Contents - Index

Estimating Expected Return:  CAPM
©2009 OS Financial Trading System



FTS DJIA School Case:  Estimating Expected Return for a Stock Using CAPM


Question:  How do you estimate the expected return for a stock using CAPM?

Note:  This lesson is the same as the lesson on estimating the cost of equity capital for a stock using CAPM.  The reason is that in a CAPM equilibrium the expected return equals the cost of of equity capital which in turn also equals the investors' required rate of return.

This relationship however is most important to Modern Portfolio Theorey.  The reason is that in a naive application of Markowitz and modern portfolio theory the expected return is often estimated from the average of the past 60-months realized returns.  Many problems immediately arise.  First, the averagecan be negative which immediately implies that if shortselling is permitted then this stock is likely to attract a short weighting.  However, in finance we are interested in "ex ante" i.e., the future not "ex post" the past and so even though the realized average is negative this implies the stock price has fallen but we still expect that at current (low) price levels it is trading with a positive expected return.  Similarly, a high flying stock is likely to have the opposite problem -- a large positive realized return average that is unlikely to be sustained over the next 5-years.  The CAPM estimate for expected return overcomes these problems plus it is based upon an equilibrium result that exploits information in the current yield curve as well as information from a long times series of stock returns (via it's estimate for the equity premium).  For these reasons it is a preferred estimate over a naive historical average estimate for an individual stock using a short time horizon.  In the default support for the RTFTS trader client FTS uses their own CAPM estimates for expected return in the default support data.  The following lesson describes how this number is arrived at.

The required parts of this project are at the end of the write-up.

Background

The single period CAPM model is covered in all standard finance textbooks.  This model describes a linear equilibrium relationship between the risk and expected return from a stock as follows:



The three major terms in the above equation are the risk free rate (rf), risk (?i), and the equity premium (excess of market return over the risk free rate) (E(RM - rf).  That is, all firm specific characteristics of the stock are captured by beta.

Why is this relationship predicted?

Formally, in CAPM all risk averse investors are predicted to invest some proportion of their money in an index fund that is designed to mimic the general market.  As a result, the risk of each security is how much the security contributes to the volatility of the general market.  In words, CAPM asserts (loosely) that a producer of rain boots is less risky if all other producers in an "island" economy produce beach going related equipment.  That is, the more correlated with the economy a stock's returns are the "riskier" the stock is, because the it contributes more to the total volatility of the market as a whole.  This is measured by a statistical concept called "covariance" and beta is a convenient scaling of covariance such that greater than 1 (more risky than the market in general) and less than 1 (less risky than the market in general) has economic meaning.  

Formally, this is defined as the covariance of the stock i, with the market M (s(i,M)) scaled by the variance of the market returns (s(M,M)).  This is provided in the following equation.



A further relationship implied from CAPM is that the expected return from a stock equals the investors' required rate of return which in turn equals the cost of equity capital for a stock.



Application:   How to Estimate the Cost of Equity Capital from CAPM in Practice

Three major inputs are required.  

A.  Risk Free Rate

First, an estimate of the risk free rate?  This can be obtained from the yield curve.  For example, if you go to the Federal Reserve Bank web site you will see current Treasury yield curve information:  

<http://www.federalreserve.gov/Releases/H15/update/>





Which is relevant?


CAPM is a one period model and so this time question is left unanswered formally.  However, in practice we will select the risk free rate that approximates our investment horizon.  For example, suppose this is 10-years then the appropriate risk free rate from the above is:  2.95% (third column is the most recent on the Federal Reserve Bank site).


B. Beta


You cannot directly observe beta in the market, instead it must be estimated.  Traditionally, beta is estimated from a regression analysis that regresses historical returns from a stock against historical returns from some proxy index for the market.  As a result, both the time period to regress returns over and the proxy you select will matter.  In addition, the way you define returns also matters.  This could be daily, weekly, and monthly.  You are encouraged to use Excel and it's regression function to provide your own estimate of beta including performing some sensitivity analysis to changing measurement rules.  You can download historical price data directly from Yahoo or MSN from their charts including different proxy indexes such as S&P500, Russell Indexes and even a Global index.  

For the purpose of this current exercise estimates of beta are freely available from all of the major financial web sites.  For example, consider one of the DJIA index stocks Wal-Mart (WMT).  We can check three major financial web sites (Yahoo, MSN and Google) February, 2009.

MSN Investor:  



Yahoo



Google



The three sites provide a variety of estimates --- 0.29, 0.35, and 0.32.  Recall, it is not unusual to come up with different estimates for beta given the different available methods for estimating beta.

The default FTS Real Time Trader at this time had a beta estimate of 0.18 see below:



Now suppose you want to override beta with your personal estimate, you would use the Parameters menu item in the FTS Real Time Trader.

The screen shot below illustrates how you can override the default beta with for example, the beta estimate provided by Google (beta = 0.32 for WMT).

Tip:  There is a simple way for entering Parameters if you have a bunch of changes --- see appendix 1.



With the check box use these values in my analytical support checked then the FTS Real Time Client overrides the default estimate with your personal estimate.  See analytical support below:



C.  Equity Premium

The final input into CAPM is an estimate of the equity premium.  First, the equity premium depends upon what time horizon you have selected for the risk free rate.  For example, if you selected a 3-month T-bill the historical equity premium is around 7.5% but if you select a 10-year time horizon as per the above example, then the historical equity premium is more like 5.5%.  

