Financial Market III: Risk Premium Theories 2- Market Risk

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

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”

3. 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),sA)

4. 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.

5. Mean-Variance Approach of a Single Asset

6. 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. -

7. 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) = S 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

8. 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?

9. SD comes from variance : • s2A • = S (rA.i – E rA)2 pri • = (rA.1 – E rA)2 pr1 + (rA.2 – E rA)2 pr2+….. • + (rA.n– E rA)2 prn

10. * 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% • s 2 = 1-/3(4- 6)2 + 1/3(6-6)2 + 1/3(8-6)2 = 8/3 Thus s = (8/3)1/2 B ~ (6, (8/3)1/2 ) * Note that here time sequence does not matter.

11. *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

12. * Stylized fact: Risk and Returns re rstock i rbond i rPicasso rT-bill i s

13. 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

14. 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 loans

15. 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

16. 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”.

17. 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:

18. 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.

19. 2) Return and Risk for 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?

20. Return of Portfolio Return: E(rp) = w1 E (rA) + w2 E(rB) Simple weighted average of two assets’ individual average rate of return

21. Risk *rA B is the correlation coefficient of rA and rB. *sA B is the covariance coefficient of rA and rB.

22. Recall • rA B = sA B/ (sA . sB) • sA B = = S (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

23. Numerical Example • Click here for a practice question

24. Depending on r A B,, there are 3 different impacts on the combined risk:

25. Case 1. rAB = 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

26. 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

27. 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

28. 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

29. *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 w1sA - w2 sB I = 0 w1 + w2 = 1 Solve for w1 and w2:

30. Case 3. –1< rAB< 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

31. **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

32. *Prove sp < w1sA + w2sBin general case of rAB <1 • Square sp and w1sA + w2sB It is now, sp2 versus (w1sA + w2sB)2 • Compare the size of the left and the right side. First, left-hand side is sp2 • Recall sp2= w12sA2 + w22sB2 + 2 w1 w2rAB sA sB • Recall rAB is less than 1. Second,-right hand side- • w12sA2 + w22sB2 + 2 w1 w2 sA sB • = w12sA2 + w22sB2 + 2 w1 w2x1x sA sB The comparison boils down to rAB versus 1. • Thus, the left-hand side is equal to or less than the right-hand side.

33. 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’.

34. 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.

35. 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.”

36. 3) Efficient Frontier: the upper part of investment opportunity set is superior to the lower part Minimum Variance Portfolio

37. *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

38. ****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

39. ***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: w2sA - This is on a straight line between Risk free asset and Asset A

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

41. 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

42. * Application Question 2: How much of foreign stocks a Canadian should include in his portfolio? 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

43. *Application Question 3: As you are mixing more and more assets, the Mean-Variance Risk of the portfolio falls: Total risk sp Unique (Diversifiable) Risk Market (Systematic) Risk # of assets

44. * Appliation Example: XYZ Fund

45. 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

46. Returns and Risks of the Art • Investment on Art, especially, on Picasso’s prints. r rstock i rbond i rT-bill i rArt Prints s

47. *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.

48. *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.

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

50. 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.