Risk and Return: The Basics Basic return concepts Basic ris

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CHAPTER 4 Risk and Return The Basics

• Basic return concepts
• Basic risk concepts
• Stand-alone risk
• Portfolio (market) risk
• Risk and return CAPM/SML

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HW CHAPTER 4

• ST1, pg 164 BE
• 4-3, 4-7, 4-8, 4-9, 4-13 pg 166-167 BE

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What are investment returns?

• Investment returns measure the financial results
  of an investment.
• Returns may be historical or prospective
  (anticipated).
• Returns can be expressed in
• Dollar terms.
• Percentage terms.

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What is the return on an investment that costs
1,000 and is soldafter 1 year for 1,100?

• Dollar return

Received - Invested 1,100 -
1,000 100.

• Percentage return

Return/ Invested 100/1,000
0.10 10.
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What is investment risk?

• Typically, investment returns are not known with
  certainty.
• Investment risk pertains to the probability of
  earning a return less than that expected.
• The greater the chance of a return far below the
  expected return, the greater the risk.

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Selected Realized Returns, 1926 2001

• Average Standard
• Return Deviation
• Small-company stocks 17.3 33.2
• Large-company stocks 12.7 20.2
• L-T corporate bonds 6.1 8.6
• L-T government bonds 5.7 9.4
• U.S. Treasury bills 3.9 3.2
• Source Based on Stocks, Bonds, Bills, and
  Inflation (Valuation Edition) 2002 Yearbook
  (Chicago Ibbotson Associates, 2002), 28.

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Probability distribution
Stock X
Stock Y
Rate of return ()
50
15
0
-20

• Which stock is riskier? Why?

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Assume the FollowingInvestment Alternatives
Economy Prob. T-Bill Alta Repo Am F. MP
Recession 0.10 8.0 -22.0 28.0 10.0 -13.0
Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0
Average 0.40 8.0 20.0 0.0 7.0 15.0
Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0
Boom 0.10 8.0 50.0 -20.0 30.0 43.0
1.00
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What is unique about the T-bill return?

• The T-bill will return 8 regardless of the state
  of the economy.
• Is the T-bill riskless? Explain. (nominal)

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Do the returns of Alta Inds. and Repo Men move
with or counter to the economy?

• Alta Inds. moves with the economy, so it is
  positively correlated with the economy. This is
  the typical situation.
• Repo Men moves counter to the economy. Such
  negative correlation is unusual.

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Calculate the expected rate of return on each
alternative.

r expected rate of return.

rAlta 0.10(-22) 0.20(-2) 0.40(20)
0.20(35) 0.10(50) 17.4.
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r
Alta 17.4
Market 15.0
Am. Foam 13.8
T-bill 8.0
Repo Men 1.7

• Alta has the highest rate of return.
• Does that make it best?

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What is the standard deviationof returns for
each alternative?
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Standard Deviation Another View
Why doesnt this formula have Probability in
it? All states assumed equally likely. What does
the -1 in the denominator tell us? This is the
calculation assuming a sample.
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Alta Inds ? ((-22 - 17.4)20.10 (-2 -
17.4)20.20 (20 - 17.4)20.40 (35 -
17.4)20.20 (50 - 17.4)20.10)1/2 20.0.
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Prob.
T-bill
Am. F.
Alta
0
8
13.8
17.4
Rate of Return ()
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• Standard deviation measures the stand-alone risk
  of an investment.
• The larger the standard deviation, the higher
  the probability that returns will be far below
  the expected return.
• Coefficient of variation is an alternative
  measure of stand-alone risk.

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Comments on standard deviation as a measure of
risk

• Standard deviation (si) measures total, or
  stand-alone, risk.
• The larger si is, the lower the probability that
  actual returns will be closer to expected
  returns.
• Larger si is associated with a wider probability
  distribution of returns.
• Difficult to compare standard deviations, because
  expected return has not been accounted for
  -comparing two risky propositions, without any
  idea of the payoff.