The size of the equity premium fluctuates over time.  For example, in the 1990's it was commonly speculated that the equity premium had declined and some estimates were as low as 3.5%.  For example, interested readers are encouraged to read the speech by Allan Greenspan "Measuring Financial Risk in the 21st Century"

<http://www.federalreserve.gov/BOARDDOCS/SPEECHES/1999/19991014.htm>


Similarly, an interesting paper by Pablo Fernandez at the University of Navarra, has extensively surveyed textbooks and the equity premium estimates in finance:


<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=933070>


As a first pass we will use a historical average of around 5.5% relative to a 10-year investment horizon.
Assuming 5.5% we are now ready to estimate the cost of equity capital for Wal-Mart for the above example using a beta equal to 0.32:


E(Return Wal-Mart) = ke = Cost of Equity Capital = 0.0295 + 0.32*0.055 = 0.0471 or 4.71%.


Modifying the FTS Real Time Client's E(Return) Estimate for Wal-Mart


Select the Parameters menu item and then click on List All to bring up all the stocks in the drop down beside Security.  Select Wal-Mart Stores Inc and then select Expected Return.  Enter the value 0.0471 and click on Submit.  You now have stored on the server your values for Wal-Mart plus by ensuring the checkbox ("Use these values in my analytical support") is checked as illustrated below:




 
Then the analytical support will reflect the checked values:





Implications for Modern Portfolio Theory:  Expected Return 


Observe in the above screen that by changing our estimate for the cost equity capital via beta in CAPM also changes our estimate for the Expected Return from Wal-Mart.  This has immediate portfolio implications.  
For example, in standard Investment Textbooks you learn about Markowitz diversification with and without short selling.  The FTS Real Time Client in the Stocks Index Model analytical support automatically provides the optimal Markowitz weights with and without short selling for benchmark purposes.  
To explore the implications from changing Wal-Mart's beta from 0.18 to 0.32 consider the following two screen shots:


Changed beta, changed Expected Return:





Original Beta, Expected Return





Observe a substantial change in the portfolio weight with short selling is implied for Wal-Mart.  The weight changed from 0.2862 to 0.4366 (both high).  Similarly, with no short selling the weight changed from 0.5260 to 0.6040.


In both cases the weight increased because in this exercise we changed both the risk (beta) and return of Wal-Mart but the return side dominated the additional risk.  
Aside:  We only adjusted risk in an partial manner because in reality if we re-assess beta up from 0.18 to 0.32 we would probably also reassess the volatility of Wal-Mart up as well.
The above exercise provides some rich insight into how the textbook treatments of CAPM work in practice.  It also provides insight into one of the common criticisms of Markowitz diversification that in practice it often generates portfolios that do not appear to be "diversified" in an intuitive sense.  That is, 52% of the portfolio is currently Wal-Mart in the above DJIA example.


A later FTS exercise provides a step by step approach to how you can over come this criticism of Markowitz by additional portfolio weight constraints.


Required:  Complete the following steps


Step 1:  Select a high beta, middle beta and low beta stock from among the DJIA stocks.


Step 2:  Go to the Federal Reserve Bank web site:


<http://www.federalreserve.gov/Releases/H15/update/>


Record the current risk free rate associated with your investment horizon.
Alternatively, you may want to check the latest yield curve at (or equivalent):


<http://www.bloomberg.com/markets/rates/index.html>


Step 3:  For the stocks selected look up and record from the web the betas from MSN Investor, Yahoo Investor and Google Finance or some other site of interest.  Compare the different betas and then select one for each stock.  You should then enter your preferred betas into the FTS Real Time Trader via the Parameters menu item as described in the Wal-Mart example above.


Step 4:  Assess what you think the current equity premium is.  This is the excess return from the market over the risk free rate that you selected in Step 2.


Step 5:  Use the information above to compute the Expected Return from each of the three stocks using CAPM.


Step 6:  Given steps 1-5 what is your best estimate of the cost of equity capital for each of the three stocks you selected.  Provide brief reasons in support of your answer.

____________________________________________________________________

Appendix 1:  Entering all Parameter changes at once to populate analytical support with your own estimates

For control and convenience it is easiest to keep all personal parameter overrides for the Parameter support in Excel.  This is because you can then copy and paste from Excel to immediately update all your parameters without requiring any typing.

Step 1:  Format is as follows: (adjacent columns) and enter the values of your parameters or link to other cells that may be computing/receiving personal support values in Excel

Column 1:  Security Name
Column 2:  Field Name
Column 3:  Value

Major Tip:  It is most important to use precisely the same security name and field name as is used by the FTS Real Time Trader.  As a result, make use of the Copy Security Names and Copy Field Names in the Edit menu item and first paste them into the Excel spreadsheet that you are working with.  This will ensure that all names are identical.

Step 2:  Once you have a contiguous block of Security Names, Field Names and Values then mark and copy from Excel to the windows clipboard, give focus to the Parameters window in FTS Real Time Client and select menu item Edit, Paste and Submit Values 

Now your personal parameters are immediately used.