f520 asset valuation and strategy

F520 – Portfolio Concepts 1

F520 Asset Valuation and Strategy

Overview

Risk and Return


Overview of Market Participants and Financial Innovation

• What Types of Risk does a Corporation or a Financial Intermediary Encounter?


Overview (Cont.)

• How can Financial Products or Intermediaries reduce these risks


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


Outline - Cont.

• More on Systematic Risk 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


Measuring Risk - Single Period

PPPr

D

0

011

P1 = the market value at the end of the intervalP0 = the market value at the beginning of the intervalD = the cash distributions during the interval


Measuring Return - Multiple PeriodsArithmetic

• Assumes no reinvestment of cash flows at the end of each period

N

R

a

N

ii

R 1

^


Measuring Return - Multiple PeriodsGeometric

• Also referred to as Time-Weighted Rate of Return

• Assumes reinvestment of cash flows at the end of each period.

1)]1)...(1)(1)(1[( /1

321 N

pNpppt RRRRR


Measuring Return - Multiple PeriodsInternal 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

ND

NN

DDD R

VC

R

C

R

C

R

CV

)1(...

)1()1()1( 33

22

11

0


Example of Return Calculations: Growth of$1 investment

assumingPeriod Price Dividend Return reinvestment

0 10 1.00 1 20 0 100% 2.00 2 10 -50% 1.00

IRR Cash FlowsArithmetic Return: 25.00% Cash flows Shares Cash flowsGeometric Return 0.00% for IRR no owned with for IRR withIRR without reinvestment 0.00% reinvestment reinvestment reinvestmentIRR with reinvestment 0.00% -10 1 -10

0 1.00 010 1.00 10

Comparing Return CalculationsWithout Dividend (Income) Cash Flows


Example of Return Calculations: Growth of$1 investment

assumingPeriod Price Dividend Return reinvestment

0 10 1.00 1 18 2 100% 2.00 2 9 -50% 1.00

IRR Cash FlowsArithmetic Return: 25.00% Cash flows Shares Cash flowsGeometric Return 0.00% for IRR no owned with for IRR withIRR without reinvestment 5.39% reinvestment reinvestment reinvestmentIRR with reinvestment 0.00% -10 1 -10

2 1.11 09 1.11 10

Comparing Return CalculationsWith Dividend (Income) Cash Flows


Measuring Total RiskVariance of actual returns

• Measures of the dispersion of returns

• Standard Deviation (STD)Standard deviation measures dispersion in percents

VarianceN tr r r

t

n

( )

1

1

2

1

Variance


Historical Returns, Standard Deviations, and Frequency Distributions: 1926-2009

• Frequency distribution is a histogram of yearly returns

Example Frequency Distribution

Goal: Select the lowest risk portfolio

• 0% stock, 100% bond








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.


Constructing PortfoliosPortfolio 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 1w1 + w2 + ... + wn = 1

RwR ip

n

ii

1


Constructing PortfoliosPortfolio Variance

• Two Asset CaseVar(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

abbabbaa wwww 22222

ghg

G

hhg

G

g

G

gg www

111

2 2

hggh

G

hhg

G

g

G

gg pwww g

111

2 2

hg

gh

ghp


Portfolio VarianceUsing 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

hggh

G

hhg

G

g

G

ggVAR pwwwR gp

111

2)(2

hg

gh

ghp

baabbabbaapxxxxRVAR p 2)( 2222


Efficient FrontierCorrelation = 1

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%

Exp

ect

ed

Re

turn

Standard Deviation

Efficeint Frontier (Corr = 1)

Correlation = 1

Input Data A B

Return 12% 16%

Std. Dev. 10% 20%

Correlation 1.00

bbaa

bbaa

xx

xx

pxxxx

p

p

baabbabbaap

22

22222

2


Efficient FrontierCorrelation = -1

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%

Exp

ect

ed

Re

turn

Standard Deviation

Efficeint Frontier (Corr = -1)

Correlation = -1

Input Data A B

Return 12% 16%

Std. Dev. 10% 20%

Correlation -1.00

bbaa

bbaa

xx

xx

xxxx

pxxxx

p

p

bababbaap

baabbabbaap

22

22222

22222

2

2



10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%

Exp

ect

ed

Re

turn

Standard Deviation

Efficeint Frontier (Corr = 0)

Correlation = -1 Correlation = 1 Correlation = 0

Input Data A B

Return 12% 16%

Std. Dev. 10% 20%

Correlation 0.00

2222

22222

22222

2

bbaap

bbaap

baabbabbaap

xx

xx

pxxxx


Portfolio DiversificationAverage annualstandard deviation (%)

Number of stocksin portfolio

Diversifiable risk

Nondiversifiablerisk

49.2

23.9

19.2

1 10 20 30 40 1000


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.


