1 Financial Market III: Risk Premium Theories 2- Market Risk J. D. Han King’s College, UWO.

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Financial Market III:Risk Premium Theories 2- Market Risk

J. D. Han

King’s College, UWO

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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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1. Mean-Variance ApproachMarket Risk and Return for a Single Asset

- How to characterize an asset over time?

With Time-series data of the rates of return on it, get

Expected Returns = average/mean value of rates of return; and Market Risk = standard deviation

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

• Case of a Single Financial Asset:

risk is measured by standard deviation(SD) of a single financial asset.

• Case of Multiple Financial Asset in a Portfolio

variance of the portfolio is non-linear combination of SDs of each individual asset and covariance among them.

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Mean-Variance Approach of a Single Asset

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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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Statistically, • Suppose that there are n possible outcomes for rA.

And each event/outcome has probability of pr1, pr2, …..prn.

Mean Value, or rA bar

= Expected Value E(rA)

= rA.i pri

= rA.1 pr1 + rA.2 pr2.+…..+ rA.n prn

where

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

pri = 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 pri

= (rA.1 – E rA)2 pr1 + (rA.2 – E rA)2 pr2+…..

+ (rA.n– E rA)2 prn

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

Bond A: Time series data of r over 3 years are 4%, 6%, and 8%: then• E (r ) = (4 + 6 + 8)/3 = 6% Thus

Note that here time sequence does not matter.

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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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* Stylized fact: Risk and Returns

re

rT-bill i

rbond i

rstock i

rPicasso

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• 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

• Mean-Variance of Multiple Assets in a Portfolio:

- case without risk-free asset

- case with risk-free asset with return rf

free access at rf for deposits and loans14

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Diversified Portfolio: Multiple Assets

• Mixing Two or More Assets for Investment in the way to minimize the resultant SD of the portfolio

We will see• First:

Combine Two (or more) Risky Assets• Second:

Risky Assets and Risk-Free Asset

• First we will examine the combination of two risky assets, and then move onto

• The combination of multiple risky assets and the risk-free asset – here comes Tobin’s Separation Theorem saying “The best combination portfolio of risk assets is the same for everybody”.

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1) Why Diversification?• Suppose that we have two assets A and B, shown by two dots

• Diversification = Mixing the two at different rates gives the lines of return-risk profile.

• We can see the advantage of diversification could be either

i) Expanded Opportunity Set: More Options for different combinations of returns and risk; or

ii) Taking advantage of some reduced risk or smaller SD than is given by the liner aggregation:

• Of course, the second one is better. Whether the second one is available depends on the covariance/correlation between Asset A(‘s rates of return) and Asset B(‘s rates of return) over time.

• Unless the two are perfectly correlated, the second one is available.

• Even if the two are perfectly correlated, diversification means different options of combinations of assets A and B.

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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 covariance coefficient of rA and rB.

• Recall / ( .

rA.i – E rA)rB.i – E rB) pri

= (rA.1 – E rA) rB.1 – E rB) pr1 + (rA.2 – E rA)rB.2– E rB)pr2.+….

+ (rA.n – E rA)rB.n – E rB) prn

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Numerical Example

• Click here for a practice question

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Depending on there are 3 different impacts on the combined risk:

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Case 1. AB = 1 :rA and rB are perfectly positively correlated

• Return: E(rp )= w1 E(rA) + w2 E(rB)

• Portfolio Risk = weighted average of risks 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. –1< AB< 1 :Imperfect Correlation between A and B’s returns – General Case

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

• Risk< weighted average of two risks

B2A1 p

BAAB 2 1B22

2A22

1P

ww

:note

ww 2ww

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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 case of

• Square p and w1 w2

It is now, p2

versus (w1 w2

• Compare the size of the left and the right side.

First, left-hand side is p2

Recall p2

= w12 w2

2 w1 w2 Recall is less than 1.

