Description
It also includes markowitz theory, CAPM model, arbitrage pricing theory, comparison between arbitrage pricing theory and CAPM
Risk and Return
Topics Covered
• • • • Measuring Risk Portfolio Risk Beta and Unique Risk Diversification
2
First Principles
• Invest in projects that yield a return greater than the minimum acceptable hurdle rate.
– The hurdle rate should be higher for riskier projects and reflect the financing mix used - owners’ funds (equity) or borrowed money (debt) – Returns on projects should be measured based on cash flows generated and the timing of these cash flows; they should also consider both positive and negative side effects of these projects.
• Choose a financing mix that minimizes the hurdle rate and matches the assets being financed. • If there are not enough investments that earn the hurdle rate, return the cash to stockholders.
– The form of returns - dividends and stock buybacks - will depend upon the stockholders’ characteristics.
What is a investment or a project?
• Any decision that requires the use of resources (financial or otherwise) is a project. • Broad strategic decisions
– Entering new areas of business – Entering new markets – Acquiring other companies
• Tactical decisions • Management decisions
– The product mix to carry – The level of inventory and credit terms
• Decisions on delivering a needed service
– Lease or buy a distribution system – Creating and delivering a management information system
Return
• Return = D1/P0 + (P1-P0)/P0 • Return = Dividend yield + Capital gain or loss
Returns
• Expected Return
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
• Required Return
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
• Required Return - the return that an investor requires on an asset given its risk and market interest rates.
Expected Return
• Expected return is the mean, the average of a set of values, of the probability distribution of possible returns. i.e., sales projections • Future returns are not known with certainty.
10
Expected Return
• Expected return is the mean, or average, of the probability distribution of possible future returns. • To calculate expected return, compute the weighted average of possible returns
where m = Expected return Vi = Possible value of return during period i Pi = Probability of V occurring during period i
11
m = S(Vi x Pi)
Expected Return Calculation
Example: You are evaluating Zumwalt Corporation’s common stock. You estimate the following returns given different states of the economy
State of Economy Economic Downturn Zero Growth Moderate Growth High Growth
Probability .10 .20 .40 .30 1.00
Return –5% 5% 10% 20%
= = = = k=
– 0.5% 1.0% 4.0% 6.0% 10.5%
Expected rate of return on the stock is 10.5%
12
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% Calculate Expected return.
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% k = P(k1)*k1 + P(k2)*k2 + ...+ P(kn)*kn k (OU) = .2 (4%) + .5 (10%) + .3 (14%) = 10%
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% k = P(k1)*k1 + P(k2)*k2 + ...+ P(kn)*kn k (OT) = .2 (-10%)+ .5 (14%) + .3 (30%) = 14%
Have you considered
RISK?
What is Risk?
• Risk, in traditional terms, is viewed as a ‘negative’. • Webster’s dictionary, for instance, defines risk as “exposing to danger or hazard”. • The Chinese symbols for risk, reproduced below, give a much better description of risk
What is Risk?
• The Chinese symbols for risk, reproduced below, give a much better description of risk
• The first symbol is the symbol for “danger”, while the second is the symbol for “opportunity”, making risk a mix of danger and opportunity.
What is Risk?
• It is the variability of actual return from expected returns associated with that Security. • The possibility that an actual return will differ from our expected return. • Uncertainty in the distribution of possible outcomes.
What is Risk?
• Uncertainty in the distribution of possible outcomes.
What is Risk?
• Uncertainty in the distribution of possible outcomes.
Company A
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4 8 12
return
What is Risk?
• Uncertainty in the distribution of possible outcomes.
Company A
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4 8 12
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -10 -5 0 5 10 15 20 25 30
Company B
return
return
Instrument & Risk
• Treasury bills are about as safe an investment as you can make. No risk of default. Their short maturity means that the prices of Treasury bills are relatively stable. • Long term G-Bonds – investor acquires an asset whose price fluctuates as interest rates vary. Bond prices fall when interest rates increase and vice versa • Corporate Bonds – accept additional default risk • Common Stocks – direct share in the risk of the enterprise
Relation between Risk & Return
• How is the relationship risk and return shown graphically?
It depends on your tolerance for risk!
Return
Risk
Remember, there’s a tradeoff between risk and return.
What is the Required Rate of Return?
• The return on an investment required by an investor given market interest rates and the investment’s risk.
For a Treasury security, what is the required rate of return?
For a Treasury security, what is the required rate of return?
Required rate of return
=
Risk-free rate of return
Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the riskfree rate of return.
For a corporate stock or bond, what is the required rate of return?
For a corporate stock or bond, what is the required rate of return?
Required rate of return
=
Risk-free rate of return
+
Risk premium
How large of a risk premium should we require to buy a corporate security?
Average Market Risk Premia (1999-2000)
Risk premium, %
11 10 9 8 7 6 5 4 3 2 1 0
4.3
5.1
7.1 7.5 6 6.1 6.1 6.5 6.7
8
11 9.9 9.9 10 8.5
Ire
Aus
Swi
Can
Spa
Neth
USA
Den
Ger
Country
Swe
Jap
Fra
UK
Bel
It
Required rate of return
=
Risk-free rate of return
+
Risk premium
Required rate of return
=
Risk-free rate of return
+
Risk premium
market risk
companyunique risk
Required rate of return
=
Risk-free rate of return
+
Risk premium
market risk
companyunique risk
can be diversified away
Required rate of return
Let’s try to graph this relationship!
Beta
Required rate of return
12%
.
security market line (SML)
Risk-free rate of return (6%)
1
Beta
This linear relationship between risk and required return is known as the Capital Asset Pricing Model (CAPM).
Required rate of return
Is there a riskless (zero beta) security?
SML
12%
.
Risk-free rate of return (6%)
Treasury securities are as close to riskless as possible.
0
1
Beta
Required rate of return
Where does the BSE 500 fall on the SML?
SML
12%
.
The BSE 500 is a good approximation for the market 0
1
Beta
Risk-free rate of return (6%)
Required rate of return
SML Utility Stocks
12%
.
Risk-free rate of return (6%)
0
1
Beta
Required rate of return
High-tech stocks
SML
12%
.
Risk-free rate of return (6%)
0
1
Beta
Summary
• The stock market is risky because there is a spread of possible outcomes. • The usual measure of this spread is std. dev or variance • This risk of any stock can be broken into 2 parts: – Unique risk – that is particular to that stock – Market risk – that is associated with market-wide variations. • Investors can eliminate unique risk by holding a well diversified portfolio but they cannot eliminate market risk. • All the risk of a well diversified portfolio is market risk • A stock’s contribution to the risk of a full diversified portfolio depends on its sensitivity to market changes. This sensitivity is generally known as beta. • A security having beta of 1 has average market risk – well diversified portfolio of such securities have the same std. dev as the market index.
The notion of a benchmark
• Since financial resources are finite, there is a hurdle that projects have to cross before being deemed acceptable. • This hurdle will be higher for riskier projects than for safer projects. • A simple representation of the hurdle rate is as follows: Hurdle rate = Riskless Rate + Risk Premium • The two basic questions that every risk and return model in finance tries to answer are:
– How do you measure risk? – How do you translate this risk measure into a risk premium?
• Rm the past – receive the same normal rates of return • Rm = Rf + risk premium • Assumption – there is a normal stable risk premium on market portfolio, so that the expected future risk premium can be measured by average past risk premium. • Even with 75 years data, we cannot exactly estimate the market risk premium nor be sure that the investors today are demanding the same reward for the risk. • Risk premium – some measure the average
Risk Premium will be like – market return – assume that the future
difference between the stock returns and yield on long term bonds – difference between the compound rate of growth on stocks and interest rates.
Risk
• Reasons why history may overstate the risk premium investors demand today
Risk
Reasons why history may overstate the risk premium investors demand today: • Over the past 75 years stock prices have outpaced dividends – long term decline in dividend yield. • Focusing on returns from the country could have a bias – fortunes of countries change. • It is difficult to accurately estimate what the investor expects. • Rise in average annual returns in 2000s can be attributed to lower risk premium
How do we Measure Risk?
• A more scientific approach is to examine the stock’s standard deviation of returns. • Standard deviation is a measure of the dispersion of possible outcomes. • The greater the standard deviation, the greater the uncertainty, and therefore , the greater the risk.
Measuring Risk
Variance - Average value of squared deviations from mean. A measure of volatility. Standard Deviation - Average value of squared deviations from mean. A measure of volatility.
Measuring Risk
• • • • Variance & Standard deviation – ?2 = (x – x~)2 or ? = sq.rt.( x – x~) Variance = 1/(N-1) ? (Rm~ - Rm)2 Uncertainty – more things can happen than will • Std. deviation & variance are natural indices of risk
Standard Deviation
sS
n
=
i=1
(ki - k)2 P(ki)
s=S
n
i=1
(ki -
2 k)
P(ki)
Orlando Utility, Inc. ( 4% - 10%)2 (.2) = 7.2 (10% - 10%)2 (.5) = 0 (14% - 10%)2 (.3) = 4.8 Variance = 12 Stand. dev. = 12 = 3.46%
Orlando Technology, Inc. (-10% - 14%)2 (.2) = 115.2 (14% - 14%)2 (.5) = 0 (30% - 14%)2 (.3) = 76.8 Variance = 192 Stand. dev. = 192 = 13.86%
Which stock would you prefer? How would you decide?
Coefficient of Variance
• Coefficient of Variance: measure of relative risk or measure of risk per unit of expected returns. • It converts std. dev of expected values of returns into relative values to enable comparison of risks associated with securities having different expected values.
Summary
Orlando Utility
Expected Return Standard Deviation 10% 3.46%
Orlando Technology
14% 13.86%
Risk
Standard Deviation
• • • • Std. deviation is in the same unit as the rate of return. Std. deviation for certain outcome is? It is positive when we don’t know what will happen. Reasonable to assume that portfolio with histories of high variability also have the least predictability. • The longer the security/ portfolio you hold the more risk you have taken. • Variance is approximately proportional to the length of time interval over which a security or portfolio is measured. • Std. deviation is proportional to the square root of the interval.
Measuring Risk
Coin Toss Game-calculate variance and standard deviation
Percent Rate of Return + 40 + 10 + 10 - 20
Measuring Risk
Coin Toss Game-calculate variance and standard deviation
(1) (2) (3)
Percent Rate of Return Deviation from Mean Squared Deviation + 40 + 30 900 + 10 0 0 + 10 - 20 0 - 30 0 900
Variance = average of squared deviations = 1800 / 4 = 450 Standard deviation = square of root variance = 450 = 21.2%
Market returns
Histogram of Annual Stock Market Returns
# of Years
13 12 11 10 9 8 7 6 5 4 3 2 1 0
13
11
13
12
13
Return %
1
-50 to -40
1
-40 to -30
2
-30 to -20
4
-20 to -10 -10 to 0 0 to 10 10 to 20 20 to 30 30 to 40
3
40 to 50
2
50 to 60
Assignment: Calculate for Indian Market
Risk
Diversification
Risk
Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments. Unique Risk - Also called “diversifiable risk.” Market Risk
Diversification
• Can calculate the variability for individual securities and portfolio of securities • Individual stocks are more variable than the market indices • Even a little diversification can provide substantial reduction in variability • Diversification works because prices of different stocks do not move exactly together • Stock prices are less perfectly correlated
Portfolio Return
Portfolio rate fraction of portfolio = x of return in first asset + fraction of portfolio x
( (
in second asset
)( )(
rate of return on first asset
rate of return
on second asset
) )
Standard Deviation & Portfolio size
Portfolio standard deviation
0 5 10 15 Number of Securities
Standard Deviation & Portfolio size
Portfolio standard deviation
Unique risk Market risk
0 5 10 15 Number of Securities
Standard Deviation & Portfolio size
• On many occasions a decline in value of one stock is offset by a rise in the price of another. • An opportunity exists to reduce the risk by diversification. • The risk that can be eliminated by diversification is called unique risk. • Many of the perils that surround an individual company are peculiar to that company and its immediate competitors. • There is some risk that you cannot avoid regardless of how much you diversify – market risk – they are economy wide that threaten all businesses. • There is a tendency for stocks to move together. • That is why investors are exposed to market uncertainties. • For a reasonably diversified portfolio only market risk matters.