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Expected Return versus Risk
Expected
Security return Risk, ?
Alta Inds. 17.4 20.0
Market 15.0 15.3
Am. Foam 13.8 18.8
T-bills 8.0 0.0
Repo Men 1.7 13.4
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Standardized Risk
Coefficient of Variation is a measure of relative
variability.
Shows risk per unit of return. (pain/gain ratio)
Should you take the investment with the lowest
coefficient of variation (small CV is generally
better)?
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Coefficient of VariationCV Standard
deviation/expected return

• CVT-BILLS 0.0/8.0 0.0.
• CVAlta Inds 20.0/17.4 1.1.
• CVRepo Men 13.4/1.7 7.9.
• CVAm. Foam 18.8/13.8 1.4.
• CVM 15.3/15.0 1.0.

sigma(Portfolio) w1 sigma1 w2 sigma2 ,
ie weighted average of individual StDevs. If and
only If -gt ?12 1 CV has sigma in the numerator,
hence for the reason above, taking a wt. avg. of
CV will involve taking a wt. avg . of sigma (or
variance) - which is not allowed.
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Expected Return versus Coefficient of Variation
Expected Risk Risk
Security return ? CV
Alta Inds 17.4 20.0 1.1
Market 15.0 15.3 1.0
Am. Foam 13.8 18.8 1.4
T-bills 8.0 0.0 0.0
Repo Men 1.7 13.4 7.9
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Return vs. Risk (Std. Dev.) Which investment is
best?
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Portfolio Risk and Return
Assume a two-stock portfolio with 50,000 in Alta
Inds. and 50,000 in Repo Men.

Calculate rp and ?p.
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Portfolio Return, rp


rp is a weighted average
n


rp ??wiri?
i 1

rp 0.5(17.4) 0.5(1.7) 9.6.



rp is between rAlta and rRepo.
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Alternative Method
Estimated Return
Economy Prob. Alta Repo Port.
Recession 0.10 -22.0 28.0 3.0
Below avg. 0.20 -2.0 14.7 6.4
Average 0.40 20.0 0.0 10.0
Above avg. 0.20 35.0 -10.0 12.5
Boom 0.10 50.0 -20.0 15.0

rp (3.0)0.10 (6.4)0.20 (10.0)0.40
(12.5)0.20 (15.0)0.10 9.6.
(More...)
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• ?p ((3.0 - 9.6)20.10 (6.4 - 9.6)20.20
  (10.0 - 9.6)20.40 (12.5 - 9.6)20.20 (15.0
  - 9.6)20.10)1/2 3.3.
• ?p is much lower than
• either stock (20 and 13.4).
• average of Alta and Repo (16.7).
• The portfolio provides average return but much
  lower risk. The key here is negative (or less
  than perfect) correlation.
• Var(P) w12 Var1w2 2 Var22w1w2Cov(1,2)
• And Cov(1,2) corr(1,2)(Var1Var2)1/2
• If corr(1,2) 1 then Var(P) w1 ?1 w2 ?2 2

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• StDev(P) w1 ?1 w2 ?2 , ie weighted average
  of individual StDevs. IFF -gt ?12 1
• s w1 s1 w2 s2 , IFF -gt ?12 s12/s1s2 1
• If the correlation coefficient is -1 then
  portfolio standard deviation is equal s w1 s1 -
  w2 s2 and it is possible to achieve the zero
  portfolio standard deviation by varying the
  proportion of assets weights w1 and w2 in the
  portfolio. Practically impossible since very few
  assets are perfectly negatively correlated.

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Correlation Coefficient

• Correlation coefficients (?) range from
• -1 to 1
• ? -1 implies
• perfectly negative correlation
• ? 1 implies
• perfectly positive correlation
• ? 0 implies
• variables are not related
• Do most stocks have positive, negative, or zero
  correlations with each other?
• Positive, but not perfectly so
• What is correlation of any security with riskless
  asset (T-bill is it riskless?)?
• Zero

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Two-Stock Portfolios

• Two stocks can be combined to form a riskless
  portfolio if r -1.0.
• Risk is not reduced at all if the two stocks have
  r 1.0.
• In general, stocks have r ? 0.65, so risk is
  lowered but not eliminated.
• Investors typically hold many stocks.
• What happens when r 0?

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General comments about risk

• Most stocks are positively correlated with the
  market (?k,m ? 0.65).
• s ? 35 for an average stock. (what is range in 2
  of 3 years? E 12 from market)
• Combining stocks in a portfolio generally lowers
  risk.

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What would happen to therisk of an average
1-stockportfolio as more randomlyselected
stocks were added?