Second,-right hand side- w1

2 w22 w1 w2

w12 w2

2 w1 w2 x x The comparison boils down to versus 1. Thus, the left-hand side is equal to or less than the right-hand side.

• This general case includes the one where

the rates of returns on two assets are completely independent of each other;

• Still the risk of the portfolio will be smaller than the risk of the less risky asset of the two components.

• The arched-out part of the lower part of the locus(curve) of portfolio has lower risk,and the upper arched-part is ‘efficient’.

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Suppose that the two assets are independent of each other.If you start with less risky asset, the risk falls as you include some risky asset first, and, past H point, the risk starts increasing. The arrow line shows the locus. The blue arrow indicates the efficient portfolios, and the red arrows are not efficient.

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• The principle of choice of assets for portfolio:

- The smaller the correlation between the component assets, the larger the benefits of reduced risk of the portfolio.”

We search for assets whose returns are hopefully less-positively-correlated and more-negatively-correlated.

- The curve of return-risk will be arched to the left to the maximum.”

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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

Application 4. Buying Art for portfolio diversification

• An inferior single asset can be a great element, if taken in a small amount, in the portfolio.

• It lowers the rate of return of the portfolio, but it may lower the risk even more so.

Click here for J. Pesando’s paper

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Returns and Risks of the Art• Investment on Art, especially, on Picasso’s prints.

r

rT-bill i

rbond i

rstock i

rArt Prints

*Remark

• The art prints have the lower rate of return at a given risk, compared with other financial assets. In other words, the art prints seem to be inferior: For the same risk, the returns are lower.

*Would we include these prints in our portfolio?

• The answer:

- Not as a single investment item.

- But, we may include them in the portfolio.

Why? Let’s explain.

The Art Prints have a very desirable property in terms of portfolio diversification: a Negative Correlation Coefficient with some Financial Assets

Prints Stocks Bonds T-Bills Inflation

Prints 1 0.3 -0.10

(-0.17)

-0.21

(-0.27)

0.03

(0.08)

stocks 1 0.46 0.27 -0.31

bonds 1 0.73 -0.56

T-Bills 1 -0.73

Inflation 1

• The prints could provide an attractive investment as their small amount of inclusion in a portfolio of traditional financial assets may reduce the mean return a little but it may reduce the entire risk by a substantially larger margin.

Returns and Risks• When T bills and prints are mixed at the

ratio of 94 to 6(%), the portfolio has the minimum variance.

r

rT-bill i

rPicasso

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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

• Note that depending on his preference an investor can end up on any point on the efficient frontier: it will be his optimal portfolio.

• However, regardless of preferences, the combination of the risk assets is the same for everybody, and it is called here ‘market portfolio’.

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• Tobin’s Separation Theorem

Investment decision(of choosing the right combination of risky assets), and

Financing decision(of depositing or borrowing from banks at the risk-free rate) are independent of each other.

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*Tangent Portfolio=market portfolio = Optimum-risk portfolio

• It is not ‘overall-Optimum portfolio.

• It is the optimum portfolio only with risky assets.

• It has the highest

‘Sharp Ratio’ = E(rp ) – rf

p

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*Importance of the unlimited access to borrowing and lending at the risk-free rate:

• Without it, the choice of (overall) optimal portfolio would be on the Curved Line of the portfolio locus.

• The curved line is in general inferior to the capital market line.

-> This smooth combination of investment(securities business) and commercial banking would be important

<- The Financial Holding company by G-L-B act in the U.S. may be justifiable in this contribution:

In practice, a portfolio manager of a securities company can coordinate with a credit officer of a commercial bank within the same FHC for a client’s loans and deposits at the risk free rate so that the client can finance his investment along the straight line of Efficient Frontier.

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Is there only one market portfolio?

• Because of different available set of assets for different financial companies, it varies.

• However, across the board, the return of the market portfolio is similar.

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Practice Question of Making your own Portfolio

Here is a detailed instruction.

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2. 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