University of Texas at Dallas
		School of Management

Finance 6301		Professor Day
Corporate Finance		Fall 1999

		Lecture 8:  Risk and Return

Expected Return and Risk

	The application of the net present value rule to the selection of capital investment projects having uncertain future cash flows requires an implicit adjustment for the associated risk.  This adjustment can be made using either of two alternative (but equivalent) methods.  Under the certainty equivalent approach, estimates of the expected of the future cash flows are reduced by a complicated risk adjustment factor to create a certainty equivalent future cash flow which is then discounted using a risk-free rate of interest.  Alternatively, under the risk-adjusted discount rate approach, the present value of the expected future cash flows is computed using a discount rate that properly account for the risk associated with the cash flows in question.  Before discussing the theory and empirical procedures used to estimate risk-adjusted discount rates, it is useful to  consider the way in which the measurement of risk has evolved over the last century.

Evolution of the Measurement of Risk in Finance

1.	Early 1900's:

	During the early part of the century, the quality of the balance sheet was the primary determinant of risk.  Firms having relatively greater debt levels and relatively low levels of current assets in relation to current liabilities were thought to be riskier firms.

2.	Graham and Dodd and Fundamental Security Analysis:

	During the 20's and 30's, Benjamin Graham and David Dodd pioneered the techniques that have come to be known as fundamental security analysis.  Graham and Dodd argued that by carefully analyzing financial statements, investors could find securities whose intrinsic value was substantially greater than the current market price.  By purchasing only securities providing a suitable “margin of safety”, Graham and Dodd suggest that investors can simultaneously minimize risk and maximize return.

3.	Harry Markowitz and the Statistical Measurement of Risk:

	During the 1950's, Markowitz argued that risk measures must explicitly accounted for the variability of asset returns, which he measured using the standard deviation of a security's return.  Markowitz's work was important (earning a Nobel Prize in Economics) because it shifted the focus of risk measurement from the risk of each security measured in isolation to the contribution of each security to the risk of a well-diversified portfolio.  We will see that it is the risk that a security adds to a well-diversified portfolio that should be used to determine the risk-adjusted rate of return used in capital budgeting.

4.	Multifactor Models of Risk and Return

	A great deal of current research in Finance concerns the development of procedures for precisely calibrating the impact of multiple risk factors (e.g., interest rates and the prices of key industrial materials such as oil) on the required returns of publicly traded stocks.

Risk Adjusted Discounted Rates and Expected Returns

	In general, a risk-adjusted discount rate may be though of as the sum of a risk-free interest rate (that is used to adjust for the differential timing of cash flows) and a risk premium that is required to compensate investors for exposure to one or more systemic risk factors.  In other words, the required return used in discounting estimates of risky future cash flows can be expressed as

			Expected	=	Bond	+	Risk
			Return		Yield	Premium

Note that since the bond yield component of the required return represents the compensation that an investor deserves for deferring consumption (without taking any risk) from today until tomorrow, one frequently used proxy for this component of the required return of return is the annualized yield on a three-month Treasury bill.  The second component of the required return, the risk premium, may be thought of as either (1) the product of a single measure of the risk associated with the cash flow in question and the associated compensation per unit of risk or (2) a sum of the risk premiums required for exposure to multiple risk factors underlying the ultimate realization of the cash flows in question.  For example, the single risk factor version of the Capital Asset Pricing Model implies that the appropriate risk-adjusted discount rate is equal to  a risk-free rate (denoted by RF ) plus a risk premium determined by the product of a measure of market risk (beta) and the differential return expected for the “Market” relative to the risk-free rates

		E(i )	=        RF     +     bi  [E( M ) -  RF ]    ,

	Note that the risk-adjusted discount rate given above is ultimately determined by the risk premium that investors earn from investing in the stock market relative to the returns for risk-free Treasury bills.  Consequently, the estimation of risk-adjusted discount rates requires explicit estimates of the compensation per unit of risk for each of the systemic factors inherent in the risk of the project.

