1. F520 Asset Valuation and Strategy
Overview
Risk and Return
F520 – Portfolio Concepts

2. Overview of Market Participants and Financial Innovation
What Types of Risk does a Corporation or a Financial Intermediary Encounter?
F520 – Portfolio Concepts

3. Overview (Cont.)
How can Financial Products or Intermediaries reduce these risks
F520 – Portfolio Concepts

4. Risk and Return - Outline
How is the return on an asset affected by the risk of the asset?
How do we measure risk and return on an asset?
Unique Risk (diversifiable, unsystematic, residual, or specific)
Market Risk (undiversifiable, systematic, or covariance)
Constructing Portfolios -- How do we measure risk and return on a portfolio of assets?
Choosing Stocks -- Development of the Efficient Frontier and use of Indifference Curves
F520 – Portfolio Concepts

5. Outline - Cont. More on Systematic Risk Obtaining Estimates of Beta
The Capital Asset Pricing Model (CAPM)
Security Market Line (SML)
Obtaining Estimates of Beta
Uses of Beta
Tests of the Capital Asset Pricing Line and Beta.
Arbitrage Pricing Theory (APT), an alternative to CAPM
F520 – Portfolio Concepts

6. Measuring Risk - Single Period
P1 = the market value at the end of the interval
P0 = the market value at the beginning of the interval
D = the cash distributions during the interval
F520 – Portfolio Concepts

7. Measuring Return - Multiple Periods Arithmetic
Assumes no reinvestment of cash flows at the end of each period
F520 – Portfolio Concepts

8. Measuring Return - Multiple Periods Geometric
Also referred to as Time-Weighted Rate of Return
Assumes reinvestment of cash flows at the end of each period.
F520 – Portfolio Concepts

9. Measuring Return - Multiple Periods Internal Rate of Return
Also referred to as Dollar-Weighted Rate of Return
Allows additions and withdrawals
When no further additions or withdrawals occur and all dividends are reinvested, the Geometric and the IRR will yield the same
F520 – Portfolio Concepts

10. Comparing Return Calculations Without Dividend (Income) Cash Flows
F520 – Portfolio Concepts

11. Comparing Return Calculations With Dividend (Income) Cash Flows
F520 – Portfolio Concepts

12. Measuring Total Risk Variance of actual returns
Measures of the dispersion of returns
Standard Deviation (STD) Standard deviation measures dispersion in percents
F520 – Portfolio Concepts

13. Historical Returns, Standard Deviations, and Frequency Distributions: 1926-2009
F520 – Portfolio Concepts

14. Example Frequency Distribution
Frequency distribution is a histogram of yearly returns

15. Goal: Select the lowest risk portfolio
0% stock, 100% bond
20% stock, 80% bond
40% stock, 60% bond
60% stock, 40% bond
80% stock, 20% bond
100% stock, 0% bond
F520 – Portfolio Concepts

16. Constructing Portfolios
Investors seek to maximize the expected return from their investment given some level of risk, or
Investors seek to minimize the risk they are exposed to given some target expected return.
F520 – Portfolio Concepts

17. Constructing Portfolios Portfolio Return
Expected Return of a Portfolio equals the weighted average return on the portfolio Rp = wa * Ra + wb * Rb
wa = weight of asset a
wb = weight of asset b
Ra = Expected return of asset a
Rb = Expected return of asset b
General Formula
Weights must add to 1 w1 + w wn = 1
F520 – Portfolio Concepts

18. Constructing Portfolios Portfolio Variance
Two Asset Case Var(Rp) = Var(wa * Ra + wb * Rb )
General Case
for h  g
since 12 = 21, each covariance term is included in this equation twice.
i is the variance of asset i
gh is the covariance between asset g and asset h
where
F520 – Portfolio Concepts

19. Portfolio Variance Using Correlation
Correlation is the covariance standardized by the standard deviation of the two variables.
p = 1, perfect positive correlation
p = -1, perfect negative correlation
p = 0, no correlation
Two Asset Case
General Case
F520 – Portfolio Concepts

20. Efficient Frontier Correlation = 1
Input Data
A B
Return % %
Std. Dev % 20%
Correlation
Efficient Frontier Correlation = 1
F520 – Portfolio Concepts

