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Objectives

A. Review and solve problems using the CAL, MPT, and the Single Index model

B. Understand the implications of capital asset pricing theory and the CAPM to compute security risk premiums

C. Understand the arbitrage pricing theory and how it works

Solve problems using the CAL, CML, MPT and Single Index Models

Capital Market Line Review

- You estimate that a passive portfolio invested to mimic the S&P 500 (an index fund) has an expected return of 13% with a standard deviation of 25%. Your portfolio has an expected return of 17% with a standard deviation of 27%. With the risk-free rate at 7%, draw the CML and your fund’s CAL on an expected return-standard deviation diagram.

A. What is the slope of the CML? Your CAL?

B. Characterize in one short paragraph the advantage of your fund over the passive fund.

Answer

Slope of the CML = (13-7)/25 = .24

Slope of your CAL = (17-7)/27 = .37

b. Your fund allows an investor a higher mean for any given standard deviation than the passive strategy.

Your Fund

17%

13%

Index Fund

7%

10%

20%

30%

MPT Review

- Suppose that for some reason you are required to invest 50% of your portfolio in bonds (sb = 12%, E(rb) = 10%) and 50% in stocks (ss = 25%, E(rs) = 17%).

A. If the standard deviation of your portfolio is 15%, what must be the correlation coefficient between stock and bond returns?

B. What is the expected rate of return on your portfolio?

C. Now suppose that the correlation between stock and bond returns is 0.22 but that you are free to choose whatever portfolio proportions you desire. Are you likely to be better or worse off that you were in part a?

Answer

- A. sp2= w12s12 + w22s22 + 2W1W2 (r1,2s1s2 )

(.15) 2 =[(.512.1212) +(.522.2522) + 2(.51.52)*(.121.252 )] * r1,2

r1,2 = .2183 or 21.8% (take my word for this)

- B. E(rp) = (.5 * .10) + (.5 * .17) = 13.5%
- C. While the current correlation is slightly lower than 22%, this implies slightly greater benefits from diversification. However, the 50% bond constraint represents a cost since you cannot choose your optimal risk-return tradeoff for your risk level. Unless you would choose to have 50% bonds anyway, you are better off with the slightly higher correlation and the ability to choose your own portfolio weights.

Factor Review

- Investors expect the market rate of return to be 10%. The expected rate of return on the stock with a beta of 1.2 is currently 12%.

If the market return this year turns out to be 8%, how would you review/change your expectations of the rate of return on the stock?

Answer

- The expected return on the stock would be your beta (1.2) times the market return or:

1.2 * 8% = 9.6%

- Likewise, you could also determine how much the return would decrease by multiplying the beta times the change in the market return or:

1.2 * (8%-10%) = -2.4% + 12% = 9.6%

Questions

Any questions of Capital Allocation Lines, Modern Portfolio Theory, or Single Index Models?

Implications of Capital Market Theory and CAPM

- What have we done this far?
- We have been concerned with how an individual or institution would select an optimum portfolio.
- If investors act as we think, we should be able to determine how investors will behave, and how prices at which markets will clear are set
- This market clearing of prices and returns has resulted in the development of so-called general equilibrium models
- These models allow us to determine the risk for any asset and the relationship between expected return and risk for any asset when the markets are in equilibrium, i.e. balance or constant state

Capital Asset Pricing Theory

- What is capital asset pricing theory?
- It is the theory behind the pricing of assets which takes into account the risk and return characteristics of the asset and the market
- What is the Capital Asset Pricing Model?
- It is an equilibrium model (i.e., a constant state model) that underlies all modern financial theory
- It provides a precise prediction between the relationship between the risk of an asset and its expected return when the market is in equilibrium
- With this model, we can identify mis-pricing of securities (in the long-run)

CAPM(continued)

- Why is it important?
- It provides a benchmark rate of return for evaluating possible investments, and identifying potential mis-pricing of investments
- For example, an analyst might want to know whether the expected return she forecast is more or less than its “fair” market return.
- It helps us make an “educated” guess as to the expected return on assets that have not yet been traded in the marketplace
- For example, how do we price an initial public offering?

