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Risk, reward, and the CAPM model -- review for my M&B students.
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Questions
• What is risk? What is return? What’s risk aversion?
• How do we measure the risk of a given asset?
• How can we use statistical measures to quantify the risk of agiven asset?
• What is the effect on risk of diversifying a portfolio of assets?
• Are there different types of risk and what are they?
• What’s the beta of an asset?
• What is the capital asset pricing model (CAPM) and what isthe security market line?
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Risk and return
Risk is the chance of gain/loss. (A loss is a negative gain.)
The future is fundamentally unknown. Therefore, there is uncer-tainty about the future consequences of choices we make today.
The past is a guide to the future only if the future looks like thepast in some way. But if the future doesn’t look like the past,then the past is not necessarily a good guide.
One way to measure risk in financial assets is to look at the variabil-ity of returns. Think of financial assets as lotteries. For example,a lottery gives you return X if the state of the world is A and areturn Y if the state of the world is non−A.
If risk is the variability of returns. What is return?
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Risk and return
Return is the total gain/loss experienced on investing in a financialasset over a period of time. Usually, we express it as a percentageof the value of the investment at the beginning of the period.
More formally,
kt =Ct + Pt − Pt−1
Pt−1(1)
where kt is the actual, expected, or required return rate (or justreturn) over period t, Ct is the cash flow received from the assetinvestment in the time period from t−1 to t, Pt is the price (value)of the asset at time t, and Pt−1 is the price (value) of the asset attime t− 1.
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Risk aversion
People have “preferences” for risk. Some like it or tolerate it betterthan others. If we look at the behavior of crowds (e.g. markets),then it’s clear that most people are risk averse, since people placea return premium on risky assets.
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Examples
Work on your assignment.
Download returns on some financial asset and find expected return(mean), distance from mean, square distance from mean,.
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Risk of one single asset
Sensitivity analysis vs. statistics (probability distributions).
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Probability distributions
The return on an investment is a random variable because, inadvance, we don’t know for sure what it will be. Its value iscontingent upon the particular state of the world realized.
Say we can list all the possible states of the world. Say thereare only three equally-possible states of the world: (a) Good, (b)Regular, and (c) Bad. And, under each of th.ese states of theworld, we know (or can guess) the return rate. Then we can formexpectations on the return.
Also, we can compute different measures of dispersion or variabilitythat could give us a measure of risk.
Discrete and continuous distributions. Show examples.
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Probability distributions
Expected return: Mean of returns.
k̄ =n∑
j=1
kjpj (2)
where k̄ is the expected return, kj for j = 1, . . . , n is the list ofreturns observed under different states of the world, and pj is theprobability that kj occurs – being the probability a number between0 and 1, where 0 means absolute impossibility of occurrence and1 means absolute certainty that it will occur.
The measures of dispersion or variability used as proxies for riskare formulas (3) the variance, (4) the standard deviation, and (5)the coefficient of variation – of the returns. The higher each ofthese is, the greater the variability of returns and the higher the
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risk.
σ2k =
n∑j=1
(kj − k̄)2pj (3)
σk =
√√√√ n∑j=1
(kj − k̄)2pj (4)
CVk =σk
k̄(5)
Portfolio risk
Correlation is the statistical measure of the association betweenany two series of numbers, e.g. returns of two assets under differ-ent states of the world.
The degree of correlation is measured by the Pearson coefficient:−1 ≤ ρ ≤ 1, where ρ = −1 means perfect negative correlation,ρ = 0 means no correlation at all, and rho = 1 means perfectpositive correlation.
In Excel, do example in p. 207.
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Diversification
However correlated the returns of two assets may be, the expectedreturn of a portfolio with the two assets will fall in some midpointbetween the returns of the two assets held in isolation.
If the assets are highly positively correlated, the risk is some mid-point between the risk of each asset in isolation.
If the assets are largely uncorrelated, the risk is some midpointbetween the risk of the most risky asset and less than the risk ofthe least risky asset, but greater than zero.
If the assets are highly negatively correlated, the risk is some mid-point between the risk of most risky asset and zero.
In no case will a portfolio be riskier than the riskiest asset includedin it.
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Types of risk
Say, we measure the risk of a portfolio by its standard deviation,σkp. Start with a portfolio with one single asset and add assetsrandomly to the portfolio, one at a time. What happens to risk asyou keep adding assets? It tends to decline towards a lower limitor baseline risk.
On average, portfolio risk approaches the lower limit when you’veadded 15-20 randomly selected securities to your portfolio. So,total risk can be viewed as the sum of two types of risk:
Total security risk = Nondiversifiable risk + Diversifiable risk.
Diversifiable risk is also called ‘unsystematic,’ because it comesfrom random factors that can be eliminated by diversifying theportfolio. Nondiversifiable risk is also known as ‘systematic’ and itcomes from market-wide factors that affect all securities. It mayalso be called ‘market’ or ‘macroeconomic’ risk.
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CAPM Model
The model links nondiversifiable risk to returns for all assets.
First, we will discuss the beta, an element of the model. Second,we will introduce the CAPM equation. Third, we will show howto use it in concrete applications.
The beta is a measure of the nondiversifiable risk. It indicates towhat extent the return on an asset responds to a change in themarket or average return of all assets.
