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Risk and Return Primer 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 2 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 3 What is risk? 4 Measuring Risk There is no universally agreed-upon measure However, variance and standard deviation are both widely accepted measures total risk 5 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 6 Different Variances 7 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) Alternatives give very different answers with small samples Ex. s=3 σ2 = p1 * (X1 – μ)2 + p2 * (X2 – μ)2 + p3 * (X3 – μ)2 8 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% 9 Calculations Expected Return = Variance = Standard Deviation = 10 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? 11 Risk Example 2 1996 1997 1998 1999 2000 20% 15% -5% 5% 10% Sample Mean Sample Variance = Standard Deviation = 12 Risk A risky asset is one in which the rate of return is uncertain. Risk is measured by ________________ 13 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 14 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? 15 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 16 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 17 The Risk-Return Tradeoff 18% Small-Company Stocks Annual Return Average 16% 14% Large-Company Stocks 12% 10% 8% 6% T-Bonds 4% T-Bills 2% 0% 5% 10% 15% 20% 25% 30% 35% Annual Return Standard Deviation 18 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? Why is the normal distribution informative? What is the difference between arithmetic and geometric averages? 19 Why we care? This is the very basics of investing General knowledge that “finance” people possess 20