Estimation of the Market Risk Premium

	Estimation of the market risk premium (or any risk premium required within a multi-factor model of risk and return) is a particularly difficult challenge since the ex ante risk and expected returns of both individual stock and well-diversified portfolios can never be observed directly.  Fortunately (under certain conditions) we are able to infer (estimate) the approximate trade-off between risk and return at the portfolio level using the historic record of asset returns.  Assuming that the level of risk for alternative investments (e.g., stocks and bonds) has been stable (more or less the same) over the recent past, we can estimate the risk premium for stocks ( as well as other asset classes) by comparing their averages returns with the average returns on Treasury bills.

		Historic Evidence on Risk Premiums

	One of the important questions in the application of financial models to the valuation of capital investment projects is the magnitude of the risk premium which investors demand for taking risk.  One of the most important benchmarks for the size of the risk premium for an investment of “average” risk is the difference between the average yearly return on the stock market (defined here as the return on the S&P 500) and the average yearly return from holding “risk-free” Treasury bills.  The most commonly referenced source for these data is the yearly publication by Roger Ibbotson and Rex Sinquefield referenced below.

	Average Total Returns 1926-1997
	Ibbotson and Sinquefield 1998 Yearbook

	Asset	Arithmetic	Risk	Standard
	Class	Mean	Premium	 Deviation

	Large Stocks	13.0%	9.2%	20.3%

	Small Stocks	17.7	13.9	33.9

	LT Corporate Bonds	6.1	2.3	8.7

	LT Government Bonds	5.6	1.8	9.2

	IT Government Bonds	5.4	1.6	5.7

	Treasury bills	3.8		3.2

	Inflation	3.2		4.5

			Selected Average Total Returns 1973-1997

	Large Stocks	14.43	7.29	16.80

	Small Stocks	16.86	9.72	26.47

	Treasury bills	7.14		2.72

	Inflation	5.53		3.38

Note that in the table above the universe of common stocks under consideration is S&P 500, while the proxy for “small stocks” is the bottom 20 percent of the NYSE firms in terms of total equity capitalization.  The corporate bond series reflects the returns on AAA corporate bonds having approximately 20 years to maturity.  The returns for the long-term government bonds series reflects the returns on government bonds having 20 years to maturity while the returns on Treasury bills represent the annualized returns from sequentially investing in 3-month Treasury bills.

Application:  Estimation of the Expected Return on the S&P 500

	The most common approach to estimating risk-adjusted discount rates involves adding a historic (or long-run average) risk premium to the current risk-free rate.  This approach is preferable to estimating the expected/required return for a given asset directly from historic averages due to the fact that the level of interest rates has fluctuated widely over the period since 1925.  As a first approximation, we can think of risk-adjusted discount rates as being determined by a (more or less constant) risk premium that floats up and down on top of a highly variable risk-free rate of interest.  Consequently, more accurate estimates of risk-adjusted discount rates can be obtained by adding a long-run estimate of the risk premium to the current yield on risk-free securities.  In effect, this approach allows our estimate for the risk-adjusted discount rate to reflect current market conditions. 

	Given the summary of historic returns reported previously, a reasonable proxy for the current expected return on the “market” (as represented by the S&P 500) is to start with the current risk-free rate of interest and then add an estimate of the risk premium based on the information given above.  On Monday October 25, the 91-day Treasury bill sold at a discount of 5.00 percent.  Based on this discount the price of the 91-day Treasury bill would be

		P	=     100    -    x discount

			=     100    -    x 5.00

			=       98.7361  .

Based on the price of the 91-day Treasury bill given above, the annualized (compound) return on 91-day Treasury bills is 5.22 percent.  If we use 9.2 percent as our proxy for the expected risk premium for stocks relative to Treasury bills, then the expected return on the S&P 500 should be

		E()	=       5.22%    +    9.20%

			=     14.44%  .