21. Efficient Frontier Correlation = -1
Input Data
A B
Return % %
Std. Dev % 20%
Correlation
Efficient Frontier Correlation = -1
F520 – Portfolio Concepts

22. Efficient Frontier Correlation = 0
Input Data
A B
Return % %
Std. Dev % 20%
Correlation
Efficient Frontier Correlation = 0
F520 – Portfolio Concepts

23. Portfolio Diversification
Average annual standard deviation (%)
49.2
Diversifiable risk
23.9
19.2
Nondiversifiable risk
Number of stocks in portfolio 1 10 20 30 40
1000
F520 – Portfolio Concepts

24. Efficient Frontier Conclusions
The covariance of two assets is important in determining the variance of a portfolio
As long as assets are not perfectly correlated, combining them in a portfolio reduces risk
Systematic risk cannot be eliminated by diversification because it is the covariance risk. Also called non-diversifiable or market risk, since it is primarily from economy wide factors.
Unsystematic risk (also called diversifiable risk, unique risk, or firm specific risk) comes from circumstances unique to the firm. This is why in a well diversified portfolio, unique risk is unimportant.
F520 – Portfolio Concepts

25. Covariance – the key to diversification Mathematical Example
Assume a Special Case: Cov(i,h) = 0
As our portfolio gets large, the variances of the portfolio gets vary small if all the covariances are 0.
If all assets have weight Yn then x = 1 / n
If the largest variance is V
As n gets large, this goes to zero.
Therefore, our portfolio choices are dominated by concern over the covariance terms. In other words, well diversified investors need only price the risk associated with the covariance of assets.
F520 – Portfolio Concepts

26. Covariance the key to diversification - Intuitive Example
F520 – Portfolio Concepts

27. Conclusions on Covariance
Question What will the addition of this asset to my portfolio do to my level of risk?
Answer: Look at the covariance of the asset with my portfolio, rather than the variance.
F520 – Portfolio Concepts

28. Choosing Stocks Investors maximize their welfare by choosing the:
Set of securities (investments) that maximize return for a given level of risk.
Set of securities (investments) that minimize risk for a given level of return.
F520 – Portfolio Concepts

29. Efficient Frontier Correlation = 0
Input Data
A B
Return % 12%
Std. Dev % 16%
Correlation
Efficient Frontier Correlation = 0
QU: How do Investors Choose a Portfolio on the Efficient Frontier?
F520 – Portfolio Concepts

30. Use Indifference Curves – measures of investor risk aversion
QU: How Does this Change when a Risk-free asset is offered?
F520 – Portfolio Concepts

31. Investors can move to a higher indifference curve – greater utility.
QU: Can you identify the important parts in the graph.
F520 – Portfolio Concepts

32. Important points on the graph.
AAL – Asset Allocation or CML – Capital Market Line
Borrowing
Lending
Market Portfolio
Risk-free
rate
QU: What is meant by two-fund separation?
F520 – Portfolio Concepts

33. Measuring Risk and Return for the CML
The risk free asset has no variance and its return is known with certainty (proxy – T-bill)
Portfolio Return on CML
Portfolio Risk on CML
Standard Deviation is a linear function of the STD of the market portfolio
F520 – Portfolio Concepts

34. Conclusions from Efficient Frontier and CML
As long as there are only risky assets, it makes sense for investors to hold a portfolio on the efficient frontier. The existence of a risk-free asset changes this. The new efficient frontier (called the capital market line) will connect the risk free asset to some risky portfolio.
The market portfolio (Rm) should be chosen because any other security will lead to a lower return for a given level of risk (Tangent portfolio).
All investors will hold some combination of the risk-free asset and the market portfolio, since this will maximize their risk-return trade-off. (called two-fund separation)
The CML portfolio chosen by an investor depends upon their risk aversion
F520 – Portfolio Concepts

35. The Capital Market Line (CML) is Rp = Rf + slope (Standard Deviation)
The CML is a linear relationship between the efficient portfolio’s standard deviation and its expected return.
QU: Can we transform the CML to another measure of
risk which only accounts for systematic risk?
F520 – Portfolio Concepts

36. SML, Beta, and CAPM
The CML shows that all investors must hold a combination of the risk-free asset and the market portfolio to maximize their utility. Furthermore, it shows that their is a linear relationship between risk and return. Knowing that two points make a line, let’s form the SML by plotting these points.
F520 – Portfolio Concepts

