Unformatted text preview: Risk and Return
Reading: Text Ch. 11 Determination of "k" Fact: All investors are naturally risk averse some more than others. Implication: In equilibrium, more risky assets must offer investors a higher expected rate of return or else no one will purchase those assets. Objective: can we come up with a mathematical relationship between risk and return? Ja n 10 20 30 40 50 60 70 0 Ja 80 Price Return & Risk: The Historical Ford Experience Monthly Stock Price: 1/31/1980 to 12/31/2006 n8 Ja 1 n8 Ja 2 n8 3 Ja n8 Ja 4 n8 Ja 5 n8 Ja 6 n8 Ja 7 n8 Ja 8 n8 Ja 9 n9 Ja 0 n9 1 Ja n9 Ja 2 n9 Ja 3 n9 Ja 4 n9 Ja 5 n9 Ja 6 n9 Ja 7 n9 8 Ja n9 Ja 9 n0 Ja 0 n0 Ja 1 n0 Ja 2 n0 Ja 3 n0 Ja 4 n0 Ja 5 n0 6 Ja n0 7 15% 25% 35% 5% 5% 25% 15% Ja Ja n8 n8 0 The Historical Ford Experience  Returns Monthly Return: 1/31/1980 to 12/31/2006 Ja 1 n8 Ja 2 n8 Ja 3 n8 Ja 4 n8 Ja 5 n8 Ja 6 n8 Ja 7 n8 Ja 8 n8 Ja 9 n9 Ja 0 n9 Ja 1 n9 Ja 2 n9 Ja 3 n9 Ja 4 n9 Ja 5 n9 Ja 6 n9 Ja 7 n9 Ja 8 n9 Ja 9 n0 Ja 0 n0 Ja 1 n0 Ja 2 n0 Ja 3 n0 Ja 4 n0 Ja 5 n0 Ja 6 n0 7 The Ford Experience Historical Returns Histogram
50 45 40 35 30 25 20 15 10 5 0
0. 1 0.2 5 75 25 5 25 75 5 0.3 0. 2 0. 1 0. 0 0.0 0.1 0.2 0.4 25 Frequency ForwardLooking Rates of Return
Assume the following:
S&P 500 = $ 1,200, dividend $ 20 Coca Cola = $ 40, dividend = $ 1 Ebay = $50, dividend = $0 3 "states of nature" will generate the following outcomes
Probability S&P kS&P Coke kCoke Ebay kEbay .2 1540 47 80 .5 .3 1420 940 47 31 60 35 State Boom Normal Recession What would be the holding period returns? Summarizing Distributions of Returns
If we know the true underlying distribution:
Expected Return: ^ k = Pri ki
i =1 n Variance: ^ = Pri ki  k
2 i =1 n ( ) 2 Standard Deviation = 2 Summarizing: Expected Return
Expected Return: S&P ^ k = Pri ki
i =1 n Coca Cola Ebay Summarizing: Variance and Standard Deviation
Variance: S&P
^2 = Pri ki  k
2 i =1 n ( ) Coca Cola Ebay Modifications When Using Historical Data
We rarely have knowledge of the full distribution of returns. Instead, we usually have some historical data and are trying to estimate the underlying distribution that produced that data. Sample Average formula: Sample Variance Formula
2 1 n 2 ^ = ( ki  k ) n  1 i =1 1 k = ki n i =1 n Portfolio Expected Return
Assume you form a portfolio by investing $4,000 in Coca Cola and $6,000 in Ebay: The portfolio's expected return will be a weighted average of individual security average returns: ^ ^ ^ k p = w1k1 + w2 k 2 Portfolio Variance and Standard Deviation
Two methods for calculating portfolio variance: 1. Treat the portfolio as a standalone security:
State Probability Coke Ebay Portfolio Boom .2 0.2 0.6 0.44 Normal .5 0.2 0.2 0.2 Recession .3 0.2 0.3 0.26 Portfolio Variance and Standard Deviation
1. Use a "special" formula p = + + 212 cov1, 2
2 2 1 2 1
n 2 2 2 2 ^ ^ cov1, 2 = i k1,i k1 k 2 ,i k 2 Pr
i= 1 ( )( ) Covariance Example
Daily Temperature 100 90 80 70 Temperature (F) 60 50 40 30 20 10 0 8/1/2004 Columbia Beaufort Amarillo Buenos_Aires 2/17/2005 9/5/2005 3/24/2006 Date 10/10/2006 4/28/2007 Portfolio Variance and Standard Deviation
Compare the weighted average of Coke and Ebay's standard deviations to the portfolio standard deviation: This will always be the case because of something known as diversification: when holding a portfolio of assets, risk is reduced since asset returns are usually not perfectly correlated. Correlation and Covariance
The final term in the 2 security portfolio variance calculation was a covariance Variance and standard deviation: measures of how much variation there is in a single security's return. ^2 = Pri ki  k
2 i =1 n ( ) Covariance: a measure of how much variation there is in security 1 and 2's returns and whether the returns are correlated. ^ ^ cov1, 2 = i k1,i k1 k 2 ,i k 2 Pr
i= 1 n ( )( ) Correlation: the degree to which two security's returns move "in sync" with one another. cov 1, 2 = 1, 2 1 2 Low Bigger Diversification Benefits
What happens to our 40/60 portfolio as the correlation () between CocaCola and Ebay changes?
