Lessons from Capital Market History

1
Lessons from Capital Market History

• Shad Valley

1
1
K. Hartviksen
2
Dollar Returns

• measured in absolute dollars.
• less meaningful than percentage returns because
  they depend on the amount of the original
  investment.
• Bonds give rise to two kinds of returns
• Capital gains or capital losses, and
• Interest
• Stock investments give rise to dollar returns as
• Capital gains or capital losses, and
• Dividends

3
Holding Period Return(For investments that yield
dividend cash flow returns)
4
Holding Period Return(For investments that yield
dividend cash flow returns)
5
Holding Period ReturnIllustrated (For
investments that yield dividend cash flow returns)
6
The Geometric Average

• Measuring Investment Returns

7
The Bias Inherent in the Arithmetic Average

• Arithmetic averages can yield incorrect results
  because of the problems of bias inherent in its
  calculation.
• Example
• Consider an investment that was purchased for
  10, rose to 20 and then fell back to 10.
• Let us calculate the HPR in both periods

8
The Bias Inherent in the Arithmetic Average

• Example Continued ...
• Consider an investment that was purchased for
  10, rose to 20 and then fell back to 10.
• Let us calculate the HPR in both periods
• The arithmetic average return earned on this
  investment was

9
The Bias Inherent in the Arithmetic Average

• Example Continued ...
• The answer is clearly incorrect since the
  investor started with 10 and ended with 10.
• The correct answer may be obtained through the
  use of the geometric average

10
Geometric Versus Arithmetic Average Returns

• Consider two investments with the following
  realized returns over the past few years

If the returns are equal over time, the
arithmetic average return will equal the
geometric average return.
11
Geometric Versus Arithmetic Average Returns
SAME ANSWER !
12
Geometric Versus Arithmetic Average Returns
Now consider volatile returns
NOT THE SAME ANSWER !
Volatility of returns over time eats away at your
realized returns!!! The greater the volatility
the greater the difference between the arithmetic
and geometric average. Arithmetic average
OVERSTATES the return!!!
13
Measuring Returns

• When you are trying to find average returns,
  especially when those returns rise and fall,
  always remember to use the geometric average.
• The greater the volatility of returns over time,
  the greater the difference you will observe
  between the geometric and arithmetic averages.
• Of course, there are limitations inherent in the
  use of geometric averages.

14
Historical Returns

• Average Standard Risk
• Return Deviation Premium
• Canadian Equities 13.20 16.62 7.16
• U.S. Equities 15.59 16.86 9.55
• Long-Term bonds 7.64 10.57 1.60
• Treasury bills 6.04 4.04 0.00
• Small cap stocks 14.79 23.68 8.75
• Inflation 4.25 3.51 -1.79

15
Historical Returns
The historical pattern of returns exhibit the
classic risk-return tradeoff
16
Capital Asset Pricing Model
Return
Required return Rf bs kM - Rf
km
Market Premium for risk
Security Market Line
Rf
Real Return
Premium for expected inflation
Beta Coefficient
BM1.0
6
17
Measurement of Risk in an Isolated Asset Case

• The dispersion of returns from the mean return is
  a measure of the riskiness of an investment.
• This dispersion can be calculated using
• Variance (an absolute measure of dispersion
  expressed in units squared)
• Standard Deviation (an absolute measure of
  dispersion expressed in the same units as the
  mean)
• Coefficient of Variation (this is a relative
  measure of dispersionit is a ratio of the
  standard deviation divided by the mean)

18
Ex Post and Ex Ante Calculations

• Returns and risk can be calculated after-the-fact
  (ie. You use actual realized return data) This
  is known as an ex post calculation.
• Or you can use forecast datathis is an ex ante
  calculation.

19
Standard Deviation

• The formula for the standard deviation when
  analyzing sample data (realized returns) is

Where k is a realized return on the stock and n
is the number of returns used in the calculation
of the mean.
20
Standard Deviation

• The formula for the standard deviation when
  analyzing forecast data (ex ante returns) is
• it is the square root of the sum of the squared
  deviations away from the expected value.

