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**CHAPTER 7 and 6 (mostly Ch 7)**

Ch 7: Optimal Risky Portfolios Ch 6: Capital Allocation to Risky Assets

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**Contents Diversification and portfolio risk**

Portfolio of two risky assets Capital allocation and Asset allocation The Markowitz portfolio selection model

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**Diversification and portfolio risk**

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**Systematic and Unsystematic Risk**

Systematic risk is risk that influences a large number of assets. Also called market risk, so it is non-diversifiable. Example: Labor strike, CEO resignation, Produce liability lawsuit, Earnings reports. Unsystematic risk is risk that influences a single company or a small group of companies. Also called unique risk or firm-specific risk, so it is diversifiable. Example: Variability of growth in the money supply, Interest rate changes, GDP changes. Total risk = Systematic risk + Unsystematic risk

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**Pop Quiz: Systematic Risk or Unsystematic Risk?**

The government announces that inflation unexpectedly jumped by 2 percent last month. Systematic Risk One of Big Widget’s major suppliers goes bankruptcy. Unsystematic Risk The head of accounting department of Big Widget announces that the company’s current ratio has been severely deteriorating. Congress approves changes to the tax code that will increase the top marginal corporate tax rate.

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

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**Portfolio Diversification**

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**Why Diversification Works**

Portfolio risk depends on the correlation between the returns of the assets in the portfolio Correlation (or covariance) The tendency of the returns on two assets to move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the portfolio standard deviation. Positively correlated assets tend to move up and down together. Negatively correlated assets tend to move in opposite directions. Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.

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**Why Diversification Works**

Covariance: Correlation Coefficient Correlation is a standardized covariance scaled by the product of two standard deviations, so that it can take a value between -1 and +1. The correlation coefficient is denoted by Corr(RA, RB) or simply, A,B.

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Correlation

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**Combining Negatively Correlated Assets to Diversify Risk**

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**Investing on Metals in Different Stage of Business Cycle**

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**Portfolio of risky assets**

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**Two-Security Portfolio: Return**

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**Two-Security Portfolio: Risk**

= Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E Variance is reduced if the covariance term is negative. Even if the covariance term is positive, the portfolio SD is still less than the weighted average of two SDs on individual asset, unless the two securities are perfectly positively correlated. Initially, portfolio risk can be expressed as follows:

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**Three-Asset Portfolio**

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**The Power of Diversification**

Remember: If we define the average variance and average covariance of the securities as: We can then express portfolio variance as:

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**The Power of Diversification**

The first term approaches zero as n becomes larger. When security returns are uncorrelated, the power of diversification to reduce portfolio risk is unlimited. When the average covariance among security returns is zero, as it is when all risk is firm-specific, portfolio variance can be driven to zero. However, in reality, securities are positively related. In this case, as n increases, portfolio variance remains positive. Although firm-specific risk, represented by the first term, is still diversified away, the second term simply to average Cov as n becomes greater. Thus, the irreducible risk of a diversified portfolio depends on the covariance of the returns of the component securities

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**Correlation Coefficients**

When ρDE = 1, there is no diversification When ρDE = -1, a perfect hedge is possible! These weights drive the standard deviation of the portfolio zero.

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**A Simulation: The Set-up**

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**A Simulation: The Result**

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

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

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**Portfolio Expected Return as a Function of Standard Deviation**

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**The Minimum Variance Portfolio**

The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation, the portfolio with least risk. When correlation is less than +1, the portfolio standard deviation may be smaller than that of either of the individual component assets. When correlation is -1, the standard deviation of the minimum variance portfolio is zero. (i.e., perfect hedge) When correlation is 1, there is no advantage from diversification because the portfolio standard deviation is the simple weighted average of the component asset standard deviation.

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Correlation Effects The amount of possible risk reduction through diversification depends on the correlation. The risk reduction potential increases as the correlation approaches -1. If r = +1.0, no risk reduction is possible. If r = 0, σP may be less than the standard deviation of either component asset. If r = -1.0, a riskless hedge is possible.

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**Opportunity Set and Optimal Portfolio**

The various combinations of risk and return available all fall on a smooth curve. This curve is called an investment opportunity set because it shows the possible combinations of risk and return available from portfolios of these two assets. Suppose now an investor wished to select the optimal portfolio from the opportunity set. The best portfolio will depend on risk aversion. Portfolio to the northeast in opportunity sets provide higher rates of return but impose greater risk Investors with greater risk aversion will prefer portfolios to the southwest, with lower expected return but lower risk. The best trade-off among these numerous choices is a matter of personal preference.

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**The Opportunity Set with Intel and Coca-Cola**

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**Capital allocation with risky asset and risk-free asset**

Some slides in this section cover chapter 6 materials. Capital allocation with risky asset and risk-free asset

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**The Investment Decision**

Top-down process with 3 steps: Capital allocation between the risky portfolio and risk-free asset Asset allocation across broad asset classes Security selection of individual assets within each asset class According to an academic study, a pension plan’s asset allocation policy explained 92% of the return earned.

