1. Return and Risk The Capital Asset Pricing Model (CAPM)

2. Important Security Characteristics
μ, which ____
σ2 and σ, which _____
Covariance and Correlation, which are _____

3. Statistics Review: Covariance
Covariance: measures how two variables move in relation to one another
Positive: The two variables move up together or down together
Ex: Height and Weight
Negative: When one moves up the other moves down
Ex: Sleep and Coffee consumption

4. Calculation Cov (X,Y) = σXY = Σ pi * (Xi – μx) * (Yi – μY)
σXY = p1*(X1 – μX)*(Y1 – μY)+p2*(X2 – μX)*(Y2 – μY)+….+pN*(XN – μX)*(YN – μY)
Note that σXX = Σ pi * (Xi – μx) * (Xi – μX)
Which implies?

5. Covariance Strength
If the covariance of asset 1 and 2 is -1,000, and between asset 1 and 3 is 500. Which asset is more closely related to 1?

6. Correlation Coefficient
Correlation Coefficient: Measures the strength of the covariance relationship
The correlation coefficient is a standardization of covariance
ρ12 = σ12 / (σ1 * σ2)

7. Correlation Coefficient Formula
ρ12 = σ12 / (σ1 * σ2)
ρ 12 –
σ12 -
σ1 -
σ2 -

8. Possible Correlation Coefficients
ρ12 has to be between -1 and 1
+1 implies that the assets are perfectly positively correlated
0 implies that the assets are not related
-1 implies that the assets are perfectly negatively correlated

9. Comparing Strength
When determining the strength of a correlation all we care about is the absolute value of the correlation coefficient
If ρ13 is -0.8, and ρ23 is 0.5, which asset is more correlated with 3?

10. Correlation Coefficient Example
σ13 is -1,000; σ23 is 500
σ1 is 10; σ2 is 1,000; σ3 is 250
Is 1 or 2 more strongly correlated with 3?

11. Portfolio Risk
So far we’ve been examining the risk of individual assets, but what happens when we combine individual assets into a portfolio?

12. Portfolio Illustration

13. Diversification
Reduces risk by combining assets that are unlikely to all move in the same direction, without sacrificing expected return
Intuition: “Don’t put all your eggs in one basket”
Stocks don’t move in exactly the same way. On a given day, while Boeing may yield positive 1% return, Microsoft may have gone down 1%. So, if you had invested a dollar each in Boeing and Microsoft, you would not have lost any money.

14. Two Types of Risk Diversifiable/Unsystematic/Unique risk
Diversifiable risk affects individual or small groups of firms (industries)
Ex: Lawsuits, Strikes
Non-Diversifiable/Systematic/Market risk
Affects all firms, economy wide risks
Ex: Business Cycle, Inflations Shocks
Which does σ measure?

15. How Diversification Works
Reduces/eliminates unsystematic risk.
Why can’t we diversify away systematic risk?

16. The Wonders of Diversifying

17. Portfolio Variances Formulas

18. Portfolio Covariance Matrix
Stock 1
Stock 2
Stock 3
Stock N
Stock 1
Var (1,1)
Cov (2,1)
Cov (3,1)
Cov (N,1)
Stock 2
Cov (1,2)
Var (2,2)
Cov (3,2)
Cov (N,2)
Cov (1,3)
Var (3,3)
Stock 3
Cov (2,3)
Cov (N,3)
Stock N
Cov (1,N)
Cov (2,N)
Cov (3,N)
Var (N,N)

19. Portfolio Variance Example
Two stocks A and B have expected returns of 10% and 20%.
In the past, A and B have had std dev of 15% and 25%, respectively, with a correlation coefficient of 0.2.
You decide to invest 30% in A and the rest in B.
Calculate the portfolio return and portfolio risk. Has diversification been of any use? Explain.

20. Calculations wA = 30% wB = A = 10% B = 20%
A = 15% B = 25% AB = 0.2
Portfolio Return =
Portfolio Variance =
Portfolio Standard Deviation =
Weighted Average Standard Deviation

21. Remarks on Diversification
Diversification reduces the p from 22% to 18.9%
What happens if AB = 1?
What happens as AB approaches -1?

22. Which Stock do you Prefer?
Stock A :
 = 10%;  = 2%
Stock B :
 = 10%;  = 3%
Stock C :
 = 12%;  = 2%

23. Fundamental Premise of Portfolio Theory
Rational investors prefer the highest expected return at the lowest possible risk
How can investors lower risk without sacrificing return?

