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Chapter 8Capital Market Theory

J. D. Han

King’s College, UWO

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1. Risk

• Credit Risk – Default Risk

• Liquidity Risk

• Inflation Risk

• Market Risk = Variability of the Rate of Return of an asset

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How to evaluate/measure Risks?

• Credit RiskCredit rating• Liquidity RiskUsually proportional to the maturity length, but not always

• Inflation RiskProportional to expected money creation rates;

• Market RiskVolatility

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(Bond) Credit Rating

• Only 5 rating companies have the Nationally Recognized Statistical Rating Organization (NRSRO) designation, and are overseen by the SEC in their assignment of credit ratings:

• Standard & Poor's (S&P), Moody's, Fitch, A. M. Best and Dominion Bond Rating Service.

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Scales of Credit Ratings                                           

    

* Below BBB, or Baa are “Junk Bonds”.

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Examples

• Dominion Bond Rating Services

http://www.dbrs.com/intnlweb/jsp/search/listResults.faces

• Standard and Poor’s in Canada

http://www2.standardandpoors.com/portal/site/sp/en/ca/page.topic/ratings_corp/2,1,3,0,0,0,0,0,0,0,4,0,0,0,0,0.html

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Risk requires Compensation for Investor, “Risk Premium”

By John Hull at Rotman School of Business

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2. How to measure Market Risk of Individual Asset?

1. Variability= Deviation from its own Average Rate of Return“Mean Variance Approach”

2. Co-movement with the Market Index = Relative Variability of Rate of Return to the Market Index

“Capital Market Pricing Model”

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3. Mean Variance Method:Market Risk and Return for a Single Asset

- How to characterize an asset?

With Expected Returns, and Market Risk

- rA ~ Distribution(E(rA), A )

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1) Expected Return: a Statistical Statement

What will be the expected return for asset A = rA for next year?

- There are many possible contingencies- Assume that history will repeat in the future

- Look back at the historical data of various ri that have hanged over time in different contigencies.

- Get the mean value (weighted average for all possible states of affairs) as the expected rate of return.

-

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Mathematically,

Mean Value, or rA bar

= Expected Value E(rA)

= rA.i prA.i

= rA.1 prA.1 + rA.2 prA.+…..+ rA.m prA.m

where

rA.i = annualized rate of returns of asset A in situation i

prA.i = probability of situation i taking place

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2) Market Risk by Standard Deviation

• Mean Variance Approach measure the risk by standard deviation:

• How mcuh do the actual rates of return deviate from its own average value over time?

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SD comes from variance

rA.i – E rA)2 prA.i

= (rA.1 – E rA)2 prA.1 + (rA.2 – E rA)2 prA.+…..

+ (rA.m – E rA)2 prA.m

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* Numerical Example: How to calculate the variance and the standard deviation?

1) Stock B: Data of r over 3 years are 4%, 6%, and 8%E (r ) = (4 + 6 + 8)/3 = 6%

Stock C: Data r: 3 times of 4, 5 times of 6, twice of 8

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*Various Assets

• Expected Rate of returns of a Stock (ith company’s stock)

: E (r s I) • Expected Rate of returns of a Bond (ith institution’s

bond): E( r b i )

• Expected Rate of returns of a T-Bill: E (r T-bill i) ) = rf (“risk free asset”)

• Expected Rate of returns of the Market Portfolio: E( rm)• Expected Rate of returns of gold: E(rg)• Expected Rate of returns of Picasso Print: rpicasso

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***Risk and Returns

re

rT-bill i

rbond i

rstock i

rPicasso

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**Stylized Fact

• The Higher the Standard Deviation, the Higher the Average Rate of Returns

- The Higher the Market Risk, the Higher the Risk Premium

an Asset should pay to the investor.

Otherwise, no investor will hold this asset

• However, the Risk Premium does NOT rise in proportion to the Market Risk

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4. Portfolio Diversification: Multiple Assets

• Mixing Two or More Assets for Investment• Spreading Investment over two or more assets

We will see• First:

Combine Two (or more) Risky Assets• Second:

Risky Assets and Risk-Free Asset

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1) Why Diversification?• Expanded Opportunity Set: More Options

for different combination of returns and risk;

or

• Taking advantage of non-linear trade-off between returns and risk

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2) Return and Risk for Combining Two Risky Assets

• Asset A ~( E(rA), A)

• Asset B ~ (E(rB), B)

• Suppose we mix A and B at ratio of w1 to w2for a portfolio

Resultant Portfolio P’s

Expected Rate of Return?

Market Risk?

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Return of Portfolio

Return: E(rp) = w1 E (rA) + w2 E(rB)

Simple weighted average of two assets’ individual average rate

of return

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BAABBA ww w w2 2 1

222

221

p

w w2 2 122

222

1 p ABBA ww

Risk

* is the correlation coefficient of rA and rB. * is the correlation coefficient of rA and rB.