CAPM(continued)

- How was it derived?
- Derived using principles of diversification with very simplified (i.e. somewhat unrealistic) assumptions
- Does it work, i.e. withstand empirical tests in real life?
- Not totally
- But it does offer insights that are important and its accuracy may be sufficient for some applications
- Do we use it?
- Yes, but with knowledge of its limitations

CAPM Assumptions

- What does the model assume (some are unrealistic)?
- Individual investors are price takers (cannot affect prices)
- Single-period investment horizon (an its identical for all)
- Investments are limited to traded financial assets
- No taxes, and no transaction costs (costless trading)
- Information is costless and available to all investors
- Investors are rational mean-variance optimizers
- Investors analyze information in the same way, and have the same view, i.e., homogeneous expectations

Resulting Equilibrium Conditions

- Based on the previous assumptions:
- All investors will hold the same portfolio for risky assets – the market portfolio (M)
- The market portfolio (M) contains all securities and the proportion of each security is its market value as a percentage of total market value
- The risk premium on the market depends on the average risk aversion of all market participants
- The risk premium on an individual security is a function of its covariance (correlation and sssm) with the market

Capital Market Line

E(r)

M = Market portfolio rf = Risk free rate

E(rM) - rf = Market risk premium

[E(rM) - rf]/sM= Market price of risk

CML

M

E(rM)

rf

The efficient frontier without lending or borrowing

s

sm

Expected Return and Risk of Individual Securities

- What does this imply?
- The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio
- Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

CAPM Key Thoughts

- Key statements:
- Portfolio risk is what matters to investors, and portfolio risk is what governs the risk premiums they demand
- Non-systematic, or diversifiable risk can be reduced through diversification.
- Investors need to be compensated for bearing only non-systematic risk (risk that cannot be diversified away)
- The contribution of a security to the risk of a portfolio depends only on its systematic risk, as measured by beta. So the risk premium of the asset is proportional to its beta. (ß = [COV(ri,rm)] / m2)

Expected Return – Beta Relationship

Expected return - beta relationship of CAPM:

E(rM) - rf = E(rs) - rf

1.0 bs

In other words, the expected rate of return of an asset exceeds the risk-free rate by a risk premium equal to the asset’s systematic risk (its beta) times the risk premium of the market portfolio. This leads to the familiar re-arrangement of terms to give (memorize this):

E(rs) = rf + bs [E(rM) - rf ]

The Security Market Line

Notice that instead of using standard deviation, the Security Market Line uses Beta

SML Relationships

ß = [COV(ri,rm)] / sm2

Slope SML = E(rm) – rf = market risk

premium

E(r)

SML

E(rM)

rf

SML = rf + ß[E(rm) - rf]

ß

ß

= 1.0

M

Differences Between the SML and CML

- What are the differences?
- The CML graphs risk premiums of efficient portfolios , i.e. complete portfolios made up of the risk portfolio and risk-free asset, as a function of standard deviation
- The SML graphs individual asset risk premiums as a function of asset risk.
- The relevant measure of risk for individual assets is not standard deviation; rather, it is beta
- The SML is also valid for portfolios

Example: SML Calculations

- Put the following data on the SML. Are they in equilibrium?

Market data: E(rm) - rf = .08 rf = .03

Asset data: bx = 1.25 by = .60

- Calculations:

bx = 1.25 so E(r) on x =

E(rx) = .03 + 1.25(.08) = .13 or 13%

by = .60 so E(r) on y =

E(ry) = .03 + .6(.08) = .078 or 7.8%

Disequilibrium Example

- Suppose a security with a beta of 1.25 is offering expected return of 15%
- According to SML, it should be 13%
- Under priced: offering too high of a rate of return for its level of risk. Investors therefore would:
- Buy the security, which would increase demand, which would increase the price, which would decrease the return until it came back into line.

Disequilibrium Example

E(r)

The return is above the SML, so you would buy it

SML

15%

As more people bought the security, it would push the price up, which would bring the return down to the line.

Rm=11%

rf=3%

ß

1.0

1.25

CAPM and Index Models

- CAPM Problems
- It relies on a theoretical market portfolio which includes all assets
- It deals with expected returns
- To get away from these problems and make it testable, we change it and use an Index model which:
- Uses an actual index, i.e. the S&P 500 for measurement
- Uses realized, not expected returns
- Now the Index model is testable

The Index Model

- With the Index model, we can:
- Specify a way to measure the factor that affects returns (the return of the Index)
- Separate the rate of return on a security into its macro (systematic) and micro (firm-specific) components
- Components

ά = excess return if market factor is zero

ßiRm = component of returns due to movements in the overall market

ei = component attributable to company specific events

Ri = a i + ßiRm + ei

- (Notice the similarity to the Single Index model discussed earlier)

Security Characteristic Line

Excess Returns (i)

SCL

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Plot of a company’s excess return as a function of the excess return of the market

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Excess returns

on market index

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Ri = a i + ßiRm + ei

Does the CAPM hold?

