ex ante beta measurement.ppt

Measuring the Ex Ante Beta

2039

Calculating a Beta Coefficient Using Ex Ante Returns

Ex Ante means forecast… You would use ex ante return data if historical rates

of return are somehow not indicative of the kinds of returns the company will produce in the future.

A good example of this is Air Canada or American Airlines, before and after September 11, 2001. After the World Trade Centre terrorist attacks, a fundamental shift in demand for air travel occurred. The historical returns on airlines are not useful in estimating future returns.

In this slide set

The beta coefficient The formula approach to beta measurement

using ex ante returns– Ex ante returns– Finding the expected return– Determining variance and standard deviation– Finding covariance– Calculating and interpreting the beta coefficient

The Beta Coefficient

Under the theory of the Capital Asset Pricing Model total risk is partitioned into two parts:

– Systematic risk– Unsystematic risk

Systematic risk is the only relevant risk to the diversified investor

The beta coefficient measures systematic risk

Systematic Risk Unsystematic Risk

Total Risk of the Investment

The Beta Coefficient – the formula

)Var(R

)RCov(R Beta

ReturnsMarket theof Variance

market theandstock ebetween th Returns of CovarianceBeta

M

Ms

The Term – “Relevant Risk”

What does the term “relevant risk” mean in the context of the CAPM?– It is generally assumed that all investors are wealth maximizing

risk averse people– It is also assumed that the markets where these people trade are

highly efficient– In a highly efficient market, the prices of all the securities adjust

instantly to cause the expected return of the investment to equal the required return

– When E(r) = R(r) then the market price of the stock equals its inherent worth (intrinsic value)

– In this perfect world, the R(r) then will justly and appropriately compensate the investor only for the risk that they perceive as relevant…hence investors are only rewarded for systematic risk…risk that can be diversified away IS…and prices and returns reflect ONLY systematic risk.

The Proportion of Total Risk that is Systematic

Each investor varies in the percentage of total risk that is systematic

Some stocks have virtually no systematic risk.– Such stocks are not influenced by the health of the economy in

general…their financial results are predominantly influenced by company-specific factors

– An example is cigarette companies…people consume cigarettes because they are addicted…so it doesn’t matter whether the economy is healthy or not…they just continue to smoke

Some stocks have a high proportion of their total risk that is systematic

– Returns on these stocks are strongly influenced by the health of the economy

– Durable goods manufacturers tend to have a high degree of systematic risk

The Formula Approach to Measuring the Beta

)Var(R

)RCov(RBeta

M

Ms

You need to calculate the covariance of the returns between the stock and the market…as well as the variance of the market returns. To do this you must follow these steps:

• Calculate the expected returns for the stock and the market• Using the expected returns for each, measure the variance

and standard deviation of both return distributions• Now calculate the covariance• Use the results to calculate the beta

Ex ante return data (a sample)

An set of estimates of possible returns and their respective probabilities looks as follows:

Possible Future State

of the Economy Probability

Possible Returns on the Stock

Possible Returns on the Market

Boom 0.25 0.28 0.2

Normal 0.5 0.17 0.11

Recession 0.25 -0.14 -0.04

The Total of the Probabilities must equal 100%

This means that we have considered all of the possible outcomes in this discrete probability distribution

Possible Future State

of the Economy Probability

Possible Returns on the Stock

Possible Returns on the Market

Boom 0.25 0.28 0.2

Normal 0.50 0.17 0.11

Recession 0.25 -0.14 -0.04

1.00

Measuring Expected Return on the stock From Ex Ante Return Data

The expected return is weighted average returns from the given ex ante data

(1) (2) (3) (4)Possible

Future State of the

Economy Probability

Possible Returns on the Stock (4) = (2)*(3)

Boom 0.25 0.28 0.07

Normal 0.50 0.17 0.085

Recession 0.25 -0.14 -0.035

Expected return on the stock = 0.12

Measuring Expected Return on the market From Ex Ante Return Data

The expected return is weighted average returns from the given ex ante data

(1) (2) (3) (4)Possible

Future State of the

Economy Probability

Possible Returns on the Market (4) = (2)*(3)

Boom 0.25 0.2 0.05

Normal 0.50 0.11 0.055

Recession 0.25 -0.04 -0.01

Expected return on the market = 0.095

Measuring Variances, Standard Deviations from Ex Ante Return Data

Using the expected return, calculate the deviations away from the mean, square those deviations and then weight the squared deviations by the probability of

their occurrence. Add up the weighted and squared deviations from the mean and you have found the variance!