24. Possible two asset portfolios
What are the possible portfolios we can create using only stocks and bonds
The correlation coefficient is -0.99
Stock: std dev = 14.3% expected return = 11.0%
Bond: std dev = 8.2% expected return = 7.0%

25. Possible Portfolios Some portfolios are better: why? which ones?
100% stocks
100% bonds

26. Efficient Portfolio Efficient Portfolios
Can these same risk and return pairing be achieved with a single stock?

27. Including More Assets In the real world there are more than two assets
The efficient frontier is the outer most envelope of possible portfolios, given the universe of available assets
Form every possible portfolio of the various assets and the efficient frontier are the portfolios on the edge

28. Including More Assets
Possible Portfolios
return P

29. Including a Risk Free Asset
How will the ability to lend and borrow at the risk free rate affect our possible risk-return combinations?

30. Risk Free and Risky Assets
return P
Start with our risky assets

31. Which Efficient Portfolio?
Under certain assumptions everyone holds the MARKET PORTFOLIO, M
This portfolio contains every asset in the market, according to their market weights
We have simplified investing down to two assets

32. Why the market portfolio
It offers the greatest return per unit of risk
SHARPE RATIO measure the risk return trade off: (ri - rf) / I,
The price of risk
A higher Sharpe Ratio implies we receive more return per unit of risk

33. M & the Risk free asset
return
100% stocks
M, Market Portfolio rf
100% bonds 
Investors choose to combine the risk-free asset and M, to create their portfolio

34. Why only two assets?
Holding any risky asset other than M offers a lower risk adjusted return
The investor is bearing more risk than necessary
The investor is accepting too low a return
Rf is how we adjust for our risk tolerance

35. Investing on the CML: In Two Easy Steps
Find M
Determine your level of risk aversion
Risk aversion determines location on the CML
The more risk averse the investor the safer the portfolio
Closer to the risk free

36. Where on the CML to invest?
Why do people move along the CML?
More risk averse investors will hold more?
More risk tolerant investors will hold more?
How do we move along the CML?

37. How do we Lend & Borrow? When we lend $, we give $ & receive interest
We do the same when we buy T-Bills
So buying T-Bills is the same as lending $
When we borrow $, we get $ and pay interest
We do the same when we short T-Bills
So shorting T-Bills is the same as borrowing $

38. Borrowing & Lending $
Who lends $?
Who borrows $?

39. Where are we borrowing & lending
100% bonds
100% stocks rf
return 
M, Market Portfolio

40. Risk compensation
In a diversified portfolio, risk depends exclusively on the underlying securities exposure to systematic risk
Market will not compensate investors for unsystematic risk
Why?
What risk will the market compensate investors for?

41. CAPM
Market will only compensate an investor for systematic risk, which is measured by BETA (β)
β – measures the sensitivity of the stock return to the market return (ex S&P 500)
E(Ri) = Rf + βi (RM - Rf)

42. CAPM Assumptions Investors all have the same expectations
Investors are risk averse and utility-maximizing
Investors only care about mean and variance
Expected return and risk
Perfect markets
No taxes
No transactions costs
Unlimited borrowing and lending at the risk-free rate

43. β Formulas
βi = i,m / m2
βi = (ρim i )/ m

44. βi = (ρim i )/ m Interpretation
I – Measures asset ‘i’ total risk
ρim – Measures the proportion of i’s total risk that is systematic
m – Measures the total market risk
Which is???
ρimi– Measures the systematic risk of asset ‘i’
So βi:

45. Market βm
βm = m,m / m2
What is the β of the risk free asset?

46. Notes on β β – tells us how sensitive a stock is to market movements
“Average Stock” has a β of 1
Stocks with β > 1 amplify market movements
Stocks with 0 < β < 1 reduce market movements
Stocks with negative β?

47. Portfolio β The weighted average of the component stocks’ β
Example: You invested 40% of your money in asset A, βA is 1.5 and the balance in asset B, βA is 0.5. What is the portfolio beta?

48. CAPM and β
CAPM states that expected returns are proportional to an investment’s systematic risk
A stock’s expected risk premium varies in proportion to it’s β
E(Ri) = Rf + βi (RM - Rf)
Security Market Line (SML) is the graphical representation of this relation

49. Security Market Line
What is the slope of the SML?

50. Putting Stocks in the SML
In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly priced
A stock under the SML is: Under or Over priced
A stock above the SML is: Under or Over priced

51. Dis-Equilibrium SML
E(ri)
SML C B A β

52. CAPM Formula CAPM: E(Ri) = Rf + βi (RM - Rf)
Market Risk Premium: (RM - Rf)
Asset i’s Risk Premium: βi (RM - Rf)

53. Example Rf = 5% Historical average risk premium is 8.4%
β = 1.5, return =
β = 1, return =
β = 0.5, return =
β = 0, return =
β = -1, return =

54. Example 1
The stock market moves up by 10%. Assume stock A has a beta of 1.5, stock B has a beta of 0.5 and stock C has a beta of -0.5.
Predict A, B, and C’s response to the market?

55. Example 2 βA is 1.5, and A is 20% βB is 2, and B is 15%;
Which stock has a higher expected return?

56. Why We Care Basic investment rule Explains the risk return trade off
Big rewards are accompanied by large risks
Explains the risk return trade off