- There is much evidence that supports the CAPM
- There is also evidence that does not support the CAPM
- Is the CAPM useful?
- Yes. Return and risk are linearly related for securities and portfolios over long periods of time
- Yes. Investors are compensated for taking on added market risk, but not diversifiable risk
- Perhaps instead of determining whether the CAPM is true or not, we might ask: Are there better models?

Questions

- Any questions on capital asset pricing and the Capital Asset Pricing Model?

CAPM Problem

- Suppose the risk premium on the market portfolio is 9%, and we estimate the beta of Dell as bs = 1.3. The risk premium predicted for the stock is therefore 1.3 times the market risk premium of 9% or 11.7%. The expected return on Dell is the risk-free rate plus the risk premium. For example, if the T-bill rate were 5%, the expected return of Dell would be 5% +(1.3 * 9%) = 16.7%.

a. If the estimate of the beta of Dell were only 1.2, what would be Dells required risk premium?

b. If the market risk premium were only 8% and Dell’s beta was 1.3, what would be Dell’s risk premium?

Answer

- a. If Dell’s beta was 1.2 the required risk premium would be (remember the risk premium is the expected return less the risk-free rate):

E(rs) = rf + bs [E(rM) - rf ] or the expected return on Dell = 5% + 1.2 (9%) = 15.8%

Dell’s risk premium (over the risk free rate) =

15.8% - 5% = 10.8%

- b. If the market risk premium was 8%:

E(rs) = rf + bs [E(rM) - rf ]

E(r) of Dell = 5% + 1.3 (8%) = 15.4%

Dell’s new risk premium is 15.4 – 5% = 10.4%

C. Understand Arbitrage Pricing Theory (APT) and How it Works

- What is arbitrage?
- The exploitation of security mis-pricing to earn risk-free economic profits
- It rises if an investor can construct a zero investment portfolio (with a zero net investment position netting out buys and sells) with a sure profit
- Should arbitrage exist?
- In efficient markets (and in CAPM theory), profitable arbitrage opportunities will quickly disappear as more investors try to take advantage of them

Arbitrage Pricing Theory (APT) (continued)

- What is APT based on?
- It is a variant of the CAPM, and is an attempt to move away from the mean-variance efficient portfolios (the calculation problem)
- Ross instead calculated relationships among expected returns that would rule out riskless profits by any investor in a well-functioning capital market
- What is it?
- It is a another theory of risk and return similar to the CAPM.
- It is based on the law of one price: two items that are the same can’t sell at different prices

APT (continued)

- In its simplest form, it is:

Ri = a i + ßiRm + ei the same as CAPM

The only value for a which rules out arbitrage opportunities is zero. So set a to zero and you get:

Ri = ßiRm Subtract the risk-free rate and you get the well-known equation:

E(rs) = rf + bs [E(rM) - rf ] from CAPM

APT and CAPM Compared

- Differences:
- APT applies to well diversified portfolios and not necessarily to individual stocks
- With APT it is possible for some individual stocks to be mispriced – to not lie on the SML
- APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio
- APT can be extended to multifactor models, such as:

Ri = a i + ß1R1 + ß2R2 + ß3R3 + ßnRn + ei

APT and Investment Decisions

- Roll and Ross argue that APT offers an approach to strategic portfolio planning
- Investors need to recognize that a few systematic factors affect long-term average returns
- Investors should understand those factors and set up their portfolios to take those factors into account
- Key Factors:
- Changes in expected inflation
- Unanticipated changes in inflation
- Unanticipated changes in industrial production
- Unanticipated changes in default-risk premium
- Unanticipated changes in the term structure of interest rates

Questions

- Any questions on Arbitrage Pricing Theory and how it differs from CAPM?

Problem

- Suppose two factors are identified for the U.S. economy: the growth rate of industrial production (IP) and the inflation rate (IR). IP is expected to be 4% and IR 6% this year. A stock with a beta of 1.0 on IP and 0.4 on IR currently is expected to provide a rate of return of 14%. If industrial production actually grows by 5% while the inflation rate turns out to be 7%, what is your best guess on the rate of return on the stock?

Answer

- The revised estimate on the rate of return on the stock would be:
- Before
- 14% = a +[4%*1] + [6%*.4]

a = 7.6%

- With the changes:
- 7.6% + [5%*1] + [7%*.4]

The new rate of return is 15.4%

Review of Objectives

- A. Can you solve problems using the CAL, MPT, and the Single Index model?
- B. Do you understand the implications of capital asset pricing theory and the CAPM to compute security risk premiums?
- C. Do you understand arbitrage pricing theory and how it works?

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