21
Forecasting Risk and Return for the Individual
Asset
K. Hartviksen
22
A Normal Probability Distribution
The area under the curve bounded by -1 and 1 s
is equal to 68
Probability
- 1 standard deviation away from the mean
1 standard deviation away from the mean
13.2
Return on Large Cap Stocks
23
Finding the Probability of an Event using Z-value
tables

• You can find the number of standard deviations
  away from the mean that a point of interest lies
  using the following z value formula
• Once you know z then you can find the areas
  under the normal curve using the z value table
  found on the following slide.

24
Values of the Standard Normal Distribution
Function Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.0
7 0.08 0.09
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .
0279 .0319 .0359 0.1 .0398 .0438 .0478 .0517 .055
7 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .
0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.
3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443
.1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .17
36 .1772 .1808 .1844 .1879 0.5 .1915 .1950 .1985
.2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .22
57 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517
.2549 0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2
764 .2794 .2823 .2852 0.8 .2881 .2910 .2939 .2967
.2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3
186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .338
9 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .
3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .372
9 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .
3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.
3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147
.4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .42
65 .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357
.4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .44
52 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535
.4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4
608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664
.4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4
719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .476
7 2.0 .4773 .4778 .4783 .4788 .4793 .4798 .4803 .
4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .483
8 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .
4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.
3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911
.4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .49
29 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941
.4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .49
53 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963
.4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4
971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977
.4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4
982 .4982 .4982 .4984 .4984 .4985 .4985 .4986 .498
6 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .
4989 .4990 .4990 z is the number of standard
deviations from the mean. Some area tables are
set up to indicate the area to the left or right
of the point of interest. In this table, we
indicate the area between the mean and the point
of interest.
25
MPT Modern Portfolio Theory

• Shad Valley

1
1
K. Hartviksen
26
Risk and Return - MPT

• Prior to the establishment of Modern Portfolio
  Theory, most people only focused upon investment
  returnsthey ignored risk.
• With MPT, investors had a tool that they could
  use to dramatically reduce the risk of the
  portfolio without a significant reduction in the
  expected return of the portfolio.

3
27
Correlation

• The degree to which the returns of two stocks
  co-move is measured by the correlation
  coefficient.
• The correlation coefficient between the returns
  on two securities will lie in the range of 1
  through - 1.
• 1 is perfect positive correlation.
• -1 is perfect negative correlation.

10
28
Perfect Negatively Correlated Returns over Time
11
29
Ex Post Portfolio ReturnsSimply the Weighted
Average of Past Returns
14
5
K. Hartviksen
30
Ex Ante Portfolio ReturnsSimply the Weighted
Average of Expected Returns
14
5
K. Hartviksen
31
Grouping Individual Assets into Portfolios

• The riskiness of a portfolio that is made of
  different risky assets is a function of three
  different factors
• the riskiness of the individual assets that make
  up the portfolio
• the relative weights of the assets in the
  portfolio
• the degree of comovement of returns of the assets
  making up the portfolio
• The standard deviation of a two-asset portfolio
  may be measured using the Markowitz model

32
Risk of a Three-asset Portfolio
The data requirements for a three-asset portfolio
grows dramatically if we are using Markowitz
Portfolio selection formulae. We need 3 (three)
correlation coefficients between A and B A and
C and B and C.
A
?a,b
?a,c
C
B
?b,c
33
Risk of a Four-asset Portfolio
The data requirements for a four-asset portfolio
grows dramatically if we are using Markowitz
Portfolio selection formulae. We need 6
correlation coefficients between A and B A and
C A and D B and C C and D and B and D.
?a,b
?a,d
?a,c
?b,d
?b,c
?c,d
34
Diversification Potential

• The potential of an asset to diversify a
  portfolio is dependent upon the degree of
  co-movement of returns of the asset with those
  other assets that make up the portfolio.
• In a simple, two-asset case, if the returns of
  the two assets are perfectly negatively
  correlated it is possible (depending on the
  relative weighting) to eliminate all portfolio
  risk.
• This is demonstrated through the following chart.

