1. Chapter 8 Capital Market Theory
J. D. Han
King’s College, UWO

2. 1. Market Risk and Return:
How to characterize an asset?
With Returns, Market Risk
rA ~ Distribution(E(rA), sA )

3. 1) Expected Return: a Statistical Statement
What will be the expected return for asset A = rA for next year?
- look back at the historical data of ri that have hanged over time(variable).
- get the mean value (weighted average for all possible states of affairs) as the expected rate of return.
- Mathematically,
E(rA) = rA bar = S rA.i prA.i = rA.1 prA.1 + rA.2 prA.+…..+ rA.m prA.m
rA.i = annualized rate of returns of asset A in situation i
prA.i = probability of situation i taking place

4. 2) Market Risk SD comes from variance What is the market risk?
-A measure that the actual rate of return may deviate from the expected rate of return
Market risk is measured by standard deviation
sA.
SD comes from variance
s2A = S (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

5. * 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 ) = ( )/3 = 6%
s 2 = 1/3(4- 6)2 + 1/3(6-6)2 + 1/3(8-6)2 = 8/3
= (8/3)1/2
B ~ (6, (8/3)1/2 )
2) Stock C: Data r: 3 times of 4, 5 times of 6, twice of 8

6. *Various Assets Expected Rate of returns of gold: E(rg)
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

7. ***Risk and Returns re
rstock i
rbond i
rPicasso
rT-bill i s

8. **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

9. 2. Portfolio Diversification
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

10. 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

11. 3. Combining Two Risky Assets
Asset A ~( E(rA), sA)
Asset B ~ (E(rB), sB)
Suppose we mix A and B at ratio of w1 to w2for a portfolio
Resultant Portfolio P’s
Expected Rate of Return?
Market Risk?

12. Return: E(rp) = w1 E (rA) + w2 E(rB) Risk:
Portfolio ~ (E(rP), sp) which has A and B assets at the ratio of w1 and w2 (w1 + w2 = 1.0)
Return: E(rp) = w1 E (rA) + w2 E(rB)
Risk:
*rAB is the correlation coefficient of rA and rB.
*sAB is the correlation coefficient of rA and rB.
* sAB = rAB sA sB

13. Return: E(rp )= w1 E(rA) + w2 E(rB)
Case 1. rAB = 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

14. In this case, the Investment Opportunity Set looks like
E (Rp)
As B’s portion w2 rises,
E (Rp) B w2 sp
Portfolio 1= 0.9* A + 0.1*B A sp

15. Case 2. rAB = -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

16. In this case, the Investment Opportunity Set looks like
As B’s portion w2 rises,
E (Rp)
E (Rp) B
Portfolio X = a’ A + b’ B : “Perfect Hedge” sp w2
Portfolio 1= 0.9* A + 0.1*B A sp

17. *Perfect Hedge: Portfolio P which has zero market risk- At what ratio should A and B be mixed?
Two equations and two unknowns:
sp= I w1 sA - w2 sB I = 0
w1 + w2 = 1
Solve for w1 and w2:

18. Case 3. Generally –1< rAB< 1 : Imperfect Correlation between A and B’s returns
Return: E (Rp ) = w1 E( RA) + w2 E( RB )
Risk:

19. **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 r AB= 1.
E (Rp)
E (Rp) B w2
Portfolio 1= 0.9* A + 0.1*B sp A sp

20. *Prove sp < w1 sA + w2 sB
Square the both sides.
The above is, sp2 versus (w1 sA + w2 sB)2
First, left-hand side-
Recall sp2 = w12 sA2 + w22 sB2 + 2 w1 w2 rAB sA sB
Second,-right hand side-
w12 sA2 + w22 sB2 + 2 w1 w2 sA sB
= w12 sA2 + w22 sB2 + 2 w1 w2 x 1x sA sB
The comparison boils down to rAB versus 1.
Recall rAB is equal to or less than 1.
Thus, the left-hand side is equal to or less than the right-hand side.

