1. Efficient Diversification
Chapter 6

2. Portfolio Risk
So far we’ve been focusing on individual assets, but what happens when we combine individual assets into a portfolio?

3. Diversification A B
A&B
Time
Time
Time

4. Portfolio v Single Asset
Investing everything in a single asset exposes you to that investment’s total risk
A portfolio allows us to spread out our investment reducing the impact of swings in a single asset
→Assets are unlikely to all move in the same direction
Intuition: “Don’t put all your eggs in one basket”

5. Historical Non-Diversifiers
People on record for saying: “Put all your eggs in one basket and watch it closely”
Mark Twain: Died penniless
Andrew Carnegie: Died known as “The Richest Man in the World”
→Definition of Risk

6. Diversification and Portfolio Risk
Market/Systematic/Non-diversifiable Risk
Risk factors common to whole economy
Affect all firms, economy wide risk
Unique/Firm-Specific/Nonsystematic/ Diversifiable Risk
Risk that can be eliminated by diversification
Affect individual or small groups of firms (industries)
What does σ measure?

7. The Wonders of Diversifying

8. Risk versus Diversification

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

10. Aside: Why Diversified Investors Set Prices
Investor A is undiversified, while B has a diversified portfolio. Both investors want to buy a share of Facebook. Who gets the share?
Which investor uses a 10% discount rate? 20%?
Facebook will pay a constant $20 dividend
Which investor offers a higher price for the share? (Remember: Price = Div / discount rate)

11. Measuring Co-Movement
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

12. Calculation Cov (rS, rB) = σSB = Σ p(i)*(rS(i)– μS)*(rB(i) – μb)
σXY = p1*(X1 – μS)*(Y1 – μB)+p2*(X2 – μS)*(Y2 – μB)+….+pN*(XN – μS)*(YN – μB)
Note that σss = Σ p(i)*(Xs(i) – μs)*(Xs(i) – μs)
Which implies?
How does a risky asset’s return move with the risk free?

13. 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?

14. Correlation Coefficient
Measures the strength of the covariance relation
It is a standardization of covariance
Bounded by -1 & 1
1 means the two are perfectly positively correlated
-1 means the two are perfectly negatively correlated

15. Correlation Coefficient Formula
ρSB = σSB / (σS * σB)
ρSB – Correlation Coefficient
σSB – Covariance between S and B
σS – Std Dev of S
σB – Std Dev of B

16. 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?

17. 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?

18. Covariance & Correlation Aside
The covariance between a risky asset and a risk free is 0. Why?

19. Portfolio Return & Risk
Portfolio Return: weighted average of component returns
A stock’s weight is the percent of the portfolio it represents
Expected Return: weighted average of component expected returns
Variance: depends on covariance between the assets

20. Portfolio Return Example
If a portfolio comprised of Google {E(r) = 25%} and Acme {E(r) = 10%} has an E(r) = 20%, how much is invested in Google and Acme?

21. Portfolio Variance Formulas
Depends primarily on covariances between the assets
Two Stock Portfolio
σP2 = (wBσB)2+(wSσS)2+2wBwSσSB
σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB
Three Stock Portfolio
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSB + 2wBwCσBC + 2wSwCσSC
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSσBρSB + 2wBwCσBσCρBC + 2wSwCσSσCρSC

22. Portfolio Variance Formulas
Depends primarily on covariances between the assets
Two Stock Portfolio
σP2 = (wBσB)2+(wSσS)2+2wBwSσSB
σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB
Three Stock Portfolio
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSB + 2wBwCσBC + 2wSwCσSC
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSσBρSB + 2wBwCσBσCρBC + 2wSwCσSσCρSC

23. 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.

24. 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

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

26. Portfolio Example
An investor wants to maximize his return, but doesn’t want the risk of his portfolio to exceed 15%. He has two options, a stock fund with a standard deviation of 35% and T-Bills, how much does he invest in the stock fund?

