Investments 2-218-05 Lecture 3: Return, Risk, Equilibrium in Capital Markets, CAPM, Index Model and APT : Investments 2-218-05 Lecture 3: Return, Risk, Equilibrium in Capital Markets, CAPM, Index Model and APT
Measuring Returns : Measuring Returns Rate of return on any investment is Specifically, for bonds after the first coupon payment Specifically, for stocks after the first dividend payment
Expected Returns : Expected Returns The difference between EXPECTED RETURN and HISTORICAL RETURN:
Expected return is the return that the investors expect (based on their current knowledge) over a certain future period.
Historical average return is the realized return over a certain past period.
Historical Returns : Historical Returns If you have historical data on returns, i.e. R1, R2, …, Rn, then the easiest way to summarize this data is to calculate the average return.
The average return is simply the sum of the returns divided by the number of years.
Average return is just like any other average, i.e. if you pick a random period, your best guess of return in that period is the average return.
Arithmetic Average Return =
Geometric Average Return =
Historical Returns : Historical Returns Example:
R1 = 10% and R2 = 5.66%
Arithmetic Average Return = 7.83%
Geometric Average Return = 7.81%
Historical Returns : Historical Returns The geometric average is always less than the arithmetic average.
The greater the variability of returns, the greater the difference between geometric and arithmetic average.
The geometric average return is exactly the constant rate of return we would have needed to earn in each year to match actual performance over some past investment period. Hence it is an excellent measure of past performance.
However, if our focus is on future performance, then arithmetic average return is better because it is an unbiased estimate of future expected return.
Expected Returns : Expected Returns Assume that there are only three possible states of the economy (recession state, normal state and an expansion state). From past experience and your personal beliefs, you expect the economy will be in a recession state 25% of the time, in the normal state 50% of the time and in the expansion state 25% of the time. There are two stocks, A and B with the following returns in different states of the economy.
Expected Returns : Expected Returns Expected Return of Stock A Expected Return of Stock B General Formula for Expected Returns
Expected return is the weighted average of possible outcomes where the weights are the probabilities.
Historical Variance : Historical Variance The question is “How do we measure the variability of returns?”.
Although an average is a first step to summarize data, it is generally not enough.
The variance and its square root, the standard deviation, are the most commonly used measures of volatility.
Variance: The average squared deviation between the actual return and the average return.
Standard Deviation: The square root of the variance
Variance : Variance Variance of Return on Stock A Variance of Return on Stock B General Formula for Variance
Variance is the weighted average of squared deviations from the expected return where the weights are the probabilities.
Variance : Variance Variance is a measure of statistical dispersion, indicating how the possible values are spread around the expected value.
In finance, variance or standard deviation is a measure of risk.
If investors are risk averse, i.e. they prefer low risk, then they choose the investment with the lowest variance between two investment with the same level of expected returns.
Return and Risk : Which of these two stocks should you buy?
Due to the risk aversion of investor, there will generally be a trade-off between return and risk.
The greater the risk, the greater the potential reward (expected return).
The reward for the risk bearing is the risk premium.
Why do common stocks have higher average returns than t-bills?
Common stocks are generally riskier than t-bills and investors holding common stocks want to be compensated for the risk bearing. Return and Risk
Portfolios : Portfolios So far, we have concentrated on individual assets but most investors hold more than one asset.
A portfolio is when an investor invests part of his wealth in more than one asset.
The most convenient way to describe a portfolio is to list the wealth invested in individual asset as a percentage of the total portfolio’s value.
These percentages are called portfolio weights.
Returns on a Portfolio : Returns on a Portfolio Suppose we invest fraction w of our wealth in asset A and fraction (1–w) in asset B.
Suppose we have $100 in asset A and $300 in asset B, then total value of our portfolio is $400 and w is 0.25 (or 25%) and (1–w) is 0.75 (75%).
What are the expected return and variance of our portfolio?
