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Calculating and Interpreting Beta

Introduction:In 1990, William Sharpe won a Nobel Prize in Economics for his work in

developing the Capital Asset Pricing Model (CAPM). Traditionally the CAPM has been the basis for calculating the required return to the shareholder. This figure in turn has been used to calculate the economic value of the stock and the Weighted Average Cost of Capital (WACC) for capital budgeting. In recent years, the CAPM has been attacked as an incomplete model for explaining market pricing behavior, but academics and practitioners cannot agree on a good replacement. And so the CAPM remains an important model in practical investment and financial management decision making.

Calculating Beta:The most important component in calculating the required return to

shareholder (from the CAPM) is the company’s beta. The CAPM can be succinctly stated as:

sRFsRFMRFs kkkkk PremiumRisk Market [1]The original model was conceived of theoretically, and was expected to be

forward looking. Careful reading of Sharpe’s original work show that the market assesses systematic risk looking at expected future covariance of the company’s returns with that of the overall market. It is assumed that these covariances are unbiased and efficient estimates of the observed relationships ex post facto.

Traditionally the CAPM relationship is estimated using simple regressionon historical outcomes, where ks is the y variable, and kM-kRF (or the market risk premium) is the only x variable. Care must be taken that the returns plugged into the regression are all for the same period. Calculated stock returns should be annualized if the risk-free rate is an annual rate.

The market risk premium is merely the difference between the return to the market portfolio and the risk-free rate. Academics typically use a value weighted portfolio to proxy for the market portfolio, and a one-month Treasury Bill rate to proxy for the risk-free rate. Practitioners can use equally weighted portfolios (but not all do) and tend to use longer term Treasury bonds for the risk-free rate. The biggest contention between the two is over the risk-free rate proxy. Academics want a rate that is free from all risk, including interest rate risk. Practitioners want a default-risk-free instrument with a more comparable maturity to stock.

Bringing the Data into Excel:For our purposes, we will use publicly available information to estimate

our company’s beta. We do not have access to the market weighted or value

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weighted portfolios used by academics or high-priced stock analysts. Instead we will use the S&P Index. Since most students will not become academics, we will allow the beta to be estimated using the 20 year Treasury bond.

The first step in estimating your company’s beta is gathering the return data. We will gather the company’s stock returns first. This data is readily available online. The example below shows how to gather it from Yahoo:

1. So first, go to http://finance.yahoo.com, and type in your company’s ticker symbol.

2. Along the left-hand side of the screen, click on the option entitled Historical Prices.

3. Next we need to get price (return) data for the stock over the most recent few years. Traditionally either monthly or weekly data are

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used for this purpose. We will use monthly data if enough years are available, so click on Monthly. The maximum time series is the default displayed in Yahoo, make sure you have about five years of data if you are going to use monthly data. So that the data are for the same period (are synchronous) as the Treasury bond data, we must change the Date Range to be on the first of each month. Then click on the Get Prices button.

4. Although the data will be updated, it will appear as though little has changed. Scroll now to the bottom of the Prices table. Just

below the table is button that says . Click on this to get all of the data downloaded to spreadsheet. It

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will ask whether you want to save or open it. You can do either one, but you will need it in Excel to run a regression. That means that if you choose to open it, you will have to open Excel, then copy and paste the information over to Excel.

5. Now we need to collect the Treasury Bond rate. For that information, you will need to go to the Federal Reserve Bank of St. Louis (http://research.stlouisfed.org/fred2/series/GS20/). Click on the Download Data link, and follow the links at the bottom of the next page, to download the monthly 20 year bond rate.

6. Next we need the market return data. This is gathered the same way the stock price data was gathered. First go to http://finance.yahoo.com; then click on S&P 500. Then follow the

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same directions listed for the stock prices (steps 2, 3 and 4 above) to download the index levels for the S&P 500.

Adjusting the Data:The data will have to be lined up so that they cover the same month, and

so that they are return data. We will also need to adjust the data so that they are annual returns, since we want annual required return, and we will have to adjust the Treasury yield data, since 3.80 means 3.80% (which should say 0.0380). For the explanation below, it is assumed that each component downloaded (stock price data, Treasury bond rates, and S&P levels), are each on a separate sheet in Excel.