37. Where (Rm - Rf) is the slope of the line
Security Market Line
Ri = Rf + (Rm - Rf)
Where (Rm - Rf) is the slope of the line
Beta measures the risk of a stock in regards to the market portfolio (similar to the average stock).
F520 – Portfolio Concepts

38. Understanding Beta and Calculating Portfolio Betas
Beta measures the relative volatility of stock i with the market portfolio.
The beta of a portfolio is the market value weighted average of the betas in the portfolio.
F520 – Portfolio Concepts

39. Example: Portfolio Beta Calculations
Market Portfolio Stock Value Weights Beta
(1) (2) (3) (4) (3) x (4)
Haskell Mfg. $ 6,000 50%
Cleaver, Inc. 4,000 33%
Rutherford Co. 2,000 17%
Portfolio $12, %
F520 – Portfolio Concepts

40. Beta, Expected Return and the Choice of Projects (Stock)
The concept that all assets must lie on the SML can also be Shown through an arbitrage argument. Consider Assets A, B, C, and D below. What will happen to the prices and expected returns of these assets in a competitive market using diversification techniques to eliminate all unsystematic risk?
QU: How do I set up a trade to take advantage of this “mis-pricing”?
F520 – Portfolio Concepts

41. Hedge Fund Example
How should I invest in these securities to take advantage of my expectations in returns relative to the required return. (Think about a hedge fund.)
CML = 5+B(6)
Beta
E(Return)
Req. Ret A
0.6
8.6
5+.6*6 = 8.6 B
0.8
12.0
5+.8*6 = 9.8 C
1.4 10
5+1.4*6 = 13.4 D 4
F520 – Portfolio Concepts

42. Hedge Fund Example
Some may think of having a net investment of zero, but look at the returns with market movements. None of our securities moved closer to efficiency in the example below. They each just followed the market as their risk would suggest.
How can we reduce our market risk while still taking a position on our expectations?
F520 – Portfolio Concepts

43. Hedge Fund Example
How can we reduce our market risk while still taking a position on our expectations?
Wb*Bb + Wc*Bc + Wd*Bd = 0 [no market risk]
Wb + Wc + Wd = 0 [no investment for arbitrage]
Having a portfolio beta of zero immunizes the portfolio from the market changes, and allows us to profit only from the unsystematic movements in prices, which is where one would find “mis-pricing”.
Remember this still has risk (betas could be incorrect, our estimates of over- and under-pricing could be incorrect).
Controlling for market movements, you expect prices of securities with expected returns that are higher relative to the required return to increase and lower expected returns to decrease.
F520 – Portfolio Concepts

44. Hedge Fund Example
The prior example showed no profit, because we assume that the returns on the stock were exactly equal to their expected return based on the market return and their beta. What is the hedge fund correctly predicted over and undervalued stocks?
Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on a market return of 10%, we expected it to increase 8% (market * beta), but our hedge fund model prediction was correct, adding 2%, so we made a net 10%.
Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of 10%, we expected it to increase 14% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net increase of 12%. Since we were short, we lost 12%.
Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of 10%, we expected it to increase 6% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net increase of 4%. Since we were short, we lost 4%.
Our portfolio has 0 beta and made money.
F520 – Portfolio Concepts

45. Hedge Fund Example What is the market had decreased in value?
Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on a market return of -10%, we expected it to decrease 8% (market * beta), but our hedge fund model prediction was correct, adding 2%, so we made a lost 6%.
Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of -10%, we expected it to decrease 14% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net decrease of 16%. Since we were short, we made 16%.
Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of -10%, we expected it to decrease 6% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net decrease of 8%. Since we were short, we made 8%.
Our portfolio has 0 beta and made money.
As long as our hedge fund model to predict over and under-valued stocks is correct, we make money in either an up or a down market.
F520 – Portfolio Concepts

46. Uses of Beta Discount rates in capital budgeting
Discount rates for pricing assets (stocks)
Utilities often base rates on the rate of return investors demand.
Cost of capital calculations
QU: What does the SML tell about the risk that managers should be concerned with when choosing a real asset investment (specifically a capital budgeting decision)?
F520 – Portfolio Concepts