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Depending on there are 3 different cases

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Case 1. AB = 1 :

rA and rB are perfectly positively correlated

Return: E(rp )= w1 E(rA) + w2 E(rB)

Risk:Weighted average of risk of two component assets

BA

BA

BABA

BAABBA

P

ww

ww

wwww

wwww

21

221

2122

222

1

2122

222

1

2

2

)(

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In this case, the Investment Opportunity Set looks like

Portfolio 1= 0.9* A + 0.1*B

A

B

E (Rp)

p

As B’s portion w2 rises,

w2

E (Rp)

p

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Case 2. = -1:

rA and rB are perfectly negative correlated

Return: E (rp) = w1 E(rA) + w2 E(rB)

Risk:weighted difference between risks of two assets

I I

2

2

21

221

2122

222

1

2122

222

1

BA

BA

BABA

BAABBA

P

ww

ww

wwww

wwww

)(

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In this case, the Investment Opportunity Set looks like

Portfolio 1= 0.9* A + 0.1*B

A

B

E (Rp)

p

Portfolio X = a’ A + b’ B : “Perfect Hedge”

As B’s portion w2 rises, E (Rp)

p w2

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*Perfect Hedge: Portfolio P which has zero market risk- At what ratio should A and B be mixed?

wo equations and two unknowns:

p= I w1 w2

w1 + w2 = 1

Solve for w1 and w2:

BA

A2

BA

B1 w w

;

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Case 3. Generally –1< AB< 1 :Imperfect Correlation between A and B’s returns

• Return: E (Rp ) = w1 E( RA) + w2 E( RB )

• Risk:

BABA

BAABBA

P

wwww

wwww

21 p21

2 122

222

1

:seecan we

2

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**In this case, the Opportunity Set Looks Like:Note that the expected value of the portfolio is the linear function of the expected rates of returns of the assets, and the standard deviation is less than the weighted average unless = 1.

Portfolio 1= 0.9* A + 0.1*B

A

B

E (Rp)

p

E (Rp)

pw2

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*Prove p < w1 w2 in general

• Square the both sides.• The above is, p

2 versus (w1 w2

First, left-hand side- Recall p

2 = w1

2 w2

2 w1 w2

Second,-right hand side- w1

2 w2

2 w1 w2

w12

w22

w1 w2 x x

The comparison boils down to versus 1.

Recall is equal to or less than 1. Thus, the left-hand side is equal to or less than the

right-hand side.

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3) Efficient Frontier: the upper part of investment opportunity set

is superior to the lower part

Minimum Variance Portfolio

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*What if there are more than 2 risky-assets?General Case of Mean Variance Approach

• Risk or SD is given by the square root of

jijiii www . 2

22

p 2

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****What if there are more than one set of risky assets?

Step 2. Get the Best Results of Combing a pair of risky assets, and get their envelope curve for Efficient Frontier

A

B

C

D

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*** Combining Market-Risk- Free Lending/Borrowing, and Risky Asset

• Risk Free Asset ~ (rf , 0)

• Correlation coefficient with any other asset = 0

• Portfolio which mixes Risk free asset and Asset A at w1 to w2

~ return: w1 rf + w2 E(rA)

market risk: w2 A

- This is on a straight line between Risk free asset and Asset A

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*With Market-Risk-Free Borrowing/Lending, the Efficient Frontier is a Straight Line:

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•Application Question 1: Should a Canadian investment include a H.K. stock?

• H.K. has currently depressed stock market• H.K. stocks have lower rates of returns and a higher risk

(a larger value of SD) compared to the Canadian Stocks.• What would the possible benefit for a Canadian fund

including a H.K. stock(with a lower return and a higher risk)?

- surely, more comparable investment options- Maybe, a possibility of some new superior options Show this on a graph

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* Application Question 2: How much of foreign stocks a Canadian

should include in his portfolio?

100% Canadian Equities(TSE 300)

100% International Stock(MSCI World Index)

Minimum Risk Portfolio 76% of MSCI and 24% of TES 300

Source: “About 75% Foreign Content Seems Ideal for Equity Portfolio”, Gordon Powers, Globe and Mail, March 6, 1999

15.5%

14.6%

10.9%

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*Application Question 3: As you are mixing more and more assets, the Mean-Variance Risk of the portfolio falls:

# of assets

Total risk

p

Unique (Diversifiable) Risk

Market (Systematic) Risk

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* Appliation Example: XYZ Fund

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5. Choice of Optimum Portfolio for an Individual Customer Tangent Point of

Efficient Frontier of Portfolio’s Return and Risk

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Individual Customer’s Indifference Curve showing his Risk Preference (- Attitude towards Risk and Return)

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*Risk Preference of Client may vary

Risk-Averse vs Risk-Loving

Indifference Curves

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In case there is no risk-free asset, we can choose the Optimum now.

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What will be the graph of choice like for the case with Market-Risk-Free Lending/Borrowing and Risky Assets?