The first step is going to the sheet containing the Treasury yield data and putting it in the same order as the Yahoo provided data. This is rather simple:

1. Move the cellpointer to the first date in the list.

2. Click on the Data menu, and choose Sort.

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3. Leave the Sort by as Date, but change the default to Descending as in the dialog box displayed below. Then click OK.

Note, alternatively, you can just move your cellpointer as indicated in step

1 above, then click on the Sort Descending button if it is displayed.Now we need to calculate the return figures for both the stock price and

the S&P levels. These are done exactly the same way. Therefore, only the stock price will be shown. You will, however, need to do both. The monthly return is simply the percent change in Adj. Close from one month to the next. To get the annualized return we will simply multiply by 12:

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In my example, I added an extra sheet entitled Summary. On this sheet, I set the first column as the date, so that I was sure that the data are synchronous. Then the variables are simply the figures which we collected. The Treasury bond data are divided by 100 so that they are directly comparable to the return data from the other two. The risk premium is simply the difference between the market return and the risk-free rate.

Normally, the Treasury data is not as up-to-date as the stock price data. But one lost observation makes little difference.

Estimating Beta:As mentioned above, estimating beta is simply a matter of running a

simple regression model. To do that in Excel, we will use the regression capabilities described earlier when we projected our income statement:

1. Click on the Tools menu.2. Choose Data Analysis (if it doesn’t show up on the list, follow the

directions listed under the notes to income statement projections to get it to show up).

3. Choose Regression from the dialog box.

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4. If you are using monthly data, use about five years worth of data, entering the stock return data as the Y variable, and Risk Premium as the X variable. Fill in the new worksheet name, then click OK.

Graphically Estimating Beta:Excel can also estimate simple regression using graphs. For some, this

method helps them understand what the beta estimate is doing. Remember, that the observed difference between the market return and the risk-free rate is our x-variable, and the return to the stock is our y-variable. This method begins by graphing the x and y variables.

1. Highlight the first sixty (60) stock return figures on the summary

sheet. Then click on the chart wizard button .2. Choose XY Scatter plot.

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3. Click Next, then click on the Series tab at the top of the dialog box. Click in the X variable range and enter the first sixty (60) risk premium figures on the summary sheet. Be sure these are the same rows as those entered for the stock returns.

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4. After both series are entered, then click Next. I would enter the labels into the X and Y axes labels, then click Next.

5. Place this graph on its own sheet, then click Finish.

6. To get the graph to draw a regression function through the data, right click any of the data points, and choose Add Trendline. On this dialog box, click on the Options tab at the top, then click the “Display equation on chart” and “Display R-square on the chart.” Then click on OK. (This dialog box is displayed on the top of the next page.)

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Both of these methods will yield the same answer if you do them correctly. The graphical method shows more of what it is doing.

Interpreting the Regression Output:First, let’s discuss how to estimate the firm’s required return. Equation [1]

suggests that we only need to multiply the beta by the risk premium, then add the risk-free rate. As mentioned in the textbook, we might want to adjust our beta before estimating our required return. The textbook adjusts the beta for mean reversion as follows:

33.067.0 EstimatedAdjusted [2]In practice, the adjusted beta is used in equation [1]. In addition, the

expected return to the market is rarely used in practice. Instead, an estimate of the risk premium is used. Some researchers have estimated that the risk premium has been relatively stable at between 3.5% and 6.5%. Again our text likes to use an average of 5%, so we can do the same. And since we used the 20 year bond rate thus far, we can continue.