(1) (2) (3) (4) (5) (6) (7)Possible

Future State of the

Economy Probability

Possible Returns on the Stock (4) = (2)*(3) Deviations

Squared Deviations

Weighted and

Squared Deviations

Boom 0.25 0.28 0.07 0.16 0.0256 0.0064Normal 0.50 0.17 0.085 0.05 0.0025 0.00125Recession 0.25 -0.14 -0.035 -0.26 0.0676 0.0169Expected return on the stock = 0.12 Variance = 0.02455

Standard Deviation = 0.156684

Measuring Variances, Standard Deviations from Ex Ante Return Data

Now do this for the possible returns on the market

(1) (2) (3) (4) (5) (6) (7)Possible

Future State of the

Economy Probability

Possible Returns on the Market (4) = (2)*(3) Deviations

Squared Deviations

Weighted and

Squared Deviations

Boom 0.25 0.2 0.05 0.105 0.011025 0.002756Normal 0.50 0.11 0.055 0.015 0.000225 0.000113Recession 0.25 -0.04 -0.01 -0.135 0.018225 0.004556Expected return on the market 0.095 Variance = 0.007425

Standard Deviation = 0.086168

Covariance

The formula for the covariance between the returns on the stock and the returns on the market is:

Covariance is an absolute measure of the degree of ‘co-movement’ of returns. The correlation coefficient is also a measure of the degree of co-movement of returns…but it is a relative measure…this is why it is on a scale from +1 to -1.

n

t

mMsstMs RRRRPRRCov1

))(()(

Correlation Coefficient

The formula for the correlation coefficient between the returns on the stock and the returns on the market is:

The correlation coefficient will always have a value in the range of +1 to -1.

Ms

MsMs

RRCovRRCorr

)(

)(

Measuring Covariances and Correlation Coefficients from Ex Ante Return Data

Using the expected return (mean return) and given data measure the deviations for both the market and the stock and multiply them

together with the probability of occurrence…then add the products up.

(1) (2) (3) (4) (5) (6) (7) (8) "(9)

Possible Future

State of the Economy Prob.

Possible Returns on the Stock

(4) = (2)*(3)

Possible Returns on the Market (6)=(2)*(5)

Deviations from the mean for the stock

Deviations from the mean for

the market(8)=(2)(6)(7

)

Boom 0.25 28.0% 0.07 20.0% 0.05 16.0% 10.5% 0.0042Normal 0.50 17.0% 0.085 11.0% 0.055 5.0% 1.5% 0.000375Recession 0.25 -14.0% -0.035 -4.0% -0.01 -26.0% -13.5% 0.008775Expected return on the stock = 12.0% 9.5% Covariance = 0.01335

The Beta Measured Using Ex Ante Return Data

Now you can plug in the covariance and the variance of the returns on the market to find the beta of the stock:

8.1007425.

01335.

)Var(R

)RCov(RBeta

M

Ms

A beta that is greater than 1 means that the investment is aggressive…its returns are more volatile than the market as a whole. If the market returns were expected to go up by 10%, then the stock

returns are expected to rise by 18%. If the market returns are expected to fall by 10%, then the stock returns are expected to fall by

18%.

Lets Prove the Beta of the Market is 1.0

Let us assume we are comparing the possible market returns against itself…what will the beta be?

(1) (2) (3) (4) (5) (6) (6) (7) (8)

Possible Future

State of the Economy Prob.

Possible Returns on the Market

(4) = (2)*(3)

Possible Returns on the Market (6)=(2)*(5)

Deviations from the mean for the stock

Deviations from the mean for

the market(8)=(2)(6)(7

)

Boom 0.25 20.0% 0.05 20.0% 0.05 10.5% 10.5% 0.002756Normal 0.50 11.0% 0.055 11.0% 0.055 1.5% 1.5% 0.000113Recession 0.25 -4.0% -0.01 -4.0% -0.01 -13.5% -13.5% 0.004556Expected return on the market = 9.5% 9.5% Covariance = 0.007425

Since the variance of the returns on the market is = .007425 …the beta for the market is indeed equal to 1.0 !!!

Since the variance of the returns on the market is = .007425 …the beta for the market is indeed equal to 1.0 !!!

Proving the Beta of Market = 1

If you now place the covariance of the market with itself value in the beta formula you get:

0.1007425.

007425.

)Var(R

)RCov(RBeta

M

MM

How Do We use Expected and Required Rates of Return?

Once you have estimated the expected and required rates of return, you can plot them on a SML and see if the stock is under or overpriced.

% Return

Risk-free Rate = 3%

BM= 1.0

E(RM)= 4.2%

BX = 1.464

R(RX) = 4.76%

E(R) = 5.0%

SML

Since E(r)>R(r) the stock is underpriced.

How Do We use Expected and Required Rates of Return?

The stock is fairly priced if the expected return = the required return.

This is what we would expect to see ‘normally’ or most of the time.

% Return

Risk-free Rate = 3%

BM= 1.0

E(RM)= 4.2%

BX = 1.464

E(RX) = R(RX) 4.76%SML

Use of the Forecast Beta

We can use the forecast beta, together with an estimate of the risk-free rate and the market premium for risk to calculate the investor’s required return on the stock using the CAPM:

]Rr[EβR fMjf )( Return Required

Conclusions

Analysts can make estimates or forecasts for the returns on stock and returns on the market portfolio.

Those forecasts can be analyzed to estimate the beta coefficient for the stock.

The required return on a stock can be calculated using the CAPM – but you will need the stock’s beta coefficient, the expected return on the market portfolio and the risk-free rate.