The Trade-Off Between Portfolio Risk and Return

	The trade-off between required return and risk that is used to determine risk-adjusted discount rates is derived from the presumption that investors prefer higher expected returns, but seek to avoid higher levels of risk.  Based on this presumption, all investors will choose to invest in portfolios of securities that are efficient in the sense that their portfolios either

	1.	attain the maximum possible level of expected return for a given level of exposure to risk

	2.	attain the minimum level of risk for a given target level of expected return.

Portfolios which satisfy either of the above criteria are referred to as efficient portfolios.  Assuming that all investors choose efficient portfolios, we will show that the required rate of return for individual securities depends on how much risk that security contributes to the risk of a “well-diversified” portfolio.

The Capital Market Line

	The Capital Market Line (CML) defines the trade-off between risk and return for well-diversified portfolios.  Due to the fact that investors always prefer the portfolio having the highest expected return, for a given level of risk, we are able to show that all well-diversified portfolios should offer investors the same risk premium per unit of risk.  To illustrate this point, assume that investors can borrow and lend at a risk-free rate of interest denoted by RF .  Given the set of efficient portfolios (all portfolios having the highest return for a given level of risk), investors prefer to trade risk for return (by borrowing and lending) using the risky portfolio offering the most favorable rate of exchange available.  To determine which portfolio offers the most favorable risk/return trade-off, we compute the ratio of the expected return in excess of the risk-free rate ( E( P )  -  RF ) to the standard deviation of the portfolio return,

		.

	The portfolio offering the highest reward/risk ratio will then be the only risky portfolio in which investors will choose to invest.  This ratio, which is sometimes referred to as the Sharpe ratio (after Nobel Prize winner William Sharpe), is often used to measure ex post portfolio performance by using average returns in place of the expected returns given in the ratio above.

	The Sharpe Ratio can be used to examine the relative diversification of the S&P 500 and the smallest 20 percent of the firms traded on the NYSE.  The Ibbotson and Sinquefield data show that since 1926 small stocks have earned a risk premium of 13.9 percent, with a yearly standard deviation of 33.9 percent.  Therefore, small stocks offer investors a Sharpe ratio of 

			=       ,

			=       0.41

which represents a risk premium 0.41 percent per one percent of standard deviation.  By contrast, the S&P 500 offered investors a risk premium of 9.2 percent, with a yearly standard deviation of 20.3 percent.  Consequently, the S&P 500 offers investors a Sharp ratio of 

			=       ,

			=       0.45

which represents a risk premium of 0.45 percent per one percent of standard deviation.

	This result is not surprising.  Since the small stock group is comprised of the 20 percent of those NYSE stocks having the lowest market equity capitalization at the beginning of the year, the small stock group almost never includes more than 400 stocks (and often less than 150), which implies that the portfolio of low equity capitalization stocks is less diversified than the S&P 500.  This greater diversification reduces the risk per unit of return for the more diversified S&P 500.

Trading Risk and Return along the Capital Market Line

	To illustrate the ability of investors to trade risk and return using a single risky portfolio, we will simplify the previous example.  Suppose that investors who can borrow or lend at 5 percent are faced with a choice between two “well-diversified” portfolios, the S&P 500 denoted by SP and a portfolio of small stocks denoted by SM.  Portfolio SM has an expected return of 17 percent and a standard deviation of 33 percent, while portfolio SP has an expected return of 13 percent and a standard deviation of 20 percent.  The reward to variability ratio  for portfolio SM is

			=       ,

which represents a ratio of risk premium per unit of risk of 0.36 (percent return per unit of standard deviation).

	The S&P 500 offers a Sharpe (reward to variability) ratio of

			=       ,

which represents a reward to variability ratio of 0.40 (or 40 basis points per 1 percent of standard deviation).