21. ***Efficient Frontier: the upper part of investment opportunity set is superior to the lower part
Minimum Variance Portfolio

22. ****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 D B C A

23. *What if there are more than 2 risky-assets?
Diversification
Total risk sp
Unique (Diversifiable) Risk
Market (Systematic) Risk
# of assets

24. Example: XYZ Fund

25. * Example:
15.5%
100% International Stock(MSCI World Index)
14.6%
Minimum Risk Portfolio 76% of MSCI and 24% of TES 300
10.9%
100% Canadian Equities(TSE 300)
Source: “About 75% Foreign Content Seems Ideal for Equity Portfolio”, Gordon Powers, Globe and Mail, March 6, 1999

26. **Consider Preference of Client
Risk-Averse vs Risk-Loving
Indifference Curves

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

28. ** 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

29. 4. Combining Risk Free Asset 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 sA
- This is on a straight line between Risk free asset and Asset A

30. Introducing Borrowing and Lending: Diversification between Risk Free Asset and Market PortfoliosM

31. Security Market Line(SML):
Risk Premium
ri - rf
Slope of CML =( rm– rf )/ bM
= risk premium / risk
= risk premium per unit of risk
= price of (a unit of) risk
rM - rf
bM = bi

32. *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% (= )/0.30
-> “The market demands 0.20 percent of additional return for each one percent increase in a portfolio’s risk measured by its s.”

33. *** 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

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

35. 5. Capital Asset Pricing Model
Risk Premium depends on Asset’s Systematic Risk only
Systematic Risk means the Co-movement of Return on an asset and the Market Portfolio (index).

37. Security Market Line(SML):
Risk Premium
ri - rf
Slope of SML =( rM– rf )/ bM
= risk premium / risk
= risk premium per unit of risk
= price of (a unit of) systematic risk
rM - rf
bM = bi

38. -Remarks:
* b measures the degree to which an asset's returns covaries with the returns on the market: relative measure of risk.
-b <1 “Defensive”
=1 “Typical”
>1 “Aggressive”

39. ***Why is b superior than s as a measure of market risk?
Asset B
Asset A
RA and Rm over time
RB and Rm over time
sB=1
bB= -1
Extremely Desirable Asset for Portfolio Diversification
sA=1
bA=1
Typical Asset

40. *Comparison of SD and beta
Beta of CAPM model
-Measuring only the portion of fluctuations of the rate of returns which move along with the Market
-Measuring only
Systematic Risk
Standard Deviation
(<- variance)
-Measuring the entirety of fluctuations of the rate of returns over time
-Measuring
Systematic and
Non-systematic risks

41. * Two Component of Market Risk
Systematic Risk
= Market-wide Risk
= Foreseen Risk
= Non-diversifiable Risk
Non-systematic Risk
=Firm-specific Risk
=Idiosyncratic Risk
=Unforeseen Risk
=Diversifiable Risk

42. **“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

43. ***** Some Canadian Examples in the Stock Market
Cetricom
Clearnet
Air Canada
Noranda
BCE
Chapters
Bank of Nova Scotia 1.03
Bombardier
Hudson’s Bay 0.58
Loblaw
Source: Compustat, Feb 2000

44. *Security Market Line (SML): Visual Presentation of CAPM model
Required Yields or Expected Rates
E(Ri)
E(RM) Rf
bM =1 bi b

45. * Numerical Example Thus the covariance = 0.3 x 0.35 x 0.25= 0.02625
Suppose that the correlation coefficient between Inert Technologies Ltd and the stock market index is 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 = rAB sA sB
Thus the covariance = 0.3 x 0.35 x 0.25=
Beta = covariance / variance of market portfolio = /(0.25)2 =0.42
Required Rate = (0.09) = 0.117

46. *Evidence Regarding the CAPM: Ex-Post or Actual Ri may differ from ex-ante 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.

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