Portfolio
• Portfolio means combination of one or more securities. • A large number of portfolios can be formed from a set of securities. • Each portfolio has a characteristic of its own
Portfolio Risk
The variance of a two stock portfolio is the sum of these four boxes
Stock1 Stock1 Stock 2
2 2 x 1? 1
Stock 2 x 1x 2? 12 = x 1x 2? 12? 1? 2 x 2? 2 2 2
x 1x 2? 12 = x 1x 2? 12? 1? 2
Portfolio Risk
Expected Portfolio Return = (x 1 r1 ) ? ( x 2 r2 )
2 2 Portfolio Variance = x 1? 1 ? x 2? 2 ? 2( x 1x 2? 12? 1? 2 ) 2 2
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% and on Reebok it is 20%. (13.5%) Calculate the expected return on your portfolio. Assume a correlation coefficient of 1 and calculate the portfolio risk, if ? for Coca Cola is 31.5% and ? for Reebok is 58.5%.
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% x 65% = 6.5% and on Reebok it is 20% x 35% = 7.0%. The expected return on your portfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 1.
Coca - Cola Coca - Cola Reebok
2 2 x 1 ?1 = (. 65 ) 2 ? (31 .5) 2
Reebok x 1 x 2 ?12 ?1? 2 = .65 ? .35 ? 1 ? 31 .5 ? 58 .5 x 2 ? 2 = (. 35 ) 2 ? (58 .5) 2 2 2
x 1 x 2 ?12 ?1? 2 = .65 ? .35 ? 1 ? 31 .5 ? 58 .5
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% x 65% = 6.5% and on Reebok it is 20% x 35% = 7.0%. The expected return on your portfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 1.
Portfolio Valriance = [(.65)2 x(31.5)2 ] ? [(.35)2 x(58.5)2 ] ? 2(.65x.35x 1x31.5x58.5) = 1,676.1 Standard Deviation = 1,676.1 = 40.5 %
Portfolio Risk
The shaded boxes contain variance terms; the remainder contain covariance terms.
1 2 3 4 STOCK 5 6
To calculate portfolio variance add up the boxes
N 1 2 3 4 5 6 N STOCK
Portfolios
• Combining several securities in a portfolio can actually reduce overall risk. • How does this work?
Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return
kA
kB
time
What has happened to the variability of returns for the portfolio?
rate of return
kA
kB
time
What has happened to the variability of returns for the portfolio?
rate of return
kA
kB
kp
time
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified?
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified? YES! • Would you have eliminated all of your risk?
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified? YES! • Would you have eliminated all of your risk? NO! Common stock portfolios still have risk.
Diversification
• When does the greatest pay off to diversification come?
Diversification
• The greatest pay off when the 2 stocks are negatively correlated. • Unfortunately this almost never happens. • When there is perfectly negative correlation, there is always a portfolio strategy (represented by peculiar set of portfolio weights) which will completely eliminate risk. • It is too bad perfect correlation doesn’t occur between stocks.
Diversification
• Covariance between 2 stocks = ?12 = ?12 ?1 ?2, ?12 – correlation coefficient. • If the prospects of the stocks are wholly unrelated, the both the correlation coefficient and covariance would be 0. • If they move in opposite direction, the correlation coefficient and covariance will be negative. • Covariance becomes important as we add more securities in the portfolio. • Thus the variability of a well diversified portfolio reflects mainly the covariance.
Diversification
• If equal investments are made in each N stocks, • Portfolio variance=N(1/N)2 x average variance + (N2 – N) (1/N)2 x average covariance =1/N x average variance + (1-1/N) x average covariance • Notice as N increases the portfolio variance steadily approaches average covariance. • If average covariance is 0, it would be possible to eliminate risk by holding a sufficient number of securities. • Market risk – it is the average covariance which constitutes the bedrock of risk remaining after diversification has done its work. • The risk of a well diversified portfolio depends on the market risk of the securities included in the portfolio.
Diversification
• Investing in more than one security to reduce risk. • If two stocks are perfectly positively correlated, diversification has no effect on risk. • If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.
Do some firms have more market risk than others?
Yes. For example: Interest rate changes affect all firms, but which would be more affected:
a) Retail food chain b) Commercial bank
Do some firms have more market risk than others?
Yes. For example: Interest rate changes affect all firms, but which would be more affected:
a) Retail food chain b) Commercial bank
Some risk can be diversified away and some can not. • Market risk (systematic risk) is nondiversifiable. This type of risk can not be diversified away. • Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.
Risk faced by Firms
• The risk faced by a firm can be fall into the following categories –
– Project-specific; an individual project may have higher or lower cash flows than expected. – Competitive Risk, which is that the earnings and cash flows on a project can be affected by the actions of competitors. – Industry-specific Risk, which covers factors that primarily impact the earnings and cash flows of a specific industry. – International Risk, arising from having some cash flows in currencies other than the one in which the earnings are measured and stock is priced – Market risk, which reflects the effect on earnings and cash flows of macro economic factors that essentially affect all companies
Company-unique Risk
• A company’s labor force goes on strike. • A company’s top management dies in a plane crash. • A huge oil tank bursts and floods a company’s production area.
The Effects of Diversification
• On economic grounds, diversifying and holding a larger portfolio eliminates firm-specific risk for two reasons– Each investment is a much smaller percentage of the portfolio, muting the effect (positive or negative) on the overall portfolio. – Firm-specific actions can be either positive or negative. In a large portfolio, it is argued, these effects will average out to zero. (For every firm, where something bad happens, there will be some other firm, where something good happens.)
Market Risk
• Unexpected changes in interest rates. • Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle.
As you add stocks to your portfolio, company-unique risk is reduced.
portfolio risk
companyunique risk
Market risk number of stocks
Note As we know, the market compensates investors for accepting risk - but only for market risk. Company-unique risk can and should be diversified away. So - we need to be able to measure market risk.
This is why we have Beta.
Beta: a measure of market risk. • Specifically, beta is a measure of how an individual stock’s returns vary with market returns.
• It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market.
The market’s beta is 1
• A firm that has a beta = 1? • A firm with a beta > 1? • A firm with a beta < 1?
The market’s beta is 1
• A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market.
– (ex: technology firms)
• A firm with a beta < 1 is less volatile than the market.
– (ex: utilities)
Estimating Beta
• The standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) Rj = a + b R m
– where a is the intercept and b is the slope of the regression.
• The slope of the regression corresponds to the beta of the stock, and measures the riskiness of the stock.
Estimating Performance
• The intercept of the regression provides a simple measure of performance during the period of the regression, relative to the capital asset pricing model.
Rj = Rf + b (Rm - Rf) = Rf (1-b) + b Rm Rj = a + b Rm ........... ........... Capital Asset Pricing Model Regression Equation
• If a > Rf (1-b) .... Stock did better than expected during regression period a = Rf (1-b) .... Stock did as well as expected during regression period a < Rf (1-b) .... Stock did worse than expected during regression period • This is Jensen's alpha.
More on Jensen’s Alpha
If you did this analysis on every stock listed on an exchange, what would the average Jensen’s alpha be across all stocks? • Depend upon whether the market went up or down during the period • Should be zero • Should be greater than zero, because stocks tend to go up more often than down
Firm Specific and Market Risk
• The R squared (R2) of the regression provides an estimate of the proportion of the risk (variance) of a firm that can be attributed to market risk; • The balance (1 - R2) can be attributed to firm specific risk.
Setting up for the Estimation
• Decide on an estimation period
– Services use periods ranging from 2 to 5 years for the regression – Longer estimation period provides more data, but firms change. – Shorter periods can be affected more easily by significant firmspecific event that occurred during the period
• Decide on a return interval - daily, weekly, monthly
– Shorter intervals yield more observations, but suffer from more noise. – Noise is created by stocks not trading and biases all betas towards one.
• Estimate returns (including dividends) on stock
– Return = (PriceEnd - PriceBeginning + DividendsPeriod)/ PriceBeginning – Included dividends only in ex-dividend month
• Choose a market index, and estimate returns (inclusive of dividends) on the index for each interval for the period.
Beta and Unique Risk
1. Total risk = diversifiable risk + market risk 2. Market risk is measured by beta, the sensitivity to market changes - 10% Expected stock return +10% beta
+10% -10%
Expected market return
Calculating Beta: The Characteristic Line
XYZ Co. returns 15
.. .
Beta = slope = 1.20
BSE 500 returns
-15
.. . . 10 . . . . . .. . . .5 . . .. . . . . . 5 . 10 -10 -5 -5 .. . . .. . . -10 .. . . . . .-15 .
15
Summary
• We know how to measure risk, using standard deviation for overall risk and beta for market risk. • We know how to reduce overall risk to only market risk through diversification. • We need to know how to price risk so we will know how much extra return we should require for accepting extra risk.
Beta and Unique Risk
Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the BSE 500, is used to represent the market.
Beta - Sensitivity of a stock’s return to the return on the market portfolio.
Beta and Unique Risk
?
2 crucial points about security risk and portfolio risk ? market risk accounts for most of the risk of a well diversified portfolio ? ? of an individual stock measures its sensitivity to market movements
Beta and Unique Risk
?
?
?
Std. deviation of portfolio returns depend on the average number of securities in the portfolio. With more securities and therefore better diversification portfolio risk declines until all unique risk is eliminated and only the bedrock of market risk remains. Bedrock – depends on the average ? of the securities selected.
Beta and Unique Risk
s im Bi = 2 sm
Beta and Unique Risk
s im Bi = 2 sm
Covariance with the market
Variance of the market
It turns out that this covariance to variance ratio measures a stock’s contribution to portfolio risk.
Beta and Unique Risk
• Stock’s contribution to portfolio risk will depend on its relative importance in the portfolio and the average covariance with the stocks in the portfolio. • The proportion depends on the size of the holding and a measure of the effect of that holding on portfolio risk. The latter values are ?s of individual stocks relative to that portfolio. • To calculate the ? of the stock relative to the market portfolio. We calculate its covariance with the market portfolio and divide it by the variance of the market.
Fundamental Determinants of Betas
• Type of Business: Firms in more cyclical businesses or that sell products that are more discretionary to their customers will have higher betas than firms that are in non-cyclical businesses or sell products that are necessities or staples. • Operating Leverage: Firms with greater fixed costs (as a proportion of total costs) will have higher betas than firms will lower fixed costs (as a proportion of total costs) • Financial Leverage: Firms that borrow more (higher debt, relative to equity) will have higher equity betas than firms that borrow less.
Determinant 1: Product Type
• Industry Effects: The beta value for a firm depends upon the sensitivity of the demand for its products and services and of its costs to macroeconomic factors that affect the overall market.
– Cyclical companies have higher betas than noncyclical firms – Firms which sell more discretionary products will have higher betas than firms that sell less discretionary products
A Simple Test
Consider an investment in Big Bazaar. What kind of beta do you think this investment will have? • Much higher than one • Close to one • Much lower than one
Determinant 2: Operating Leverage Effects
• Operating leverage refers to the proportion of the total costs of the firm that are fixed. • Other things remaining equal, higher operating leverage results in greater earnings variability which in turn results in higher betas.
Measures of Operating Leverage
Fixed Costs Measure = Fixed Costs / Variable Costs • This measures the relationship between fixed and variable costs. The higher the proportion, the higher the operating leverage. EBIT Variability Measure = % Change in EBIT / % Change in Revenues • This measures how quickly the earnings before interest and taxes changes as revenue changes. The higher this number, the greater the operating leverage.
Determinant 3: Financial Leverage
• As firms borrow, they create fixed costs (interest payments) that make their earnings to equity investors more volatile. • This increased earnings volatility which increases the equity beta
Betas are weighted Averages
• The beta of a portfolio is always the market-value weighted average of the betas of the individual investments in that portfolio. • Thus,
– the beta of a mutual fund is the weighted average of the betas of the stocks and other investment in that portfolio – the beta of a firm after a merger is the market-value weighted average of the betas of the companies involved in the merger.
Firm Betas versus divisional Betas
• Firm Betas as weighted averages: The beta of a firm is the weighted average of the betas of its individual projects. • At a broader level of aggregation, the beta of a firm is the weighted average of the betas of its individual division.
