Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 5
Content:
• The CAPM model
• Estimation of CAPM parameters
The CAPM model
When investing we face two types of risk:
• common (or systematic) risk related to the entire market;
• independent (or specific or idiosyncratic) risk related to
single assets. This can be removed through diversification.
The CAPM model
Idiosyncratic risk can be reduced with larger portfolios
without reducing expected return because the deviations in
the return of one asset can be compensated by the return of
other assets.
We say that a portfolio is an efficient portfolio when it is
NOT possible to find another portfolio that has either:
• a higher expected return without increasing the volatility;
• or a lower volatility without lowering the expected return.
Let us consider a simple example with two stocks.
The CAPM model
The CAPM model
The lower the correlation between the assets, the lower the
volatility that we can obtain for our portfolio given a certain
return (or the higher the return given a certain volatility).
The CAPM model
• Perfectly positively correlated stocks (red line): the
volatility of the portfolio is equal to the weighted average
volatility of the two stocks: there is no diversification.
• Perfectly negatively correlated stocks (blue line): all risk is
diversified and it is possible to hold a riskless portfolio
• In intermediate cases, the lower the correlation, the
more risk can be reduced (and the more the curve bends
to the left)
In practice, assets tend to be positively correlated, or at
most slightly negatively correlated. However, even
combining positively correlated assets allows for efficiency
gains if the correlation is less than 1.
The CAPM model
Efficient frontier: the set of optimal portfolios that offer the
highest expected return for a given risk (or the lowest risk for
a given expected return). Efficiency tends to increase with
the number of assets:
The CAPM model
Law of one price: equivalent securities must have the same
price. This requires efficient markets:
• no arbitrage opportunities (or at least not large enough
to compensate for transaction costs)
• operations by individual investors cannot affect the
market prices
Under the above conditions we have that:
• idiosyncratic risk can be diversified away
• systematic risk can be eliminated only by sacrificing
expected returns.
The CAPM model
Therefore:
• the risk premium for this diversifiable risk is zero: no
compensation for holding firm-specific risk.
• the risk premium of a security is determined only by its
systematic risk, i.e. by its correlation with the market.
To measure the systematic risk of a stock, we must determine
how much of the variability of its return is due to systematic
shocks that affect the economy as a whole.
The price changes of a portfolio with only systematic risk can
reflect systematic shocks to the economy. A natural one is
the market portfolio, but it is difficult to measure it. A good
approximation is a large stock index, like the S&P 500.
The CAPM model
We can then measure the systematic risk of a security from
the sensitivity of the security’s return to the return of the
market portfolio, known as the beta (β) of the security.
• The beta of a security is the expected % change in its return
given a 1% change in the return of the market portfolio.
• By definition the market has beta equal to 1.
• A stock with beta greater than 1 variates more than the
market in response to systemic risk shocks; a stock with
beta lower than 1 move less than the market in response
to systemic risk.
The CAPM model
The Market Risk Premium is the risk premium investors earn
by holding market risk, and is defined as the difference
between the market portfolio’s expected return and the riskfree
interest rate:
𝑀𝑎𝑟𝑘𝑒𝑡 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐸[𝑅 𝑀𝑘𝑡 ] − 𝑅𝑓
The risk-free interest rate is the rate paid by a hypothetical
asset that bears no risk. In practice, the interest rate of shortterm
government bonds from financially solid countries is
typically used.
The market has β=1, so an investment with, e.g., β=2 should
require a risk premium twice the market risk premium.
The CAPM model
From this reasoning we obtain the Capital Asset Pricing
Model (CAPM), which states that the expect return of a
security is equal to the risk-free rate plus its beta times the
market risk premium:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖(𝐸 𝑅 𝑚 − 𝑅𝑓)
The CAPM model
Instead of diversifying risky investments, we can also reduce
risk by keeping some money in a risk-free investment.
Or we can increase expected return by borrowing money to
invest in the stock market.
Let’s put a fraction x of our money in an arbitrary risky
portfolio with returns 𝑅 𝑝, while leaving the remaining
fraction (1 - x) in risk-free Treasury bills with a yield of 𝑅𝑓 .
The expected return of this portfolio, whose returns we
indicate with 𝑅 𝑥𝑝, is:
𝐸[𝑅 𝑥𝑝] = (1 − 𝑥)𝑅𝑓 + 𝑥𝐸[ 𝑅 𝑝] = 𝑅𝑓 + 𝑥(𝐸[𝑅 𝑝] − 𝑅𝑓)
The CAPM model
The volatility of the portfolio is:
𝑆𝐷 𝑅 𝑥𝑝 =
1 − 𝑥 2 𝑉𝑎𝑟 𝑟𝑓 + 𝑥2 𝑉𝑎𝑟 𝑅 𝑝 + 2 1 − 𝑥 𝑥𝐶𝑜𝑣 𝑟𝑓 , 𝑅 𝑝 =
0 + 𝑥2 𝑉𝑎𝑟 𝑅 𝑝 + 0 = 𝑥2 𝑉𝑎𝑟 𝑅 𝑝 = 𝑥𝑆𝐷(𝑅 𝑝)
The blue line in the next figure illustrates combinations of
volatility and expected return for different choices of x.
The CAPM model
As the fraction x invested in P increases, both risk and risk
premium increase proportionally. Hence the line is straight
from the risk-free investment through P. If we increase x
beyond 100 we are short selling the risk-free investment, so
we must pay the risk-free return (we borrow money at 𝑅𝑓).
The CAPM model
Borrowing money to invest in stocks is referred to as buying
stocks on margin or using leverage. The portfolio obtained in
such way is known as a levered portfolio.