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**The Risk-Free Asset Only the government can issue default-free bonds.**

Risk-free in real terms only if price indexed and maturity equal to investor’s holding period. T-bills viewed as “the risk-free asset” An asset with zero standard deviation and zero correlation with all other risky assets Provides the risk-free rate of return. Will lie on the vertical axis of a portfolio graph

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**Portfolios of One Risky Asset and a Risk-Free Asset**

It’s possible to create a complete portfolio by splitting investment funds between safe and risky assets. Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the portfolio, while leaving the remaining fraction in risk-free Treasury bills. Let y = portion allocated to the risky portfolio, P 1 – y = portion to be invested in risk-free asset, F.

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**Example rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p**

(1-y) = % in rf

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Example (Ctd.) The expected return on the complete portfolio is the risk-free rate plus the weight of P times the risk premium of P. The risk of the complete portfolio is the weight of P times the risk of P: Note 1: The standard deviation is only a fraction of the volatility of the risky portfolio, based on the amount invested in the risky portfolio. Note 2: As the allocation to the risky asset increases (higher y), expected return increases, but so does volatility., so utility can increase or decrease.

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**Example (Ctd.) Rearrange and substitute y=sC/sP:**

Thus the expected return of the complete portfolio as a function of its standard deviation is a straight line, with intercept rf and slope. This line is called investment opportunity set or capital allocation line (CAL). The slope is called the reward-to-volatility ratio and also called the Sharpe Ratio.

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**Capital Allocation Line: Investment Opportunity Set**

CAL depicts all the risk-return combination available to investors. The reward-to-volatility ratio or the Sharpe Ratio is the slope of CAL equals the increase in the expected return of the complete portfolio per unit of additional standard deviation. If investors can borrow at the risk-free rate at, say, 7%, they can construct portfolios that may be plotted on the CAL to the right of P.

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The Sharpe Ratio From the investors’ perspective, their goal is to maximize the slope of the CAL for any possible portfolio, P. The objective function is the slope: The slope is also the Sharpe ratio.

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Indifference Curve

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Risk Attitude Scenario: Suppose we give you $1,000. Then, you have the following choice to make: You can receive another $500 for sure. You can flip a fair coin. If the coin-flip comes up “heads,” you get another $1,000, but if it comes up “tails,” you get nothing. If you choose A, you are risk-averse.

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**What’s Your Risk Tolerance?**

Just 60 days after you put money into an investment, its price falls 20%. Assuming none of the fundamentals have changed, what would you do? A. Sell to avoid further worry and try something else B. Do nothing and wait for the investment to come back C. Buy more. It was a good investment before; now it’s cheap investment, too.

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Utility Function U = utility E ( r ) = expected return on the asset or portfolio s2 = variance of returns ½ = a scaling factor A = coefficient of risk aversion, or the investor’s degree of risk aversion. The larger values of A, the investor will penalize risky investments more severely.

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**Risk Aversion and Indifference Curve**

Risk-averse or risk-avoiding (A >0) An investor would accept certain payoff, rather than gambling Reject investment portfolios that are fair games or worse. Risk-neutral (A=0) An investor would be indifferent between certain payoff and gambling Judge risky prospects solely by their expected rates of return. Risk-loving (A < 0) An investor would paly a gamble, rather than taking certain payoff Adjusts the expected return “upward” to take into account the “fun” of confronting the prospect’s risk.

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**Finding the Optimal Complete Portfolio Using Indifference Curves**

More risk-averse investors have steeper indifference curves than less risk-averse investors. Steeper curves mean that investors require a greater increase in expected return to compensate for an increase in portfolio risk. More risk averse

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**Risk Tolerance and Optimal Portfolio**

X Y U3 U2 U1 U3’ U2’ U1’

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**Maximizing Utility given Risk Tolerance**

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Capital Allocation Using our numerical example, rf=7%, E(rp)=15%, σp=22%, This particular investor with risk aversion of 4, will invest 41% of the investment budget in risky asset and 59% in the risk-free asset. The Sharpe Ratio = Sp = ( 10.28% - 7% ) / 9.02% = .36

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**Asset allocation with stocks, bonds, and bills**

Some slides in this section cover chapter 6 materials. Asset allocation with stocks, bonds, and bills

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

Suppose we invest in the bond and stock funds, but now we can also invest in risk-free T-bills yielding 5%. Two possible CALs are drawn from the risk-free rate to two feasible portfolios. Asset allocation with stocks, bonds, and Bills. In the previous chapter, we examined the capital allocation decision, the choice of how much of the portfolio to leave in risk-free money market securities vs. in a risky portfolio. Now we have taken a further step, specifying that the risky portfolio comprises a stock and a bond fund. We still need to show how investors can decide n the proportion of their risky portfolios to allocate to the stock versus the bond market. This is an asset allocation decision. Portfolio B dominates portfolio A

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**The Objective Function**

From the investor’s perspective, their goal is to maximize the Sharpe Ratio,

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The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