Bottom-up versus Top-down Beta
• The top-down beta for a firm comes from a regression • The bottom up beta can be estimated by doing the following:
– Find out the businesses that a firm operates in – Find the unlevered betas of other firms in these businesses – Take a weighted (by sales or operating income) average of these unlevered betas – Lever up using the firm’s debt/equity ratio
• The bottom up beta will give you a better estimate of the true beta when
– the standard error of the beta from the regression is high (and) the beta for a firm is very different from the average for the business – the firm has reorganized or restructured itself substantially during the period of the regression – when a firm is not traded
Estimating Betas for Non-Traded Assets
• The conventional approaches of estimating betas from regressions do not work for assets that are not traded. • There are two ways in which betas can be estimated for non-traded assets
– using comparable firms – using accounting earnings
Beta for InfoSoft, a Private Software Firm
The following table summarizes the unlevered betas for publicly traded software firms. Grouping Number of Beta D/E Ratio Unlevered Firms Beta All Software 264 1.45 3.70% 1.42 Small-cap Software 125 1.54 10.12% 1.45 Entertainment Software 31 1.50 7.09% 1.43 • We will use the beta of entertainment software firms as the unlevered beta for InfoSoft. • We will also assume that InfoSoft’s D/E ratio will be similar to that of these publicly traded firms (D/E = 7.09%) • Beta for InfoSoft = 1.43 (1 + (1-.42) (.0709)) = 1.49 (Assumption: tax rate of 42%)
Total Risk versus Market Risk
• Adjust the beta to reflect total risk rather than market risk. This adjustment is a relatively simple one, since the R squared of the regression measures the proportion of the risk that is market risk.
Total Beta = Market Beta / ?R squared
•
In the InfoSoft example, where the market beta is 1.10 and the average R-squared of the comparable publicly traded firms is 16%,
– Total Beta = 1.49/?0.16 = 3.725 – Total Cost of Equity = 5% + 3.725 (5.5%)= 25.49%
• This cost of equity is much higher than the cost of equity based upon the market beta because the owners of the firm are not diversified.
Diversification
• Diversification is a good thing for an investor. • If an investor can diversify on their account, they will not pay any extra firms that diversify. • And if they have sufficiently wide choice of securities, they will not pay any less because they are unable to invest separately in each factory. • Therefore diversification does not add to a firm’s value or subtract from it. • The total value is the sum of its parts. • The concept of value additivity is important. • If capital market establishes the value of asset A as PV(A) and of asset B as PV(B). • Then PV(A+B) = PV(A) + PV(B). PV(AB) is the market value of a firm holding both these assets A & B.
RISK
• How to measure risk • How to reduce risk • How to price risk
RISK
• How to measure risk (variance, standard deviation, beta) • How to reduce risk (diversification) • How to price risk (security market line, CAPM)
Assignment
You are advising a very risky software firm on the right cost of equity to use in project analysis. You estimate a beta of 2.0 for the firm and come up with a cost of equity of 18%. The CFO of the firm is concerned about the high cost of equity and wants to know whether there is anything he can do to lower his beta. How do you bring your beta down?
Should you focus your attention on bringing your beta down? • Yes • No
Assignment
• The R squared for Deutsche Bank is very high (57%), at least relative to U.S. firms. Why is that? • The beta for Deutsche Bank is 0.84.
– Is this an appropriate measure of risk? – If not, why not?
• If you were an investor in primarily U.S. stocks, would this be an appropriate measure of risk?
Question
• Returns & Std. Deviations Particular Return T-Bills 6% ABC 10 DEF 14.5 GHI 21 Calculate the portfolio of the following portfolio: • 50% T-bills 50% ABC • 50% DEF 50% GHI assuming the shares have
– Perfect +ve correlation – Perfect –ve correlation – No correlation
Std. Dev. 0% 14 28 26
Question
• • • • 7% 27% 1% 19.1%
Question
• A game offers the following odds and pay offs. Each game costs Rs. 100.
Probability 10% 50% 40% Payoff 500 100 0
What are the expected cash payoffs and expected rate of return? Calculate the variance and standard deviation of this rate of return.
132
Question
• • • • Expected cash payoffs – 100 Expected rate of return – 0 Variance – 20,000 Standard deviation – 141%
133
Question
• Ritesh Modi, Head of Aum Mutual Fund, has produced the following rates of return from 2003 to 2007. Rates of return of BSE 100 have been given for comparison Year Modi(%) BSE 100(%) 2003 16.1 23.1 2004 28.4 33.4 2005 25.1 28.6 2006 14.3 21.0 2007 -6.0 -9.1 • Calculate Modi’s average return and standard deviation. • Did he do better or worse than BSE 100 by these measures?
134
Question
• Result Average return Standard deviation Modi 15.6 12.0 BSE 100 19.4 14.9
135
Question
• State whether True or False
1. Investors prefer diversified companies because they are less risky 2. If stocks were perfectly positively correlated, diversification would not reduce risk 3. The contribution of a stock to the risk of a well diversified portfolio depends in its market risk 4. A well diversified portfolio with a beta 2 is twice as risky as the market portfolio 5. An undiversified portfolio with a beta of 2 is less than twice as risky as the market portfolio
136
Question
• • • • • F T T T F
137
Question
• Suppose stand deviation of the market return is 20%. 1. What is the standard deviation of returns of a well diversified portfolio with beta 1.3 another with beta 0. 2. A well diversified portfolio having standard deviation of 15%, what is its beta.
138
Question
1. Beta 1.3 - 26% Beta 0 – 0% 2. Standard deviation 15% Beta – 0.75
139
Question
• Hitesh Shah invests 60% of his fund in Stock 1 and the balance in stock 2. The standard deviation of 1 is 10% and of 2 is 20%. Calculate the variance of the portfolio returns if
– – – – Correlation is 1 Correlation is 0.5 Correlation is 0 Correlation is -1
140
Individual Assignment Question
• If the Beta of Company A & B are as follows: A B Beta 2.21 1.81 Std. Dev. 62.7 50.7 • Assume std dev of return on market to be 15%. • The correlation coeff is 0.66, what is the std dev of the portfolio, if equal invesment is made in A & B. • What is the std. dev. If 1/3 each is invested in A, B and T-bills • What is the std dev if invested equally between A & B, but investment is financed at 50% margin – the investor puts only 50% of the amount and borrows the balance from the broker. • What would be the std dev of a portfolio of 100 stocks having beta of A? How about 100 stocks like B?
141
Individual Assignment Question
• Some stocks have high standard deviations and relatively low betas. Sometimes it is the other way round. Why do think so? Illustrate your answer by calculating standard deviation and betas using data for 5 years data (use daily or weekly data) for the 2 companies selected by you. Also find the standard deviation and beta of a portfolio comprising of the 2 stocks in varying proportion stating from (10% to 90%, in steps of 10%).
142
Topics Covered
• • • • Markowitz Portfolio Theory Risk and Return Relationship Testing the CAPM CAPM Alternatives
Markowitz Portfolio Theory
• Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation. • Correlation coefficients make this possible. • The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfolios.
Markowitz Portfolio Theory
Price changes vs. Normal distribution
0.14
Microsoft - Daily % change 1990-2001
Proportion of Days
0.12 0.1 0.08 0.06 0.04 0.02 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Daily % Change
Returns - Distribution
• When measured over fairly short interval of time, the past rates of return on any stock conform closely to a normal distribution. • If returns are distributed normally then investors only need to consider expected returns and std. dev. • If you were to measure over a long interval, then the distribution would be skewed. Use log normal distribution.
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment A
20 18
% probability
16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment B
20 18
% probability
16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment C
20
% probability
18 16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment D
20
% probability
18 16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
? Expected Returns and Standard Deviations vary given different weighted combinations of the stocks
Reebok Expected Return (%)
35% in Reebok
Coca Cola Standard Deviation
Efficient Frontier
•Each half egg shell represents the possible weighted combinations for two stocks. •The composite of all stock sets constitutes the efficient frontier
Expected Return (%)
Standard Deviation
Markowitz Portfolio Theory
• Expected return is simply a weighted average of the expected return in the holding. • The risk of the portfolio is less than the average risk of the separate stocks. • Markowitz – efficient portfolio – along solid enveloping line • Problem of rationing – to deploy a limited amount of capital to capital investment in a mixture of projects to give the highest NPV. Here we deploy investors funds to give the highest expected return for a given std. dev. Solution using quadratic programming • Given the expected return & std. dev of each stock, as well as correlation between each pair of stocks, we can calculate a set of efficient portfolios using quadratic program • If stocks are highly correlated they offer less diversification benefits
Lending & Borrowing
• Suppose you can lend and borrow at some risk free rate. • If you invest your money T-bills and place the remainder in common stock portfolio S, you can obtain any combination of expected risk returns along the straight line joining Rf and S. • Since borrowing is negative of lending you can extend the range to the right of S by borrowing at Rf and investing in S. • Regardless of the level of risk you choose, you can get the highest expected return by a mixture of portfolio S and T-bills (lending or borrowing). S is the best efficient portfolio.
Efficient Frontier
•Lending or Borrowing at the risk free rate (rf) allows us to exist outside the efficient frontier.
S
Expected Return (%)
rf
T
Standard Deviation
Lending & Borrowing
• Start with the vertical axis at Rf and draw the steepest line or tangent to the curved heavy line of efficient portfolios. • The efficient portfolio at the tangency point is better than all others. It offers the highest ratio of risk premium to std. dev.
– Best portfolio of common stock must be selected S. – This portfolio must be blended with borrowing or lending to obtain an exposure to risk that suits the particular investors’ taste.
• Each investor must therefore put money in 2 benchmark investments – risky portfolio S and risk free loan (lending or borrowing) T-bills • Portfolio S is the point of tangency to the set of efficient portfolios. It offers the highest expected risk premium (r – Rf) per unit of std. dev.
Efficient Portfolio
• If you have better information than your rivals you will want the portfolio to include relatively large investment in the stock you think is undervalued. • But in competitive world you are unlikely to have monopoly of good ideas. • In that case there is no reason to hold different portfolio of common stocks from anybody else. • Hold market portfolio. • That is why many professional investors invest in a market index portfolio and not hold any other well diversified portfolio.
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation ? Return ?
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4%
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4%
Let’s Add stock XYZ to the portfolio
Efficient Frontier
Example Stocks Portfolio XYZ
s
28.1 30
Correlation Coefficient = .3 % of Portfolio Avg Return 50% 17.4% 50% 19%
NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%
Efficient Frontier
Example Stocks Portfolio XYZ
s
28.1 30
Correlation Coefficient = .3 % of Portfolio Avg Return 50% 17.4% 50% 19%
NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%
NOTE: Higher return & Lower risk How did we do that? DIVERSIFICATION
Efficient Frontier
Return
B
A
Risk (measured as
s)
Efficient Frontier
Return
B
AB A Risk
Efficient Frontier
Return
B
AB A X
Risk
Efficient Frontier
Return
B
ABX AB A X
Risk
Efficient Frontier
Return
Goal is to move up and left. WHY? B
ABX AB A X
Risk
Efficient Frontier
Return Low Risk High Return High Risk High Return
Low Risk Low Return
High Risk Low Return Risk
Efficient Frontier
Return Low Risk High Return High Risk High Return
Low Risk Low Return
High Risk Low Return Risk
Efficient Frontier
Return B
ABX AB A
X
Risk
Security Market Line
Return
Market Return = rm Risk Free Return =
.
Efficient Portfolio
rf
Risk
Security Market Line
Return
Market Return = rm Risk Free Return =
.
Efficient Portfolio
rf
1.0 BETA
Security Market Line
Return
.
