Understanding Risk and Return: A Primer on Expected Value and Investment Decisions

1. Risk and ReturnPrimer

2. Expectations • Expected value (μ) is weighted sum of possible outcomes • E(X) = μ = p1X1 + p2X2 + …. psXs • E(X) – Expected value of X • Xi – Outcome of X in state i • pi – Probability of state i • s – Number of possible states • Probabilities have to sum to 1 • p1 + p2 + …..+ ps = 1

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? • 1st pays $1,500 • 2nd pays $750 • 3rd pays $250

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? • 1st pays $1,500, 2nd pays $750, 3rd pays $250 • Chance of coming in 3rd: 1-0.3-0.4 = 0.3 • 0.3*1,500 + 0.4*750 + 0.3*250 = $825

5. What is risk? Uncertainty

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

7. Statistics Review: Variance • Variance (σ2) 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. Different Variances

9. Variance Calculation • Variance = σ2 = Σpi * (Xi – μ)2: Use this one • Alternative formulas you may have seen • σ2 = Σ(Xi – μ)2 / N • σ2 = Σ(Xi – μ)2 / (N-1) • All give similar answers with large samples • BUT each give very different answers with small samples • Ex. s=3 σ2 = p1 * (X1 – μ)2 +p2 * (X2 – μ)2 +p3 * (X3 – μ)2

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. Calculations Expected Return = Variance = Standard Deviation =

12. Calculations Expected Return = p1X1 + p2X2 + p3X3 = 0.2*0.20+0.3*0.05+0.5*(-0.09) = 0.01 Variance = Standard Deviation =

13. Calculations Expected Return = 0.01 Variance = p1(X1- μX)2+p2(X2-μX)2+p3(X3-μX)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. Calculations Expected Return = 0.01 Variance = 0.0127 =127 (%)2 Standard Deviation = √σ2 √0.0127 = 0.113 = 11.3%

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. 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%

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

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. 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%

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

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. 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