47. Estimating Beta – Characteristic Line
Ri = Rf + (Rm - Rf)
rearranging terms Ri = Rf + *Rm - *Rf Ri = (1- ) Rf + * Rm
Characteristic Line (also called market model) Ri = ά + * Rm + eit
Where  = covariance (Ri, Rm) / Var (Rm)
Based on the market model, we can also break down an assets total risk into systematic and unsystematic components. Total Risk = 2i = 2i 2m + 2ei
F520 – Portfolio Concepts

48. Differences in Beta Calculations
Merrill Lynch – 5 years of monthly returns
Value Line – 5 years of weekly returns
Historic Beta – Calculated with only the raw return data
Adjusted Beta – Begins with a firms historic beta and makes an adjustment for the expected future movement towards one. (Beta has been found to gradually approach 1 over time)
Fundamental Beta – Adjusts historic betas for variables such as financial leverage, sale volatility, etc.
F520 – Portfolio Concepts

49. Data For Beta Calculation – Lilly Stock Calculations in yellow, WRETD = Value weighted return,
F520 – Portfolio Concepts

50. Data For Beta Calculation – Lilly Stock
F520 – Portfolio Concepts

51. Assumptions of the CAPM (SML)
Assumptions about investor behavior
Investors use only two measures to determine their strategy, expected return and risk,
Investors will choose portfolios as a risk reduction technique,
Investors make investment decisions over some single-period investment horizon,
Homogenous expectations with respect to asset returns, variances, and correlations
Assumptions about capital markets
Perfect competition,
No transaction costs
-No bid-ask spreads, -No commissions, -No information costs, -No taxes, -No regulation, and -all assets are marketable
Investors can borrow and lend at the risk-free rate.
F520 – Portfolio Concepts

52. Test of the CAPM (SML)
Clearly the assumptions are unrealistic, but the true test of a model comes from answering two questions
Does the model change when the assumptions are changed?
How well does the model predict?
Empirical Findings
There is a significant positive relationship between realized returns and systematic risk. However, the slope is usually less than predicted by the CAPM.
The relationship between risk and return appears to be linear. No evidence of curvature has been found.
Tests assessing the importance of company specific risk after controlling for market risk are inconclusive. Econometrically controlling for market risk given its high correlation with total risk is difficult.
The CAPM should be valid for all assets; however, bonds do not track along the SML.
Betas of individual stocks are not stable over time; however, betas for portfolios are stable over time.
F520 – Portfolio Concepts

53. Anomalies with using the CAPM
Small firm effect
Price-to-Book Ratios (Growth versus value stocks)
January effect
F520 – Portfolio Concepts

54. Common Question: When using CAPM [Ri=Rf+i(Rm – Rf)], what is the Risk Premium (Rm – Rf)
What is the Rf you are using?
Should you use Large or Small Stocks?
Should you use arithmetic or geometric returns?

55. Can CAPM be used for bond? (August 9, 2013 data)
Lehman Index (ticker = AGG) Effective Duration years Average Yield to Maturity % Beta (against Standard Index) Yahoo R-squared (against Standard Index) Yahoo betas are 5-years
Lehman 1-3 year Treasury Bond Fund (ticker = SHY) Effective Duration Average Yield to Maturity 0.32% Beta (against Standard Index) Yahoo R-squared (against Standard Index) What is the standard index in this case?
So what is beta in this case? 1.86 / 5.05 = 0.36, compare to Beta?
F520 – Portfolio Concepts

56. Can CAPM be used for bond?
Lehman 7-10 Year Treasury Bond Fund (ticker = IEF) Effective Duration 7.48 Average Yield to Maturity 2.29% Beta (against Standard Index) Yahoo R-squared (against Standard Index)
So what is beta in this case? 7.48 / 5.05 = 1.48, compare to Beta?
Lehman 20+ Year Treasury Bond Fund (ticker = TLT) Effective Duration Average Yield to Maturity 3.61% Beta (against Standard Index) Yahoo R-squared (against Standard Index)
So what is beta in this case? / 5.05 = 3.25, compare to Beta?
F520 – Portfolio Concepts