	The reward to variability ratios above show that the only risky portfolio that investors would purchase is portfolio SP.  Since SP has the best reward to variability ratio, a combination of risk-free borrowing (or lending) and portfolio SP will dominate any other feasible risk/reward position.  That is, by borrowing (or lending) and investing in portfolio SP, investors can attain any position on the line running from the reward/risk position given by the risk-free asset earning RF through the reward/risk position given by portfolio SP, earning E(SP) on average and having a standard deviation of sSP.  This trade-off between reward and risk can be expressed as

		E( P )	=     ( 1 - x )  RF      +    x E( M )   , 

			=        RF     +    x [E( M ) -  RF ]   . 

Example 1:

	Suppose that the S&P 500 portfolio (portfolio SP) has an expected return of 13 percent with a standard deviation of 20 percent and the risk-free rate of return is 5 percent.  What is the risk of the portfolio containing the S&P and the risk-free asset that has an expected return of 9 percent?

	To determine the risk of a portfolio of SP and risk-free borrowing or lending, we first must determine the fraction of the portfolio that is invested in the S&P 500 (portfolio SP).  Since the portfolio weights of 1 - x in the risk-free asset and x in the S&P 500 must satisfy

		.09	=      RF    +    x [E( M )  -   RF ]  ,

			=     .05    +    x  [.13   -   .05 ]  . 

	Therefore, the fraction of the portfolio invested in portfolio SP must satisfy

		.09  - .05	=     x [.13  - .05]   , 

which implies that

		x	=      ,

			=        .

	Since the risk-free rate of interest (with a standard deviation of zero) is uncorrelated with the (actual) return on the S&P 500 (or any other portfolio of stocks), the standard deviation for a portfolio with x percent in the market and 1-x percent in the risk-free asset is given by

		sP	=     x  sSP   ,

		=     .20   ,

		=    10 percent.

	Note that the formula above is somewhat more complicated that it appears.  We will actually “derive” this formula during our discussion of the standard deviation for portfolios of individual stocks.

Example 2:

	Assuming that the expected return and risk for the S&P and the risk-free asset are the same as in the previous example, determine the risk of a portfolio of SP and the risk-free asset that has an expected return of 17 percent.

	Given that the portfolio must have an expected return of 17 percent, the fraction of the portfolio that is invested in portfolio S&P must satisfy

		.17	=      .05      +    x  [ .13   -   .05 ]  , 

so that

		.17   -  .05	=     x [ .13   -  .05 ]   , 

which implies that x is equal to .12/.08, indicating that 150 percent of the assets must be invested in the S&P (x = 1.5).  Therefore, the investor must borrow an amount equivalent to fifty percent of current wealth in order to obtain the funds required for additional investment in the risky asset.  The resulting portfolio risk is

		sP	 =     x  sM   ,

		 =     .20   ,

		 =     30 percent.

	Note the implication of this example for the comparison of the Sharpe ratios for S&P 500 and the small stock portfolio.  The small stock portfolio has a Sharpe ratio of only .36 (36 basis points per unit of standard deviation), but offers investors more return (17 percent versus 13 percent) at a cost of and additional 13 percent standard deviation (33 percent versus 20 percent).  The example above shows that if we are willing to borrow an amount equivalent to 50 percent of our investment capital and invest the proceeds of the loan in S&P 500, we can create a portfolio (of risk-free borrowing at 5 percent and the S&P 500) that has the same expected return as the small stock portfolio but has a lower standard deviation (33 percent versus 20 percent).

Example 3:

	Assuming that the expected return and risk for the Market and the risk-free asset are the same as in the previous examples, determine the expected return for a portfolio of the S&P and the risk-free asset having a portfolio standard deviation of 33 percent.

	The standard deviation of a portfolio consisting of x percent invested in the market and 1 - x percent invested in the risk-free asset is 

		.33	=     x  sSP   ,

		=     x .20   ,

which implies that x is equal to .33/sSP  (i.e., .33/.20), indicating that 165 percent of current wealth should be invested in the S&P 500.