Risk Free Return =
rf
Security Market Line (SML)
BETA
Security Market Line
SML Return
rf
1.0
BETA
SML Equation = rf + B ( rm - rf )
Risk & Return Relationship
• The return on T-bills is fixed, it is unaffected by what happens in the market. Its beta is 0. • The average market risk: beta = 1 • Wise don’t take risk just for fun. They are playing with real money. Therefore they require a higher return from the market portfolio than from T-bills. • The difference between the return on the market and T-bills is termed as market risk premium = Rm – Rf
Capital Asset Pricing Model
R = r f + B ( r m - rf )
CAPM
CAPM
• What is the expected risk premium when beta is neither 0 nor 1? • William Sharpe John Linter Jack Treynor – CAPM • In a competitive market the expected risk premium varies in direct proportion to beta. This means all investments must plot along the slopping line known as Security Market Line (SML). • Expected risk premium on stock = beta * expected risk premium on market • i.e. r – Rf = b(Rm-Rf)
Capital Asset Pricing Model
Assumptions behind the capital asset pricing model are: • Investors maximize the expected wealth at the end of one period. • Investors choose between portfolios on the basis of the mean and variance of return. This implies that returns are normally distributed. • Investors can borrow or lend at an exogenously determined risk-free rate of interest. • There are no restrictions on short selling. • Investors have homogeneous probability beliefs. • The market is perfect. Thus, all assets are marketable, there are no transaction costs or taxes, and all investors are price takers.
Capital Asset Pricing Model
• You can also use CAPM to find the discount rate for new capital investment. • Instead of investing in plant and machinery the firm could return the money to the shareholders. • The opportunity cost of investing is the return that shareholders expect to earn buying financial assets. This expected return depends on the market risk of the asset. • In practice choosing discount rate is seldom easy. • One has to adjust for the extra risk caused by borrowing & how to estimate the discount rates for projects that do not have the same risks as the company’s existing business. • There are also tax issues.
Capital Asset Pricing Model
• • • Investors like high expected return and low std. dev. Common stocks portfolio that offer the highest expected return for a given std. dev are known as efficient portfolio. If the investor can lend or borrow at risk-free rate of interest, one efficient portfolio is better than all others: the portfolio that offers the highest risk premium to std. dev. A risk-averse investor will put part of his money in this efficient portfolio and part of risk-free asset. A risk tolerant investor may borrow and put all his money in this efficient portfolio. The composition of this best efficient portfolio depends on investors assessment of expected returns, std. dev and correlations. But suppose everyone has the same information and the same assessment. If there is no superior information, each investor should hold the same portfolio as everyone else, i.e. hold market portfolio.
•
• • •
Capital Asset Pricing Model
• Don’t look at risk of a stock in isolation but at its contribution to portfolio risk. This contribution depends on the stocks’ sensitivity to the changes in the value of the portfolio. A stock’s sensitivity to changes in the value of the market portfolio is known as beta. Beta measures the marginal contribution of a stock to the risk of the market portfolio. If everyone holds the market portfolio, if beta measures each security’s contribution to the market portfolio risks, then risk premium demanded by investor would be proportional to beta. That is CAPM
• • •
Estimating Risk Premiums in Practice
• Survey investors on their desired risk premiums and use the average premium from these surveys. • Assume that the actual premium delivered over long time periods is equal to the expected premium - i.e., use historical data • Estimate the implied premium in today’s asset prices.
The Survey Approach
• Surveying all investors in a market place is impractical. • However, you can survey a few investors (especially the larger investors) and use these results. In practice, this translates into surveys of money managers’ expectations of expected returns on stocks over the next year. • The limitations of this approach are:
– there are no constraints on reasonability (the survey could produce negative risk premiums or risk premiums of 50%) – they are extremely volatile – they tend to be short term; even the longest surveys do not go beyond one year
The Historical Premium Approach
• This is the default approach used by most to arrive at the premium to use in the model • In most cases, this approach does the following
– it defines a time period for the estimation (1926-Present, 1962-Present....) – it calculates average returns on a stock index during the period – it calculates average returns on a riskless security over the period – it calculates the difference between the two – and uses it as a premium looking forward
• The limitations of this approach are:
– it assumes that the risk aversion of investors has not changed in a systematic way across time. (The risk aversion may change from year to year, but it reverts back to historical averages) – it assumes that the riskiness of the “risky” portfolio (stock index) has not changed in a systematic way across time.
Implied Equity Premiums
• If we use a basic discounted cash flow model, we can estimate the implied risk premium from the current level of stock prices. • For instance, if stock prices are determined by the simple Gordon Growth Model:
– Value = Expected Dividends next year/ (Required Returns on Stocks Expected Growth Rate) – Plugging in the current level of the index, the dividends on the index and expected growth rate will yield a “implied” expected return on stocks. Subtracting out the risk-free rate will yield the implied premium.
• The problems with this approach are:
– the discounted cash flow model used to value the stock index has to be the right one. – the inputs on dividends and expected growth have to be correct – it implicitly assumes that the market is currently correctly valued
The Market Portfolio
• Assuming diversification costs nothing (in terms of transactions costs), and that all assets can be traded, the limit of diversification is to hold a portfolio of every single asset in the economy (in proportion to market value). This portfolio is called the market portfolio. • Individual investors will adjust for risk, by adjusting their allocations to this market portfolio and a riskless asset (such as a T-Bill) Preferred risk level Allocation decision No risk 100% in T-Bills Some risk 50% in T-Bills; 50% in Market Portfolio; A little more risk 25% in T-Bills; 75% in Market Portfolio Even more risk 100% in Market Portfolio A risk hog.. Borrow money; Invest in market portfolio; • Every investor holds some combination of the risk free asset and the market portfolio.
Validity of CAPM
• Investors are principally concerned with those risks that they cannot eliminate by diversification. • If it were not so, we should find that stock prices increases whenever 2 companies merge to spread their risk. So also find that investment companies which invest in the shares of other companies are more valuable than the shares they hold. But that is not true. • Many finance managers find CAPM the most convenient tool for coming to grips with the slippery notion of risk.
Validity of CAPM
• CAPM predicts that the risk premium should increase in proportion to beta. • Actual stock returns reflect expectations but they also embody lot of ‘noise’ – the steady flow of surprises that conceal whether an average investor has received the returns they expected. This noise makes it impossible to judge whether the model holds better in one period than another. • Market portfolio should contain actually all risky investments – stocks, bonds, commodities, real estate and human capital. However, most indices contain only a sample of common stocks.
Validity of CAPM
• Fisher & Black have shown that if there are borrowing restrictions there should still exists a positive relationship between expected return and beta but the SML would be less steep. • Burgundy Line – cumulative difference between the returns on small cap shares and return on large cap share i.e. if you buy small cap and sell large cap • The small cap did not always do well but over the long run they have made substantially better than large cap
Validity of CAPM
• Value stock – high ratio of book value to market value • Growth stock – low ratio of book to market value • Historically the value stocks have shown higher returns than growth stocks. This is in contradiction to CAPM, where higher risk meant higher returns. • Small firms have higher betas, but the difference in beta is not sufficient to explain the difference in returns. There is no simple relationship between book to market ratios and betas.
Testing the CAPM
Return vs. Book-to-Market
25
Dollars
20 15 10 5 0
1928 1933 1938 1943 1948 1953 1958
High-minus low book-tomarket
Small cap minus Large cap
1963
1968
1973
1978
1983
1988
1993
1998
Limitation of CAPM
•
• •
•
T-bills are risk free – no chance of default. But don’t guarantee real returns. Uncertainty of inflation Investors can borrow money at the same rate of interest at which they lend. Generally the borrowing rates are higher than lending rates. The real important idea is that the investors are content to invest their money in a limited number of benchmark portfolios. In basic CAPM – T-bills and market portfolio are the basic benchmarks In modified CAPMs expected returns still depend on market risk, but the definition of market risk depends on the nature of the benchmark portfolio.
Limitations of the CAPM
• The model makes unrealistic assumptions • The parameters of the model cannot be estimated precisely
– - Definition of a market index – - Firm may have changed during the 'estimation' period‘
• The model does not work well
– - If the model is right, there should be
• a linear relationship between returns and betas • the only variable that should explain returns is betas
– - The reality is that
• the relationship between betas and returns is weak • Other variables (size, price/book value) seem to explain differences in returns better.
Consumption Betas vs Market Betas
Stocks
(and other risky assets)
Market risk makes wealth uncertain.
Standard CAPM
Wealth = market portfolio
Consumption Betas vs Market Betas
Stocks Stocks
(and other risky assets)
(and other risky assets)
Wealth is uncertain
Market risk makes wealth uncertain.
Standard CAPM
Wealth
Consumption CAPM
Consumption is uncertain
Wealth = market portfolio
Consumption
Consumption Betas vs Market Betas
• The standard CAPM concentrates on how stocks contribute to the level of uncertainty of investors’ wealth. • Standard CAPM assumes that investors are concerned with the amount and uncertainty of future wealth. • Market risk makes future wealth uncertain. • The consumption beta is defined as a stock’s contribution to uncertainty about consumption. The consumption CAPM connects uncertainty about stock returns directly to uncertainty about consumption. It does not have any immediate practical use as individual stocks have low or erratic consumption betas.
Arbitrage Pricing Theory
Alternative to CAPM
Expected Risk Premium = r - rf = Bfactor1(rfactor1 Return
- rf) + Bf2(rf2 - rf) + …
= a + bfactor1(rfactor1)
+ bf2(rf2) + …
Arbitrage Pricing Theory
• APT is derived assuming that stock returns are based partly on “factors” and partly on “noise.” For any individual stock there are two sources of risk, the risk based on factors and the risk based on noise. Diversification eliminates risk based on noise and only risk based on factors is relevant.
Arbitrage Pricing Theory
• It assumes that each stock’s return depends partly on pervasive macroeconomic influences or factors and partly by noise – events that are unique to that company. • Return = a + b1(Rfactor1) + b2(Rfactor2) + … + noise • The theory doesn’t say what these factors are – they could be oil prices, interest rate, etc. • Return on market portfolio may or may not serve as a factor. • Some stocks will be more sensitive to particular factor than other stocks.
Arbitrage Pricing Theory
• For any individual stock, there are 2 sources of risks:
– Pervasive macroeconomic factors, which cannot be eliminated by diversification – Risk arising from possible events that are unique to the company.
• If diversification does eliminate unique risk, then diversified investor must ignore it when deciding to buy or sell a stock.
Arbitrage Pricing Theory
• The expected risk premium on a stock is affected by macro economic risks, it is not affected by unique risk. • APT states that the expected risk premium on a stock should depend on the expected risk premium associated with each factor and the stock’s sensitivity to each factor. • There may be some macroeconomic factors that investors are simply not concerned with, e.g. inflation. Such factors would not command a risk premium and would drop out of APT. • Expected risk premium = r-Rf=b1(Rfactor1-Rf) + b2(Rfactor2-Rf) +…
Comparison between APT and CAPM
CAPM Market Portfolios APT Market portfolio does not feature, it may be taken as one of the factors. So we don’t bother measuring market portfolios Doesn’t tell us what the underlying factors are
Collapses all macroeconomic risks into a well defined single factor – return on market portfolio
Arbitrage Pricing Theory
Estimated risk premiums for taking on risk factors (1978-1990)
Factor Yield spread Interest rate Exchange rate Real GNP Inflation Mrket Estimated Risk Premium (rfactor ? rf ) 5.10% - .61 - .59 .49 - .83 6.36
APT Calculations
• APT – only if can identify a reasonable short list of macroeconomic factors, measure the expected risk premiums on each factor and measure the sensitivity of each stock to these factors. • Step 1: Identify the Macroeconomic factors • Step 2: Estimate Risk Premiums for each factor • Step3: Estimate the Factor sensitivities
FAMA French 3 Factor Model
• Stocks of small cap stocks with high book to market ration have provided with above average returns – there factors are related to company profitability and therefore may be picking up risk factors that are left out of the CAPM. • If investors do demand an extra premium for taking exposure to those factors, then we have a measure of the expected return similar to APT, as • r-Rf = bmarket(Rmarketfactor) + bsize(Rsizefactor) + bbooktomarket(Rbooktomarket)
FAMA French 3 Factor Model
• Step 1: Identify the factors
– – – – 3 factors are Market factor – Return on market index Size factor – Return on small cap less return on large cap Book to market – Return on high book to market less return on low book to market return
• Step 2: Estimate the risk Premium for each stock – Historical data • Step 3: Estimate the Factor sensitivities • Some stock are more sensitive than others to fluctuations in return on the 3 factors. • The 3 factor model provides substantially lower estimate than CAPM.
doc_422133043.pptx
It also includes markowitz theory, CAPM model, arbitrage pricing theory, comparison between arbitrage pricing theory and CAPM
Risk and Return
Topics Covered
• • • • Measuring Risk Portfolio Risk Beta and Unique Risk Diversification
2
First Principles
• Invest in projects that yield a return greater than the minimum acceptable hurdle rate.