Who knows about ‘all assets,’ but there are broad indices of se-curities, e.g. S&P 500. How do we estimate the beta of a givenasset? We need data on the returns on that asset and data on thereturns of a well-diversified portfolio representative of the marketas a whole. We use regression analysis.
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Regression analysis
Consider the plot of the returns on two assets. The horizontalaxis shows the return on an asset representative of the whole mar-ket, e.g. the S&P 500. The vertical axis shows the return on aparticular asset, e.g. the return on Google.
Show plot in Excel.
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A linear equation
Consider the simple bi-variate linear equation:
y = a+ bx (6)
The equation says that the value of the variable y (the dependentvariable) depends on the value of the variable x (the independentvariable).
The literals a and b are called, respectively, the intercept and theslope. The intercept indicates the value of y when x = 0. Theslope indicates the change in y associated to a unit change in x orb = ∆Y
∆X.
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Regression analysis
Regression is a statistical procedure to fit a line in a scatterplot.
yi = α+ βxi + εi (7)
Show regression in Excel.
The beta coefficient (slope) indicates the change in the returnon a particular asset (e.g. Google or GE) when the return on amarket portfolio (e.g. S&P 500) changes in one unit. The betashows how sensitive the return on the asset is to performance ofthe market as a whole.
What do the betas of Google and GE say?
The beta of a portfolio is the weighted average of its individualassets’ betas:
βp =n∑i=1
wiβi (8)
where∑n
i=1 = 1. The beta of a portfolio indicates how responsivethe portfolio’s return is to changes in the market return.
Example in p. 215.
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CAPM Model
The capital asset pricing model (CAPM) gives us the return thatwe would require in order to compensate for the risk involved inholding a given asset i. The model helps us determine the returnover and above a risk-free return that would offset the risk inherentto that asset. And by risk inherent to that asset, we mean the riskof holding that asset that is not diversifiable, i.e. the risk that itshares with a well-diversified market portfolio.
So, to determine that extra return or risk premium, we need tomeasure the extent to which an asset is exposed to nondiversifiablerisk. But instead of pulling that measure out of our own heads,we consider the way crowds (e.g. markets) determine that extrareturn.
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CAPM Model
We now remember that the beta of an asset tells us the change inthe return on the asset that is due to a one-percent change in themarket return. That beta can be used as an index or measure ofnondiversifiable risk, since it reflects to covariation of the returnon that asset and the market return. So, we plug the beta in thefollowing equation and get the required return, ki:
ki = RF + [βi(km −RF)] (9)
where, again, ki is the required return on asset i, RF is the risk-freerate of return (usually the return on a U.S. Treasury bill), βi is thebeta coefficient or index of nondiversifiable risk for asset i, and kmis the return on a market portfolio.
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CAPM Model
ki = RF + [βi(km −RF )] (10)
Under the CAPM model, the required return on asset i
has two parts: (1) the risk-free rate of return – some-
thing like a baseline rate of return (say, the return or
interest on a 3-month T bill) – and (2) the risk pre-
mium. In turn, (2) has two parts: (a) the beta or index
of nondiversifiable risk and (b) (km−RF ) or market risk
premium. The market risk premium represents the pre-
mium investors require for taking the average amount
of risk associated with holding a well-diversified market
portfolio of assets.
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CAPM Model
The graphical representation of the CAPM model is called the se-curity market line (SML). We plot our measure of nondiversifiablerisk on the horizontal axis and the required return on the verticalaxis. Note that beta is our independent variable.
Suppose the risk-free return (RF) is 7% and the expected marketreturn (km) is 11%. Then, (km − RF) = 4%. The intercept ofour graph will be RF . The slope will indicate the change in ourrequired return when we vary beta in one unit. The slope is givenby (km −RF) = 4%.
Example in textbook, p. 217. If beta goes from 1 to 1.5, i.e.∆β = 0.5, the required return increases from 11% to 13% or2% = 0.5 × 4%.
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CAPM Model
In brief, the CAPM model provides us with a way to determine areturn adequate to the (nondiversifiable) risk involved in holdinga given asset. And determining an adequate return is essentialto value assets. It translates (nondiversifiable) risk into a riskpremium or additional return that would compensate (accordingto the market) for our taking the risk of holding that given asset.
The CAPM is not foolproof. We can only use historical data toestimate the betas. But past variability may not reflect futurevariability. So, we have to be careful and make adjustments if wehave additional information.
The CAPM, if it is to function well, requires that the market thatprices assets and, therefore, determines returns, be competitiveand efficient – in the sense of being made up by many buyersand sellers, and capable of absorbing available information quickly.It is also assumed that government or other types of restrictionsdon’t exist, that there are no taxes or transaction costs, and thatinvestors are rational. Real life is more complicated. Still, theCAPM is a nice, useful benchmark.
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The CAPM performs well with assets such as stocks of highlytraded companies. It doesn’t do as well with real corporate assets,such as buildings, equipment, etc. But that may be due to thelack of efficient markets.
All in all, the CAPM is a good starting point. And it is widely usedby finance managers in large companies.
What have we learned?
• What is risk? What is return? What’s risk aversion?
• How do we measure the risk of a given asset?
• How can we use statistical measures to quantify the risk of agiven asset?
• What is the effect on risk of diversifying a portfolio of assets?
• Are there different types of risk and what are they?
• What’s the beta of an asset?
21
• What is the capital asset pricing model (CAPM)?
• What is the security market line?