Covariance – the key to diversificationMathematical 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.

p i ii

n

x2 2 2

1

n

i

j

j

n

ip

nn 1

2

22

1

2

21

pi

n V nV V

nn n2

2 21

ihi

G

hhi

G

i

G

ii www

111

2 2


Covariance the key to diversification- Intuitive Example

# of Assets in the Portfolio

# of Variance Terms

# of Covariance terms

1 1 02 2 13 3 34 4 65 5 10

10 10 4520 20 19050 50 1225

100 100 4950


Conclusions on Covariance

• QuestionWhat 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.


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.



0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%

Exp

ecte

d R

etur

n

Standard Dev iation

Efficeint Frontier

Correlation = 0

Input Data

A BReturn 6.5% 12%

Std. Dev. 7.1% 16%

Correlation 0.00

QU: How do Investors Choose a Portfolio on the Efficient Frontier?


Use Indifference Curves – measures of investor risk aversion

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%

Exp

ecte

d R

etur

n

Standard Dev iation

Efficeint Frontier

Correlation = 0

QU: How Does this Change when a Risk-free asset is offered?


Investors can move to a higher indifference curve – greater utility.

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%

Exp

ecte

d R

etu

rn

Standard Deviation

Efficeint Frontier

Correlation = 0

QU: Can you identify the important parts in the graph.


Important points on the graph.

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%

Expe

cted

Ret

urn

Standard Deviation

Efficeint Frontier

Correlation = 0

Market Portfolio

Borrowing

Lending

AAL – Asset Allocation orCML – Capital Market Line

Risk-freerate

QU: What is meant by two-fund separation?


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

RxRxR mmFRFp

bbp

bbp

bbp

baabbabbaap

baabbabbaap

wRwR

wRwwwwR

pwwwwR

STD

STD

VAR

VAR

VAR

)(

)(

)(

)(

)(

22

22

2222

2222

0020

2

Standard Deviation is a linear function of the STD of the market portfolio


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


• 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.

pFp

M

FM RRRR

QU: Can we transform the CML to another measure of risk which only accounts for systematic risk?


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.

Return

Beta1.00.0

Rf

Rm


• 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).

Return

Beta1.00.0

Rf

Rm

Security Market Line


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.

n

iiip BxB

1


Example: Portfolio Beta Calculations

Market PortfolioStock Value Weights Beta

(1) (2) (3) (4) (3) x (4)

Haskell Mfg. $ 6,000 50% 0.90 0.450

Cleaver, Inc. 4,000 33% 1.10 0.367

Rutherford Co. 2,000 17% 1.30 0.217

Portfolio $12,000 100% 1.034


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?

Return

Beta 1.0 0.0

Rf

Rm

B

A C

D QU: How do I set up a trade to take advantage of this “mis-pricing”?


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.)

Return

Beta 1.0 0.0

Rf

Rm

B

A C

D

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 0.6 4 5+.6*6 = 8.6


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.

Beta E(Return)Req. Ret Invest

Portfolio Beta

Market +10%

Profit (Loss)

Market -10%

Profit (Loss)

A 0.6 8.6 8.6 0 - 6.00% -$ -6.00% -$ B 0.8 12 9.8 2000 0.40 8.00% 160$ -8.00% (160)$ C 1.4 10 13.4 -1000 (0.35) 14.00% (140)$ -14.00% 140$ D 0.6 4 8.6 -1000 (0.15) 6.00% (60)$ -6.00% 60$

4000 (0.10) -1.00% (40)$ 1.00% 40$ Absolute

• How can we reduce our market risk while still taking a position on our expectations?


Hedge Fund Example


S=-1 L=+1

Net Position

Portfolio Beta

Market +10%

Profit (Loss)

Market -10%

A 0.6 8.6% 8.6 0 - - 0.00% -$ 0.00%B 0.8 12.0% 9.8 2000 1 2,000 0.40 8.00% 160$ -8.00%C 1.4 10.0% 13.4 500 -1 (500) (0.17) -14.00% (70)$ 14.00%D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) -6.00% (90)$ 6.00%

4000 0 0.00 0.00% 0$ 0.00%

• 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.