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*Answer: Choice depending on Preference in case where risk-free lending and borrowing is possible

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*** Tobin’s Separation Theorem• The investment decision of which portfolio of risky

assets to hold is separate from the financing decision of how to allocate investment between the risky assets and the risk-free assets.

• In other words, there is one “optimal” portfolio of risk assets for all investors.

• Of course, a risk-loving person will hold more of risk-free assets, and a risk-averse person will hold more of risky assets. However, for both, the best relative combination of different risky assets is the same

- financial advisors should recommend the same proportion of risky assets in clients’ portfolio

- In reality, this is not the case

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5. Capital Asset Pricing Model

• Improve on Mean-Variance Approach

• Risk Premium depends on Asset’s Systematic Risk only

• Systematic Risk is measured by Co-movement of Return on an asset and

the Market Portfolio (index).

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• Asset A • Asset B

1) Why is a superior measure of market risk than Mean-Variance

Typical Asset

xtremely Desirable Asset for Portfolio Diversification

RA and Rm over time RB and Rm over time

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*Comparison of SD and • Standard Deviation (<- Mean-variance)

-Measuring the entirety of fluctuations of the rate of returns over time

-Measuring Systematic andNon-systematic risks

• Beta of CAPM model

-Measuring only the portion of fluctuations of the rate of returns which move along with the Market

-Measuring onlySystematic Risk

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* Two Component of Market Risk

“Systematic Risk”= changes in price of an

asset when the entire market (prices) moves.

= Market-wide Risk= Foreseen Risk= Non-diversifiable Risk= risk premium for it.

“Non-systematic Risk”= unrelated to the entire

market movement=Firm-specific Risk=Idiosyncratic Risk=Unforeseen Risk=Diversifiable Risk=No risk premium for

this

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**“Market Pays Risk Premium only on Systematic Risk”Why?

• Anybody can remove unsystematic risk by portfolio diversification

-> positive deviation of one asset may offset negative deviation of another asset

• If the market pays risk premium on non-systematic risk, nobody would try hard to diversify his portfolio

-> risk premium on non-systematic risk would discourage ‘due diligence’ for portfolio diversification

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measures the degree to which an asset's returns covaries with the returns on the overall market, or the relative market risk of an asset to the typical market to the market portfolio (market index) as a whole

means that this asset has twice as much as variation in price as the market index as a whole.

Thus this asset is twice as risky as the market portfolio.

“Defensive” “Typical” “Aggressive”

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***** Some Canadian Examples in the Stock Market

• Cetricom 2.92• Clearnet 1.77• Air Canada 1.66• Noranda 1.57• BCE 1.22• Chapters 1.01• Bank of Nova Scotia 1.03• Bombardier 0.68• Hudson’s Bay 0.58• Loblaw 0.35Source: Compustat, Feb 2000

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MM

iM

i

.

),

M

iM

r of Variance

r(r Covariance

2) Market Risk by

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r E(r[ r (r E fMfi ])) i

3) Risk Premium

Beta x Market Portfolio’s Risk Premium

4) Required Rate of Return on this Asset

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5) Security Market Line(SML):

M =1 i

Slope of SML =( rM– rf )/ M

= risk premium / risk

= risk premium per unit

of risk

= price of (a unit of)

systematic risk0

rM - rf

Risk Premium

ri - rf

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*Intuition:the slope of the CML

indicates the market price of risk

Suppose that the Market Portfolio has 12% of expected returns and 30% of standard deviation. The risk free rate on a 30-day T-Bills is 6%. What is the slope of the CML?

->Answer: 20% (=0.12-0.06)/0.30

-> “The market demands 0.20 percent of additional return for each one percent increase in a portfolio’s risk measured by its ”

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*Security Market Line (SML): Visual Presentation of CAPM model

Required Yields or Expected Rates

=1

E(RM)

Rf

E(Ri)

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* Numerical Example• Suppose that the correlation coefficient between Inert Technologies Ltd

and the stock market index is 0.30. The rate of return on a 30-day T-Bill is 8%. Overall, the rates of return on stocks are 9% higher than the rate of return on T-Bills. The standard deviation of the stock market index is 0.25, and the standard deviation of the returns to Inert Technologies Ltd is 0.35.

• What is the required rate of return on a Inert Technologies Ltd stock?

: Covariance = AB

Thus the covariance = 0.3 x 0.35 x 0.25= 0.02625

Beta = covariance / variance of market portfolio = 0.02625/(0.25)2 =0.42

Required Rate = 0.08 + 0.42 (0.09) = 0.117

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6) Evidence Regarding the CAPM: Ex-Post or Actual Ri may differ from ex-ante or required Ri or E (Ri )

• Note that e is random unexpected error, or unsystematic risk, idiosyncratic risk.

• e has an average value of 0: it is diversifiable risk• The market does not pay any risk premium for this

as it cannot be anticipated and it can be diversified.

ei ])fMf

i R[E(R R R

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* Undervalued?

Suppose that X is observed ex-post as having the following rate of return and risk. What does this mean?:

X

Security Market LineX