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Lessons from Capital Market History

- Shad Valley

1

1

K. Hartviksen

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

Holding Period Return(For investments that yield

dividend cash flow returns)

Holding Period Return(For investments that yield

dividend cash flow returns)

Holding Period ReturnIllustrated (For

investments that yield dividend cash flow returns)

The Geometric Average

- Measuring Investment Returns

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

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

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

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.

Geometric Versus Arithmetic Average Returns

SAME ANSWER !

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!!!

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.

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

Historical Returns

The historical pattern of returns exhibit the

classic risk-return tradeoff

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

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)

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.

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.

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.

Forecasting Risk and Return for the Individual

Asset

K. Hartviksen

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

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.

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.

MPT Modern Portfolio Theory

- Shad Valley

1

1

K. Hartviksen

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

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

Perfect Negatively Correlated Returns over Time

11

Ex Post Portfolio ReturnsSimply the Weighted

Average of Past Returns

14

5

K. Hartviksen

Ex Ante Portfolio ReturnsSimply the Weighted

Average of Expected Returns

14

5

K. Hartviksen

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

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

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

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.

Example of Portfolio Combinations and Correlation

Perfect Positive Correlation no diversification

Example of Portfolio Combinations and Correlation

Positive Correlation weak diversification

potential

Example of Portfolio Combinations and Correlation

No Correlation some diversification potential

Lower risk than asset A

Example of Portfolio Combinations and Correlation

Negative Correlation greater diversification

potential

Example of Portfolio Combinations and Correlation

Perfect Negative Correlation greatest

diversification potential

Risk of the portfolio is almost eliminated at 70

asset A

Diversification of a Two Asset Portfolio

Demonstrated Graphically

An Exercise using T-bills, Stocks and Bonds

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

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.

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.

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

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

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.

Risk

- Risk is the chance of harm or loss danger.
- We know that various asset classes have yielded

very different returns in the past

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

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.

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

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!

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

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

Standard Deviation

- The formula for the standard deviation when

analyzing population data (realized returns) is

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.

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.

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.

Systematic Risk

- We know that the economy goes through economic

cycles of expansion and contraction as indicated

in the following

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

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)

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).

Characteristic Line for Imperial Tobacco

Characteristic Line for Imperial Tobacco

Returns on Imperial Tobacco

Returns on the Market (TSE 300)

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.

Characteristic Line General Motors

Characteristic Line for GM (high R2)

Returns on General Motors

Returns on the Market (TSE 300)

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.

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!)

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.

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.

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.

Plot the Risk-Free Rate

Return

Rf

Beta Coefficient

1.0

Plot Expected Return on the Market Portfolio

Return

km 12

Rf 4

Beta Coefficient

1.0

Draw the Security Market Line

Return

SML

km 12

Rf 4

Beta Coefficient

1.0

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

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

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

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

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

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.

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

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