	Therefore, the portfolio has an expected return of 

		E()	=     .05    +    1.65  [.13  -  .05]

			=     18.2 percent  (i.e., .182).

	The difference in the Sharpe ratios for S&P 500 and Small Stock portfolios actually helps us predict how much extra return we receive if we use the S&P 500 and risk-free borrowing to create a portfolio having the same standard deviation as the small stock portfolio (i.e., 33 percent).  Since the S&P 500 has a Sharpe ratio of 0.40 (40 basis points per unit of standard deviation) whereas the small stock portfolio has a Sharpe ratio of only .3636 (only 36.36 basis points per unit of standard deviation) we are getting an additional 3.64 basis points per unit of standard deviation if we use the S&P 500 (rather than the small stock portfolio) as our core holding in creating our preferred exposure to market risk.  If we are willing to accept a standard deviation of 33 percent for our investment holdings, then by using the S&P to create our overall exposure to market risk we get an extra 3.64 basis points per unit of standard deviation, which implies that we generate an additional 120 basis points (3.64 basis points times the 33 units of standard deviation for our final portfolio exposure) or 1.20 percent in expected return by investing in the S&P 500 (utilizing risk-free borrowing) rather than investing in the small stock portfolio.

Calculation of Portfolio Risk and Return

	Consider a portfolio with 75 percent invested in Bristol-Myers Squib (beta of 0.80 ) at an expected return of 12.5 percent and 25 percent invested in Genentech (beta of 1.4 ) at an expected return of 18.5 percent.  The expected return on this portfolio is given by

		E()	=     .75 x 12.5%     +     .25 x 18.5%

			=       14.0%   .

In general, the expected return for a portfolio is simply a weighted average of the expected returns of the component securities,

		E()	=     ,

where xrepresents the fraction of the portfolio invested in security i and the constraint that the portfolio be fully invested requires that the portfolio weights sum to 1,

		1	=     .

Portfolio Risk:

	Although the average standard deviation of the returns on the stocks traded on the New York Stock Exchange is 50 percent, the standard deviation of the rate of return on the S&P 500 is only 20 percent.  In other words, the standard deviation for the portfolio is substantially less than the average of the standard deviations for the individual stocks included in the portfolio.

	Although there is a general formula for the risk of a portfolio consisting of an arbitrarily large number of component assets (say N), the meaning of this formula is best understood by examining the risk of a portfolio containing only two risky assets.  The variance of the return for a portfolio having two risky assets is given by 

		s	=     xVar()   +   xVar()   +    2 xxCov(,) 

which can be expressed as

		s	=     xs+   xs+    2 xxr ss

since r, the correlation between the returns on security 1 and security 2 is defined by

		r     =     
with -1 < r < 1.

	To motivate the origins of this formula, note that

		 s=     Var(x+  x)

can be thought of in terms of the square of the argument of Var(*),

	(x+  x)=     x+   x+    2 xx.

If we interpret and as the variances of the respective asset returns and as the covariance of the returns on the two securities then we have a simple (but somewhat imprecise) derivation of the formula for the variance of the return for a portfolio having two risky assets.  

	Note that portfolio risk is most often expressed in terms of the standard deviation (which expressed in units of return ) so that the “risk” for a portfolio of securities is given by

		s=    .

	To illustrate the computation of portfolio risk, consider the following examples.  In each case, the return for asset 1 has a standard deviation of 40 percent, while the return for asset 2 has a standard deviation of 60 percent.  To avoid the use of decimals, we will express a standard deviation of 60 percent (0.60) as 60 (in units of percent).

Example 4:  Risk Averaging

	Assume that the returns on asset 1 and asset 2 are perfectly correlated ( r = 1).  Then the standard deviation of the return on a portfolio having 3/4s of its value (x) invested in asset 1 and 1/4 of its value (x) invested in asset 2 is given by

		s	=    ,

			=         xs+   xs,

			=        40%  +   60%   ,

which is equal to 45 percent.  Note that when the returns on two securities are perfectly correlated, the standard deviation of the portfolio is simply an average of the standard deviations for the returns of the component securities.  In other words, there is an averaging of risks but no risk reduction through diversification.

