1. Risk and Return Primer

2. Expectations Expected value (μ) is weighted sum of possible outcomes E(X) = μ = p 1 X 1 + p 2 X 2 + …. p s X s  E(X) – Expected value of X  X i – Outcome of X in state i  p i – Probability of state i  s – Number of possible states  Probabilities have to sum to 1 p 1 + p 2 + …..+ p s = 1 2

3. 3 Horse Race There are three horse racing in the Finance Derby. Your horse is “Love of NPV”. If your horse has a 30% chance of coming in first, and a 40% chance of coming in second. How much do you expect your horse to win?  1 st pays $1,500  2 nd pays $750  3 rd pays $250

4. 4 Horse Race There are three horse racing in the Finance Derby. Your horse is “Love of NPV”. If your horse has a 30% chance of coming in first, and a 40% chance of coming in second. How much do you expect your horse to win?  1 st pays $1,500, 2 nd pays $750, 3 rd pays $250 Chance of coming in 3 rd : 1-0.3-0.4 = 0.3 0.3*1,500 + 0.4*750 + 0.3*250 = $825

5. 5 What is risk? Uncertainty

6. 6 Measuring Risk There is no universally agreed-upon measure  However, variance and standard deviation are both widely accepted measures of total risk

7. 7 Statistics Review: Variance Variance ( σ 2 ) measures the dispersion of possible outcomes around μ Standard deviation ( σ ) is the square root of variance Higher variance (std dev), implies a higher dispersion of possible outcomes  More uncertainty

8. 8 Different Variances

9. 9 Variance Calculation Variance = σ 2 = Σp i * (X i – μ) 2 : Use this one  Alternative formulas you may have seen σ 2 = Σ(X i – μ) 2 / N σ 2 = Σ(X i – μ) 2 / (N-1)  All give similar answers with large samples  BUT each give very different answers with small samples Ex. s=3 σ 2 = p 1 * (X 1 – μ) 2 + p 2 * (X 2 – μ) 2 + p 3 * (X 3 – μ) 2

10. 10 Risk Example Economy is “Good” with 20% probability DJIA will return 20% Economy is “Fair” with 30% probability DJIA will return 5% Economy is “Bad” with 50% probability DJIA will return -9%

11. 11 Calculations Expected Return = Variance = Standard Deviation =

12. 12 Calculations Expected Return = p 1 X 1 + p 2 X 2 + p 3 X 3 = 0.2*0.20+0.3*0.05+0.5*(-0.09) = 0.01 Variance = Standard Deviation =

13. 13 Calculations Expected Return = 0.01 Variance = p 1 (X 1 - μ X ) 2 +p 2 (X 2 -μ X ) 2 +p 3 (X 3 -μ X ) 2 = 0.2*(0.20-0.01) 2 + 0.3*(0.05-0.01) 2 + 0.5*(-0.09-0.01) 2 = 0.0127 =127 (%) 2 Standard Deviation =

14. 14 Calculations Expected Return = 0.01 Variance = 0.0127 =127 (%) 2 Standard Deviation = √ σ 2 √0.0127 = 0.113 = 11.3%

15. 15 Historical Data In practice we do not know all of the possible states of the world, so we use historical data to form expectations  Idea: Look at what has happened in the past and we can calculate the mean and variance What is each states probability of occurring?

16. 16 Risk Example 2 Sample Mean = 0.2*0.20+0.2*0.15+0.2*(-0.05)+0.2*0.05+0.2*0.10 = 0.09 = 9% Sample Variance = = 0.2*(0.20-0.09) 2 + 0.2*(0.15-0.09) 2 + 0.2*(-0.05-0.09) 2 + 0.2*(0.05-0.09) 2 + 0.2*(0.10-0.09) 2 = 74% 2 Standard Deviation = √0.0074 = 0.086 = 8.6% 19961997199819992000 20%15%-5%5%10%

17. 17 Risk A risky asset is one in which the rate of return is uncertain. Risk is measured by ________________

18. 18 Risk A risky asset is one in which the rate of return in uncertain. Risk is measured by standard deviation. higher σ → more uncertainty

19. 19 General Securities T-bills are a very safe investment  No default risk, short maturity  Risk free asset Stocks are much riskier Bond’s riskiness is between T-bills and Stocks

20. Why Do We Demand a Higher Return Investors seem to dislike risk (ex. insurance)  Risk Averse If the expected return on T-Bills (risk-free), is 10%, and the expected return for Ford is 10%, which would you buy?  The 10% offered by T-Bills is guaranteed while this is not the case for Ford  A guaranteed 10% dominates a possible 10% 20

21. 21 Return Breakdown A risky asset’s return has two components:  Risk free rate + Risk premium Risk free rate: The return one can earn from investing in T-Bills Risk Premium: The return over and above the risk free rate  Compensation for bearing risk

22. Average Risk Premiums (1926-2005) Small company stocks :  17.4% – 3.8% = 13.6% Large company stocks :  12.3% – 3.8% = 8.5% Long-term corporate bonds :  6.2% – 3.8% = 2.4% The more risk the larger the risk premium 22

23. 23 The Risk-Return Tradeoff Highest Risk & Return: Small Cap Stocks, Large Cap Stocks, L.T. Corp bonds, L.T. Gov Bonds, U.S. T-Bills

24. 24 Quick Quiz Which of the investments discussed has had the highest average return and risk premium? Which of the investments discussed has had the highest standard deviation?

25. Why we care? This is the very basics of investing General knowledge that “finance” people possess 25