Hedge Fund Example


S=-1 L=+1

Net Position

Portfolio Beta

Exp Ret, Mkt +10%

Actual Return

Profit (Loss)

A 0.6 8.6% 8.6 0 - - 0.00% 0.00% -$ B 0.8 12.0% 9.8 2000 1 2,000 0.40 8.00% 10.00% 200$ C 1.4 10.0% 13.4 500 -1 (500) (0.17) -14.00% -12.00% (60)$ D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) -6.00% -4.00% (60)$

4000 0 0.00 0.00% 80$

• 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.


Hedge Fund Example


S=-1 L=+1

Net Position

Portfolio Beta

Exp Ret, Mkt -10%

Actual Return

Profit (Loss)

A 0.6 8.6% 8.6 0 - - 0.00% 0.00% -$ B 0.8 12.0% 9.8 2000 1 2,000 0.40 -8.00% -6.00% (120)$ C 1.4 10.0% 13.4 500 -1 (500) (0.17) 14.00% 16.00% 80$ D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) 6.00% 8.00% 120$

4000 0 0.00 0.00% 80$

• 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.


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)?


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 = 2

i = 2i 2

m + 2ei

Biim

m

2


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.


Data For Beta Calculation – Lilly StockCalculations in yellow, WRETD = Value weighted return,

DATE DIVAMT PRC CFACPR Adj Prc Adj Div Return VWRETD31-Jan-95 65.875 4 16.46875 028-Feb-95 0.645 67 4 16.75 0.16125 0.026869 0.039622831-Mar-95 73.125 4 18.28125 0 0.091418 0.026982328-Apr-95 74.75 4 18.6875 0 0.022222 0.024882831-May-95 0.645 74.625 4 18.65625 0.16125 0.006957 0.034146530-Jun-95 78.5 4 19.625 0 0.051926 0.030840731-Jul-95 78.25 4 19.5625 0 -0.003185 0.0406674

31-Aug-95 0.645 81.875 4 20.46875 0.16125 0.054569 0.009342729-Sep-95 89.875 4 22.46875 0 0.09771 0.036390531-Oct-95 96.625 4 24.15625 0 0.075104 -0.01114530-Nov-95 0.685 99.5 4 24.875 0.17125 0.036843 0.04297129-Dec-95 0 56.25 2 28.125 0 0.130653 0.015399931-Jan-96 57.25 2 28.625 0 0.017778 0.028087429-Feb-96 0.3425 60.625 2 30.3125 0.17125 0.064934 0.016058229-Mar-96 65 2 32.5 0 0.072165 0.011202130-Apr-96 59.125 2 29.5625 0 -0.090385 0.025125331-May-96 0.3425 64.25 2 32.125 0.17125 0.092474 0.026722128-Jun-96 65 2 32.5 0 0.011673 -0.00765931-Jul-96 56 2 28 0 -0.138462 -0.05339

30-Aug-96 0.3425 57.25 2 28.625 0.17125 0.028438 0.032222430-Sep-96 64.5 2 32.25 0 0.126638 0.052991831-Oct-96 70.5 2 35.25 0 0.093023 0.013937729-Nov-96 0.3425 76.5 2 38.25 0.17125 0.089965 0.065729631-Dec-96 73 2 36.5 0 -0.045752 -0.01136231-Jan-97 87.125 2 43.5625 0 0.193493 0.053039528-Feb-97 0.36 87.375 2 43.6875 0.18 0.007001 -0.00088931-Mar-97 82.25 2 41.125 0 -0.058655 -0.04439630-Apr-97 87.875 2 43.9375 0 0.068389 0.042480230-May-97 0.36 93 2 46.5 0.18 0.062418 0.07126330-Jun-97 109.3125 2 54.65625 0 0.175403 0.044199431-Jul-97 113 2 56.5 0 0.033734 0.0763158

29-Aug-97 0.36 104.625 2 52.3125 0.18 -0.070929 -0.03645630-Sep-97 121 2 60.5 0 0.156511 0.058009431-Oct-97 0 67.0625 1 67.0625 0 0.108471 -0.034116


Data For Beta Calculation – Lilly StockSUMMARY OUTPUT

Total PercentRegression Statistics Systematic Risk 7.6811E-05 0.00806 R-squared

Multiple R 0.089765 Unsystematic Risk 0.00945575 0.99194 1- R-squaredR Square 0.008058 0.008057732 Total Risk 0.00953256Adjusted R Square -0.006318 0.00953256 1 Should add to 1