The return on our portfolio is
Returns on a Portfolio : Returns on a Portfolio Then, the expected return and variance on our portfolio are The general formula for the expected return and variance on a portfolio of N assets with returns Ri and weights wi are
Total, Systematic and Specific Risk : Total, Systematic and Specific Risk The “umbrella and resort” example
Consider the Bounty Island which have only two businesses – see the table
The profit of the two are highly risky
What if you invest $1 in both companies?
Your return would be
(50%-25%)/2=12.5% if it rains
However, it would be the same if the sun is shining Share
holder
=12.5%
=0%
Total, Systematic and Specific Risk : Total, Systematic and Specific Risk This small example illustrates the advantage of diversification
In particular, what made it work is the fact that although both companies were risky they were effected differently by weather conditions.
We say that the returns they generated had a negative covariance or correlation.
The fact that the correlation was “perfect” allowed us to completely eliminate risk.
However, less than perfect negative correlation will help too.
Total, Systematic and Specific Risk : Total, Systematic and Specific Risk What about real life?
This plot shows the monthly return on Nortel from April 1996 through September 2005
The annualized standard deviation is 291%
Suppose you instead had a portfolio of Nortel, Canadian Tire and Alcan – see the figure
The annualized standard deviation of this portfolio is 115%
Mean Variance Graph : Mean Variance Graph
Mean Variance Graph : Mean Variance Graph The important thing in this graph is the correlation.
Correlation is the covariance divided by the product of standard deviations.
Correlation is a measure of strength and direction of a linear relationship between the returns of two stocks.
If ρ = 1, then diversification would not decrease the risk.
If ρ < 1, then diversification would decrease the risk.
If ρ = - 1, then diversification might totally eliminate the risk.
Total, Systematic and Specific Risk : Total, Systematic and Specific Risk In real world, can we create a perfectly diversified portfolio by adding enough stocks?
As we add more and more stocks to our portfolio, the standard deviation decreases.
However, there are limits to diversification.
The remaining risk is due to the extent which all stocks move together because of underlying economic conditions.
Total, Systematic and Specific Risk : Total, Systematic and Specific Risk
Diversification : Diversification First of all, part of the total risk can be eliminated by diversification, i.e. by investing in many different assets.
This part of total risk that can be eliminated by diversification is called diversifiable risk.
Secondly, there is a minimum level of risk that cannot be eliminated simply by diversifying.
This part of total risk that cannot be eliminated by diversification is called nondiversifiable risk.
Diversification : Diversification Why can some part of total risk be eliminated by diversification whereas the other part cannot?
The answer hinges on the distinction between systematic and unsystematic risks.
If we hold a large enough portfolio, unsystematic risk of individual companies would cancel each other and only the systematic risk associated with each company will be the risk of the portfolio.
Capital Market Line (CML) : Capital Market Line (CML) Tangent (or Market) Portfolio Risk Averse Investor Less Risk Averse Investor Borrowing at rf Mean-Variance Efficient Frontier Global Minimum Variance Portfolio
Two Types of Risk : Two Types of Risk To summarize our discussion about risk, there are two types of risk:
Systematic Risk = Nondiversifiable Risk = Market Risk
Diversifiable Risk = Idiosyncratic Risk = Firm-specific Risk = Unsystematic risk
Idiosyncratic risk can be easily eliminated by diversification.
Systematic risk is common to all stocks and cannot be easily eliminated by diversification.
Risk and Risk Premium : Risk and Risk Premium What determines the size of the risk premium on a risky asset?
The answer depends on the distinction between systematic and idiosyncratic risk.
Since idiosyncratic risk can be easily eliminated at virtually no cost (by diversifying), there is no reward for bearing idiosyncratic risk.
Hence, the risk premium on a risky asset depends only on the systematic risk, not the total risk. (The systematic risk principle)
The Capital Asset Pricing Model -Introduction : The Capital Asset Pricing Model -Introduction The Capital Asset Pricing Model, always referred to as the CAPM, is a centerpiece in modern financial economy
The model gives us a precise prediction of the relationship that we should observe between the risk of an asset and its expected return
The model is useful especially in two situations
Evaluate the “fair” return a traded security should offer
Evaluate the “expected” return on new assets, like IPO’s
The Capital Asset Pricing Model -Assumptions : The Capital Asset Pricing Model -Assumptions If the security markets are ideal in the sense that:
Investments are limited to an universe of publicly traded financial assets, such as stocks and bonds, and to risk-free borrowing and lending arrangements.