By combining the risk-free asset with a portfolio higher on
the efficient frontier than P, we get a line that is steeper
than the line through P. If the line is steeper, then for any
level of volatility we will earn a higher expected return.
The slope of the line through a given portfolio P is often
referred to as the Sharpe ratio of the portfolio:
𝑆𝑅 =
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦
=
𝐸 𝑅 𝑝 −𝑅 𝑓
𝑆𝐷(𝑅 𝑝−𝑅 𝑓)
The CAPM model
The optimal portfolio to combine with the risk-free asset is
the one with the highest Sharpe ratio, where the line with
the risk-free investment is tangent to the efficient frontier of
risky investments. The portfolio that generates this tangent
line is known as the tangent portfolio, and it provides the
biggest reward per unit of volatility of any portfolio available.
The CAPM model
• The tangent portfolio is efficient. Once we include the riskfree
investment, all efficient portfolios are combinations
of the risk-free investment and the tangent portfolio.
• Every investor should invest in the tangent portfolio
independently of his preferences, which will determine
only how much to invest in the tangent portfolio versus
the risk-free investment.
• This allows the investor to earn the highest possible
expected return for any level of volatility he is willing to
bear.
The CAPM model
The CAPM model rests on three assumptions:
1. Investors can buy and sell all securities at competitive
market prices (without incurring taxes or transactions
costs) and can borrow and lend at the risk-free rate.
2. Investors hold only efficient portfolios of traded
securities, that is, portfolios that yield the maximum
expected return for a given level of volatility.
3. Investors have homogeneous expectations regarding the
volatilities, correlations, and expected returns of
securities.
The CAPM model
If investors have homogeneous expectations, they will all
identify the same portfolio as having the highest Sharpe
ratio in the economy, and will all demand the same efficient
portfolio of risky securities (that is, the tangent portfolio)
adjusting only their investment in risk-free securities.
If every investor is holding the tangent portfolio, then the
combined portfolio of risky securities of all investors must
also equal the tangent portfolio. And because every
security is owned by someone, the sum of all investors’
portfolios must equal the portfolio of all risky securities
available in the market, that is, the market portfolio.
The CAPM model
Therefore, the efficient, tangent portfolio of risky securities
(the portfolio that all investors hold) must equal the market
portfolio.
If a security is not part of the efficient portfolio, no investor
wants to own it, and its demand does not equal its supply.
Its price falls, causing its expected return to rise until it
becomes an attractive investment: prices adjust so that the
efficient portfolio and the market portfolio coincide.
The CAPM model
When the tangent line goes through the market portfolio, it
is called capital market line (CML). According to CAPM, all
investors should choose a portfolio on the CML, by holding a
combination of the risk-free asset and the market portfolio.
The CAPM model
As the beta of a security 𝑖 expresses the fraction of its
volatility that is common with the market, we can write:
𝛽𝑖 =
𝑆𝐷 𝑅𝑖 ∗ 𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅 𝑀𝑘𝑡)
𝑆𝐷(𝑅 𝑀𝑘𝑡)
and replacing 𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅 𝑀𝑘𝑡) with
𝐶𝑜𝑣(𝑅 𝑖,𝑅 𝑀𝑘𝑡)
𝑆𝐷 𝑅 𝑖 𝑆𝐷(𝑅 𝑀𝑘𝑡)
we get:
𝛽𝑖 =
𝐶𝑜𝑣(𝑅𝑖, 𝑅 𝑀𝑘𝑡)
𝑉𝑎𝑟(𝑅 𝑀𝑘𝑡)
To determine the appropriate risk premium for any
investment, we must rescale the market risk premium by the
amount of market risk present in the security’s returns,
measured by its beta with the market.
The CAPM model
CAPM implies that there is a linear relationship between a
stock’s beta and its expected return.
The security market line (SML) is the line along which all
individual securities should lie when plotted according to
their expected return and beta.
• The capital market line shows no clear relationship
between an individual stock’s volatility and its expected
return because total volatility also includes diversifiable
risk that receives no compensation.
• The relationship between risk and return for individual
securities becomes evident only when we measure
market risk rather than total risk.
The CAPM model
The CAPM model
The CAPM model
Because the security market line applies to all tradable
investment opportunities, we can also apply it to portfolios.
The beta of a portfolio is simply the weighted average beta
of the securities in the portfolio:
𝛽 𝑝 = ෍
𝑖
𝑤𝑖 𝛽𝑖
Estimation of CAPM parameters
To apply the CAPM we need an estimate of the risk-free
rate, of the market risk premium, and of betas.
The yields on U.S. Treasury securities are generally used
as proxy for the risk-free rate.
The risk premium can be estimated from past data or
using some fundamental analysis approach (e.g. assuming
a certain expected growth model for the market).
A stock’s beta is generally estimated from its past
sensitivity to market risk.
Estimation of CAPM parameters
One can exploit the fact that the beta corresponds to the
slope of the best-fitting line in the plot of the security’s
excess returns versus the market excess return.
Estimation of CAPM parameters
To identify the best-fitting line through the set of points
we can use the linear regression:
𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 𝑅 𝑀𝑘𝑡 − 𝑅𝑓 + 𝜀𝑖
where 𝜀𝑖 is the error (or residual) term, which on average
is zero.
The intercept 𝛼𝑖 is called Jensen’s alpha, and it measures
the historical performance of the security relative to the
expected return predicted by the security market line.
According to the CAPM, 𝛼𝑖 should not be significantly
different from zero.