Module 2: Capital Asset Pricing Model (CAPM)

In the Markowitz efficient frontier in case if there are only risky assets the optimal portfolio
is determined at the point where one of the investor's indifference curves (risk-return profile)
is tangential to the frontier of efficient portfolios. This portfolio is optimal because it provides
the best combination of risk and return to suit their preferences. But what if risk-free
investments (such as government stocks) are included in portfolios? Presumably, investors
who are totally risk-averse would opt for a riskless selection of financial and government
securities, including cash. Those who require an element of liquidity would construct a mixed
portfolio that combines risk and risk-free investments to satisfy their needs. Thus, what we
require is a more sophisticated model than that initially offered by Markowitz, whereby the
returns on new investments (risk-free or otherwise) can be compared with the risk of the
market portfolio. Fortunately, John Tobin (1958) developed such a model, built on
Markowitz efficiency and the perfect capital market assumptions.

Tobin’s Separation Theorem and Capital Market Line

To understand what is now termed Tobin's Separation Theorem, suppose every stock market
participant invests in all the market's risky securities, with their expenditure in each
proportionate to the market's total capitalization. Every investor's risky portfolio would now
correspond to the market portfolio (S) with a market return (R m) and market standard
deviation ( σ m) respectively. Remember market portfolio (S) is determined by the tangency
point between the efficient frontier and the Utility Indifference Curve. Now, assume that all
market participants can not only choose risky investments they also have the option of
investing in risk-free securities (such as short-term government stocks) at a risk-free rate, R f
percent as shown in Exhibit 9.8. According to their aversion to risk and their desire for
liquidity, we can now separate their preferences, (hence the term Separation Theorem).
Investors may now opt for a riskless portfolio, or a mixed portfolio, which comprises any
preferred combination of risk and risk-free securities. If he lends a portion of his funds at R f
and invests the balance in S (S is the point on the efficient frontier of risky portfolios where
the straight line emanating from Rf is tangential to the efficient frontier or risky portfolios),
he can obtain any combination of risk and return along the line that connects R f and S.
Further, if he borrows some money at a risk free rate and invests it along with his own funds,
he can reach a point like G, beyond S, as shown in Exhibit 9.8. Thus with the opportunity of
lending and borrowing, the efficient frontier changes. It is no longer AFSX. Rather, it
becomes RfSG because RfSG, as is clear from Exhibit 9.8, dominates AFSX. For every point
on AFSX, there is at least one point on RfSG which is superior to the point on AFSX.

For example, compared to C on AFSX, D on R fSG offers a higher expected return for the
same standard deviation; likewise, compared to B on AFSX, E on R fSG offers the same
expected return with a lower standard deviation; and so on. Since R fSG dominates AFSX,
every investor would do well to choose some combination of R f and S. A conservative
investor may choose a point like U, whereas an aggressive investor may choose a point like
V. However, note that both of them choose some combination of R f and S. While the
conservative investor weights Rf more in his portfolio, the aggressive investor weights S
more in his portfolio. The line (Rf -S-G) in Exhibit 9.8 is a new portfolio "efficiency frontier"
for all investors, termed the Capital Market Line (CML).
The Capital Market Line Equation

R m−R f
R p =R f +( )σ p
σm

Rp : Return on Portfolio R f : Risk free return R m: Return on Market portfolio

σ m: Standard deviation or risk on market portfolio σ p : Standard deviation of the Portfolio
Thus the return on portfolio is the risk free rate of return plus the trade-off between risk and
R m−R f
return ( ¿ also known as the Sharpe ratio -multiplied by the standard deviation of
σm
portfolio. The slope of the Capital market Line is give by the Sharpe ratio which is also called
the price of risk. It tells us how much the expected rate of return of the portfolio (R p )
increase if the standard deviation of the portfolio (σ p ¿ increases by 1 unit.

Difference Between Capital Allocation Line (CAL) Capital Market Line (CML)
Often the Capital market Line (CML) is confused with Capital Allocation Line (CAL). While
the CAL are infinite number of lines plotting the possible combinations of the risk free asset
and a portfolio of risky assets- depending on investor’s return expectations. The CML is the
specific instance where the risky portfolio is the market portfolio. The CML is that CAL with
the highest Sharpe ratio (slope). The greater the value of the Sharpe ratio, the more attractive
will be risk adjusted return.
The Capital Asset Pricing Model (CAPM)

In 1964 William Sharpe (won the Nobel prize in Economics), John Lintner, Jack Treynor and
Jan Mossin developed the capital asset pricing model (CAPM). The model predicts the
relationship between the risk of an asset and its expected return.

The CAPM is based on the following assumptions:

1. Investors are risk averse.
2. Investors have homogeneous expectations -they have identical subjective estimates of
the means, variances, and covariances among returns.
3. All investors’ decisions are based on a single time period.
4. Investors can borrow and lend freely at a riskless rate of interest.
5. Market is perfect with competition, no taxes, transaction costs and perfectly divisible
securities.