Standard Error 0.09864 due to the degrees of freedom inObservations 71 regression these are not exactly

equal to each otherANOVA

df SS MS F Significance FRegression 1 0.00545357 0.00545357 0.56049986 0.456603Residual 69 0.671358409 0.009729832Total 70 0.676811978

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 0.027862 0.01243276 2.241034391 0.02824458 0.00306 0.05266491 0.00305957 0.052664912X Variable 1 0.196282 0.262175336 0.748665388 0.45660343 -0.326744 0.71930692 -0.3267437 0.719306923

RESIDUAL OUTPUT

Observation Predicted Y Residuals SS(residuals)1 0.035639 -0.008770407 7.692E-05

Beta estimate

Alpha

Percent of total variation explained =MS(reg)/SS(total)


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.


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.


Anomalies with using the CAPM

• Small firm effect

• Price-to-Book Ratios (Growth versus value stocks)

• January effect

Common Question:

When using CAPM [Ri=Rf+i(Rm – Rf)],

what is the Risk Premium (Rm – Rf)

F301_CH12-54

What is the Rf you are using?

Should you use Large or Small Stocks?

Should you use arithmetic or geometric returns?

Can CAPM be used for bond?(August 9, 2013 data)

• Lehman Index (ticker = AGG)http://www.ishares.com/product_info/fund/overview/AGG.htm

Effective Duration 5.05 years Average Yield to Maturity 2.16% http://finance.yahoo.com/q/rk?s=AGG+Risk

Beta (against Standard Index) 1.01 Yahoo R-squared (against Standard Index) 98.88

Yahoo betas are 5-years• Lehman 1-3 year Treasury Bond Fund (ticker = SHY)

http://www.ishares.com/product_info/fund/overview/SHY.htm

Effective Duration 1.86 Average Yield to Maturity 0.32% http://finance.yahoo.com/q/rk?s=SHY+Risk

Beta (against Standard Index) 0.14 YahooR-squared (against Standard Index) 24.31What is the standard index in this case?

– So what is beta in this case?1.86 / 5.05 = 0.36, compare to Beta?


Can CAPM be used for bond?• Lehman 7-10 Year Treasury Bond Fund (ticker = IEF)

http://www.ishares.com/product_info/fund/overview/IEF.htm

Effective Duration 7.48Average Yield to Maturity 2.29% http://finance.yahoo.com/q/rk?s=IEF+Risk

Beta (against Standard Index) 1.70 YahooR-squared (against Standard Index) 69.52


• Lehman 20+ Year Treasury Bond Fund (ticker = TLT)http://www.ishares.com/product_info/fund/overview/TLT.htm

Effective Duration 16.43Average Yield to Maturity 3.61% http://finance.yahoo.com/q/rk?s=TLT

Beta (against Standard Index) 3.37 Yahoo R-squared (against Standard Index) 53.81


F520 – Portfolio Concepts56

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: http://investing.money.msn.com/investments/etf-management?symbol=ewo For Ishare Japan Fund: http://investing.money.msn.com/investments/etf-management?symbol=ewj Standard Index is MSCI EAFE NDTR_DEAFE 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)http://investing.money.msn.com/investments/etf-management?symbol=ijr For Ishare NAREIT Industrial/Office Index Fund (uses MSCI World)http://investing.money.msn.com/investments/etf-management?symbol=fnio



Multifactor CAPM

• Multi-Factor CAPME(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


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


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


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.


Commonly used Portfolio Performance Criteriaare based on the Efficient Frontier or CAPM concepts

Global Tech Fund: Return: +37.2%, 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%

Return

Beta1.00.0

Rf

Rm

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%

Exp

ecte

d R

etu

rn

Standard Deviation

Efficeint Frontier

Correlation = 0


This fund has a beta of 1.29, substantially greater than the market beta of 1.0. 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) = 1.0Solve for x.x = 1/1.29 = .78 portion of the portfolio in the Global Tech Fund1-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.

M-Squared Measure (Modigliani and Modigliani)Global Tech Fund: Return: +37.2%, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%

f520 asset valuation and strategy

Documents

percentsf520 portfolio

portfolio of assets

1110f520 portfolio concepts

0010f520 portfolio concepts

cash flowsarithmetic

return outlinehow

cash distributions

f520 asset valuation