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? Capital Asset Pricing Model
Then: The relationship between systematic risk and required return is linear and given by the following equation: k i = k rf + i ( k M  k rf )
where i = cov i ,mkt 2 mkt 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 Calculating the Rate of Return
Holding Period Return (HPR) End Price  Beg. Price + Div HPR = Beg. Price Example: You purchase Ford at $50 per share. The annual dividend is $1.08 and after 1 year you are able to sell for $55 per share. What is your HPR? 55  50 + 1.08 HPR = = 0.1216 50 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
State Boom Normal Recession Probability S&P kS&P Coke kCoke Ebay kEbay .2 1540 47 80 .5 .3 1420 940 47 31 60 35 What would be the holding period returns? Summarizing Return Distributions 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 ( .2 .3) + ( .5 .2) + ( .3 .2) = .10 Coca Cola ( .2 .2) + ( .5 .2) + ( .3 .2) = .08 Ebay ( .2 .6) + ( .5 .2) + ( .3 .3) = .13 Summarizing: Variance and Standard Deviation
Variance: S&P
2 ^2 = Pri ki  k
2 i =1 n ( ) .2 ( .3  .1) + .5 ( .2  .1) + .3 (  .2  .1) = .04
2 2 Coca Cola
2 2 2 .2 ( .2  .08) + .5 ( .2  .08) + .3 (  .2  .08) = .0336 Ebay
2 2 2 .2 ( .6  .13) + .5 ( .2  .13) + .3 (  .3  .13) = .1021 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
= ( .4 .08) + ( .6 .13) = .11 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 .2 ( .44  .11) + .5 ( .2  .11) + .3 (  .26  .11) = .0669
2 2 2 p = .0669 =.2587 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 ( )( ) cov = .2 ( .2  .08)( .6  .13) + .5 ( .2  .08)( .2  .13) + .3 (  .2  .08)(  .3  .13) = .0516
2 p = (.4 2 .0336) + (.6 2 .1021) + ( 2 .4 .6 .0516 ) = .0669 2 p = p = 0.2587 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 variances to the portfolio variance: Wtd Avg = ( .4 .1833) + ( .6 .3195) = 0.265 0.265 > 0.259
While portfolio expected return was calculated as a weighted average of individual security expected returns, portfolio variance will almost always be less than the weighted average of individual security variances. Cov term in calculation is < average 2 term...why? This reduction in risk is due to Diversification 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
Standard Deviation 0.16 0.12 0.08 0.04 0
1 0. 75 0. 5 0. 25 0 0 .2 5 0 .5 0 .7 5 1 0.4 0.3 0.2 0.1 0 Correlation Expected Return Standard Deviation Diversification & MultiSecurity Portfolio Assume you have a portfolio of 3 securities: a,b,c p = + + + 2 a b cov a ,b + 2 a c cov a ,c + 2 b c covb ,c
2 2 a 2 a 2 b 2 b 2 c 2 c N variance terms and N2N covariance terms. The "weights" always sum to 1 ... relative importance of covariance terms increases with N while variance terms become inconsequential. Average covariance term is small relative to average variance term since average between any two U.S. stocks 0.5 Portfolio variance < variance of average security. Add stock N+1 to portfolio. What determines whether portfolio will become more or less risky? 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. 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 Required Return, but, what exactly is the relationship? CAPM: Assume...
1. 2. 3. 4. All investors try to maximize expected returns and minimize variance. All investors have the same expectations (information) regarding future security returns. All investors can borrow and lend at the krf. Markets are "perfect": no barriers or costs to buying and selling any security. Capital Asset Pricing Model
Then: The relationship between systematic risk and required return is linear and given by the following equation: k i = k rf + i ( k M  k rf )
where i = cov i ,mkt 2 mkt CAPM in our Simple World
State Boom Normal Recession Probability kS&P kCoke .2 .3 .2 .5 .3 .2 .2 .10 .04 .2 .2 .08 kEbay .6 .2 .3 .13 k i = k rf + i ( k M  k rf )
i =
covi ,mkt Expected Return Variance 2 mkt .0336 .1021 cov C , S & P = .2( .3  .1)( .2  .08) + .5( .2  .1)( .2  .08) + .3(  .2  .1)(  .2  .08) = .036 Coke = .036 = .9 .04 k Coke = .04 + .9( .06 ) = .094 cov e , S & P = .2( .3  .1)( .6  .13) + .5( .2  .1)( .2  .13) + .3(  .2  .1)(  .3  .13) = .061 Ebay = .061 = 1.525 .04 k Ebay = .04 + 1.525( .06 ) = .1315 CAPM: Security Market Line of the risk free asset: = 0 of the market portfolio: = 1 of the average investment: = 1 Real World: Market Risk Premium Source: Goetzman and Jorion, 1999, A century of global stock markets, Journal of Finance 54, 953980. Real World: Beta
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 CAPM Examples Assume the market risk premium is 5% and the riskfree rate of interest is 4%. Calculate required rates of return using Betas from the prior k CP = 4% + .6 ( 5% ) = 7%
k Merck = 4% + 1.1 ( 5% ) = 9.5% k Intel = 4% + 1.7 ( 5% ) = 12.5% Understanding and Systematic Risk cov i ,mkt i ,mkt i mkt i ,mkt i i = = = 2 2 mkt mkt mkt
=1: >1: Nextel, Iomega, Intel <1: Carolina Power, Barrick Gold, General Mills <0? for a bond: .1 to .2 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? D1 $4.50 = Dividend Discount Model: P0 = k g k ki = k rf + i ( k m  k rf ) = 5 % + 0 .5 ( 5 % ) = 7 .5 % $4.50 P0 = = $60 .075 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/ 15 /0 5 7/ 22 /0 5 7/ 29 /0 5 8/ 5/ 05 8/ 12 /0 5 8/ 19 /0 5 8/ 26 /0 5 7/ 1/ 05 7/ 8/ 05 Open High Low Close Do Jumps in Stock Prices Violate CAPM?
Gateway
4.5 4 Price 3.5 3 2.5
8/ 22 /0 5 8/ 29 /0 5 8/ 15 /0 5 8/ 1/ 05 8/ 8/ 05 Open High Low Close Date 3 Degrees of Market Information Efficiency
Weak form efficiency: Past price history is fully incorporated into today's pricing. SemiStrong form efficiency: Weak form efficiency plus assumes all current public information is incorporated into today's pricing. Strong form efficiency: Weak & SemiStrong form efficiency plus assumes all current private information is incorporated into today's pricing. ...
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 Fall '10
 Masoudie
 Finance, Variance, Capital Asset Pricing Model, Modern portfolio theory

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