35
Example of Portfolio Combinations and Correlation
Perfect Positive Correlation no diversification
36
Example of Portfolio Combinations and Correlation
Positive Correlation weak diversification
potential
37
Example of Portfolio Combinations and Correlation
No Correlation some diversification potential
Lower risk than asset A
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Example of Portfolio Combinations and Correlation
Negative Correlation greater diversification
potential
39
Example of Portfolio Combinations and Correlation
Perfect Negative Correlation greatest
diversification potential
Risk of the portfolio is almost eliminated at 70
asset A
40
Diversification of a Two Asset Portfolio
Demonstrated Graphically
41
An Exercise using T-bills, Stocks and Bonds
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Results Using only Three Asset Classes
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The Capital Market Line
Capital Market Line
Expected Return on the Portfolio
12
8
4
Risk-free rate
0
10
20
30
40
0
Standard Deviation of the Portfolio
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The Capital Market Line and Iso Utility Curves
A risk-takers utility curve
The risk-takers optimal portfolio combination
Expected Return on the Portfolio
12
Capital Market Line
8
4
Risk-free rate
0
10
20
30
40
0
Standard Deviation of the Portfolio
48
CML versus SML

• Please notice that the CML is used to illustrate
  all of the efficient portfolio combinations
  available to investors.
• It differs significantly from the SML that is
  used to predict the required return that
  investors should demand given the riskiness
  (beta) of the investment.

49
Data Limitations

• Because of the need for so much data, MPT was a
  theoretical idea for many years.
• Later, a student of Markowitz, named William
  Sharpe worked out a way around thatcreating the
  Beta Coefficient as a measure of volatility and
  then later developing the CAPM.

50
CAPM

• The Capital Asset Pricing Model was the work of
  William Sharpe, a student of Harry Markowitz at
  the University of Chicago.
• CAPM is an hypothesis

51
Capital Asset Pricing Model
Return
Required return Rf bs kM - Rf
km
Market Premium for risk
Security Market Line
Rf
Real Return
Premium for expected inflation
Beta Coefficient
BM1.0
6
52
CAPM

• This model is an equilibrium based model.
• It is called a single-factor model because the
  slope of the SML is caused by a single measure of
  risk the beta.
• Although this model is a simplification of
  realityit is robust (it explains much of what we
  see happening out there) and it enjoys widespread
  use in a great variety of applications.
• Although it is called a pricing model there are
  not prices on that graph.only risk and return.
• It is called a pricing model because it can be
  used to help us determine appropriate prices for
  securities in the market.

53
Risk

• Risk is the chance of harm or loss danger.
• We know that various asset classes have yielded
  very different returns in the past

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Historical Returns and Standard Deviations1948 -
941

• Average Return Standard Deviation
• Canadian common stock 12.73 16.81
• U.S. common stock (Cdn ) 14.09 16.60
• Long term bonds 7.01 10.20
• Small cap stocks 15.67 24.40
• Inflation 4.52 3.54
• Treasury bills 6.15 4.17
• ___________________
• 1The Alexander Group

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Risk and Return

• The foregoing data point out that those asset
  classes that have offered the highest rates of
  return, have also offered the highest risk levels
  as measured by the standard deviation of returns.
• The CAPM suggests that investors demand
  compensation for risks that they are exposed
  toand these returns are built into the
  decision-making process to invest or not.

56
Capital Asset Pricing Model
Return
Required return Rf bs kM - Rf
km
Market Premium for risk
Security Market Line
Rf
Real Return
Premium for expected inflation
Beta Coefficient
BM1.0
6
57
CAPM

• The foregoing graph shows that investors
• demand compensation for expected inflation
• demand a real rate of return over and above
  expected inflation
• demand compensation over and above the risk-free
  rate of return for any additional risk
  undertaken.
• We will make the case that investors dont need
  compensation for all of the risk of an investment
  because some of that risk can be diversified
  away.
• Investors require compensation for risk they
  cant diversify away!

58
Beta Coefficient

• The beta is a measure of systematic risk of an
  investment.
• Systematic risk is the only relevant risk to a
  diversified investor according to the CAPM since
  all other risk may be diversified away.
• Total risk of an investment is measured by the
  securities standard deviation of returns.
• According to the CAPM total risk may be broken
  into two partssystematic (non-diversifiable) and
  unsystematic (diversifiable)
• TOTAL RISK SYSTEMATIC RISK UNSYSTEMATIC RISK
• The beta can be determined by regressing the
  holding period returns (HPRs) of the security
  over 30 periods against the returns on the
  overall market.

7
59
Measuring Risk of the Individual Security

• Risk is the possibility that the actual return
  that will be realized, will turn out to be
  different than what we expect (or have forecast).
• This can be measured using standard statistical
  measures of dispersion for probability
  distributions. They include
• variance
• standard deviation
• coefficient of variation

60
Standard Deviation

• The formula for the standard deviation when
  analyzing population data (realized returns) is

61
Standard Deviation

• The formula for the standard deviation when
  analyzing forecast data (ex ante returns) is
• it is the square root of the sum of the squared
  deviations away from the expected value.

