chapter 5 risk and rates of return
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CHAPTER 5 Risk and Rates of Return. Stand-alone risk Portfolio risk Risk & return: CAPM / SML. Investment returns. The rate of return on an investment can be calculated as follows:. - PowerPoint PPT PresentationTRANSCRIPT
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CHAPTER 5 Risk and Rates of Return
Stand-alone risk Portfolio risk Risk & return: CAPM / SML
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Investment returns
The rate of return on an investment can be calculated as follows:
For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: ($1,100 - $1,000) / $1,000 = 10%.
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What is investment risk? Investment risk is related to the probability of
earning a low or negative actual return. The greater the chance of lower than expected or negative returns, the riskier the investment.
Two types of investment risk Stand-alone risk: all our money is tied to a
single asset Portfolio risk : Asset is held as one of many
assets in the portfolio
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Probability distributions
A listing of all possible outcomes, and the probability of each occurrence.
Can be shown graphically.
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Selected Realized Returns, 1926 – 2001
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T-bills T-bills will return the promised return,
regardless of the economy. T-bills do not provide a risk-free return,
as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time.
T-bills are also risky in terms of reinvestment rate risk.
T-bills are risk-free in the default sense of the word.
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Example: Investment alternatives
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How do the returns of HT and Coll. behave in relation to the market?
HT – Moves with the economy, and has a positive correlation. This is typical.
Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.
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Expected Return Weighted average Weights are probabilities Weights add up to 1
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Measuring Stand-alone Risk Calculating the standard deviation for each alternative
standard deviation is the square root of variance. It has same unit as expected return
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Comparing standard deviations
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Comments on standard deviation as a measure of risk
Standard deviation (σi) measures total, or stand-alone, risk.
The larger σi is, the lower the probability that actual returns will be closer to expected returns.
Larger σi is associated with a wider probability distribution of returns.
Difficult to compare standard deviations, because return has not been accounted for.
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Investor attitude towards risk
Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.
Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.
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Portfolio construction: Risk and return
Assume a two-stock portfolio is created with $50,000 invested in HT and $50,000 invested in Collections.
Expected return of a portfolio is a weighted average of each of the component assets of the portfolio.
Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.
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portfolio expected return
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An alternative method for determining portfolio expected return
In recession, $ return from HT $50,000*(-22%)= -$11,000$ return from Coll $50,000*(28%)= $14,000Total $ return is $3,000 so % return is 3%
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Calculating portfolio standard deviation
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Comments on portfolio risk measures σp = 3.3% is much lower than the σi of either
stock (σHT = 20.0%; σColl. = 13.4%).
σp = 3.3% is lower than the weighted average of HT and Coll.’s σ (16.7%).
Portfolio provides average return of component stocks, but lower than average risk.
Why? Negative correlation between stocks.
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Calculation of covariance and correlation
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Combination line
The risk of a portfolio is very different from a simple average of the risk of individual assets in the portfolio. There is a risk reduction from holding a portfolio of assets if assets do not move in perfect unison.
Combination line shows how the expected return and risk of a two-security portfolio changes as weights are changed.
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Combination line
PARAMETERS
E.RETURN A 0.1
VARIANCE A 0.0028
E.RETURN B 0.07
VARIANCE B 0.0049
CORRELATION A,B -0.5
COVAB -0.002
B
A
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0.00% 2.00% 4.00% 6.00% 8.00%
standard deviation
expe
cted
retu
rn
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General comments about risk
Most stocks are positively correlated with the market (ρk,m 0.65).
σ 35% for an average stock. Combining stocks in a portfolio
generally lowers risk.
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Illustrating diversification effects of a stock portfolioTo examine the relationship between portfolio size and portfolio risk, consider average annual standard deviations for equally-weighted portfolios that contain different numbers of randomly selected NYSE securities.
Number of stocksin portfolio
Average standard deviation
Of annual portfolio returns123
….1020
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diversification
Principle of diversification: Spreading an investment across many assets will eliminate some of the risk.
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Breaking down sources of risk
Stand-alone risk = Market risk + Firm-specific risk
Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.
Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.
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In other words Firm-specific or diversifiable risk is caused by
such random events as lawsuits, strikes and other events that are unique to a particular firm. Since these events are random, their effects on a portfolio can be eliminated by diversification-bad events in one firm will be offset by good events in another.
Market or non-diversifiable risk stems from factors that systematically affect most firms: war, inflation, recessions, and high interest rates. Since most stocks are negatively affected by these factors, market risk cannot be eliminated by diversification.
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single index model
Models return generating process. It offers significant new insights
into the nature of systematic vs. firm-specific risk
ri - rf = i + i(rM-rf) + i single index model
Single-index (factor) model assumes
Var(ri - rf )= Var(i + i(rM-rf) + i)= i2Var(rM-rf) + Var(i)
Total risk = market risk + firm-specific risk
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Well-diversified portfolio
Well-diversified portfoliorp= Σwiri = Σwi(rf +i + i(rM-rf) + i) =rf + Σwii + (Σwii)(rM-rf) + Σwii
Last term becomes zerorp-rf = Σwii + (Σwii)(rM-rf) = p + p(rM-rf)
So well-diversified portfolio does not have firm-specific component
Var(rp-rf )= p2Var(rM-rf)
Total risk of well-diversified portfolio is proportional to total risk of market portfolio
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For the pricing of risk what is the relevant measure?
If investors are primarily concerned with the riskiness of their portfolios rather than the riskiness of the individual securities in the portfolio, how should the riskiness of an individual stock be measured?
CAPM states that the relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified portfolio.
