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risk_utility - Risk and Utility 1 Risk and Risk Aversion...

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) Emma Rasiel, 2009 1 Risk and Utility 2 You have $100 to invest. Choose between a riskless investment earning 5% and two alternative risky investments that are available in the market: Risk and Risk Aversion $100 riskless investment X= $105 risky investment X= $150 X= $80 p=0.5 1-p=0.5 E[ X ] = $115 115/(1+ r 1 ) = 100 r 1 = 15% E[ X ] = $105 105/(1.05) = 100 very risky investment X= $200 X= $60 p=0.5 1-p=0.5 E[ X ] = $130 130/(1+ r 2 ) = 100 r 2 = 30% Aug 01 Aug 02 PV What is the relation between the discount rate r and risk? What do we mean by "risk" in this context?
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) Emma Rasiel, 2009 3 Example 1 : Suppose you buy two lottery tickets, each costing $5. The prize is a $12,000 car. Eight thousand tickets in total are sold. What is the Expected Value of your winnings? E( X ) = (-10) + (11,990) = -7.00 Example 2 : Suppose you purchase a $10,000 annual insurance policy for full repayment (no deductible) in the event of an outcome which is known to occur roughly twice in every 100 years. How much would you pay for this policy? E( X ) = 0.02 * 10,000 = $200 Expected Values 7998 8000 2 8000 Is this consistent with risk- aversion? 4 If we toss a coin three times, how many heads do we expect to get? E( X ) = Σ x xp(x) = µ For the coin-tossing experiment: E( X ) = 0(0.125) + 1(0.375) + 2(0.375) + 3(0.125) = 1.5 Notes : E( X ) is not necessarily one of the x i E( X ) can be thought of as the long-run average The Mean of a Distribution
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) Emma Rasiel, 2009 5 Estimating Distributions x p(x) 1.00 0.75 0.50 0.25 0.00 p(x) x What is the difference between a histogram and a distribution? 1.00 0.75 0.50 0.25 0.00 6 Histogram of Bond Returns Histogram of Daily Bond returns between 1995-2000 (10 "buckets") 0 20 40 60 80 100 120 -1.76% -1.00% -0.23% 0.53% 1.30% Weekly Return Frequency What is the connection between the frequency (on the y-axis) and p(x) ? Frequency divided by total # observations is the probability of that “bucket”
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) Emma Rasiel, 2009 7 Variance: Graphic Representation x p(x) x p(x) 0 50 100 0 50 100 What is E(X) = µ for these two distributions? 50 What does the area under the curve represent? Cumulative probability Do these distributions have the same Variance? No! 8 Variance & Standard Deviation: Definitions The Variance, Var(X) or σ 2 of a random variable X is: σ 2 = Σ x [ x – µ ] 2 p(x) In words : The variance is the probability-weighted average of the squares of the deviations of the x i from the mean. The Standard Deviation, σ , is the square root of the Variance. σ = { Σ x [ x – µ ] 2 p(x) } 1/2 In finance, we use the term Volatility in place of Standard Deviation. σ = Volatility
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) Emma Rasiel, 2009 9 Variance and Volatility: Examples Example 1 : Example 2 : x i p(x i ) y i p(y i ) 49 0.05 0 0.05 50 0.90 50 0.90 51 0.05 100 0.05 µ x = 49(0.05) + 50(0.9) + 51(0.05) = 50 σ x = {(49 – 50) 2 0.05 + (50 – 50) 2 0.9 + (51 – 50) 2 0.05} 1/2 = 0.32 µ y = 0(0.05) + 50(0.9) + 100(0.05) = 50 σ y = {(0 – 50) 2 0.05 + (50 – 50) 2 0.9 + (100 – 50) 2 0.05} 1/2 = 15.8 10 Estimating Volatility in Financial Markets With financial market data over a long time period, we estimate volatility on returns , not prices (why)? For a “time series” of n +1 daily prices S i (i = 0,1, … , n) the convention for calculating the volatility is as follows: Step 1 : Calculate the daily continuously compounded rate of return: u i = ln(S i / S i-1 ) Step 2 : Calculate the mean return: û = Step 3: Calculate the variance as: σ n 2 = (1/n) Σ i u i Σ i ( u i – û ) 2 1 (n-1) What assumption are we making about probabilities?
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