Investors pay no taxes on returns and no transaction costs (comissions and service charges) on trades in securities.
The security markets are large, and investors are price-takers
Additionally:
Investors are single period planners (myopic or short-sighted)
They are all alike and share the same beliefs about the state of the economy
They care only about the tradeoff between the expected return and variance. They are willing to accept risk only when rewarded with higher return, i.e. they are mean-variance optimizers, i.e. they use Markowitz portfolio selection model.
The Capital Asset Pricing Model -Implications : The Capital Asset Pricing Model -Implications Then:
Investors hold identical risky portfolios
The market portfolio is the optimal portfolio
All investments should offer the same reward to risk ratio and hence:
The Capital Asset Pricing Model -Implications : The Capital Asset Pricing Model -Implications Implication 1:
Since all investors are alike and share the same beliefs about the market they end up choosing to hold a portfolio of assets with the same weights
Implication 2:
Summing over all investors’ portfolios we obtain the market portfolio
All traded assets will be included in this portfolio through price adjustment
The Capital Asset Pricing Model -Derivation : The Capital Asset Pricing Model -Derivation So in the plot from before the portfolio is now the Market portfolio, M
Any point along the “A” region can be reached by mixing M with the risk-free asset
However, any point along “B” can also be reached by short selling the risk-free asset
Points along the line may be described by
The Capital Asset Pricing Model -Derivation : The Capital Asset Pricing Model -Derivation Implication 3:
In equilibrium all investments should offer the same reward to risk ratio
For the P’th asset, the contribution to the premium is wP(E(rP)-rf)
The contribution to variance can be measured by the covariance between the asset and the market portfolio, wPCov(rP,rM)
Thus for portfolios P and Q we have
The Capital Asset Pricing Model -Derivation : The Capital Asset Pricing Model -Derivation Ad 3:
Moreover for the market we have that this ratio is given by:
This is called the market price of risk
It follows that in equilibrium we have
The Capital Asset Pricing Model -Derivation : The Capital Asset Pricing Model -Derivation The most familiar expression of the CAPM to practitioners is the expected return-beta relationship
where
The left figure plots this line which is known as the Security Market Line, SML.
The Capital Asset Pricing Model -Results : The Capital Asset Pricing Model -Results More Implications:
One implication of the CAPM is that the passive strategy of investing in a market index portfolio is efficient
Furthermore, the risk premium of any particular individual security is the product of the risk premium on the market portfolio and the contribution of the security to the variance of the market portfolio relative to the total variance measure by the “beta”
Looking at the formula above we may talk about the term β*(E(rM)-rf) as the risk premium
CAPM : σ E(R) E(Rm) σm Portfolios on the CML are combinations of the risk free asset and the market portfolio.
The market portfolio is where a line from the risk free rate is tangent to the efficient frontier. The point M is the Market Portfolio and it includes all risky assets in the economy with weights corresponding to the ratio of their market value to total market value.
All investor independent of their risk aversion would hold the same combination of risky assets and the sum of their portfolios is the market portfolio.
Part A : The investor hold the market portfolio and the risk free asset.
Point M : The investor holds just the market portfolio.