27. Individual Stock Allocation
Your offer a portfolio comprised of 70% stock A and 30% stock B. If an investor has half their wealth invested in your portfolio, how much of her wealth is in stock A?

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

29. Fundamental Premise of Portfolio Theory
Rational investors prefer the highest expected return at the lowest possible risk
Investors want Lower Risk & Higher Returns → Mean-Variance Criterion
If E(rA) ≥ E(rB) and σA ≤ σB
Portfolio A dominates portfolio B
How can investors lower risk without sacrificing return?

30. Investment Opportunity Set
The set of possible risk-return pairings offered by the portfolios that can be formed from the available securities

31. Investment Opportunity Set Example

32. Investment Opportunity Set Continued
return
Stock
Min Var.
Bonds P

33. Changing the Correlation Coefficient
Artificial Risk Free Asset (-1)
How does the correlation coefficient affect the diversification benefit?

34. What portfolios do we prefer?
return
Efficient Portfolios:
-Maximize risk premium for any level of standard deviation
-Minimize standard deviation for any level of return
-Maximize Sharpe ratio for any standard deviation or risk premium P

35. Including More Assets Possible Portfolios Efficient Portfolios return

36. Including a risk free asset?
Capital Allocation Line, Capital Market Line
A risk free asset changes our investment opportunity set and our efficient portfolio
Optimal Risky Portfolio (O)
return rf P

37. Which Efficient Portfolio?
O, the Optimal Efficient Portfolio
It offers the best risk return trade off
Highest SHARPE RATIO (ri - rf) / i
All portfolios on the CAL will have the same Sharpe Ratio as O

38. Optimal Risky Portfolio
Given two risky asset

39. Capital Allocation Line
The set of efficient portfolios available once we include a risk free asset
Why only two assets?
O dominates all other risky assets
All other are either too risky or offer to low a return
Rf is used to adjust for risk tolerance

40. Tobin’s Separation Property
Implies investing can be separated into two tasks
Find O (Optimal risky portfolio)
Determining where we want to be on the CAL?
Known as Asset Allocation

41. Asset Allocation How do we allocate our portfolio across asset classes
Bogle: This the most fundamental investment question
Basically: How much risk do we want?
We can adjust our risk level through our risk free investments

42. Finding Your Spot
Notionally there is a formula for finding an investors preferred location on the CAL, which is based on an investors level of risk aversion
Risk Aversion measures how investors feel about risk
An investor that is more risk averse will hold more of the risk free asset
As risk aversion decreases (investor becomes more risk tolerant, risk loving) the investor will hold more O

43. Measuring Risk Aversion
We measure risk aversion with A
The risk premium an investor demands for investing everything in a portfolio given its risk
The Sharpe Ratio an investor requires to put everything into a single diversified portfolio

44. How Much O Should an Investor Hold
Y is the amount of an investors portfolio invested in the Optimal Risky Portfolio
To find Y divide the Available Sharpe Ratio (which is offered by O) by that demanded by the investors (A)

45. What Does Our Portfolio Look Like? Where are we on the CAL?
What risk premium do you demand to invest all your money in a diversified portfolio with a σ of 20%?
If you have $250,000 to invest, how much of your total investment portfolio is invested in O? O’s risk premium is 11% and its σ is 18%
Where is the rest of your portfolio invested?

46. Moving on the CAL As risk averse increases Y falls → Hold more T-Bills
Purchasing T-Bills is lending the government money
As investors become more risk tolerant (loving) Y increase → Hold more of O
We can move past O by shorting T-Bills
Borrow T-Bills → Sell them →Use proceeds to buy O
Latter return the T-Bill plus interest

47. Composition of our portfolio (C)

48. Our Portfolio (C)

49. Alternative: Passive Investing
Based on the idea that securities are fairly priced
Steps involved
Buy a portfolio of assets that tracks the broad economy
Relax
Advantages: Simple, Low Cost, Outperform the average active strategy