Example 5:  Risk Reduction

	If the correlation between the returns for asset 1 and asset 2 is 0.667 (2/3) (assuming that sand sas before) the standard deviation of the return on a portfolio having 3/4s of its value (x) invested in asset 1 and 1/4 of its value (x) invested in asset 2 is 

		s	=      ,

			=      ,

is equal to 41.5  percent.  Note that although the portfolio weights in this example are identical to those in Example 1, the fact that these assets are less than perfectly correlated allows genuine risk reduction from 45 percent to 41.5 percent.

Example 6:  More Risk Reduction 

	Suppose now that the correlation between the returns for asset 1 and asset 2 is 0.333 (1/3).  Then given the same portfolio weights as in the previous examples,

		s	=      ,

			=      ,

which is equal to 37.75  percent.  When r is equal to 0.33 the risk of the portfolio is less than the risk of either of the component securities included in the portfolio.  While the example depends critically on our assumptions, it reflects the potential for risk reduction that is actually available in the financial markets.  As noted previously,  the standard deviation of the return on the average security is approximately 50 percent while the standard deviation of the return on the S&P 500 and the New York Stock Exchange Composite Index are both approximately 20 percent.

	The previous examples can be generalized to portfolios having large numbers of securities (say N) to show that

	1.	All mean-variance efficient portfolios will contain large numbers of securities (which in turn implies that xbecomes close to zero),

	2.	The risk of an efficient portfolio will depend primarily on the covariances between the returns of the component securities,

	3.	Since each investor will insist on holding an efficient portfolio and since the “Market Portfolio” is a portfolio comprised of the efficient portfolios of individual investors, the Market Portfolio must also be an efficient portfolio.

Risk of Individual Securities

	If the risk of any security or portfolio of securities can be adequately described by the standard deviation (or variance) of the future returns, and if investors insist on holding well-diversified portfolios, then risk may be decomposed into systematic and unsystematic components.  In other words,

		Total	=        Systematic      +      Unsystematic
		 Risk	               Risk                          Risk

In the expression above, the unsystematic component of risk may be thought of as the risk that is eliminated through diversification.  The remaining risk is systematic in that it arises from the common component of security returns.  That is systematic or “market” risk arises from the tendency of securities to move together (either positively or negatively).  This is the covariance risk that cannot be eliminated from the standard deviation of a “well-diversified” portfolio.

	The decomposition of risk into systematic and unsystematic components highlights the fact that there are two risk/return trade-offs which are of interest in financial market:

1.	the risk/return trade-off for “well-diversified” portfolios given by the Capital Market Line,

2.	the risk/return trade-off for individual securities given by the Security Market Line.

	The slope of the capital market line is important for two reasons.  First, since broad market aggregates such as the Standard & Poor's 500 Index and the New York Stock Exchange Composite Index represent well-diversified portfolios, the slope of the capital market line reflects the risk premium that investors receive (relative to the returns on shot-term Treasury securities) per unit of standard deviation.  The Ibbotson and Sinquefield data show us that investors in these proxies for the stock market have received a risk premium on the order of 9.2 percent per year for accepting a yearly standard deviation of approximately 20 percent per year.  Note that sophisticated investors would tend to exclude a stock from a well-diversified portfolio if the ratio of the risk premium to the risk that the security adds to the portfolio is less than the reward to risk ratio (the slope of the capital market line) for a well-diversified portfolio.  Therefore, in order to determine the risk/return trade-off for individual securities (the slope of the security market line), we first need to know the slope of the capital market line.