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**Optimal Complete Portfolio**

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**Determination of the Optimal Overall Portfolio**

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**The Markowitz portfolio selection model**

We can generalize the portfolio construction problem to the case of many risky securities and a risk-free asset. The Markowitz portfolio selection model

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**The Minimum-Variance Frontier of Risky Assets**

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**The Markowitz Efficient Frontier**

Investors can create numerous portfolios combining various assets. Suppose the investment universe consists of 20,000 stocks, 7,000 bonds, 50,000 derivatives, numerous choices of arts, diamonds, real estate properties, and so on. Then, we can create almost infinite number of portfolios. The efficient frontier is a graphical presentation of showing these millions of possibilities. The Markowitz Efficient frontier is the set of portfolios with the maximum return for a given risk AND the minimum risk given a return. For the plot, the upper left-hand boundary is the Markowitz efficient frontier. All the other possible combinations are not efficient. The undesirable portfolios are said to be dominated or inefficient. That is, investors would not hold these portfolios because they could get either more return for a given level of risk, or less risk for a given level of return.

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**Markowitz Portfolio Selection Model**

The first step is to determine the risk-return opportunities available. The portfolio manager needs to estimate the expected return and covariance matrix – n estimates of the expected returns and n2 estimates for variance, and (n2-n)/2 covariances. Next, we identify the optimal portfolio of risky assets by finding the portfolio weight that result in the steepest CAL, or the highest the reward-to-variability ratio. All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations. Finally, we choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio.

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**Capital Allocation Lines with Various Portfolios from the Efficient Set**

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**Capital Allocation and the Separation Property**

The separation property tells us that the portfolio choice problem may be separated into two independent tasks Determination of the optimal risky portfolio is purely technical. Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference.

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**Capital Allocation and the Separation Property, cont’d**

In theory, Different portfolio managers will offer the same P. Investors with varying degrees of risk aversion would be satisfied with a universe of only two mutual funds: a money market fund for risk-free investments and a mutual fund that holds the optimal risky portfolio, P. Everyone invests in P, regardless of their degree of risk aversion. More risk averse investors put more in the risk-free assets Less risk averse investors put more in P. In practice, however, Different managers will estimate different input lists (i.e., estimates for expected returns and covariances), thus deriving different efficient frontiers. The source of the disparity lies in the security analysis.

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**The Markowitz Efficient Frontier with Stocks and Bonds**

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**Efficient Frontier with Ten Stocks Versus Three Stocks**

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**Risk and Return with Multiple Assets**

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**The Efficient Frontier with Two Stocks**

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**The Risk–Return Combinations from Combining a Risk-Free Investment and a Risky Portfolio**

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**The Tangent or Efficient Portfolio**

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**A Simple Spreadsheet Exercise**

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**A simple Numerical illustration for diversification**

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Expected Return Suppose there are two stocks for one year investment horizon: Starcents Jpod Investors agree that the expected return: for Starcents is 25 percent for Jpod is 20 percent Why would anyone want to hold Jpod shares when Starcents is expected to have a higher return? The answer depends on risk. Starcents is expected to return 25 percent But the realized return on Starcents could be significantly higher or lower than 25 percent. Similarly, the realized return on Jpod could be significantly higher or lower than 20 percent.

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**Calculating Expected Returns**

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**Example: Calculating Expected Returns and Variances - Equal State Probabilities**

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**Expected Returns and Variances, Starcents and Jpod**

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**Portfolios: Expected Returns**

The expected return on a portfolio is a linear combination, or weighted average, of the expected returns on the assets in that portfolio. The formula, for “n” assets, is: In the formula: E(RP) = expected portfolio return wi = portfolio weight in portfolio asset i E(Ri) = expected return for portfolio asset i

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**Variance of Portfolio Expected Returns**

Note: Unlike returns, portfolio variance is generally not a simple weighted average of the variances of the assets in the portfolio. If there are “n” states, the formula is: In the formula, VAR(RP) = variance of portfolio expected return ps = probability of state of economy, s E(Rp,s) = expected portfolio return in state s E(Rp) = portfolio expected return Note that the formula is like the formula for the variance of the expected return of a single asset.

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**Example: Calculating Portfolio Expected Returns**

Note that the portfolio weight in Jpod = 1 – portfolio weight in Starcents.

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Variance of Portfolio

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**It is possible to construct a portfolio of risky assets with zero portfolio variance!**

What? How? (Open this spreadsheet, scroll up, and set the weight in Starcents to 2/11ths. Note: This is a continuation of the spreadsheet in slide 14. You can set the weights as =2/11 for Starcents, and you will see that the return is the same in both states, i.e., riskless! (as shown by the standard deviation)

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**Example: Correlation and the Risk-Return Trade-Off, Two Risky Assets**

If you open this spreadsheet, you can see how the correlation affects the shape of the efficient set. Enter correlations as integers, ranging from -100 to +100.

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A Simple Example of Diversification using Historical Returns: ExxonMobil (XOM) and Panera Bread (PNRA)

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