Correlation and Benefits of Diversification 0.2
Expected Return 0.5 0.3 0.2 0.1 0
Standard Deviation 0.16 0.12 0.08 0.04 0
1 0. 75 0. 5 0. 25 0 0 .2 5 0 . 0 5 .7 5 1 0.4 Expected Return Correlation Standard Deviation Key Points on Diversification
Diversification reduces portfolio risk (variance) Makes individual variance terms relatively unimportant while making covariance terms important. Average covariance term is < average variance term since stocks are not perfectly correlated. Since the average correlation coefficient for U.S. stocks is about 0.5, a diversified portfolio has less risk than the individual securities in the portfolio. Diversification does not "hurt" expected return. Diversification and Types of Risk
Total Risk ( or 2) Unsystematic Risk
Driven by company specific factors. Uncorrelated with similar factors in other firms. Systematic Risk (cov)
Driven by common macroeconomic factors. Regulatory issues. Common events across firms or industries. Diversification eliminates nonsystematic risk! Systematic & NonSystematic Risk
The canonical picture of total risk versus diversification Systematic & NonSystematic Risk
Thumbrule: "20" securities achieves most benefits. Exposure to nonsystematic risk is not associated with higher expected returns. WHY? Exposure to systematic risk is associated with higher expected returns. WHY? Systematic Risk & Expected Return
High Systematic Risk High Expected Return, but, what exactly is the relationship? CAPM: Assume...
1. All investors try to maximize expected returns and minimize variance. 2. All investors have the same expectations (information) regarding future security returns. 3. All investors can borrow and lend at the krf. 4. Markets are "perfect": no barriers or costs to buying and selling any security. Capital Asset Pricing Model
Then: The relationship between systematic risk and expected return is linear and given by the following equation: ki = k rf + i ( k M  k rf )
where i = cov i ,mkt 2 mkt CAPM: Security Market Line of the risk free asset: = 0 of the market portfolio: = 1 of the average investment: = 1 Market Risk Premium Source: Goetzman and Jorion, 1999, A century of global stock markets, Journal of Finance 54, 953980. CAPM Example Suppose the assumptions of the CAPM hold. Stock A: = 1.5, Stock B: = 0.6. km = 9% and krf = 4%. What are the expected returns for stocks A and B? Calculating Beta in Practice
What What information do we really want? information do we actually have? What do we do with this? Some Actual Betas
Stock Carolina Power Barrick Gold ExxonMobil General Mills Merck IBM Nextel Iomega Intel .60 .70 .85 .85 1.10 1.15 1.65 1.70 1.70 Understanding and Systematic Risk cov i ,mkt i ,mkt i mkt i ,mkt i i = = = 2 2 mkt mkt mkt =1: >1: <1: <0? for a bond: .1 to .2 Nextel & Iomega Duke, Barrick Gold, General Mills Making Use of CAPM Output
Assume a share of preferred stock promises a dividend of $4.50 per year. eta = 0.5, rrf = 5%, Market Risk Premium = 5% Current price = $50 .... is it fairly priced? Dividend Discount Model: D1 P0 = k g Criticism of CAPM
Prediction of CAPM: diversification will ensure that returns are explained by exposure to systematic risk alone no other risk factors should matter. Some apparent violations after controlling for : Small firms seem to have higher returns. Low Markettobook firms seem to have higher returns. Why? Capital Market imperfections. Irrational investors. Poor testing methods Do Jumps in Stock Prices Violate CAPM?
DaimlerChrysler around 7/28/05
54 52 50 48 46 44 42 40
7/ 1/ 20 05 7/ 8/ 20 05 7/ 15 /2 00 5 7/ 22 /2 00 5 7/ 29 /2 00 5 8/ 5/ 20 05 8/ 12 /2 00 5 8/ 19 /2 00 5 8/ 26 /2 00 5 Open High Low Close Do Jumps in Stock Prices Violate CAPM?
Gateway
4.5 4 Price 3.5 3 2.5
Open High Low Close 8/22/2005 Date 8/29/2005 8/1/2005 8/8/2005 8/15/2005 ...
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 Fall '10
 Masoudie
 Finance, Variance, Capital Asset Pricing Model, Modern portfolio theory, Ja Ja

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