock**

Case 1: Weights Bond = .5 Stock = .5 What is the E(rp) and σp E(rp) = 8% σp = 13.87% Average std = (25+12)/2 = 18.5 By combining stocks, get average return, but the risk is lower than average

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**Numerical Example: Bond and Stock**

Case 1: Weights Bond = .75 Stock = .25 What is the E(rp) and σp E(rp) = 7% σp = 10.96% By combining you get higher return than bond but lower risk than bond This is power of diversification

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**Two Asset Portfolio St Dev – Stock and Bond**

Std of the portfolio is always smaller than the weighted average of the 2 std in the portfolio. Std of the portfolio is maximized when the correlation = 1, and is minimized when correlation = -1 No diversification benefit when the correlation = 1

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**Figure 6.3 Investment Opportunity Set for Stock and Bonds**

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**Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations**

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**6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET**

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**Extending to Include Riskless Asset**

The optimal combination becomes linear A single combination of risky and riskless assets will dominate

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**Figure 6.5 Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines**

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**Dominant CAL with a Risk-Free Investment (F)**

CAL(O) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of O & F dominate

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**Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills**

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**Figure 6.7 The Complete Portfolio**

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**Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem**

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**6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS**

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**Extending Concepts to All Securities**

The optimal combinations result in lowest level of risk for a given return The optimal trade-off is described as the efficient frontier These portfolios are dominant

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**Figure 6.9 Portfolios Constructed from Three Stocks A, B and C**

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**Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets**

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**Portfolio Selection Asset allocation Security selection**

These two are separable!

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Asset Allocation “Asset allocation accounts for 94% of the differences in pension fund performance” Identify investment opportunities (risk-return combinations) Choose the optimal combination according to investor’s risk attitude

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**Optimal Portfolio Construction**

Step 1: Using available risky securities (stocks) to construct efficient frontier. Step 2: Find the optimal risky portfolio using risk-free asset Step 3: Now We have a risk-return tradeoff, choose your most favorable asset allocation

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**Feasible and Efficient Portfolios**

Expected Portfolio Return, rp Efficient Set Feasible Set Risk, p Feasible and Efficient Portfolios

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**An efficient portfolio is one that offers:**

The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks. An efficient portfolio is one that offers: the most return for a given amount of risk, or the least risk for a give amount of return. The collection of efficient portfolios is called the efficient set or efficient frontier.

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**Optimal Portfolios Expected Return, rp Risk p IB2 IB1 IA2 IA1**

Investor B IA2 IA1 Optimal Portfolio Investor A Risk p Optimal Portfolios

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**What impact does rRF have on the efficient frontier?**

When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets. The straight line connecting rRF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier.

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**Efficient Set with a Risk-Free Asset**

Expected Return, rp Z . B M ^ rM . The Capital Market Line (CML): New Efficient Set . A rRF M Risk, p

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**What is the Capital Market Line?**

The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M. Portfolios below the CML are inferior. The CML defines the new efficient set. All investors will choose a portfolio on the CML.

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**. . Expected Return, rp rRF c M CML I1 M ^ rM C ^ rR C = Optimal**

Portfolio rRF Risk, p c M

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**Disadvantages of the efficient frontier approach**

The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990. However, the approach involved too many inputs, calculations If a portfolio includes only 2 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need? n variances n(n-1)/2 covariances

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Single index model

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Single index model When we diversify, all the specific risk will go away, the only risk left is systematic risk component Now, all we need is to estimate beta1, beta2, ...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before

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Estimate beta Run a linear regression according to the index model, the slope is the beta For simplicity, we assume beta is the measure for market risk Beta = 0 Beta = 1 Beta > 1 Beta < 1

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**Advantages of the Single Index Model**

Reduces the number of inputs for diversification Easier for security analysts to specialize

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