57. Cont.
The concept of Beta, used by Yahoo Finance and MSN Money for bonds is not the same concept of beta referred to in stocks. When a bond index is used as the standard index, we obtain a relative measure of duration. When a stock index is used, we obtain the traditional measure of systematic risk.
When using public betas, identify the index used to interpret the concept of beta reported. For many companies/funds, they state a “Standard Index”, to properly interpret the measures, you must clearly identify the index. (MSN Money provides identification, Yahoo does not.)
For Ishare Austria Fund: For Ishare Japan Fund: Standard Index is MSCI EAFE NDTR_D EAFE stands for Europe, Australasia, and Far East. The index has stocks from 21 developed markets, excluding the U.S. and Canada.
For Ishare S&P Small Cap 600 Index, (uses S&P500) For Ishare NAREIT Industrial/Office Index Fund (uses MSCI World)
F520 – Portfolio Concepts

58. Multifactor CAPM
Multi-Factor CAPM E(Ri) = Rf + i,M[E(RM) - Rf] + i,f1[E(Rf1) - Rf]+
i,f2[E(Rf2) - Rf] +…+ i,fn[E(Rfn) - Rf]
By rearranging terms we get the multiple regression typically used. E(Ri) = Rf + i,M*E(RM) - i,M*Rf + i,f1*E(Rf1) - i,f1*Rf i,f2*E(Rf2) - i,f2*Rf +…+ i,fn*E(Rfn) - i,fn*Rf E(Rit) =  + i,Mt*E(RMt) + i,f1*E(Rf1t) + i,f2*E(Rf2t) +… i,fn*E(Rfnt) + eit where  = Rf - i,M*Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf Rf = Riskfree Rate Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts

59. Arbitrage Pricing Theory (APT), an alternative to the CAPM
E(Ri) = Rf + i,f1[E(Rf1) - Rf] + i,f2[E(Rf2) - Rf] +…+
i,fn[E(Rfn) - Rf]
By rearranging terms we get the multiple regression typically used. E(Rit) = + i,f1*E(Rf1t) + i,f2*E(Rf2t) +…+ i,fn*E(Rfnt) + eit where  = Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf Rf = Risk-free Rate Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts

60. Assumptions of APT
APT assumes returns are a function of several factors, not just one as in the CAPM
Suggested factors (Roll & Ross 1983)
Index of Industrial Production,
Changes in the default risk premium on bonds,
Changes in the yield curve,
Unanticipated inflation
Other factors frequently considered
Factors for size
Factors for book-to-market value
F520 – Portfolio Concepts

61. Principles to Take Away from the APT and CAPM
Investing has two dimensions, risk and return.
It is inappropriate to look at the risk of an individual asset when deciding whether it should be included in a portfolio. What is important is how the inclusion of an asset into a portfolio will affect risk of the portfolio (covariance and/or beta must be considered).
Risk can be divided into two categories, systematic and unsystematic
Investors should only be concerned about systematic risks.
F520 – Portfolio Concepts

62. Commonly used Portfolio Performance Criteria are based on the Efficient Frontier or CAPM concepts Global Tech Fund: Return: %, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%
Sharpe Ratio
= (Rp – Rf)/σp
= (37.2 – 5)/25 = 1.29
Treynor Ratio
= (Rp – Rf)/Bp
= (37.2 – 5)/1.29 = 25.0%
Jensen’s alpha (αp)
= Rp – CAPM
= Rp – [Rf+Bp(RM – Rf)]
= 37.2 – [5+1.29(6)] = 24.5%
F520 – Portfolio Concepts

63. M-Squared Measure (Modigliani and Modigliani) Global Tech Fund: Return: %, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%
This fund has a beta of 1.29, substantially greater than the market beta of To compare it to the market, we must determine what portion can be invested in the risk-free rate and what portion invested in the Global Tech Fund to have the same risk as the market.
Let x = the percent invested in the Global Tech Fund, subsequently (1-x) is the percent in the risk-free asset.
Note that the beta of the risk-free asset is equal to zero.
(1-x)(0) + x(1.29) = 1.0
Solve for x.
x = 1/1.29 = .78 portion of the portfolio in the Global Tech Fund
1-x = .22 portion invested in the risk-free asset
Now calculate your risk-adjusted return:
The Global Tech Fund earned 37.2% and the risk-free asset over this 3-year period earned 5%. The proportions in each asset are calculated above.
.22(5%) + .78(37.2%) = 30.1%
This value can be compared to what the market earned during this period, since it has a beta of 1.
F520 – Portfolio Concepts