• ?p would decrease because the added stocks would
  not be perfectly correlated, but rp would remain
  relatively constant.

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Prob.
Large
2
1
0
15
Return
?1 ??35 ?Large ??20.
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?p ()
Company Specific (Diversifiable) Risk
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Stand-Alone Risk, ?p
20 0
Market Risk
10 20 30 40 2,000
Stocks in Portfolio
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Stand-alone Market Diversifiable
.
risk risk
risk
Market risk is that part of a securitys
stand-alone risk that cannot be eliminated by
diversification. Firm-specific, or diversifiable,
risk is that part of a securitys stand-alone
risk that can be eliminated by diversification.
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Two Components of Risk

• Company-specific (Diversifiable) risk
• Unique to specific firms.
• Results from random or uncontrollable events.
• What are some examples?
• Natural disasters, accidents, strikes, lawsuits,
  death of CEO, etc.
• Market (systematic) risk
• Relates to forces affecting all investments.
• What are some examples?
• Inflation, recession, war, yield inversion etc.

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Conclusions

• As more stocks are added, each new stock has a
  smaller risk-reducing impact on the portfolio.
• ?p falls very slowly after about 40 stocks are
  included. The lower limit for ?p is about 20
  ?M .
• By forming well-diversified portfolios, investors
  can eliminate about half the riskiness of owning
  a single stock.

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Can an investor holding one stock earn a return
commensurate with its risk?

• No
• Stand-alone risk is not important to a
  well-diversified investor because it vaporizes
• Rational, risk-averse investors are concerned
  with sp, which is based upon market risk.
• There can be only one price (the market return)
  for a given security.
• No compensation should be earned for holding
  unnecessary, diversifiable risk.

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How is market risk measured for individual
securities?

• Market risk, which is relevant for stocks held in
  well-diversified portfolios, is defined as the
  contribution of a security to the overall
  riskiness of the portfolio.
• It is measured by a stocks beta coefficient.
  For stock i, its beta is
• bi covi,m/varm
• si sm riM / sm2
• (riM si) / sM

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Betas.?

• In addition to measuring a stocks contribution
  of risk to a portfolio, beta also measures the
  stocks volatility relative to the market.
• Shows how the price of a security responds to
  changes in the overall stock market (not just its
  variance)
• Stocks Beta is the only relevant measure of risk
  from a portfolio standpoint in a well
  diversified portfolio there is no USR so beta
  gives the stocks response to a change in the
  market (syst. risk)

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Using a Regression to Estimate Beta

• Run a regression with returns on the stock in
  question plotted on the Y axis and returns on the
  market portfolio plotted on the X axis.
• The slope of the regression line, which measures
  relative volatility, is defined as the stocks
  beta coefficient, or b.

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Calculation of Beta
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Use the historical stock returns to calculate the
beta for PQU.
Year Market PQU
1 25.7 40.0
2 8.0 -15.0
3 -11.0 -15.0
4 15.0 35.0
5 32.5 10.0
6 13.7 30.0
7 40.0 42.0
8 10.0 -10.0
9 -10.8 -25.0
10 -13.1 25.0
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Calculating Beta for PQU
r
pqu
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Show in Excel, and how to get return series on
stocks, market
20
r
0
M
-40
-20
0
20
40
-20
r
0.83r
0.03
PQU
M
-40
2
R
0.36
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What is beta for PQU?

• The regression line, and hence beta, can be found
  using a calculator with a regression function or
  a spreadsheet program. In this example, b 0.83.

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Calculating Beta in Practice

• Many analysts use the SP 500 to find the market
  return.
• Analysts typically use four or five years of
  monthly returns to establish the regression line.
  
• Some analysts use 52 weeks of weekly returns.

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How is beta interpreted?

• If b 1.0, stock has average risk.
• If b gt 1.0, stock is riskier than average.
• If b lt 1.0, stock is less risky than average.
• Most stocks have betas in the range of 0.5 to
  1.5.
• Can a stock have a negative beta?
• Beta of .5 If
• Market goes up 1,
• stock only goes up .5.
• But if market goes down 1,
• stock drops just .5.
• Beta of 0
• No correlation with market

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Finding Beta Estimates on the Web

• http//finance.yahoo.com/q/ks?sC
• http//www.investor.reuters.com/StockEntry.aspx
• http//new.quote.com/stocks/company.action?symC
• change the ticker in the URL Address bar,
  directly
• http//finance.google.com/finance?qC
• Changes in estimation window length, return
  frequency, choice of market and statistical
  adjustments will result in varying betas from
  different sources. It may be best to calculate
  your own you can fix the variables and
  parameters.