– The hurdle rate should be higher for riskier projects and reflect the financing mix used - owners’ funds (equity) or borrowed money (debt) – Returns on projects should be measured based on cash flows generated and the timing of these cash flows; they should also consider both positive and negative side effects of these projects.
• Choose a financing mix that minimizes the hurdle rate and matches the assets being financed. • If there are not enough investments that earn the hurdle rate, return the cash to stockholders.
– The form of returns - dividends and stock buybacks - will depend upon the stockholders’ characteristics.
What is a investment or a project?
• Any decision that requires the use of resources (financial or otherwise) is a project. • Broad strategic decisions
– Entering new areas of business – Entering new markets – Acquiring other companies
• Tactical decisions • Management decisions
– The product mix to carry – The level of inventory and credit terms
• Decisions on delivering a needed service
– Lease or buy a distribution system – Creating and delivering a management information system
Return
• Return = D1/P0 + (P1-P0)/P0 • Return = Dividend yield + Capital gain or loss
Returns
• Expected Return
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
• Required Return
Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
• Required Return - the return that an investor requires on an asset given its risk and market interest rates.
Expected Return
• Expected return is the mean, the average of a set of values, of the probability distribution of possible returns. i.e., sales projections • Future returns are not known with certainty.
10
Expected Return
• Expected return is the mean, or average, of the probability distribution of possible future returns. • To calculate expected return, compute the weighted average of possible returns
where m = Expected return Vi = Possible value of return during period i Pi = Probability of V occurring during period i
11
m = S(Vi x Pi)
Expected Return Calculation
Example: You are evaluating Zumwalt Corporation’s common stock. You estimate the following returns given different states of the economy
State of Economy Economic Downturn Zero Growth Moderate Growth High Growth
Probability .10 .20 .40 .30 1.00
Return –5% 5% 10% 20%
= = = = k=
– 0.5% 1.0% 4.0% 6.0% 10.5%
Expected rate of return on the stock is 10.5%
12
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% Calculate Expected return.
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% k = P(k1)*k1 + P(k2)*k2 + ...+ P(kn)*kn k (OU) = .2 (4%) + .5 (10%) + .3 (14%) = 10%
Expected Return
State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession .20 4% -10% Normal .50 10% 14% Boom .30 14% 30% k = P(k1)*k1 + P(k2)*k2 + ...+ P(kn)*kn k (OT) = .2 (-10%)+ .5 (14%) + .3 (30%) = 14%
Have you considered
RISK?
What is Risk?
• Risk, in traditional terms, is viewed as a ‘negative’. • Webster’s dictionary, for instance, defines risk as “exposing to danger or hazard”. • The Chinese symbols for risk, reproduced below, give a much better description of risk
What is Risk?
• The Chinese symbols for risk, reproduced below, give a much better description of risk
• The first symbol is the symbol for “danger”, while the second is the symbol for “opportunity”, making risk a mix of danger and opportunity.
What is Risk?
• It is the variability of actual return from expected returns associated with that Security. • The possibility that an actual return will differ from our expected return. • Uncertainty in the distribution of possible outcomes.
What is Risk?
• Uncertainty in the distribution of possible outcomes.
What is Risk?
• Uncertainty in the distribution of possible outcomes.
Company A
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4 8 12
return
What is Risk?
• Uncertainty in the distribution of possible outcomes.
Company A
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4 8 12
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -10 -5 0 5 10 15 20 25 30
Company B
return
return
Instrument & Risk
• Treasury bills are about as safe an investment as you can make. No risk of default. Their short maturity means that the prices of Treasury bills are relatively stable. • Long term G-Bonds – investor acquires an asset whose price fluctuates as interest rates vary. Bond prices fall when interest rates increase and vice versa • Corporate Bonds – accept additional default risk • Common Stocks – direct share in the risk of the enterprise
Relation between Risk & Return
• How is the relationship risk and return shown graphically?
It depends on your tolerance for risk!
Return
Risk
Remember, there’s a tradeoff between risk and return.
What is the Required Rate of Return?
• The return on an investment required by an investor given market interest rates and the investment’s risk.
For a Treasury security, what is the required rate of return?
For a Treasury security, what is the required rate of return?
Required rate of return
=
Risk-free rate of return
Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the riskfree rate of return.
For a corporate stock or bond, what is the required rate of return?
For a corporate stock or bond, what is the required rate of return?
Required rate of return
=
Risk-free rate of return
+
Risk premium
How large of a risk premium should we require to buy a corporate security?
Average Market Risk Premia (1999-2000)
Risk premium, %
11 10 9 8 7 6 5 4 3 2 1 0
4.3
5.1
7.1 7.5 6 6.1 6.1 6.5 6.7
8
11 9.9 9.9 10 8.5
Ire
Aus
Swi
Can
Spa
Neth
USA
Den
Ger
Country
Swe
Jap
Fra
UK
Bel
It
Required rate of return
=
Risk-free rate of return
+
Risk premium
Required rate of return
=
Risk-free rate of return
+
Risk premium
market risk
companyunique risk
Required rate of return
=
Risk-free rate of return
+
Risk premium
market risk
companyunique risk
can be diversified away
Required rate of return
Let’s try to graph this relationship!
Beta
Required rate of return
12%
.
security market line (SML)
Risk-free rate of return (6%)
1
Beta
This linear relationship between risk and required return is known as the Capital Asset Pricing Model (CAPM).
Required rate of return
Is there a riskless (zero beta) security?
SML
12%
.
Risk-free rate of return (6%)
Treasury securities are as close to riskless as possible.
0
1
Beta
Required rate of return
Where does the BSE 500 fall on the SML?
SML
12%
.
The BSE 500 is a good approximation for the market 0
1
Beta
Risk-free rate of return (6%)
Required rate of return
SML Utility Stocks
12%
.
Risk-free rate of return (6%)
0
1
Beta
Required rate of return
High-tech stocks
SML
12%
.
Risk-free rate of return (6%)
0
1
Beta
Summary
• The stock market is risky because there is a spread of possible outcomes. • The usual measure of this spread is std. dev or variance • This risk of any stock can be broken into 2 parts: – Unique risk – that is particular to that stock – Market risk – that is associated with market-wide variations. • Investors can eliminate unique risk by holding a well diversified portfolio but they cannot eliminate market risk. • All the risk of a well diversified portfolio is market risk • A stock’s contribution to the risk of a full diversified portfolio depends on its sensitivity to market changes. This sensitivity is generally known as beta. • A security having beta of 1 has average market risk – well diversified portfolio of such securities have the same std. dev as the market index.
The notion of a benchmark
• Since financial resources are finite, there is a hurdle that projects have to cross before being deemed acceptable. • This hurdle will be higher for riskier projects than for safer projects. • A simple representation of the hurdle rate is as follows: Hurdle rate = Riskless Rate + Risk Premium • The two basic questions that every risk and return model in finance tries to answer are:
– How do you measure risk? – How do you translate this risk measure into a risk premium?
• Rm the past – receive the same normal rates of return • Rm = Rf + risk premium • Assumption – there is a normal stable risk premium on market portfolio, so that the expected future risk premium can be measured by average past risk premium. • Even with 75 years data, we cannot exactly estimate the market risk premium nor be sure that the investors today are demanding the same reward for the risk. • Risk premium – some measure the average
Risk Premium will be like – market return – assume that the future
difference between the stock returns and yield on long term bonds – difference between the compound rate of growth on stocks and interest rates.
Risk
• Reasons why history may overstate the risk premium investors demand today
Risk
Reasons why history may overstate the risk premium investors demand today: • Over the past 75 years stock prices have outpaced dividends – long term decline in dividend yield. • Focusing on returns from the country could have a bias – fortunes of countries change. • It is difficult to accurately estimate what the investor expects. • Rise in average annual returns in 2000s can be attributed to lower risk premium
How do we Measure Risk?
• A more scientific approach is to examine the stock’s standard deviation of returns. • Standard deviation is a measure of the dispersion of possible outcomes. • The greater the standard deviation, the greater the uncertainty, and therefore , the greater the risk.
Measuring Risk
Variance - Average value of squared deviations from mean. A measure of volatility. Standard Deviation - Average value of squared deviations from mean. A measure of volatility.
Measuring Risk
• • • • Variance & Standard deviation – ?2 = (x – x~)2 or ? = sq.rt.( x – x~) Variance = 1/(N-1) ? (Rm~ - Rm)2 Uncertainty – more things can happen than will • Std. deviation & variance are natural indices of risk
Standard Deviation
sS
n
=
i=1
(ki - k)2 P(ki)
s=S
n
i=1
(ki -
2 k)
P(ki)
Orlando Utility, Inc. ( 4% - 10%)2 (.2) = 7.2 (10% - 10%)2 (.5) = 0 (14% - 10%)2 (.3) = 4.8 Variance = 12 Stand. dev. = 12 = 3.46%
Orlando Technology, Inc. (-10% - 14%)2 (.2) = 115.2 (14% - 14%)2 (.5) = 0 (30% - 14%)2 (.3) = 76.8 Variance = 192 Stand. dev. = 192 = 13.86%
Which stock would you prefer? How would you decide?
Coefficient of Variance
• Coefficient of Variance: measure of relative risk or measure of risk per unit of expected returns. • It converts std. dev of expected values of returns into relative values to enable comparison of risks associated with securities having different expected values.
Summary
Orlando Utility
Expected Return Standard Deviation 10% 3.46%
Orlando Technology
14% 13.86%
Risk
Standard Deviation
• • • • Std. deviation is in the same unit as the rate of return. Std. deviation for certain outcome is? It is positive when we don’t know what will happen. Reasonable to assume that portfolio with histories of high variability also have the least predictability. • The longer the security/ portfolio you hold the more risk you have taken. • Variance is approximately proportional to the length of time interval over which a security or portfolio is measured. • Std. deviation is proportional to the square root of the interval.
Measuring Risk
Coin Toss Game-calculate variance and standard deviation
Percent Rate of Return + 40 + 10 + 10 - 20
Measuring Risk
Coin Toss Game-calculate variance and standard deviation
(1) (2) (3)
Percent Rate of Return Deviation from Mean Squared Deviation + 40 + 30 900 + 10 0 0 + 10 - 20 0 - 30 0 900
Variance = average of squared deviations = 1800 / 4 = 450 Standard deviation = square of root variance = 450 = 21.2%
Market returns
Histogram of Annual Stock Market Returns
# of Years
13 12 11 10 9 8 7 6 5 4 3 2 1 0
13
11
13
12
13
Return %
1
-50 to -40
1
-40 to -30
2
-30 to -20
4
-20 to -10 -10 to 0 0 to 10 10 to 20 20 to 30 30 to 40
3
40 to 50
2
50 to 60
Assignment: Calculate for Indian Market
Risk
Diversification
Risk
Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments. Unique Risk - Also called “diversifiable risk.” Market Risk
Diversification
• Can calculate the variability for individual securities and portfolio of securities • Individual stocks are more variable than the market indices • Even a little diversification can provide substantial reduction in variability • Diversification works because prices of different stocks do not move exactly together • Stock prices are less perfectly correlated
Portfolio Return
Portfolio rate fraction of portfolio = x of return in first asset + fraction of portfolio x
( (
in second asset
)( )(
rate of return on first asset
rate of return
on second asset
) )
Standard Deviation & Portfolio size
Portfolio standard deviation
0 5 10 15 Number of Securities
Standard Deviation & Portfolio size
Portfolio standard deviation
Unique risk Market risk
0 5 10 15 Number of Securities
Standard Deviation & Portfolio size
• On many occasions a decline in value of one stock is offset by a rise in the price of another. • An opportunity exists to reduce the risk by diversification. • The risk that can be eliminated by diversification is called unique risk. • Many of the perils that surround an individual company are peculiar to that company and its immediate competitors. • There is some risk that you cannot avoid regardless of how much you diversify – market risk – they are economy wide that threaten all businesses. • There is a tendency for stocks to move together. • That is why investors are exposed to market uncertainties. • For a reasonably diversified portfolio only market risk matters.
