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Risk and Return Learning Module

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**Expected Return The future is uncertain.**

Investors do not know with certainty whether the economy will be growing rapidly or be in recession. Investors do not know what rate of return their investments will yield. Therefore, they base their decisions on their expectations concerning the future. The expected rate of return on a stock represents the mean of a probability distribution of possible future returns on the stock.

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Expected Return The table below provides a probability distribution for the returns on stocks A and B State Probability Return On Return On Stock A Stock B % % % % % % % % % % % % The state represents the state of the economy one period in the future i.e. state 1 could represent a recession and state 2 a growth economy. The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%. The last two columns present the returns or outcomes for stocks A and B that will occur in each of the four states.

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Expected Return Given a probability distribution of returns, the expected return can be calculated using the following equation: N E[R] = S (piRi) i=1 Where: E[R] = the expected return on the stock N = the number of states pi = the probability of state i Ri = the return on the stock in state i.

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Expected Return In this example, the expected return for stock A would be calculated as follows: E[R]A = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5% Now you try calculating the expected return for stock B!

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**Expected Return Did you get 20%? If so, you are correct.**

If not, here is how to get the correct answer: E[R]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20% So we see that Stock B offers a higher expected return than Stock A. However, that is only part of the story; we haven't considered risk.

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Measures of Risk Risk reflects the chance that the actual return on an investment may be different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns. We will once again use a probability distribution in our calculations. The distribution used earlier is provided again for ease of use.

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**Measures of Risk E[R]A = 12.5% E[R]B = 20% Probability Distribution:**

State Probability Return On Return On Stock A Stock B % % % % % % % % % % % % E[R]A = 12.5% E[R]B = 20%

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Measures of Risk Given an asset's expected return, its variance can be calculated using the following equation: N Var(R) = s2 = S pi(Ri – E[R])2 i=1 Where: N = the number of states pi = the probability of state i Ri = the return on the stock in state i E[R] = the expected return on the stock

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Measures of Risk The standard deviation is calculated as the positive square root of the variance: SD(R) = s = s2 = (s2)1/2 = (s2)0.5

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Measures of Risk The variance and standard deviation for stock A is calculated as follows: s2A = .2( )2 + .3( )2 + .3( )2 + .2( )2 = sA = ( )0.5 = = 5.12% Now you try the variance and standard deviation for stock B! If you got .042 and 20.49% you are correct.

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Measures of Risk If you didn’t get the correct answer, here is how to get it: s2B = .2( )2 + .3( )2 + .3( )2 + .2( )2 = .042 sB = (.042)0.5 = = 20.49% Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

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**Portfolio Risk and Return**

Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away. From our previous calculations, we know that: the expected return on Stock A is 12.5% the expected return on Stock B is 20% the variance on Stock A is the variance on Stock B is the standard deviation on Stock A is 5.12% the standard deviation on Stock B is 20.49%

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**Portfolio Risk and Return**

The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The weights reflect the proportion of the portfolio invested in the stocks. This can be expressed as follows: N E[Rp] = S wiE[Ri] i=1 Where: E[Rp] = the expected return on the portfolio N = the number of stocks in the portfolio wi = the proportion of the portfolio invested in stock i E[Ri] = the expected return on stock i

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**Portfolio Risk and Return**

For a portfolio consisting of two assets, the above equation can be expressed as: E[Rp] = w1E[R1] + w2E[R2] If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is: E[Rp] = .50(.125) + .50(.20) = 16.25%

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**Portfolio Risk and Return**

The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient. Covariance is a measure that combines the variance of a stock’s returns with the tendency of those returns to move up or down at the same time other stocks move up or down. Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance.

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**Portfolio Risk and Return**

The Covariance between the returns on two stocks can be calculated as follows: N Cov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB]) i=1 Where: sA,B = the covariance between the returns on stocks A and B N = the number of states pi = the probability of state i RAi = the return on stock A in state i E[RA] = the expected return on stock A RBi = the return on stock B in state i E[RB] = the expected return on stock B

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**Portfolio Risk and Return**

The Correlation Coefficient between the returns on two stocks can be calculated as follows: sA,B Cov(RA,RB) Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB) Where: rA,B=the correlation coefficient between the returns on stocks A and B sA,B=the covariance between the returns on stocks A and B, sA=the standard deviation on stock A, and sB=the standard deviation on stock B

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**Portfolio Risk and Return**

The covariance between stock A and stock B is as follows: sA,B = .2( )(.5-.2) + .3( )(.3-.2) + .3( )(.1-.2) +.2( )(-.1-.2) = The correlation coefficient between stock A and stock B is as follows: -.0105 rA,B = (.0512)(.2049) =

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**Portfolio Risk and Return**

Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows: s2p = (wA)2s2A + (wB)2s2B + 2wAwBrA,B sAsB OR s2p = (wA)2s2A + (wB)2s2B + 2wAwB sA,B The Standard Deviation of the Portfolio equals the positive square root of the the variance.

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**Portfolio Risk and Return**

Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B: s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)= sp = = = 1.28% Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated.

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**Capital Asset Pricing Model (CAPM)**

If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured? In important tool is the CAPM. CAPM concludes that the relevant risk of an individual stock is its contribution to the risk of a well-diversified portfolio. CAPM specifies a linear relationship between risk and required return. The equation used for CAPM is as follows: Ki = Krf + bi(Km - Krf) Where: Ki = the required return for the individual security Krf = the risk-free rate of return bi = the beta of the individual security Km = the expected return on the market portfolio (Km - Krf) is called the market risk premium This equation can be used to find any of the variables listed above, given the rest of the variables are known.

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CAPM Example Find the required return on a stock given that the risk-free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2. Ki = Krf + bi(Km - Krf) Ki = 8% + 2(12% - 8%) Ki = 16% Note that you can then compare the required rate of return to the expected rate of return. You would only invest in stocks where the expected rate of return exceeded the required rate of return.

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Another CAPM Example Find the beta on a stock given that its expected return is 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%. 12% = 4% + bi(10% - 4%) bi = 12% - 4% 10% - 4% bi = 1.33 Note that beta measures the stock’s volatility (or risk) relative to the market.

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Chapter Outline Expected Returns and Variances of a portfolio

Chapter Outline Expected Returns and Variances of a portfolio

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