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Expected Return versus Market Risk
Expected
Security return Risk, b
Alta 17.4 1.29
Market 15.0 1.00
Am. Foam 13.8 0.68
T-bills 8.0 0.00
Repo Men 1.7 -0.86

• Which of the alternatives is best?

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Use the SML to calculate eachalternatives
required return.

• The Security Market Line (SML) is part of the
  Capital Asset Pricing Model (CAPM).
• How do we find a securitys required return ki?
• SML ri rRF (rM - rRF)bi .
• Assume rRF 8 rM rM 15.
• RPM (rM - rRF) 15 - 8 7.
• Higher beta means higher market risk and thus
  higher expected or required not realized
  return.

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Security Market Line

• What is the proxy for the risk-free rate?
• U.S. Treasury Bills
• What is the proxy for the return on the market?
• Typically use the SP500 Index ticker is spx or
  spx or spy

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Required Rates of Return
rAlta 8.0 (7)(1.29) 8.0 9.0
17.0.
rM 8.0 (7)(1.00) 15.0. rAm. F. 8.0
(7)(0.68) 12.8. rT-bill 8.0
(7)(0.00) 8.0. rRepo 8.0
(7)(-0.86) 2.0.
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Expected versus Required Returns

r r
Alta 17.4 17.0 Undervalued
Market 15.0 15.0 Fairly valued
Am. F. 13.8 12.8 Undervalued
T-bills 8.0 8.0 Fairly valued
Repo 1.7 2.0 Overvalued
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Buy Alta Sell Repo
SML ri rRF (RPM) bi ri 8
(7) bi
ri ()
.
Alta
Market
.
rM 15 rRF 8
.
T-bills
.
Repo
Risk, bi
-1 0 1 2
SML and Investment Alternatives
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Portfolio Beta

• Portfolio beta is just a weighted average of the
  security betas
• b1.8, b21, b31.2 wi?.2,.3,.5
• b?w .16.30.601.06
• Note that Swi 1

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Calculate beta for a portfolio with 50 Alta and
50 Repo
bp Weighted average 0.5(bAlta)
0.5(bRepo) 0.5(1.29) 0.5(-0.86) 0.22.
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What is the required rate of returnon the
Alta/Repo portfolio?
rp Weighted average r 0.5(17) 0.5(2)
9.5. Or use SML rp rRF (RPM) bp
8.0 7(0.22) 9.5.
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Impact of Inflation Change on SML

• What if investors raise inflation expectations by
  3? (reqd return on all risky assets increases by
  3, Prices drop)
• If Expected Return increases -gtCurrent Price
  drops
• If Realized Return increases -gt Final Price rises

Required Rate of Return r ()
? I 3
New SML
SML2
SML1
18 15 11 8
Original situation
0 0.5 1.0 1.5 2.0
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Impact of Risk Aversion Change
What if investors risk aversion increased,
causing the market risk premium to increase by 3
- rise/run3/1? (km rises from 15 to 18, so ki
for b.5 rises by 1.5, and for b2 rises by 6)
Required Rate of Return ()
SML2
After increase in risk aversion
rM 18 rM 15
SML1
18 15
? RPM 3
8
Original situation
Risk, bi
1.0
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Capital Asset Pricing Model Security Market Line

• Does a higher required return mean that the
  actual return you get will be higher?
• No, may lose all of your money on the stock.
• If the expected return on a risky asset is lt0,
  what would your weight in the asset be? But all
  assets with bgt0 have a positive expected return.
  So what would happen when the CAPM is empirically
  tested?

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CAPM Issues

• Investors required returns are based on
  future risk, but betas are calculated with
  historical data. Will a companys beta be the
  same this year and next year?
• No Non-Stationarity Problem
• Assumes markets are efficient
• There have been studies that both support and
  dispute the CAPM
• Still used in practice provides a conceptual
  framework useful for linking risk and return in
  financial decisions

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Next Chapter (5)Portfolio Risk