Portfolio
• Portfolio means combination of one or more securities. • A large number of portfolios can be formed from a set of securities. • Each portfolio has a characteristic of its own
Portfolio Risk
The variance of a two stock portfolio is the sum of these four boxes
Stock1 Stock1 Stock 2
2 2 x 1? 1
Stock 2 x 1x 2? 12 = x 1x 2? 12? 1? 2 x 2? 2 2 2
x 1x 2? 12 = x 1x 2? 12? 1? 2
Portfolio Risk
Expected Portfolio Return = (x 1 r1 ) ? ( x 2 r2 )
2 2 Portfolio Variance = x 1? 1 ? x 2? 2 ? 2( x 1x 2? 12? 1? 2 ) 2 2
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% and on Reebok it is 20%. (13.5%) Calculate the expected return on your portfolio. Assume a correlation coefficient of 1 and calculate the portfolio risk, if ? for Coca Cola is 31.5% and ? for Reebok is 58.5%.
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% x 65% = 6.5% and on Reebok it is 20% x 35% = 7.0%. The expected return on your portfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 1.
Coca - Cola Coca - Cola Reebok
2 2 x 1 ?1 = (. 65 ) 2 ? (31 .5) 2
Reebok x 1 x 2 ?12 ?1? 2 = .65 ? .35 ? 1 ? 31 .5 ? 58 .5 x 2 ? 2 = (. 35 ) 2 ? (58 .5) 2 2 2
x 1 x 2 ?12 ?1? 2 = .65 ? .35 ? 1 ? 31 .5 ? 58 .5
Portfolio Risk
Example Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% x 65% = 6.5% and on Reebok it is 20% x 35% = 7.0%. The expected return on your portfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 1.
Portfolio Valriance = [(.65)2 x(31.5)2 ] ? [(.35)2 x(58.5)2 ] ? 2(.65x.35x 1x31.5x58.5) = 1,676.1 Standard Deviation = 1,676.1 = 40.5 %
Portfolio Risk
The shaded boxes contain variance terms; the remainder contain covariance terms.
1 2 3 4 STOCK 5 6
To calculate portfolio variance add up the boxes
N 1 2 3 4 5 6 N STOCK
Portfolios
• Combining several securities in a portfolio can actually reduce overall risk. • How does this work?
Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return
kA
kB
time
What has happened to the variability of returns for the portfolio?
rate of return
kA
kB
time
What has happened to the variability of returns for the portfolio?
rate of return
kA
kB
kp
time
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified?
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified? YES! • Would you have eliminated all of your risk?
Portfolios
• If you owned a share of every stock traded on the BSE and NSE, would you be diversified? YES! • Would you have eliminated all of your risk? NO! Common stock portfolios still have risk.
Diversification
• When does the greatest pay off to diversification come?
Diversification
• The greatest pay off when the 2 stocks are negatively correlated. • Unfortunately this almost never happens. • When there is perfectly negative correlation, there is always a portfolio strategy (represented by peculiar set of portfolio weights) which will completely eliminate risk. • It is too bad perfect correlation doesn’t occur between stocks.
Diversification
• Covariance between 2 stocks = ?12 = ?12 ?1 ?2, ?12 – correlation coefficient. • If the prospects of the stocks are wholly unrelated, the both the correlation coefficient and covariance would be 0. • If they move in opposite direction, the correlation coefficient and covariance will be negative. • Covariance becomes important as we add more securities in the portfolio. • Thus the variability of a well diversified portfolio reflects mainly the covariance.
Diversification
• If equal investments are made in each N stocks, • Portfolio variance=N(1/N)2 x average variance + (N2 – N) (1/N)2 x average covariance =1/N x average variance + (1-1/N) x average covariance • Notice as N increases the portfolio variance steadily approaches average covariance. • If average covariance is 0, it would be possible to eliminate risk by holding a sufficient number of securities. • Market risk – it is the average covariance which constitutes the bedrock of risk remaining after diversification has done its work. • The risk of a well diversified portfolio depends on the market risk of the securities included in the portfolio.
Diversification
• Investing in more than one security to reduce risk. • If two stocks are perfectly positively correlated, diversification has no effect on risk. • If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.
Do some firms have more market risk than others?
Yes. For example: Interest rate changes affect all firms, but which would be more affected:
a) Retail food chain b) Commercial bank
Do some firms have more market risk than others?
Yes. For example: Interest rate changes affect all firms, but which would be more affected:
a) Retail food chain b) Commercial bank
Some risk can be diversified away and some can not. • Market risk (systematic risk) is nondiversifiable. This type of risk can not be diversified away. • Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.
Risk faced by Firms
• The risk faced by a firm can be fall into the following categories –
– Project-specific; an individual project may have higher or lower cash flows than expected. – Competitive Risk, which is that the earnings and cash flows on a project can be affected by the actions of competitors. – Industry-specific Risk, which covers factors that primarily impact the earnings and cash flows of a specific industry. – International Risk, arising from having some cash flows in currencies other than the one in which the earnings are measured and stock is priced – Market risk, which reflects the effect on earnings and cash flows of macro economic factors that essentially affect all companies
Company-unique Risk
• A company’s labor force goes on strike. • A company’s top management dies in a plane crash. • A huge oil tank bursts and floods a company’s production area.
The Effects of Diversification
• On economic grounds, diversifying and holding a larger portfolio eliminates firm-specific risk for two reasons– Each investment is a much smaller percentage of the portfolio, muting the effect (positive or negative) on the overall portfolio. – Firm-specific actions can be either positive or negative. In a large portfolio, it is argued, these effects will average out to zero. (For every firm, where something bad happens, there will be some other firm, where something good happens.)
Market Risk
• Unexpected changes in interest rates. • Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle.
As you add stocks to your portfolio, company-unique risk is reduced.
portfolio risk
companyunique risk
Market risk number of stocks
Note As we know, the market compensates investors for accepting risk - but only for market risk. Company-unique risk can and should be diversified away. So - we need to be able to measure market risk.
This is why we have Beta.
Beta: a measure of market risk. • Specifically, beta is a measure of how an individual stock’s returns vary with market returns.
• It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market.
The market’s beta is 1
• A firm that has a beta = 1? • A firm with a beta > 1? • A firm with a beta < 1?
The market’s beta is 1
• A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market.
– (ex: technology firms)
• A firm with a beta < 1 is less volatile than the market.
– (ex: utilities)
Estimating Beta
• The standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) Rj = a + b R m
– where a is the intercept and b is the slope of the regression.
• The slope of the regression corresponds to the beta of the stock, and measures the riskiness of the stock.
Estimating Performance
• The intercept of the regression provides a simple measure of performance during the period of the regression, relative to the capital asset pricing model.
Rj = Rf + b (Rm - Rf) = Rf (1-b) + b Rm Rj = a + b Rm ........... ........... Capital Asset Pricing Model Regression Equation
• If a > Rf (1-b) .... Stock did better than expected during regression period a = Rf (1-b) .... Stock did as well as expected during regression period a < Rf (1-b) .... Stock did worse than expected during regression period • This is Jensen's alpha.
More on Jensen’s Alpha
If you did this analysis on every stock listed on an exchange, what would the average Jensen’s alpha be across all stocks? • Depend upon whether the market went up or down during the period • Should be zero • Should be greater than zero, because stocks tend to go up more often than down
Firm Specific and Market Risk
• The R squared (R2) of the regression provides an estimate of the proportion of the risk (variance) of a firm that can be attributed to market risk; • The balance (1 - R2) can be attributed to firm specific risk.
Setting up for the Estimation
• Decide on an estimation period
– Services use periods ranging from 2 to 5 years for the regression – Longer estimation period provides more data, but firms change. – Shorter periods can be affected more easily by significant firmspecific event that occurred during the period
• Decide on a return interval - daily, weekly, monthly
– Shorter intervals yield more observations, but suffer from more noise. – Noise is created by stocks not trading and biases all betas towards one.
• Estimate returns (including dividends) on stock
– Return = (PriceEnd - PriceBeginning + DividendsPeriod)/ PriceBeginning – Included dividends only in ex-dividend month
• Choose a market index, and estimate returns (inclusive of dividends) on the index for each interval for the period.
Beta and Unique Risk
1. Total risk = diversifiable risk + market risk 2. Market risk is measured by beta, the sensitivity to market changes - 10% Expected stock return +10% beta
+10% -10%
Expected market return
Calculating Beta: The Characteristic Line
XYZ Co. returns 15
.. .
Beta = slope = 1.20
BSE 500 returns
-15
.. . . 10 . . . . . .. . . .5 . . .. . . . . . 5 . 10 -10 -5 -5 .. . . .. . . -10 .. . . . . .-15 .
15
Summary
• We know how to measure risk, using standard deviation for overall risk and beta for market risk. • We know how to reduce overall risk to only market risk through diversification. • We need to know how to price risk so we will know how much extra return we should require for accepting extra risk.
Beta and Unique Risk
Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the BSE 500, is used to represent the market.
Beta - Sensitivity of a stock’s return to the return on the market portfolio.
Beta and Unique Risk
?
2 crucial points about security risk and portfolio risk ? market risk accounts for most of the risk of a well diversified portfolio ? ? of an individual stock measures its sensitivity to market movements
Beta and Unique Risk
?
?
?
Std. deviation of portfolio returns depend on the average number of securities in the portfolio. With more securities and therefore better diversification portfolio risk declines until all unique risk is eliminated and only the bedrock of market risk remains. Bedrock – depends on the average ? of the securities selected.
Beta and Unique Risk
s im Bi = 2 sm
Beta and Unique Risk
s im Bi = 2 sm
Covariance with the market
Variance of the market
It turns out that this covariance to variance ratio measures a stock’s contribution to portfolio risk.
Beta and Unique Risk
• Stock’s contribution to portfolio risk will depend on its relative importance in the portfolio and the average covariance with the stocks in the portfolio. • The proportion depends on the size of the holding and a measure of the effect of that holding on portfolio risk. The latter values are ?s of individual stocks relative to that portfolio. • To calculate the ? of the stock relative to the market portfolio. We calculate its covariance with the market portfolio and divide it by the variance of the market.
Fundamental Determinants of Betas
• Type of Business: Firms in more cyclical businesses or that sell products that are more discretionary to their customers will have higher betas than firms that are in non-cyclical businesses or sell products that are necessities or staples. • Operating Leverage: Firms with greater fixed costs (as a proportion of total costs) will have higher betas than firms will lower fixed costs (as a proportion of total costs) • Financial Leverage: Firms that borrow more (higher debt, relative to equity) will have higher equity betas than firms that borrow less.
Determinant 1: Product Type
• Industry Effects: The beta value for a firm depends upon the sensitivity of the demand for its products and services and of its costs to macroeconomic factors that affect the overall market.
– Cyclical companies have higher betas than noncyclical firms – Firms which sell more discretionary products will have higher betas than firms that sell less discretionary products
A Simple Test
Consider an investment in Big Bazaar. What kind of beta do you think this investment will have? • Much higher than one • Close to one • Much lower than one
Determinant 2: Operating Leverage Effects
• Operating leverage refers to the proportion of the total costs of the firm that are fixed. • Other things remaining equal, higher operating leverage results in greater earnings variability which in turn results in higher betas.
Measures of Operating Leverage
Fixed Costs Measure = Fixed Costs / Variable Costs • This measures the relationship between fixed and variable costs. The higher the proportion, the higher the operating leverage. EBIT Variability Measure = % Change in EBIT / % Change in Revenues • This measures how quickly the earnings before interest and taxes changes as revenue changes. The higher this number, the greater the operating leverage.
Determinant 3: Financial Leverage
• As firms borrow, they create fixed costs (interest payments) that make their earnings to equity investors more volatile. • This increased earnings volatility which increases the equity beta
Betas are weighted Averages
• The beta of a portfolio is always the market-value weighted average of the betas of the individual investments in that portfolio. • Thus,
– the beta of a mutual fund is the weighted average of the betas of the stocks and other investment in that portfolio – the beta of a firm after a merger is the market-value weighted average of the betas of the companies involved in the merger.
Firm Betas versus divisional Betas
• Firm Betas as weighted averages: The beta of a firm is the weighted average of the betas of its individual projects. • At a broader level of aggregation, the beta of a firm is the weighted average of the betas of its individual division.