Part B : The investor borrows at the risk free rate to invest more in stocks (i.e. leveraging). M Rf A B CAPM
The Capital Asset Pricing Model -Results : The Capital Asset Pricing Model -Results What does this beta mean – basically:
If a stock has a beta of 1 it has the same risk as the market
If it has a beta less than the market it is less risky than the market
Finally if the beta is larger than one the stock is more risky than the market
Stocks with high betas are said to be aggressive investments whereas low beta stocks are considered defensive
The Capital Asset Pricing Model -Results : The Capital Asset Pricing Model -Results But what does this mean – more specifically
If a particular stock has a beta of 2 it means that on average the returns are twice as volatile as the market
On the other hand, a stock with a beta of say, 0.5, could e.g. be a stock that only falls with 5% whenever the market falls 10%
The CAPM model tells us what stocks with a given beta value should yield in excess of the market
If the security beta is 1.5 the stock should yield an excess return of 1.5 times that of the market
The Capital Asset Pricing Model -Results : The Capital Asset Pricing Model -Results Example: Assume that ABC has a beta of 1.5, that the risk free rate is 6%, and that the market return is expected to be 12%
Then the required and fair expected return on the stock should be
6%+1.5*(12%-6%)=15%
If the return is expected to be larger then this fair value it constitutes a good buy
The difference is sometimes called the alpha!
And we may talk about positive alpha stocks as those currently trading at too low prices
The Capital Asset Pricing Model -Results : The Capital Asset Pricing Model -Results More generally, the CAPM can be useful in capital budgeting decisions
E.g. when a firm is considering new projects the CAPM can provide a return that the project should yield for it to be acceptable
The CAPM may also be used to set the rate of return that a regulated utility is allowed to earn on its investment in equipment
The Capital Asset Pricing Model -Limitations and extensions : The Capital Asset Pricing Model -Limitations and extensions However, while the risk free interest rate may be easy to obtain what about this market portfolio
The most obvious index to use here would be the TSX composite
But remember that the CAPM market portfolio was “all risky asset publicly traded”
Canadian investors could invest in U.S. stocks – and they should be included
What about European securities – they should perhaps also be included
In reality the market portfolio may be unobservable
The Capital Asset Pricing Model -Limitations and extensions : The Capital Asset Pricing Model -Limitations and extensions The assumptions of the CAPM have been relaxed in a number of directions
Restricted borrowing
The risk free rate is replaced by the zero-beta return
Lifetime consumption and bequest plans
The simple CAPM holds
Liquidity cost can be incorporated
Index models and the APT -Index models : Index models and the APT -Index models Suppose that we group all the unanticipated factors which moves the security market as a whole into the factor F
Assume that these factors influence the return on asset i by βi*F
Further assume that asset i is influenced by firm specific factors, ei
Then the actual return on asset i may be written as
ri=E(ri)+ βi*F+ei
This is known as the single-factor model for stock returns
Index models and the APT -Index models : Index models and the APT -Index models A broad index is often used as a proxy for the factor
In particular we take RM=rM-rf as a proxy for the unanticipated component
If we further let Ri=ri-rf, we may write the single-factor model, which we call the single-index model as
Ri=αi+ βi*RM+ei
where αi is the stock excess return if the market is neutral
The single-index model says that each security has two sources of risk
Market or systematic risk, RM
Firm specific risk, ei
Index models and the APT -Index models : Index models and the APT -Index models Example: Assume that you have analysed the data for ABC and found that
RABC=0.04+1.4*RTSX+eABC
Now consider the portfolio, T, of 1.4 share of the TSX index and -0.4 in the risk free asset with an excess return of
RT=1.4*rTSX-0.4rf-rf= 1.4*RTSX
If this portfolio is sold the return on the combined investment is
RC=RABC-RT =(0.04+1.4*RTSX+eABC)-(1.4*RTSX)
=0.04+eABC
Portfolio T is called the tracking portfolio because it tracks the systematic component of the ABC’s return.