CAPM Formula and Calculation

CAPM is calculated according to the following formula:

E ( Ri ) =Rf + βi ¿

Where:

E ( Ri )= Expected return on a security R f = Risk-free rate β i = Beta of the security

E(R m) = Expected return of the market portfolio

Thus the CAPM formula is used for calculating the expected returns of a risky asset E ( Ri ) .
It equals the risk free rate (Rf) plus an expected risk premium β i ¿ or investing in the risky
asset.

Expected Return : E ( Ri ) represents the expected return of a capital asset over time, given
all of the other variables in the equation. “Expected return” is a long-term assumption about
how an investment will play out over its entire life.
Risk-Free Rate : R f is the risk-free rate, which is typically equal to the yield on a
government bond (usually 10-year government bond).

Beta : β i is a measure of a asset’s risk (volatility of returns) reflected by measuring the

fluctuation of its price changes relative to the overall market. In other words, it is the asset’s
sensitivity to market risk.

Market Risk Premium (E(R ¿¿ m)−R f )¿: The Market Risk Premium represents the
additional return over and above the risk-free rate, which is required to compensate investors
for investing in a riskier asset . The risk premium component of the equation does not
consider unsystematic risk, which can be eliminated through proper diversification. The risk
premium captures only systematic risk (otherwise known as non-diversifiable risk) that
investors need to be compensated for. A risk premium is a rate of return greater than the risk-
free rate.

CAPM
https://youtu.be/-fCYZjNA7Ps
https://youtu.be/IJeYwx-cXyc
Calculation of Expected Return
The goal of the CAPM formula is to evaluate whether a stock is valued fairly when its risk
and the time value of money are compared to its expected return.

Example 1: An investor is contemplating a stock worth $100 per share today that pays a 3%
annual dividend. The stock has a beta compared to the market of 1.3, which means it is riskier
than a market portfolio. Also, assume that the risk-free rate is 3% and this investor expects
the market to rise in value by 8% per year. The expected return of the stock based on the
CAPM formula is 9.5%:

9.5%=3%+1.3×(8%−3%)

The expected return of the CAPM formula is used to discount the expected dividends and
capital appreciation of the stock over the expected holding period (discount rate =9.5%). If
the discounted value of those future cash flows is equal to $100 then the CAPM formula
indicates the stock is fairly valued relative to risk.

Example 2: Suppose the following information is known:

  A company trades on the BSE and its operations are based in India. Current yield on a
10-year government Bond is 2.5%. The average excess historical annual return for
BSE stocks is 7.5%. The beta of the stock of HDFC Ltd is 1.25 (meaning its average
return is 1.25x as volatile as the S&P500 over the last 2 years). What is the expected
return of the security using the CAPM formula?

Solution: Let’s break down the answer using the formula from above in the article:
 Expected return = Risk Free Rate + [Beta x Market Return Premium]
 Expected return = 2.5% + [1.25 x 7.5%]
 Expected return = 11.9%

Uses of CAPM

1. The CAPM formula is widely used in the finance industry. It is vital in calculating the
weighted average cost of capital (WACC), as CAPM computes the cost of equity.
2. The underlying concepts of CAPM and the associated efficient frontier can help
investors understand the relationship between expected risk and reward as they make
better decisions about adding securities to a portfolio.

Beta of An Asset and Beta of a Portfolio

Beta of an Asset: We know that risk has two parts systematic and unsystematic.
Unsystematic risk can be eliminated by diversification But systematic risk cannot be
eliminated. Therefore market will pay a premium only for the systematic risk since it is non
diversifiable. Therefore in CAPM a measure of systematic risk arising due to market risk
premium is Beta/Beta Coefficient/Sensitivity Coefficient.

E ( Ri ) =Rf + βi ¿

Beta of an asset is calculated by dividing the covariance of the individual asset's returns and
the market's returns by the variance of the market's returns over a specified period.

Cov ( Ri , Rm )
β=
Var ( Rm )

where: Ri = the return on an individual asset Rm = the return on the overall market

Covariance=how changes in an asset’s returns are related to changes in the market’s returns
Variance=how far the market’s data points spreadout from their average value

Uses of Beta of an asset

1. Helps investors understand whether a stock moves in the same direction as the rest of
the market.
2. Provides insights about how volatile–or how risky–a stock is relative to the rest of the
market.
3. Helps to gauge how much risk a stock is adding to a portfolio. While a stock that
deviates very little from the market doesn’t add a lot of risk to a portfolio, it also
doesn’t increase the potential for greater returns.

Types of Beta Values

1. Beta Value Equal to 1.0: If a stock has a beta of 1.0, it indicates that its price activity
is strongly correlated with the market. A stock with a beta of 1.0 has systematic risk.
2. Beta Value Less Than One: A beta value that is less than 1.0 means that the security is
theoretically less volatile than the market. It is a defensive security. Including this
stock in a portfolio makes it less risky than the same portfolio without the stock.
3. Beta Value Greater Than One: A beta that is greater than 1.0 indicates that the
security's price is theoretically more volatile than the market, called an aggressive
security. For example, if a stock's beta is 1.2, it is assumed to be 20% more volatile
than the market. Technology stocks and small cap stocks tend to have higher betas
than the market benchmark. This indicates that adding the stock to a portfolio will
increase the portfolio’s risk, but may also increase its expected return.
4. Negative Beta Value: Some stocks have negative betas. A beta of -1.0 means that the
stock is inversely correlated to the market benchmark. This stock could be thought of
as an opposite, mirror image of the benchmark’s trends.

