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Class Business

Class Business. Personal Data Sheets Groups Stock-Trak Upcoming Homework. Probability Models. Suppose price to play = $0.85 We can draw a model of net returns: Two-state probability model Two states Two returns Two probabilities. Expected Return.

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Class Business

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  1. Class Business • Personal Data Sheets • Groups • Stock-Trak • Upcoming Homework

  2. Probability Models • Suppose price to play = $0.85 • We can draw a model of net returns: • Two-state probability model • Two states • Two returns • Two probabilities

  3. Expected Return • The expected return to playing this game once is • In general, the expected return of any two-state probability model is • p1 and p2 are the probabilities of the two states • r1 and r2 are the returns received in the two states

  4. Expected Return: Example • Coin flipping game: • Cost: $1 • If heads: $2 • If tails: $1 • Probability of heads: 0.75 • What is expected return from playing?

  5. Expected Return • Suppose • We don’t know the true probability model • But we can observe past data from the game • Then we could estimate the expected return • Find simple average: add-up all values and divide • by the number of values you observe • With many observations, this would be very close to • expected return derived from the probability model 100% 100% 0% 100% 0% 0% 100%

  6. Probability Models • Realistic probability models are very complex and involve an infinite # of possible outcomes. • Example: the normal distribution • To get an estimate of the expected return, it is usually easiest to just estimate simple mean from past data if available. • Simple probability models with only two possible outcomes, though unrealistic, help us understand finance theory.

  7. Uncertainty • Game 1: • 10% return with 50% probability • 20% return with 50% probability • Game 2: • 0% return with 50% probability • 30% return with 50% probability • Which game do you prefer?

  8. Uncertainty • We need a measure of uncertainty. • Both games have expected return of 15%. • How about expected deviation from mean? • Game 1 Deviations from mean: • 10%-15%=-5% with 50% probability • 20%-15%=5% with 50% probability • Expected deviation from mean is zero.

  9. Uncertainty • Game 2 Deviations from mean: • 0%-15%=-15% with 50% probability • 30%-15%=15% with 50% probability • Expected deviation from mean is zero. • The expected deviation from mean will always be zero for any probability model. • Need a more helpful measure

  10. Uncertainty • How about expected squared deviation from mean? • Game 1 squared deviations • (-5%)2=0.0025 • (5%)2= 0.0025 • Expected squared deviation from mean is 0.0025.

  11. Uncertainty • Game 2 squared deviations • (-15%)2=0.0225 • (15%)2= 0.0225 • Expected squared deviation from mean is 0.0225. • Expected squared deviations: • Game 1: 0.0025 • Game 2: 0.0225

  12. Uncertainty • VARIANCE: • Expected squared deviation from mean • STANDARD DEVIATION: • Square-root of the variance

  13. Uncertainty: Example • Coin flipping game: • Cost: $1 • If heads: $2 • If tails: $1 • Probability of heads: 0.75 • What is variance of this game? • What is the standard deviation?

  14. Uncertainty • Suppose • We don’t know the true probability model • But we can observe past data from the game • The we could estimate the variance by • Estimating expected return (simple average) • Finding squared deviation for each outcome • Take simple average of squared deviations • We could estimate the standard deviation as • Square-root of estimated variance

  15. Uncertainty • Example: Suppose for coin flipping game we observe the following outcomes: • 100%, 0%, 100%, 0% • Estimated expected return: 50% • Deviations: • 50%, -50%, 50%, -50% • Squared Deviations: • 0.25, 0.25, 0.25, 0.25 • Estimated Variance: 0.25 • Std. Deviation: .50 • From True probability model: • Expected return=75% • Variance = 0.1875 • Std. Deviation: .4330

  16. Variance • We often use • s2 to represent variance • s to represent standard deviation • Later in the course we will look at how risk is measured for portfolios that will include covariation as well as standard deviation

  17. What does Standard Deviation Tell Us? • Helps us measure likelihood of extreme outcomes. Prob(return < 1 standard deviation from mean) = 16%

  18. Probability of Extreme Bad Events • Example: Your portfolio has an expected return of 10% with a standard deviation of 0.16 over the next year. • What is probability that realized return is <-22%?

  19. Probability of Extreme Bad Events 1. How many standard deviations is outcome from mean? -.22 .10 2 Standard Deviations (.16)

  20. Probability of Extreme Bad Events 2. Use excel function normsdist(z) This function gives probability of getting z standard deviations from mean or less. normsdist(-2) = 0.02275 = 2.275%

  21. Data vs. Probability Model • Note that probability models are forward looking. They tell us about what we should expect in the future. • Estimates of means and variances from historical data are backward looking. They tell us about what happened in the past. • The hope is that the past will be indicative of the future.

  22. The Historical Record Arith. Stan. Series Mean% Dev.% Lg. Stk 12.49 20.30 Sm. Stk 18.29 39.28 LT Gov 5.53 8.18 T-Bills 3.85 3.25 Inflation 3.15 4.40

  23. Real Rates of Return • Suppose at the beginning of the year, the cost of a pizza is $10.00. You have $100 in cash. You could buy 10 pizzas, but instead, you invest the $100 in a long term gov. bond. The return on the bond is 5%. Inflation over the year is 3%. • The investment provides you a nominal income at year end of 100(1.05) = $105. • At year end, the cost of a pizza is 10.00(1.03)=$10.30. • At year end, you could buy 10.19 pizzas (105/10.3)=10.19. • Your real return is therefore only ____?%

  24. Real Rates of Return • C = amount of cash at beginning of period • P = price of a good at beginning of period • rn = nominal rate of return, rr = real return • i = inflation rate • The real (gross) rate of return was found above by solving the following equation • Since

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