This corresponds to the activity of many hedge funds
Index models and the APT -Arbitrage : Index models and the APT -Arbitrage Definition: Arbitrage
The exploitation of security mispricing in such a way that risk-free economic profits may be earned is called arbitrage
Example: IBM stock is traded for $48 at Nasdaq and for $50 at NYSE
Result: Buy an “infinite” amount of IBM at Nasdaq and sell it instantaneously at NYSE earning “a lot”
This is also known as the Law of One Price
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory The APT was developed by Stephen Ross in 1976 and relies on the following three assumptions
Security returns can be described by a factor model
Sufficient securities exist so as to diversify away idiosyncratic risk
Well-functioning security markets do not allow the persistence of arbitrage opportunities
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory Assume that each security has returns given by
ri=E(ri)+βi*F+ei
Now form a well-diversified portfolio of such securities which will have a return of
rP=E(rP)+βP*F+eP
However, since the portfolio is well diversified eP=0 and we have that
rP=E(rP)+βP*F
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory Now consider an equally well-diversified portfolio Q for which
rQ=E(rQ)+βQ*F
Form a new portfolio, Z, of P and Q with weights wP=-βQ/(βP-βQ) and wQ=βP/(βP-βQ)
The portfolio Z has
βZ= βP*wP+βQ*wQ
= βP* (-βQ/(βP-βQ))+βQ*(βP/(βP-βQ))=0
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory Thus, Z is riskless and to rule out arbitrage must earn only the risk free interest rate
E(rZ) =wP*E(rP)+wQ*E(rQ)
= -βQ/(βP-βQ)*E(rP)+βP/(βP-βQ)*E(rQ)=rf
This may equivalently be written as
[E(rP)-rf]/ βP=[E(rQ)-rf]/ βQ
Finally, let the portfolio Q correspond to the market portfolio from which it follows that βQ=1 and hence
E(rP)=rf+βP[E(rM)-rf]
It can be shown that this relation will hold for almost all individual securities
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory In contrast to the CAPM, derivation of the APT does not rely on an inherently unobservable market portfolio
In the APT any well diversified portfolio may serve as the benchmark portfolio
Furthermore, even if the chosen portfolio is not a precise proxy for the true market the SML relationship holds true
However, whereas the CAPM holds for all securities the APT cannot rule out violations for any particular asset
Index models and the APT -Arbitrage Pricing Theory : Index models and the APT -Arbitrage Pricing Theory The APT may be generalized to multiple factors
However, the theory does not give any guidance as to which are the relevant factors
One example is the Fama and French three-factor model which has
rit=αi+βiM*RMt+βiSMB*SMBt+βiHML*HMLt+eit
where SMB and HML are differences in returns of portfolios with different characteristics.
CAPM, Index models, and the APT -Comparison : CAPM, Index models, and the APT -Comparison CAPM
Assumptions:
The security markets are large
Trading can be done at fair prices
Investors are equal and share the same opinions
Result:
E(ri)=rf+βi[E(rM)-rf]
Advantages:
The CAPM is derived by appealing to equilibrium
The CAPM holds for all securities in the market
Limitations:
The CAPM is based on a number of simplifying assumptions although some may be relaxed! APT
Assumptions:
Security return follows the factor model
All nonsystematic risk can be diversified
No arbitrage opportunities exists
Result:
E(rP)=rf+βP[E(rM)-rf]
Advantages:
The APT does not rely on an unobservable market portfolio
Any well diversified portfolio may serve as a benchmark
Limitations:
Strictly speaking we do not know if the model holds for all individual stocks!
CAPM, Index models, and the APT : CAPM, Index models, and the APT One issue we have not discussed until now is where β comes from?
According to the CAPM it is given by
Given historical values this could be estimated
According to the APT the following relationship exists
This looks like a regression equation and you (may) know how to estimate this
However, for our purpose we may find estimates of provided by brokerage companies, online, etc
Example : Example According to CAPM and CML, calculate the expected return and the beta of a portfolio composed of 25% market portfolio and 75% risk-free asset.
Discuss whether stocks A and D are overpriced, underpriced or correctly priced and calculate their alphas. A b E(r) 1,0 2,0 0,7 M D 8% 12,5% SML 8,5% 18% 8,0%
Example : Example Using CAPM, find the values for ??.
Summary : Summary In this session we have talked about risk
Total risk, systematic risk, and specific risk
We talked about the trade-off between risk and return for portfolios
We have “derived” the CAPM model
The results as well as the limitations were discussed
We also introduced the Index models and APT