The Capital Asset Pricing Model:

	The Capital Asset Pricing Model is based on the assumptions that  (1)  	investors may borrow or lend at a single risk-free rate of RF and (2) all investors choose mean/variance efficient portfolios.  Under these assumptions, the expected return on any security (or portfolio) can be expressed as the sum of the risk-free rate of interest (RF) plus a risk premium that reflects bi units of risk with an expected compensation per unit of risk of [E(M) - RF ] .  Formally, the expected rate of return on security i is given by

		E(i )	=        RF     +     bi  [E( M ) -  RF ]    ,

	where

		bi     =     .

Derivation of the Capital Asset Pricing Model

	The Capital Asset Pricing Model is an implication of the assumption that investors allocate the marginal dollar invested in each security to generate the same incremental return per incremental unit of risk of a well-diversified portfolio.  To illustrate this point (prove that the CAPM is true), consider the case where an investor's risky portfolio is composed entirely of the “well-diversified” portfolio offering the highest ratio of risk premium per unit of standard deviation.  In other words, the investor in question holds the “Market” portfolio with a risk premium of E(M) - RF and a standard deviation of

		s.

The reward to variability (Sharpe) ratio for this investor is

		.

	Now suppose that this investor decides to take a very small fraction of the money invested in the market portfolio (denote this fraction by x) and invest a little more in some security that we will denote by i.  The standard deviation of this rebalanced portfolio will be

		s	=    [ xs+  (1 - x)s+   2 x (1 - x) r ss]   .

Note that by rebalancing the portfolio we change both the risk and the expected return for the portfolio.  The change in the portfolio's expected return is simply

		E(i)  -  E(M)  .

The change in the portfolios standard deviation is

	[ xs+  (1 - x)s+   2 x (1 - x) r ss]      -      s

Since x represents a very small shift away from the market, the change in the standard deviation of the portfolio is approximately equal to

		r s-   s.

	The change in portfolio risk has a straightforward explanation based on the value of diversification.  As noted previously, the risk that a security contributes to a well-diversified portfolio depends only on the extent to which the return that security covaries with the returns on the other securities included in the portfolio.  In the expression above, this covariation is measured by the term r s.  Since rrepresents the percentage of the dependent variable (e.g., the return on a stock) that is explained by an independent variable (e.g., the market portfolio) in a regression, the variation in the return on an individual security that cannot be diversified away is measured by  rs.  Therefore, by taking a dollar out of the market and investing in security i, the overall portfolio standard deviation simultaneously reduced by sand increased by rs.

	Given the change in portfolio risk, the ratio of the increase in expected return from shifting away from the market to the resulting increase in risk is

		.

	If this ratio is greater than the reward to variability ratio for the market, then investors will continue to shift resources away from the market and toward securities such as stock i.  If on the other hand, this ratio is less than the reward to variability ratio for the market, then investors will no longer wish to own security i.  In order for the supply and demand for securities to equalize, it must be the case that

		=       .

	Rearranging the expression above gives us the Capital Asset Pricing Model.  Note that the importance of the derivation of the CAPM is not in the mathematics but rather in the intuition that when investors evaluate individual securities that they focus on how an incremental dollar invested in that security alters the return and risk of a “well-diversified” portfolio.

Example 7:

	To illustrate the application of the Capital Asset Pricing Model, assume that the current risk-free rate of interest is 6 percent and that the best available estimate for the risk premium on the market (E(M) - RF) is 8 percent.  Then consider the following three securities.

	Texas Utilities, which produces electric power in North Texas, is typical of public utilities (which usually have betas between .60 and .85) with a beta of .70.  Since Texas Utilities has a beta less than 1.0, the beta for the market, Texas Utilities would be considered to be a defensive security.  The expected return for Texas Utilities is   

			E(TU )	=      RF     +      bTU  [ E( M )  -  RF ]  ,

				=     .06     +    .70  x .08   ,

				=      11.6 percent.  

