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Chapter 6

Risk and Risk Aversion. Chapter 6. Risk - Uncertain Outcomes. p = .6. W 1 = 150 Profit = 50. W = 100. 1-p = .4. W 2 = 80 Profit = -20. E(W) = pW 1 + (1-p)W 2 = 6 (150) + .4(80) = 122 s 2 = p[W 1 - E(W)] 2 + (1-p) [W 2 - E(W)] 2 = 0.6 (150-122) 2 + 0.4(80-122) 2 = 1,176,000

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Chapter 6

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  1. Risk and Risk Aversion Chapter 6

  2. Risk - Uncertain Outcomes p = .6 W1 = 150 Profit = 50 W = 100 1-p = .4 W2 = 80 Profit = -20 E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122 s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 = 0.6 (150-122)2 + 0.4(80-122)2 = 1,176,000 s = 34.293

  3. Risky Investments with Risk-Free p = .6 W1 = 150 Profit = 50 Risky Inv. 1-p = .4 100 W2 = 80 Profit = -20 Risk Free T-bills Profit = 5 Expected profit on risky is: 0.6*50 + 0.4*(-20) = 22 Risk Premium = 22 – 5 = 17 (on a 100 investment)

  4. Risk Aversion & Utility • Investor’s view of risk • Risk Averse – must be paid to take a fair bet • Risk Neutral – is indifferent to a fair bet • Risk Seeking – will pay you to take a fair bet • Utility – a measure of gain from taking an investment (rather than measuring $) • Utility Function (many possible) U = E ( r ) - 0.005 As2 A = measure of the degree of risk Aversion 0.005 is a scaling factor to make the range of A more pleasing when expressing the inputs as percent rather than decimals

  5. Certainty Equivalent • A Certainty Equivalent is the amount you would take for sure, in place of a risky bet. • It is simply the value of the utility, for the function noted above • If the certainty equivalent is lower than the risk free rate, the investor should choose the risk free asset • A risk lover would have a negative value for A in the Utility formula

  6. Risk Aversion and Value: U = E (r) ― 0.005 A s2 = 22 ― .005 A (34) 2 Risk Aversion A Value High 5 -6.90 3 4.66 Low 1 16.22 T-bill = 5% For T-bill: U = E(r) = 5

  7. Mean Variance Dominance D dominates C if: E(rD) ≥ E(rC) and σD≤ σC And at least one inequality is strict (i.e. two equal things cannot dominate one another)

  8. Dominance Principle Expected Return 4 2 3 1 Variance or Standard Deviation •2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return

  9. Utility and Indifference Curves • Represent an investor’s willingness to trade-off return and risk. • Example (for A=4) Exp RetSt DeviationU=E ( r ) - .005As2 10 20.0 2 15 25.5 2 20 30.0 2 25 33.9 2

  10. Indifference Curves Expected Return Note: Where the indifference curve hits the y axis is the certain equivalent Increasing Utility Standard Deviation

  11. Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

  12. Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

  13. Return on a Portfolio Rule 3: The expected rate of return on a portfolio is a weighted average of the expected rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. Example: A two asset portfolio E(rp ) = w1E(r1)+ w2E(r2) w1 = Proportion of funds in Security 1 w2 = Proportion of funds in Security 2 E(r1) = Expected return on Security 1 E(r2) = Expected return on Security2 Note: w1 + w2 = 1

  14. Portfolio Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, Rule 5 shows that the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.

  15. Covariance and Correlation • Covariance measures the tendency for two random variable to move together • For example, it makes sense that the returns on Ford and the returns on GM will usually move together (Why?) • Calculated like a variance, but for two assets

  16. Covariance and Correlation • Covariance can be positive or negative and from plus to minus infinity, so often one chooses to standardize it to the -1 though 1 interval using a measure called covariance

  17. Portfolio Risk Rule 5: When two risky assets with variances s12 and s22, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by: p2= w1212 + w2222 + 2w1ww2 Cov(r1r2) Cov(r1r2) = Covariance of returns for Security 1 and Security 2

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