62
Using Forecasts to Estimate Beta

• The formula for the beta coefficient for a stock
  s is
• Obviously, the calculate a beta for a stock, you
  must first calculate the variance of the returns
  on the market portfolio as well as the covariance
  of the returns on the stock with the returns on
  the market.

63
Systematic Risk

• The returns on most assets in our economy are
  influenced by the health of the system
• Some companies are more sensitive to systematic
  changes in the economy. For example durable
  goods manufacturers.
• Some companies do better when the economy is
  doing poorly (bill collection agencies).
• The beta coefficient measures the systematic risk
  that the security possesses.
• Since non-systematic risk can be diversified
  away, it is irrelevant to the diversified
  investor.

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Systematic Risk

• We know that the economy goes through economic
  cycles of expansion and contraction as indicated
  in the following

65
Canadas Business cycles from
1873-1992 Trough to ExpansionPeak to
Contraction (months from trough to peak)(months
from peak to trough) Nov 1873 66 May
1879 38 July 1882 32 Mar 1885 23 Feb 1887 12 Feb
1888 29 July 1890 9 Mar 1891 23 Apr 1893 13 Mar
1894 17 Aug 1895 12 Aug 1896 44 Apr 1900 10 Feb
1901 22 Dec 1902 18 June 1904 30 Dec 1906 19 July
1908 20 Mar 1910 16 July 1911 16 Nov 1912 25 Jan
1915 36(WWI) Jan 1918 15 Apr 1919 14 June
1920 15 Sep 1921 21 June 1923 14 Aug 1924 56 Apr
1929 47 (Depression) Mar 1933 52 July 1937 15
(Depression) Oct 1938 80(WWII) June 1945 8 Feb
1946 33 Oct 1948 11 Sep 1949 44(Korean War) May
1953 14 July 1954 31 Feb 1957 12 Feb 1958 26 Apr
1960 10 Feb 1961 160 June 1974 10 Apr
1975 58 Feb 1980 6 July 1980 12 July 1981 6 Nov
1982 89 Apr 1990 22 Feb 1992
66
Companies and Industries

• Some industries (and by implication the companies
  that make up the industry) move in concert with
  the expansion and contraction of the economy.
• Some lead the overall economy. (stock market)
• Some lag the overall economy. (ie. automotive
  industry)

67
Amount of Systematic Risk

• Some industries may find that their fortunes are
  positively correlated with the ebb and flow of
  the overall economybut that this relationship is
  very insignificant.
• An example might be Imperial Tobacco. This firm
  does have a positive beta coefficient, but very
  little of the returns of this company can be
  explained by the beta. Instead, most of the
  variability of returns on this stock is from
  diversifiable sources.
• A Characteristic line for Imperial Tobacco would
  show a very wide dispersion of points around the
  line. The R2 would be very low (.05 5 or
  lower).

68
Characteristic Line for Imperial Tobacco
Characteristic Line for Imperial Tobacco
Returns on Imperial Tobacco
Returns on the Market (TSE 300)
69
High R2

• An R2 that approaches 1.00 (or 100) indicates
  that the characteristic (regression) line
  explains virtually all of the variability in the
  dependent variable.
• This means that virtually of the risk of the
  security is systematic.
• This also means that the regression model has a
  strong predictive ability. if you can predict
  what the market will dothen you can predict the
  returns on the stock itself with a great deal of
  accuracy.

70
Characteristic Line General Motors
Characteristic Line for GM (high R2)
Returns on General Motors
Returns on the Market (TSE 300)
71
Diversifiable Risk(non-systematic risk)

• Examples of this type of risk include
• a single company strike
• a spectacular innovation discovered through the
  companys RD program
• equipment failure for that one company
• management competence or management incompetence
  for that particular firm
• a jet carrying the senior management team of the
  firm crashes
• the patented formula for a new drug discovered by
  the firm.
• Obviously, diversifiable risk is that unique
  factor that influences only the one firm.