The risk that remains after diversifying is market risk, or the risk that is inherent in the market, and it can be measured by the degree to which a given stock tends to move up or down with the market.
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contribution of a stock to the riskiness of market portfolio
If there are N securities in the market, it can be shown that
M2=w1M1M+w2M2M+.. wNMNM
So covariance of a security is directly related to the total risk of the
market portfolio. The higher a securities covariance with market,
the higher is security’s contribution to total risk of market.
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contribution of a single stock to the riskiness of a well-diversified portfolio.
Well-diversifiedportfolio
Stock A
Has only market riskBut no firm-specific
risk
Has both market and firm-specific risks
It is only the market risk of stock A that will affect the risk of the well-diversified portfolio.
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Capital Asset Pricing Model (CAPM)Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification.
Assumptions: All investors employ Markowitz portfolio theory to find
portfolios in the efficient set and then based on their individual risk preferences invest in one of the portfolios in the efficient set
All investors have the same planning horizon and identical beliefs about the distributions of security returns
No barriers to flow of capital or information. E.g. no transaction costs, no taxes etc.
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Standard CAPM risk-free lending and borrowing and short sales
Recall the efficient set •The tangency portfolio and Rf span the efficient set. •Since investors have homogenous expectations, everybody comes up with the same tangency portfolio.•A risk-averse investor (investor A) will invest positive amounts in the risk-free asset (lends at Rf) and the tangency portfolio. •A less risk-averse investor (investor B) will short risk-free asset (borrows at Rf) and buy the tangency portfolio.
In equilibrium, the prices for all assets must adjust so that aggregate amount of borrowing equals aggregate amount of lending.
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CAPMAll efficient portfolios are combinations of market with risk free securityrp=wMP rM+wrfP rf
E(rP)=wMP E(rM)+(1-wMP) rf = rf + wMP [E(rM)-rf]
P2=wMP
2M2 or P=wMPM
Efficient portfolios’ total risk depends on market risk
[E(rP)- rf]/[E(rM)-rf]=P/M
E(rP)= rf +P [E(rM)-rf]/M
The last relation is called Capital Market Line.We need a relation between risk and return that holds for all securities/portfolios. This is call Security Market Line in CAPM.
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A measure of market risk: Beta
The tendency of a stock to move up or down with the market is reflected in its beta coefficient.
Indicates how risky a stock is if the stock is held in a well-diversified portfolio.
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Calculating betas
Run a regression of past returns of a security against past returns on the market.
The slope of the regression line (sometimes called the security’s characteristic line) is defined as the beta coefficient for the security.
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Illustrating the calculation of beta
Returns of asset i and market are excess returns. Practitioners often use total rather than excess returns. This practice is most common when daily data is used where total and excess returns are almost indistinguishable.
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Comments on beta If beta = 1.0, the security is just as risky as the average stock. If beta > 1.0, the security is riskier than average. If beta < 1.0, the security is less risky than average. Most stocks have betas in the range of 0.5 to 1.5.
Can the beta of a security be negative? Yes, if the correlation between Stock i and the market is
negative (i.e., ρi,m < 0). If the correlation is negative, the regression line would
slope downward, and the beta would be negative. However, a negative beta is highly unlikely.
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Beta coefficients for HT, Coll, and T-Bills
Slope of the regression line is given by the following formula:
Given our payoff matrix we can calculate:
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Comparing expected return and beta coefficients
Security Exp. Ret. Beta HT 17.4% 1.30Market 15.0 1.00USR 13.8 0.89T-Bills 8.0 0.00Coll. 1.7 -0.87
Riskier securities have higher returns, so the rank order is OK.
?
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The Security Market Line (SML): Calculating required rates of returnRecall that CAPM is based upon concept that a stock’s requiredrate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification.
SML equation states that the risk premium is the product of risk andextra compensation per unit of risk. Risk is measured by beta, and extra compensation by excess return on market portfolio.
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What is the market risk premium?
Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.
Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion.
Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.
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Calculating required rates of return
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Expected vs. Required returns
Ceteris paribus as price rises expected return falls
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Illustrating the Security Market Line
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Calculating portfolio betaExample: Equally-weighted two-stock portfolio
Create a portfolio with 50% invested in HT and 50% invested in Collections.
The beta of a portfolio is the weighted average of each of the stock’s betas.
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Calculating portfolio required returns The required return of a portfolio is the weighted average of each of the stock’s required returns.
kP = wHT kHT + wColl kColl kP = 0.5 (17.1%) + 0.5 (1.9%)kP = 9.5%
Or, using the portfolio’s beta, CAPM can be used to solve for required return.
kP = kRF + (kM – kRF) βP kP = 8.0% + (15.0% – 8.0%) (0.215)kP = 9.5%
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Factors that change the SML
What if investors raise inflation expectations by 3%,
what would happen to the SML?
recall that kRF =k*+IPkRF will increase by 3%, RPM stays constant since kM also increases by the same amount
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Increase in risk aversionWhat if investors’ risk aversion increased, causing the
market risk premium to increase by 3%, what would happen to the SML?
Investors would require higher risk premium per unit of risk
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how can we derive the formula for assuming that SML holds (by using single index model)
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decomposition of total (stand-alone) risk
Again you may assume that kA and kM show excess returns.
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Useful formulas
recall: cov(x,y)=E(xy) – E(x) E(y)
if cov(x,y)=0 then E(xy) = E(x) E(y)
therefore E(xy) = 0 if E(x)=0 or E(y)=0
so since cov(kM, A)=0 and E( A)=0 then E(kM A)=0