For example, if our regression estimated a beta of 1.5548, and the current 20 year bond rate were 4.97%, then the required return to the shareholder would be calculated as follows (note in the Excel sheet where the beta estimate is found):

3717.133.05548.167.0 Adjusted

%8286.113717.1%5%97.4PremiumRisk Market sRFs kk

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But all of this analysis is based on simple regression. Remember that in finance, we define risk as variation. Also remember that the CAPM says that the only risk for which the investor is compensated is systematic risk. Systematic risk is the risk that is correlated with the return to the market; when the return to the market goes up, systematic return should also increase. But in addition to systematic risk, stock prices will reflect risks that are unique to the firm. The fluctuation in returns brought about by non-market information is called non-systematic, or company specific risk. The CAPM says that total risk is simply a sum of the systematic and unsystematic risk. And all risk is measured by the variance of the return.

In the income statement notes, we discussed the measure called R-squared. We pointed out that the proportion of sum of squares that was explained by the regression was called R-squared. Since the largest component of the variance is the sum of squares, many say that the R-squared is the proportion of variance that is explained by the regression model. When we translate this approximation to the CAPM model, then the R-squared is an approximate measure of the amount of systematic risk contained in the total variation. According to the CAPM the non-systematic risk can be diversified away. If my regression analysis results in a R-squared of 0.1588, then about 15.88% of all risk in this stock is systematic, meaning non-diversifiable. That also means that 84.12% of the risk displayed in the past five years of returns appears to be diversifiable.

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In addition to discussing the proportion of variation explained by the model, you should also discuss the regression’s statistical significance. If the p-value of the F-statistic or the t-statistic is not less than 0.05, then the model is probably not estimated with sufficient confidence to use. This means that over the estimation period, the price for the firm has been driven by much company specific news. That news has overwhelmed the systematic component in the estimation.

As a final check of the accuracy of your estimate, you can compare your beta estimate to that of online investing information like Finance Yahoo or MSN Money.

Finding the Price:All economic value is based on the present value of expected future cash

flows. There are two approaches we can use to estimate the appropriate economic value of the stock. The first is a dividend discount model. The second is a cash flow discount model. The problem with taking present values ofexpected cash flows is that the dividends or cash flow streams are expected to continue for a very long time. In all instances, simplifying assumptions are made so that these calculations can be made.

Dividend Discount Models:The dividend discount model approach was pioneered by Myron Gordon

in the late 1950s and early 1960s. To solve the problem of estimating a large number of expected dividends, he made a simplifying assumption; he assumed that all future dividends would grow at a pre-specified constant rate. The present value would then be:

n

n

k

gD

k

gD

k

gD

k

gDP

1

1

1

1

1

1

1

1 03

30

2

200

0 [3]

In this model, the only dividend in the model is the last observed dividend (D0), which grows at a constant rate (g), and is discounted at the required rate of return (k). To solve for P0, we will first multiply both sides of equation 3 by (1+g)/(1+k):

1

10

4

40

3

30

2

20

01

1

1

1

1

1

1

1

1

1

n

n

k

gD

k

gD

k

gD

k

gD

k

gP [4]

Now if you subtract equation 4 from equation 3, you are left with the following equality:

1

100

01

1

1

1

1

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n

n

k

gD

k

gD

k

gP [5]

At this point we will make two observations. First, note that as long as k>g, then as n gets large, the final term gets very small. In fact, the limited of the last term as n approaches infinity, is zero. Therefore, we will ignore it from here on out. The second observation concerns the left side of the equality:

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k

gk

k

g

k

k

k

g

11

1

1

1

1

11 [6]

Therefore, equation 5 can be restated in the following form:

k

gD

k

gkP

1

1

10

0 [7]

Now if we solve for P0, we get the following equality, which is usually called the Gordon Constant Growth Model:

gk

D

gk

gDP

100

ˆ1 [8]

In the final form, the numerator can either be the last dividend, allowed to grow at the constant rate, or it could be restated as the expected dividend in the next period, since the two are identical under Gordon’s simplifying assumptions.

For many applications, this form assumes too much. For many firms, the implied growth is unlikely. Remember, that the growth rate is expected to continue forever. Unless the growth rate is rather low, most firms could not expect constant growth in dividends that is much higher than inflation1.