72
Partitioning Risk under the CAPM

• Remember that the CAPM assumes that total risk
  (variability of a securitys returns) can be
  separated into two distinct components
• Total risk systematic risk unsystematic risk
• 100 40 60 (GM)
• or
• 100 5 95 (Imperial Tobacco)
• Obviously, if you were to add Imperial Tobacco to
  your portfolio, you could diversify away much of
  the risk of your portfolio. (Not to mention the
  fact that Imperial has realized some very high
  rates of return in addition to possessing little
  systematic risk!)

73
Using the CAPM to Price Stock

• The CAPM is a fundamental analysts tool to
  estimate the intrinsic value of a stock.
• The analyst needs to measure the beta risk of the
  firm by using either historical or forecast risk
  and returns.
• The analyst will then need a forecast for the
  risk-free rate as well as the expected return on
  the market.
• These three estimates will allow the analyst to
  calculate the required return that rational
  investors should expect on such an investment
  given the other benchmark returns available in
  the economy.

74
Required Return

• The return that a rational investor should demand
  is therefore based on market rates and the beta
  risk of the investment.
• To find this, you solve for the required return
  in the CAPM
• This is a formula for the straight line that is
  the SML.

75
Security Market Line

• This line can easily be plotted.
• Draw Cartesian coordinates.
• Plot the yield on 91-day Government of Canada
  Treasury Bills as the risk-free rate of return on
  the vertical axis.
• On the horizontal axis set a scale that includes
  Beta1 (this is the beta of the market)
• Plot the point in risk-return space that
  represents your expected return on the market
  portfolio at beta 1
• Draw a straight line to connect the two points.
• Plot the required and expected returns for the
  stock at its beta.

76
Plot the Risk-Free Rate
Return
Rf
Beta Coefficient
1.0
77
Plot Expected Return on the Market Portfolio
Return
km 12
Rf 4
Beta Coefficient
1.0
78
Draw the Security Market Line
Return
SML
km 12
Rf 4
Beta Coefficient
1.0
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Plot Required Return(Determined by the formula
Rf bskM - Rf
Return
SML
R(k) 13.6
km 12
R(k) 4 1.28 13.6
Rf 4
Beta Coefficient
1.0
1.2
80
Plot Expected ReturnE(k) weighted average of
possible returns
Return
SML
R(k) 13.6
R(k) 4 1.28 13.6
km 12
E(k)
Rf 4
Beta Coefficient
1.0
1.2
81
If Expected Required ReturnThe stock is
properly (fairly) priced in the market. It is in
EQUILIBRIUM.
Return
SML
R(k) 13.6
R(k) 4 1.28 13.6
km 12
E(k)
Rf 4
Beta Coefficient
1.0
1.2
82
If E(k) lt R(k)The stock is over-priced. The
analyst would issue a sell recommendation in
anticipation of the market becoming efficient
to this fact. Investors may short the stock to
take advantage of the anticipated price decline.
Return
SML
R(k) 13.6
R(k) 4 1.28 13.6
km 12
E(k)
E(k) 9
Rf 4
Beta Coefficient
1.0
1.2
83
Lets Look at the Pricing Implications

• In this example
• E(k) 9
• R(k) 13.6
• If the market expects the company to pay a
  dividend of 1.00 next year, and the stock is
  currently offering an expected return of 9, then
  it should be priced at
• But, given the other rates in the economy and our
  judgement about the riskiness of this investment
  we think that this stock should be worth

84
Practical Use of the CAPM

• Regulated utilities justify rate increases using
  the model to demonstrate that their shareholders
  require an appropriate return on their
  investment.
• Used to price initial public offerings (IPOs)
• Used to identify over and under value securities
• Used to measure the riskiness of
  securities/companies
• Used to measure the companys cost of capital.
  (The cost of capital is then used to evaluate
  capital expansion proposals).
• The model helps us understand the variables that
  can affect stock pricesand this guides
  managerial decisions.

85
Rf rises
SML2
Return
SML1
ks2
ks1
Rising interest rates will cause all required
rates of return to increase and this will force
down stock and bond prices.
Rf2
Rf1
Beta Coefficient
Bs1.2
86
The Slope of The SML rises(indicates growing
pessimism about the future of the economy)
SML2
Return
SML1
ks2
Growing pessimism will cause investors to demand
greater compensation for taking on riskthis will
mean prices on high beta stocks will fall more
than low beta stocks.
ks1
Rf1
Beta Coefficient
Bs1.2