Bottom-up versus Top-down Beta
• The top-down beta for a firm comes from a regression • The bottom up beta can be estimated by doing the following:
– Find out the businesses that a firm operates in – Find the unlevered betas of other firms in these businesses – Take a weighted (by sales or operating income) average of these unlevered betas – Lever up using the firm’s debt/equity ratio
• The bottom up beta will give you a better estimate of the true beta when
– the standard error of the beta from the regression is high (and) the beta for a firm is very different from the average for the business – the firm has reorganized or restructured itself substantially during the period of the regression – when a firm is not traded
Estimating Betas for Non-Traded Assets
• The conventional approaches of estimating betas from regressions do not work for assets that are not traded. • There are two ways in which betas can be estimated for non-traded assets
– using comparable firms – using accounting earnings
Beta for InfoSoft, a Private Software Firm
The following table summarizes the unlevered betas for publicly traded software firms. Grouping Number of Beta D/E Ratio Unlevered Firms Beta All Software 264 1.45 3.70% 1.42 Small-cap Software 125 1.54 10.12% 1.45 Entertainment Software 31 1.50 7.09% 1.43 • We will use the beta of entertainment software firms as the unlevered beta for InfoSoft. • We will also assume that InfoSoft’s D/E ratio will be similar to that of these publicly traded firms (D/E = 7.09%) • Beta for InfoSoft = 1.43 (1 + (1-.42) (.0709)) = 1.49 (Assumption: tax rate of 42%)
Total Risk versus Market Risk
• Adjust the beta to reflect total risk rather than market risk. This adjustment is a relatively simple one, since the R squared of the regression measures the proportion of the risk that is market risk.
Total Beta = Market Beta / ?R squared
•
In the InfoSoft example, where the market beta is 1.10 and the average R-squared of the comparable publicly traded firms is 16%,
– Total Beta = 1.49/?0.16 = 3.725 – Total Cost of Equity = 5% + 3.725 (5.5%)= 25.49%
• This cost of equity is much higher than the cost of equity based upon the market beta because the owners of the firm are not diversified.
Diversification
• Diversification is a good thing for an investor. • If an investor can diversify on their account, they will not pay any extra firms that diversify. • And if they have sufficiently wide choice of securities, they will not pay any less because they are unable to invest separately in each factory. • Therefore diversification does not add to a firm’s value or subtract from it. • The total value is the sum of its parts. • The concept of value additivity is important. • If capital market establishes the value of asset A as PV(A) and of asset B as PV(B). • Then PV(A+B) = PV(A) + PV(B). PV(AB) is the market value of a firm holding both these assets A & B.
RISK
• How to measure risk • How to reduce risk • How to price risk
RISK
• How to measure risk (variance, standard deviation, beta) • How to reduce risk (diversification) • How to price risk (security market line, CAPM)
Assignment
You are advising a very risky software firm on the right cost of equity to use in project analysis. You estimate a beta of 2.0 for the firm and come up with a cost of equity of 18%. The CFO of the firm is concerned about the high cost of equity and wants to know whether there is anything he can do to lower his beta. How do you bring your beta down?
Should you focus your attention on bringing your beta down? • Yes • No
Assignment
• The R squared for Deutsche Bank is very high (57%), at least relative to U.S. firms. Why is that? • The beta for Deutsche Bank is 0.84.
– Is this an appropriate measure of risk? – If not, why not?
• If you were an investor in primarily U.S. stocks, would this be an appropriate measure of risk?
Question
• Returns & Std. Deviations Particular Return T-Bills 6% ABC 10 DEF 14.5 GHI 21 Calculate the portfolio of the following portfolio: • 50% T-bills 50% ABC • 50% DEF 50% GHI assuming the shares have
– Perfect +ve correlation – Perfect –ve correlation – No correlation
Std. Dev. 0% 14 28 26
Question
• • • • 7% 27% 1% 19.1%
Question
• A game offers the following odds and pay offs. Each game costs Rs. 100.
Probability 10% 50% 40% Payoff 500 100 0
What are the expected cash payoffs and expected rate of return? Calculate the variance and standard deviation of this rate of return.
132
Question
• • • • Expected cash payoffs – 100 Expected rate of return – 0 Variance – 20,000 Standard deviation – 141%
133
Question
• Ritesh Modi, Head of Aum Mutual Fund, has produced the following rates of return from 2003 to 2007. Rates of return of BSE 100 have been given for comparison Year Modi(%) BSE 100(%) 2003 16.1 23.1 2004 28.4 33.4 2005 25.1 28.6 2006 14.3 21.0 2007 -6.0 -9.1 • Calculate Modi’s average return and standard deviation. • Did he do better or worse than BSE 100 by these measures?
134
Question
• Result Average return Standard deviation Modi 15.6 12.0 BSE 100 19.4 14.9
135
Question
• State whether True or False
1. Investors prefer diversified companies because they are less risky 2. If stocks were perfectly positively correlated, diversification would not reduce risk 3. The contribution of a stock to the risk of a well diversified portfolio depends in its market risk 4. A well diversified portfolio with a beta 2 is twice as risky as the market portfolio 5. An undiversified portfolio with a beta of 2 is less than twice as risky as the market portfolio
136
Question
• • • • • F T T T F
137
Question
• Suppose stand deviation of the market return is 20%. 1. What is the standard deviation of returns of a well diversified portfolio with beta 1.3 another with beta 0. 2. A well diversified portfolio having standard deviation of 15%, what is its beta.
138
Question
1. Beta 1.3 - 26% Beta 0 – 0% 2. Standard deviation 15% Beta – 0.75
139
Question
• Hitesh Shah invests 60% of his fund in Stock 1 and the balance in stock 2. The standard deviation of 1 is 10% and of 2 is 20%. Calculate the variance of the portfolio returns if
– – – – Correlation is 1 Correlation is 0.5 Correlation is 0 Correlation is -1
140
Individual Assignment Question
• If the Beta of Company A & B are as follows: A B Beta 2.21 1.81 Std. Dev. 62.7 50.7 • Assume std dev of return on market to be 15%. • The correlation coeff is 0.66, what is the std dev of the portfolio, if equal invesment is made in A & B. • What is the std. dev. If 1/3 each is invested in A, B and T-bills • What is the std dev if invested equally between A & B, but investment is financed at 50% margin – the investor puts only 50% of the amount and borrows the balance from the broker. • What would be the std dev of a portfolio of 100 stocks having beta of A? How about 100 stocks like B?
141
Individual Assignment Question
• Some stocks have high standard deviations and relatively low betas. Sometimes it is the other way round. Why do think so? Illustrate your answer by calculating standard deviation and betas using data for 5 years data (use daily or weekly data) for the 2 companies selected by you. Also find the standard deviation and beta of a portfolio comprising of the 2 stocks in varying proportion stating from (10% to 90%, in steps of 10%).
142
Topics Covered
• • • • Markowitz Portfolio Theory Risk and Return Relationship Testing the CAPM CAPM Alternatives
Markowitz Portfolio Theory
• Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation. • Correlation coefficients make this possible. • The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfolios.
Markowitz Portfolio Theory
Price changes vs. Normal distribution
0.14
Microsoft - Daily % change 1990-2001
Proportion of Days
0.12 0.1 0.08 0.06 0.04 0.02 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Daily % Change
Returns - Distribution
• When measured over fairly short interval of time, the past rates of return on any stock conform closely to a normal distribution. • If returns are distributed normally then investors only need to consider expected returns and std. dev. • If you were to measure over a long interval, then the distribution would be skewed. Use log normal distribution.
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment A
20 18
% probability
16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment B
20 18
% probability
16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment C
20
% probability
18 16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment D
20
% probability
18 16 14 12 10 8 6 4 2 0
-50 0 50
% return
Markowitz Portfolio Theory
? Expected Returns and Standard Deviations vary given different weighted combinations of the stocks
Reebok Expected Return (%)
35% in Reebok
Coca Cola Standard Deviation
Efficient Frontier
•Each half egg shell represents the possible weighted combinations for two stocks. •The composite of all stock sets constitutes the efficient frontier
Expected Return (%)
Standard Deviation
Markowitz Portfolio Theory
• Expected return is simply a weighted average of the expected return in the holding. • The risk of the portfolio is less than the average risk of the separate stocks. • Markowitz – efficient portfolio – along solid enveloping line • Problem of rationing – to deploy a limited amount of capital to capital investment in a mixture of projects to give the highest NPV. Here we deploy investors funds to give the highest expected return for a given std. dev. Solution using quadratic programming • Given the expected return & std. dev of each stock, as well as correlation between each pair of stocks, we can calculate a set of efficient portfolios using quadratic program • If stocks are highly correlated they offer less diversification benefits
Lending & Borrowing
• Suppose you can lend and borrow at some risk free rate. • If you invest your money T-bills and place the remainder in common stock portfolio S, you can obtain any combination of expected risk returns along the straight line joining Rf and S. • Since borrowing is negative of lending you can extend the range to the right of S by borrowing at Rf and investing in S. • Regardless of the level of risk you choose, you can get the highest expected return by a mixture of portfolio S and T-bills (lending or borrowing). S is the best efficient portfolio.
Efficient Frontier
•Lending or Borrowing at the risk free rate (rf) allows us to exist outside the efficient frontier.
S
Expected Return (%)
rf
T
Standard Deviation
Lending & Borrowing
• Start with the vertical axis at Rf and draw the steepest line or tangent to the curved heavy line of efficient portfolios. • The efficient portfolio at the tangency point is better than all others. It offers the highest ratio of risk premium to std. dev.
– Best portfolio of common stock must be selected S. – This portfolio must be blended with borrowing or lending to obtain an exposure to risk that suits the particular investors’ taste.
• Each investor must therefore put money in 2 benchmark investments – risky portfolio S and risk free loan (lending or borrowing) T-bills • Portfolio S is the point of tangency to the set of efficient portfolios. It offers the highest expected risk premium (r – Rf) per unit of std. dev.
Efficient Portfolio
• If you have better information than your rivals you will want the portfolio to include relatively large investment in the stock you think is undervalued. • But in competitive world you are unlikely to have monopoly of good ideas. • In that case there is no reason to hold different portfolio of common stocks from anybody else. • Hold market portfolio. • That is why many professional investors invest in a market index portfolio and not hold any other well diversified portfolio.
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation ? Return ?
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4%
Efficient Frontier
Example Stocks ABC Corp Big Corp
s
28 42
Correlation Coefficient = .4 % of Portfolio Avg Return 60% 15% 40% 21%
Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4%
Let’s Add stock XYZ to the portfolio
Efficient Frontier
Example Stocks Portfolio XYZ
s
28.1 30
Correlation Coefficient = .3 % of Portfolio Avg Return 50% 17.4% 50% 19%
NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%
Efficient Frontier
Example Stocks Portfolio XYZ
s
28.1 30
Correlation Coefficient = .3 % of Portfolio Avg Return 50% 17.4% 50% 19%
NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%
NOTE: Higher return & Lower risk How did we do that? DIVERSIFICATION
Efficient Frontier
Return
B
A
Risk (measured as
s)
Efficient Frontier
Return
B
AB A Risk
Efficient Frontier
Return
B
AB A X
Risk
Efficient Frontier
Return
B
ABX AB A X
Risk
Efficient Frontier
Return
Goal is to move up and left. WHY? B
ABX AB A X
Risk
Efficient Frontier
Return Low Risk High Return High Risk High Return
Low Risk Low Return
High Risk Low Return Risk
Efficient Frontier
Return Low Risk High Return High Risk High Return
Low Risk Low Return
High Risk Low Return Risk
Efficient Frontier
Return B
ABX AB A
X
Risk
Security Market Line
Return
Market Return = rm Risk Free Return =
.
Efficient Portfolio
rf
Risk
Security Market Line
Return
Market Return = rm Risk Free Return =
.
Efficient Portfolio
rf
1.0 BETA
Security Market Line
Return
.