	Phillips Electronics has a beta of 1.20.  This level of systematic risk is typical of stocks whose sales revenues and profits vary with the business cycle (cyclical stocks).  The expected return for Phillips Electronics would is

			E(PHG )	=     .06     +   1.20  x .08   ,

				=      15.6 percent.  

	Micron Electronics is a manufacturer of computer chips, has a beta of 1.70.  The expected return for Micron Electronics is 

			E(ME )	=     .06     +   1.70  x .08   ,

				=      19.6 percent.  

Portfolio Risk and the Capital Asset Pricing Model:

	The Capital Asset Pricing Model implies that the measure of risk that is relevant in determining the required rate of return on an individual security (or a portfolio of securities) is the risk that the security (or portfolio) adds to a well-diversified portfolio.  As we have seen, for an individual security, this measure of risk is given by a security's beta.  This is also true for a portfolio of securities.  The beta for a portfolio is given by

		b=     ,

where xi is the fraction of the portfolio invested in security i and N is the number of securities included in the portfolio.

	To illustrate the application of the formula above, consider a portfolio of the three securities in our last example where 30 percent of the portfolio is invested in Texas Utilities, 30 percent in Phillips Electronics, and 40 percent invested in Micron Electronics.  The beta for the portfolio is

		b	=     .30 x .70   +   .30 x 1.20   +   .40 x 1.70   ,

			=        1.25   .

Note that this gives us two ways to compute the expected return on the portfolio.  Given the respective expected returns on the three securities ( 11.6, 15.6 and 19.6 percent), the expected return on the portfolio is a simple weighted average of the expected returns of the component securities.  However, given the portfolio beta of 1.25, we can also determine the expected return for the portfolio using the CAPM relation

			E(P)	=     .06     +   1.25  x .08   ,

				=     16.00  percent.

Total Risk versus Systematic Risk:

	The CAPM makes a clear distinction between the total risk or standard deviation for an individual security and the systematic component of risk or beta which determines the required rate of return for the security.  To illustrate this distinction, assume that the standard deviation of the return on the market is 20 percent and consider the two stocks with the following standard deviations and correlations with the market

		Correlation of	Standard Deviation 
		Stock	Stock i with Market	for Stock i	
		A	0.60	.40	
		B	0.25	.60	

	Although it might be tempting to argue that security A is the riskier security since it has a larger standard deviation, the answer to the question “Which security is riskier?” depends on whether we are considering total risk or systematic risk.  To determine the systematic risk for the two securities given above, note that the beta of any security denoted is defined as

		bi     =     

		     =        

		     =       riM 

where riM represents the correlation between the return on security A and the return on the market index M, and where si and sM  respectively denote the standard deviation for security i and the standard deviation for the market index.  Therefore, the betas for securities A and B are respectively

		bA      =       .60  

		=      1.20   ,

	and

		bB      =       .25  

		=       .75   .

Note that although security B has significantly more total risk, the relatively low correlation of security B with the market portfolio causes security B to have a much lower level of systematic or market risk.

Decomposition of Total Risk:

To determine the standard deviation for the return on securities 1 and 3 , we first need to determine the standard deviation of the return for the market.  Note that the total risk for any security (security 2 for instance) can be decomposed into its systematic and unsystematic (firm specific) components according to

		s=     bs+    s

where sis the total risk for security 2 , sthe variance of the return on the market, and srepresents the firm specific (unsystematic) or residual variance for security 2.

	Given a residual variance for security 2 ( s) of .04 and a standard deviation of .25 the standard deviation of the return on the Market is defined by

		(.25)	=     (.75)s+    .04   

	which implies that 

		s	=       

		=              .04  .

Therefore, the standard deviation of the return on the Market ( s) is 20 percent.

	The variances of the total return for securities 1 and 3 are respectively

		s=     (2.0)+    .10   ,

		=      .26  .

		s=     (.50).04      +    .17  , 

		=      .18  .

	In other words, the return for security one has a standard deviation ( s) of 51 percent while the standard deviation ( s)of the return for security 3 is  42.43 percent.