Portfolio Beta: Portfolio beta is a measure of the overall systematic risk of a portfolio of
investments. Portfolio beta equals the sum of products of individual investment weights and
beta coefficient of those investments. Beta coefficient represents the systematic risk of the
portfolio.

Portfolio Beta βp = wA × βA + wB × βB + ... + wN × βN

Where βp is the portfolio beta coefficient, wA is the weight of the first investment, βA is the
beta coefficient of first investment; wB is the weight of the second investment, βB is the beta
coefficient of second investment; wn is the weight of the nth investment, βn is the beta
coefficient of nth investment and so on.

Remember: Portfolio Beta vs Portfolio Standard deviation

Portfolio beta is a measure of systematic risk which cannot be diversified. Diversifying a

portfolio does not reduce the portfolio beta.

Portfolio standard deviation is a measure of total risk of a portfolio including systematic and
unsystematic risks. Including more than one asset in a portfolio reduces the diversifiable risk
and hence lowers portfolio standard deviation.

Security market line/Characteristic Line

The security market line, also known as the ‘characteristic line’ is the graphical
representation of the CAPM equation. It is a graph that plots the expected return on
investments with reference to its beta coefficient, a measure of systematic risk. Note that the
slope of the SML is the market risk premium. There is a linear relationship between expected
return and Beta of an asset. This relationship is called the security market line (SML). The
slope if the SML is the market risk premium ¿.

E ( Ri ) =Rf + βi ¿

cov ( Ri , Rm )
E ( Ri ) =Rf +
var ( Rm )
( E ( Rm ) −Rf )

σℑ
E ( Ri ) =Rf +
σ 2m
( E ( R M ) −Rf )

where E(Ri) is the expected return on security i, Rf is the risk-free return, E(Rm) is the
expected return on market portfolio, σ 2mis the variance of return on market portfolio, and
σ ℑ is the covariance of return between security i and market portfolio.
Assets which are fairly priced plot exactly on the SML. Under-priced securities plot above
the SML ( as they have higher required return than the fair value) and over-priced securities
plot under the SML (as they have lower required return than the fair value).

What is the difference between SML and CML?

SML represents Security Market Line and CML represents Capital Market Line.

Point of Difference Capital Market Line Security Market Line

CML indicates the risk and returns SML is used to show the risk and
Meaning
for a portfolio of securities return for individual securities
Risk factor metric Standard Deviation of the portfolio Beta of a security
Total Risk (systematic plus
Risk considered Systematic Risk
unsystematic)
Depicts all market portfolios,
Representation Only depicts efficient portfolios
efficient or non-efficient
CAPM model in Investment

An investor would want to know that can CAPM help with investment decisions. The answer
to this question may not be simple. The CAPM assumes that the solution to the Markowitz
problem is that the market portfolio is the one fund (and only fund) of risky assets that
anyone needs to hold. This fund is supplemented only by the risk-free asset. Thus, the
investment recommendation that follows this argument is that an investor should purchase a
little bit of every asset that is available with the proportions determined by the relative
amounts that are issued in the market as a whole. If the world of equity securities is taken as
the set of available assets, then each person should purchase some shares in every available
stock, in proportion to the stock’s monetary share of the total of all stocks outstanding. It is
not necessary to go to the trouble of analysing individual issues and computing a Markowitz
solution. One should just buy the market portfolio (closest concept is of a Mutual fund).

CAPM assumes that everyone has identical information about the (uncertain) returns of all
assets. In reality that is not the case. Therefore an investor who feels has access to better
information can form a portfolio which is better than the market portfolio. But one important
point one must remember that it is not at all easy to obtain accurate data and information to
use in a Markowitz model. Therefore, the results computed may be non-sensical. In other
words while constructing a portfolio one should begin with the market portfolio and alter it
systematically rather than attempting to solve the entire Markowitz problem from scratch.
These designs will be thus extensions of the basic CAPM idea.
CAPM as a pricing formula

The CAPM is a pricing model. However, the standard CAPM formula does not contain prices
explicitly. It calculates on expected rates of return. To see why CAPM is called a pricing
model let us go back to the definition of return.

Suppose that an asset is purchased at a price “P’ and later sold at price ‘Q’. The rate of return
is then r = (Q-P)/P. Here P is known, and Q is random. Putting this in CAPM formula, we
have

Q−P
=R f + β ( E ( Rm ) −R f ) whereQ=E ( Q )
P

Q
Or P=
1+ R f + β ( E ( Rm )−R f )

This gives the price of the asset according to the CAPM. In the deterministic case the future
payment is discounted at interest rate R f using a factor 1/(1+Rf). In a random case (which is
more general) the future value is discounted at the interest rate R f + β ( E ( Rm ) −R f ) using a
1
factor . The interest rate is the general case is the risk-adjusted interest
1+ R f + β ( E ( R m )−R f )
rate.

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