Risk Free Return =
rf
Security Market Line (SML)
BETA
Security Market Line
SML Return
rf
1.0
BETA
SML Equation = rf + B ( rm - rf )
Risk & Return Relationship
• The return on T-bills is fixed, it is unaffected by what happens in the market. Its beta is 0. • The average market risk: beta = 1 • Wise don’t take risk just for fun. They are playing with real money. Therefore they require a higher return from the market portfolio than from T-bills. • The difference between the return on the market and T-bills is termed as market risk premium = Rm – Rf
Capital Asset Pricing Model
R = r f + B ( r m - rf )
CAPM
CAPM
• What is the expected risk premium when beta is neither 0 nor 1? • William Sharpe John Linter Jack Treynor – CAPM • In a competitive market the expected risk premium varies in direct proportion to beta. This means all investments must plot along the slopping line known as Security Market Line (SML). • Expected risk premium on stock = beta * expected risk premium on market • i.e. r – Rf = b(Rm-Rf)
Capital Asset Pricing Model
Assumptions behind the capital asset pricing model are: • Investors maximize the expected wealth at the end of one period. • Investors choose between portfolios on the basis of the mean and variance of return. This implies that returns are normally distributed. • Investors can borrow or lend at an exogenously determined risk-free rate of interest. • There are no restrictions on short selling. • Investors have homogeneous probability beliefs. • The market is perfect. Thus, all assets are marketable, there are no transaction costs or taxes, and all investors are price takers.
Capital Asset Pricing Model
• You can also use CAPM to find the discount rate for new capital investment. • Instead of investing in plant and machinery the firm could return the money to the shareholders. • The opportunity cost of investing is the return that shareholders expect to earn buying financial assets. This expected return depends on the market risk of the asset. • In practice choosing discount rate is seldom easy. • One has to adjust for the extra risk caused by borrowing & how to estimate the discount rates for projects that do not have the same risks as the company’s existing business. • There are also tax issues.
Capital Asset Pricing Model
• • • Investors like high expected return and low std. dev. Common stocks portfolio that offer the highest expected return for a given std. dev are known as efficient portfolio. If the investor can lend or borrow at risk-free rate of interest, one efficient portfolio is better than all others: the portfolio that offers the highest risk premium to std. dev. A risk-averse investor will put part of his money in this efficient portfolio and part of risk-free asset. A risk tolerant investor may borrow and put all his money in this efficient portfolio. The composition of this best efficient portfolio depends on investors assessment of expected returns, std. dev and correlations. But suppose everyone has the same information and the same assessment. If there is no superior information, each investor should hold the same portfolio as everyone else, i.e. hold market portfolio.
•
• • •
Capital Asset Pricing Model
• Don’t look at risk of a stock in isolation but at its contribution to portfolio risk. This contribution depends on the stocks’ sensitivity to the changes in the value of the portfolio. A stock’s sensitivity to changes in the value of the market portfolio is known as beta. Beta measures the marginal contribution of a stock to the risk of the market portfolio. If everyone holds the market portfolio, if beta measures each security’s contribution to the market portfolio risks, then risk premium demanded by investor would be proportional to beta. That is CAPM
• • •
Estimating Risk Premiums in Practice
• Survey investors on their desired risk premiums and use the average premium from these surveys. • Assume that the actual premium delivered over long time periods is equal to the expected premium - i.e., use historical data • Estimate the implied premium in today’s asset prices.
The Survey Approach
• Surveying all investors in a market place is impractical. • However, you can survey a few investors (especially the larger investors) and use these results. In practice, this translates into surveys of money managers’ expectations of expected returns on stocks over the next year. • The limitations of this approach are:
– there are no constraints on reasonability (the survey could produce negative risk premiums or risk premiums of 50%) – they are extremely volatile – they tend to be short term; even the longest surveys do not go beyond one year
The Historical Premium Approach
• This is the default approach used by most to arrive at the premium to use in the model • In most cases, this approach does the following
– it defines a time period for the estimation (1926-Present, 1962-Present....) – it calculates average returns on a stock index during the period – it calculates average returns on a riskless security over the period – it calculates the difference between the two – and uses it as a premium looking forward
• The limitations of this approach are:
– it assumes that the risk aversion of investors has not changed in a systematic way across time. (The risk aversion may change from year to year, but it reverts back to historical averages) – it assumes that the riskiness of the “risky” portfolio (stock index) has not changed in a systematic way across time.
Implied Equity Premiums
• If we use a basic discounted cash flow model, we can estimate the implied risk premium from the current level of stock prices. • For instance, if stock prices are determined by the simple Gordon Growth Model:
– Value = Expected Dividends next year/ (Required Returns on Stocks Expected Growth Rate) – Plugging in the current level of the index, the dividends on the index and expected growth rate will yield a “implied” expected return on stocks. Subtracting out the risk-free rate will yield the implied premium.
• The problems with this approach are:
– the discounted cash flow model used to value the stock index has to be the right one. – the inputs on dividends and expected growth have to be correct – it implicitly assumes that the market is currently correctly valued
The Market Portfolio
• Assuming diversification costs nothing (in terms of transactions costs), and that all assets can be traded, the limit of diversification is to hold a portfolio of every single asset in the economy (in proportion to market value). This portfolio is called the market portfolio. • Individual investors will adjust for risk, by adjusting their allocations to this market portfolio and a riskless asset (such as a T-Bill) Preferred risk level Allocation decision No risk 100% in T-Bills Some risk 50% in T-Bills; 50% in Market Portfolio; A little more risk 25% in T-Bills; 75% in Market Portfolio Even more risk 100% in Market Portfolio A risk hog.. Borrow money; Invest in market portfolio; • Every investor holds some combination of the risk free asset and the market portfolio.
Validity of CAPM
• Investors are principally concerned with those risks that they cannot eliminate by diversification. • If it were not so, we should find that stock prices increases whenever 2 companies merge to spread their risk. So also find that investment companies which invest in the shares of other companies are more valuable than the shares they hold. But that is not true. • Many finance managers find CAPM the most convenient tool for coming to grips with the slippery notion of risk.
Validity of CAPM
• CAPM predicts that the risk premium should increase in proportion to beta. • Actual stock returns reflect expectations but they also embody lot of ‘noise’ – the steady flow of surprises that conceal whether an average investor has received the returns they expected. This noise makes it impossible to judge whether the model holds better in one period than another. • Market portfolio should contain actually all risky investments – stocks, bonds, commodities, real estate and human capital. However, most indices contain only a sample of common stocks.
Validity of CAPM
• Fisher & Black have shown that if there are borrowing restrictions there should still exists a positive relationship between expected return and beta but the SML would be less steep. • Burgundy Line – cumulative difference between the returns on small cap shares and return on large cap share i.e. if you buy small cap and sell large cap • The small cap did not always do well but over the long run they have made substantially better than large cap
Validity of CAPM
• Value stock – high ratio of book value to market value • Growth stock – low ratio of book to market value • Historically the value stocks have shown higher returns than growth stocks. This is in contradiction to CAPM, where higher risk meant higher returns. • Small firms have higher betas, but the difference in beta is not sufficient to explain the difference in returns. There is no simple relationship between book to market ratios and betas.
Testing the CAPM
Return vs. Book-to-Market
25
Dollars
20 15 10 5 0
1928 1933 1938 1943 1948 1953 1958
High-minus low book-tomarket
Small cap minus Large cap
1963
1968
1973
1978
1983
1988
1993
1998
Limitation of CAPM
•
• •
•
T-bills are risk free – no chance of default. But don’t guarantee real returns. Uncertainty of inflation Investors can borrow money at the same rate of interest at which they lend. Generally the borrowing rates are higher than lending rates. The real important idea is that the investors are content to invest their money in a limited number of benchmark portfolios. In basic CAPM – T-bills and market portfolio are the basic benchmarks In modified CAPMs expected returns still depend on market risk, but the definition of market risk depends on the nature of the benchmark portfolio.
Limitations of the CAPM
• The model makes unrealistic assumptions • The parameters of the model cannot be estimated precisely
– - Definition of a market index – - Firm may have changed during the 'estimation' period‘
• The model does not work well
– - If the model is right, there should be
• a linear relationship between returns and betas • the only variable that should explain returns is betas
– - The reality is that
• the relationship between betas and returns is weak • Other variables (size, price/book value) seem to explain differences in returns better.
Consumption Betas vs Market Betas
Stocks
(and other risky assets)
Market risk makes wealth uncertain.
Standard CAPM
Wealth = market portfolio
Consumption Betas vs Market Betas
Stocks Stocks
(and other risky assets)
(and other risky assets)
Wealth is uncertain
Market risk makes wealth uncertain.
Standard CAPM
Wealth
Consumption CAPM
Consumption is uncertain
Wealth = market portfolio
Consumption
Consumption Betas vs Market Betas
• The standard CAPM concentrates on how stocks contribute to the level of uncertainty of investors’ wealth. • Standard CAPM assumes that investors are concerned with the amount and uncertainty of future wealth. • Market risk makes future wealth uncertain. • The consumption beta is defined as a stock’s contribution to uncertainty about consumption. The consumption CAPM connects uncertainty about stock returns directly to uncertainty about consumption. It does not have any immediate practical use as individual stocks have low or erratic consumption betas.
Arbitrage Pricing Theory
Alternative to CAPM
Expected Risk Premium = r - rf = Bfactor1(rfactor1 Return
- rf) + Bf2(rf2 - rf) + …
= a + bfactor1(rfactor1)
+ bf2(rf2) + …
Arbitrage Pricing Theory
• APT is derived assuming that stock returns are based partly on “factors” and partly on “noise.” For any individual stock there are two sources of risk, the risk based on factors and the risk based on noise. Diversification eliminates risk based on noise and only risk based on factors is relevant.
Arbitrage Pricing Theory
• It assumes that each stock’s return depends partly on pervasive macroeconomic influences or factors and partly by noise – events that are unique to that company. • Return = a + b1(Rfactor1) + b2(Rfactor2) + … + noise • The theory doesn’t say what these factors are – they could be oil prices, interest rate, etc. • Return on market portfolio may or may not serve as a factor. • Some stocks will be more sensitive to particular factor than other stocks.
Arbitrage Pricing Theory
• For any individual stock, there are 2 sources of risks:
– Pervasive macroeconomic factors, which cannot be eliminated by diversification – Risk arising from possible events that are unique to the company.
• If diversification does eliminate unique risk, then diversified investor must ignore it when deciding to buy or sell a stock.
Arbitrage Pricing Theory
• The expected risk premium on a stock is affected by macro economic risks, it is not affected by unique risk. • APT states that the expected risk premium on a stock should depend on the expected risk premium associated with each factor and the stock’s sensitivity to each factor. • There may be some macroeconomic factors that investors are simply not concerned with, e.g. inflation. Such factors would not command a risk premium and would drop out of APT. • Expected risk premium = r-Rf=b1(Rfactor1-Rf) + b2(Rfactor2-Rf) +…
Comparison between APT and CAPM
CAPM Market Portfolios APT Market portfolio does not feature, it may be taken as one of the factors. So we don’t bother measuring market portfolios Doesn’t tell us what the underlying factors are
Collapses all macroeconomic risks into a well defined single factor – return on market portfolio
Arbitrage Pricing Theory
Estimated risk premiums for taking on risk factors (1978-1990)
Factor Yield spread Interest rate Exchange rate Real GNP Inflation Mrket Estimated Risk Premium (rfactor ? rf ) 5.10% - .61 - .59 .49 - .83 6.36
APT Calculations
• APT – only if can identify a reasonable short list of macroeconomic factors, measure the expected risk premiums on each factor and measure the sensitivity of each stock to these factors. • Step 1: Identify the Macroeconomic factors • Step 2: Estimate Risk Premiums for each factor • Step3: Estimate the Factor sensitivities
FAMA French 3 Factor Model
• Stocks of small cap stocks with high book to market ration have provided with above average returns – there factors are related to company profitability and therefore may be picking up risk factors that are left out of the CAPM. • If investors do demand an extra premium for taking exposure to those factors, then we have a measure of the expected return similar to APT, as • r-Rf = bmarket(Rmarketfactor) + bsize(Rsizefactor) + bbooktomarket(Rbooktomarket)
FAMA French 3 Factor Model
• Step 1: Identify the factors
– – – – 3 factors are Market factor – Return on market index Size factor – Return on small cap less return on large cap Book to market – Return on high book to market less return on low book to market return
• Step 2: Estimate the risk Premium for each stock – Historical data • Step 3: Estimate the Factor sensitivities • Some stock are more sensitive than others to fluctuations in return on the 3 factors. • The 3 factor model provides substantially lower estimate than CAPM.
doc_422133043.pptx