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**Chapter 6 Trade-Off Between Risk & Return**

Risk, Return, and the CAPM

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**Today’s Chapter 6 & 7 Topics**

Historical Trade-Off between Risk and Return Historical Risk Premiums Calculation of Historical Return and Risk Portfolio Return and Risk Calculation of Probabilistic Expected Return & Risk Risk Diversification Unsystematic & Systematic Risk

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Risk and Return Valuing risky assets - a task fundamental to financial management Three-step procedure for valuing a risky asset 1. Determine the asset’s expected cash flows 2. Choose discount rate that reflects asset’s risk 3. Calculate present value (PV cash inflows - PV outflows) The three-step procedure is called discounted cash flow (DCF) analysis.

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**Quick Review: Financial Return**

Total return: the total gain or loss experienced on an investment over a given period of time Components of the total return Income stream from the investment Capital gain or loss due to changes in asset prices Total return can be expressed either in dollar terms or in percentage terms.

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**Quick Review: Dollar and Percentage Returns**

Total dollar return = income + capital gain or loss Percentage return: total dollar return divided by the initial investment

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**Percentage Returns on Bills, Bonds, and Stocks, 1900 - 2003**

Difference between average return of stocks and bills = 7.6% Difference between average return of stocks and bonds = 6.5% Risk premium: the difference in returns offered by a risky asset relative to the risk-free return available

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**Variability of Stock Returns**

Normal distribution can be described by its mean and its variance. Variance (2) - the expected value of squared deviations from the mean Units of variance (%-squared) - hard to interpret, so calculate standard deviation, a measure of volatility equal to square root of 2

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**Volatility of Asset Returns**

Asset classes with greater volatility pay higher average returns. Average return on stocks is more than double the average return on bonds, but stocks are 2.5 times more volatile.

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**Average Returns and St. Dev. for Asset Classes, 1900-2003**

Stocks Bills Bonds Standard Deviation (%) Investors who want higher returns have to take more risk The incremental reward from accepting more risk seems constant

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**Probabilistic Expected Return**

Expected Rate of Return given a probability distribution of possible returns (ri): E(r) n E(R) = S Pi Ri i=1

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**Probabilistic Standard Deviation**

Relevant Risk Measure for single asset Variance = s2 = S pi( ri - E(r))2 Standard Deviation = Square Root of Variance

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**Example: Exp. Return and s**

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**Example: Standard Deviation**

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**Portfolio Risk and Return**

E(rp) = S wiE(ri) = weighted average of the expected return of each asset in the portfolio In our example, MAD E(r) = 33.5% and CON E(r) = 7.5% What is the expected return of a portfolio consisting of 70% MAD and 30% CON?

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**Risk and Diversification**

E(rp) = S wiE(ri) = .7(33.5%) + .3(7.5%) = 25.7% 19

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Portfolio Risk Looking at a 2-asset portfolio for simplicity, the riskiness of a portfolio is determined by the relationship between the returns of each asset over different scenarios or over time. This relationship is measured by the correlation coefficient( r ): -1<= r < =+1 Lower r = less portfolio risk compared to the weighted average of the standard deviations.

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**Example 70% MAD, 30% CON Portfolio s**

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**Average Return and St. Dev. for Individual Securities, 1994-2003**

For various asset classes, a trade-off arises between risk and return. Does the trade-off appear to hold for all individual securities?

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**Average Return and St. Dev. for Individual Securities, 1994-2003**

Wal-Mart Anheuser-Busch American Airlines Archer Daniels Midland Standard Deviation (%) No obvious pattern here

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Diversification Most individual stock prices show higher volatility than the price volatility of portfolio of all common stocks. How can the standard deviation for individual stocks be higher than the standard deviation of the portfolio? Diversification: investing in many different assets reduces the volatility of the portfolio. The ups and downs of individual stocks partially cancel each other out.

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**The Impact of Additional Assets on the Risk of a Portfolio**

Number of Stocks Systematic Risk Portfolio of 11 stocks AMD Unsystematic Risk AMD + American Airlines AMD + American Airlines + Wal-Mart Portfolio Standard Deviation

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**Systematic and Unsystematic Risk**

Diversification reduces portfolio volatility, but only up to a point. Portfolio of all stocks still has a volatility of 21%. Systematic risk: the volatility of the portfolio that cannot be eliminated through diversification. Unsystematic risk: the proportion of risk of individual assets that can be eliminated through diversification What really matters is systematic risk….how a group of assets move together.

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**Systematic and Unsystematic Risk**

The tradeoff between standard deviation and average returns that holds for asset classes does not hold for individual stocks. Because investors can eliminate unsystematic risk through diversification, market rewards only systematic risk. Standard deviation contains both systematic and unsystematic risk.

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Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 14.1 Value at Risk Chapter 14.

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 14.1 Value at Risk Chapter 14.

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