Smaller, younger companies may have a high growth period in the corporate life cycle. That high growth period may be followed by a slower growth period. To accommodate this growth model, Gordon’s original model has been adjusted for two growth phases as follows (without derivation):

2

2011

1

100

1

1

1

1

11

1

gk

gD

k

g

k

g

gk

gDP

TT

[9]

Cash Flow Discount Models:The biggest practical problem with the dividend discount method is that

many firms do not pay dividends. Merton Miller and Franco Modigliani showed mathematically that it shouldn’t matter whether the cash flow comes to the shareholder in the form of a dividend or reinvested in the company, as long the required return is earned. That simply means that if we replace the dividends in the above models with cash flow (or more precisely free cash flow), we can use the above models to value firms not offering dividends.

Practical Application of Valuation Models: Most analysts will not simply apply the above formulas to the firms they

analyze. Instead, they would project the financial statements of the firm over the short-run, using company and market data and assumptions about the effectiveness of the firm’s strategic and marketing plans. After those short-term

1 It should be noted at this juncture that the growth estimated here is nominal growth in dividends, not real growth. That means that a growth rate equal to the average expected inflation rate would allow for a constant dividend in real terms, and should be applied later to the cash flows as well when the same models are used to value companies using cash flows.

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projections, the analyst would have to make some kind of simplifying assumptions like that proposed by Gordon.

We will demonstrate how this can be done. Earlier in the semester we projected the income for the firm. We have since then discussed different estimates of cash flow. For this exercise, we will use what we called the quick and dirty cash flow estimate (net income + depreciation expense). This cash flow estimate was for cash flow to the shareholder and is easy to calculate2.

To calculate the stock price per share, we will need to restate the cash flow on a per share basis. That means that an assumption must be made about the number of shares for any given period. At least to begin with, we can hold these figures constant at the most recently disclosed number of base shares outstanding. Taking total cash flow (net income + depreciation expense) and dividing it by the expected number of shares outstanding should yield a good estimate of per share cash flow.

At the end of this cash flow stream, we will have to project the value of the remaining cash flows outstanding. We will do that by applying a growth model to the remaining cash flow streams. This will yield a present value figure at the time of the last projected cash flow stream.

The example we used in class is for Rocky Shoe and Boot. Since they have never paid a dividend and don’t look like they will anytime soon, we cannot discount their dividends. Instead, we take their projected income, and add back depreciation expense. Then we divide by the number of shares outstanding. From the last projected year of cash flows, we need to calculate the present value of the remaining cash flows. Since the previous cash flows have included quarterly data, we restate the above Gordon Growth Model [equation 8] using quarterly rates:

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413

12 gk

gDP

rdYrAvg

thQtr [10]

This “price” represents the value in time 12, of all future cash flows. Since it is denominated in time 12 dollars, we will need to take its present value. The easiest way of accomplishing that, is to add it to the periodic cash flow and take them both together. Therefore, another line is added to sum all period cash flows (even though the only period with a “sum” is the final one. The screen capture below demonstrates the general spreadsheet layout.

2 Note that the example company, Rocky Shoe and Boot, does not list depreciation on its income statement. We used the change in accumulated depreciation from the balance sheet to estimate the period by period depreciation expense.

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In order to calculate the estimated economic value of the stock, estimates for all inputs must be made. Earlier in these notes we presented the method for estimating the required return to the stockholder. In addition to those estimates, we must guesstimate the long-term annual growth rate in cash flows. If dividends are used, then the long-term annual dividend growth rate should be guesstimated.

After all of these rates are estimated (or guesstimated), then we will be ready to estimate the price of the stock, which is simply the present value of our expected future cash flows. The easiest way to calculate the present value is to use Excel’s NPV function. Remember that the NPV function in Excel does not take into consideration the cash flow at time zero (see the Excel output at the top of the next page).

After a price is estimated, our work truly begins. The price may or may not be close to the current market price. In the example given it is significantly higher than the current stock price. That usually means that the underlying assumptions are not the same (or even similar) to those used by the marginal investors in the market. If the present value calculated is significantly different, double check your assumptions: income projections, cash flow projections, required return calculations, Gordon Growth model calculations, etc.

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