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Risk and Return – Introduction Chapter 9 For 9.220,

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Introduction It is important to understand the relation between risk and return so we can determine appropriate risk-adjusted discount rates for our NPV analysis. At least as important, the relation between risk and return is useful for investors (who buy securities), corporations (that sell securities to finance themselves), and for financial intermediaries (that invest, borrow, lend, and price securities on behalf of their clients).

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What is risk? Definition: risk is the potential for divergence between the actual outcome and what is expected. In finance, risk is usually related to whether expected cash flows will materialize, whether security prices will fluctuate unexpectedly, or whether returns will be as expected.

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Measuring Performance: Returns Dollar return (over one period): = Dividends + End of Period Price – Beginning of Period Price Percentage return (over one period): =Dollar return/Beginning of Period Price =(Dividends + End of Period Price)/Beginning of Period Price -1 Notating Percentage Returns Unless otherwise indicated, the word “return” refers to % returns (either in % or decimal form). Dividend at time = t Capital gain Dividend yield Capital gain yield

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Q.Suppose a stock had an initial price of $42 per share, paid a dividend of $0.84 per share during the year, and had an ending price of $46.2. Calculate: a. Percentage total return b. Dividend yield c. Capital gains yield Historic returns - Example 1

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Returns - Example 1 Outflows Total inflows = $47.04 Dividends = $0.84 Ending Market Value = $46.20 t – $42.00 Time: t - 1

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A. a. percentage total return Find R????? b. Find dividend yield ???? c. Findcapital gains yield = ????? Historic returns - Example 1

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uHolding Period Return HPR = (1 + R 1 )(1 + R 2 )... (1+ R T ) -1 u(Geometric) Average Return GAR = [(1 + R 1 )(1 + R 2 )... (1+ R T )] 1/T -1 = [1 + HPR] 1/T -1 u(Arithmetic) Average Return Returns

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Q. The following are TSE 300 returns for the 1994-1997 period: Year (t)Return (R t ) 1994-0.18% 199514.53 199628.35 199714.98 Calculate: a. holding period return (HPR) b. geometric average return (GAR) c. arithmetic average return (R) Historic returns - Example 2

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A. a. holding period return _______ b. geometric average return _________ c. arithmetic average return (mean) R(bar) =______ Historic returns - Example 2

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The risk premium Definition: the risk premium is the return on a risky security minus the return on a risk-free security (often T-bills are used as the risk-free security) Another name for a security’s risk premium is the excess return of the risky security. The market risk premium is the return on the market (as a whole) minus the risk-free rate of return. We may talk about the past observed risk premium, the average risk premium, or the expected risk premium.

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Risk measures Studies of stock returns indicate they are approximately normally distributed. Two statistics describe a normal distribution, the mean and the standard deviation (which is the square root of the variance). The standard deviation shows how spread out is the distribution. For stock returns, a more spread out distribution means there is a higher probability of returns being farther away from the mean (or expected return). For our estimate of the expected return, we can use the mean of returns from a sample of stock returns. For our estimate of the risk, we can use the standard deviation or variance calculated from a sample of stock returns.

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While return measures reward, we need some measure of uncertainty (variability) associated with that return Source: William M. Mercer Ltd. The average Risk Premium for Common Stocks in Canada from 1948 to 1997 was 7% Do you see evidence of risk?

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uHistoric Return Variance: Average value of squared deviations from the mean. A measure of volatility. uHistoric Standard Deviation: Also measures volatility. Other return statistics

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uHistoric Return Variance: uHistoric Standard Deviation: root of variance Other return statistics - An example

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Calculating Variance with a Table Expected return Deviation from Mean Squared Deviation ---------------------------------------------------------------- ______________________________________________ Mean = ____ Divide sqd dev with (n-1) to get ______ Then Find _____

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How to interpret the standard deviation as a measure of risk Given a normal distribution of stock returns … there is about a 68.26% probability that the actual return will be within 1 standard deviation of the mean. there is about a 95.44% probability that the actual return will be within 2 standard deviations of the mean. There is about a 99.74% probability that the actual return will be within 3 standard deviations of the mean.

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Using Return Statistics The Normal Distribution (based on TSE 300 1994-1997 return data) Probability Return on TSE300 stocks 68.26% 95.44% >99.74% – 3 - 20.56% – 2 - 8.90% – + 2.76% + 14.42% + + 26.08% + 2 + 37.74% + 3 + 49.4% Prob. (-8.90% r 37.74%) = 95.44% Prob. (r 37.74%) = 4.56%

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ArithRiskStandard MeanPremiumDeviation U.S. Common stocks13.0%9.2% 20.3% Long-term U.S. corporate bonds6.1 2.3 8.7 Long-term U.S. government bonds5.4 1.6 5.7 U.S. Treasury bills3.80 3.2 Risk premium Arithmetic (relative to U.S. Standard Series mean Treasury bills) deviation -90%90%0% Modified from Stocks, Bonds, Bills and Inflation: 1998 Yearbook, TM annual updates work by Roger C. Ibbotson and Rex A. Sinquefield (Chicago: Ibbotson Associates). All rights reserved. -90%90%0% -90%90%0% -90%90%0% U.S. Historical Return Statistics Note: as risk increases, so does the excess return on a risky asset

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Summary and conclusions We can easily calculate $, %, holding period, geometric average, and mean returns from a sample of returns data. We can also do the same for a security’s risk premium. The mean and standard deviation calculated from sample returns data are often used as estimates of expected returns and the risk measure for a security or for the market as a whole.

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