3. Investment Planning. 13. Asset Pricing Models Flashcards
Did you know that a $100 investment in Microsoft’s shares in 1986 was worth about $29,000 just twelve years later? Those who expected large returns and made this investment were indeed very fortunate. Everybody likes to get in on the ground floor of an emerging growth company, such as Intel or Microsoft. Though buying a company’s common stock may be riskier than bonds or preferred stock, it nevertheless gives the investor a stake in the company’s future - for better or worse.
What is the exact value of a stock? This is a very difficult question to answer. Nevertheless, investors must at least try to compute how much a stock is worth. Perhaps it will help them find the Microsofts of the future. Even if it doesn’t, it will make them more informed investors. Professional analysts use certain techniques to value stocks. An investor can apply these asset pricing models to value securities and options.
The Asset Pricing Models module, which should take approximately four hours to complete, will explain the theories that are used to calculate the expected returns of securities and options.
Upon completion of this module you should be able to:
* State the workings of the Capital Asset Pricing Model (CAPM),
* Describe the Arbitrage Pricing Theory (APT),
* Explain the Binomial Option Pricing Model (BOPM),
* Describe the Black-Scholes-Merton call option pricing model, and
* Apply the asset pricing formulas to data and solve equations.
Module Overview
An economic equilibrium occurs whenever supply equals demand. As a result, prices have no tendency to change. The capital asset pricing model and the arbitrage pricing theory are two similar but different equilibrium portfolio theories. These theories are similar because they are both grounded in portfolio theory and they have parallel asset pricing implications. They both use quantitative risk surrogates like variance and covariance and the concepts of diversifiable risk and undiversifiable risk.
The capital asset pricing model (CAPM) provides an intuitive way of thinking about the return that an investor should require from an investment, given the asset’s systematic risk. It suggests that investors need not worry about the market portfolio. They only need to decide how much systematic risk they wish to accept. Market forces will ensure that any stock can be expected to yield the appropriate return.
The arbitrage pricing theory is an alternative theory that has gained acceptance in the financial community. Under this theory, a security’s price is explained by multiple economic factors (known as a multi-factor model) rather than the single systematic risk factor.
The behavioral pricing model (BAPM) was developed to improve upon CAPM. At the heart of the model is the study of behavioral finance, which acknowledges the contributions of standard finance, but argues that people are “normal” instead of “rational.”
The binomial pricing model and the Black-Scholes-Merton pricing model provide formulas for determining the price of options, that is, their premiums. Binomial option pricing models are mathematically simple models that have been developed to deal with a broad class of valuation problems that include options, stocks, bonds and other risky financial claims. The Black-Scholes-Merton model was the first closed-form option-pricing model.
To ensure that you have a solid understanding of asset pricing models, the following lessons will be covered in this module:
* Capital Asset Pricing Model
* Arbitrage Pricing Theory
* Behavioral Pricing Model
* Option Pricing Models
AUDIO:
- Capital asset pricing model (CAPM) – states that the return of an asset is related to one risk factor – the beta
- Arbitrage pricing theory – asset’s returns are affected by more than 1 risk factor
- Both equilibrium models of security prices meant to determine a security’s return based on risk premiums
- Options pricing models – determine the price of call and put options
- Asset pricing models –understand the consequences to the asset prices as the variables in the model change
Section 1 – Capital Asset Pricing Model (CAPM)
The capital asset pricing model (CAPM) enters the realm of positive economics by presenting a descriptive model of how assets are priced. The significant implication of the CAPM is that the expected return of an asset is related to the measure of market risk for that asset known as beta. The capital asset pricing model provides a formula that calculates the expected return on a security based on its level of risk. The capital asset pricing model formula is the risk-free rate plus beta times the difference between the return on the market and the risk-free rate.
The formula for CAPM is:
ri is the CAPM expected return
rf is the risk-free rate
rm is the market return
ßi is the security’s beta
(rm - rf) is often referred to as the “market premium.”
This model provides the intellectual basis for a number of the current practices in the investment industry. Although many of these practices are based on various extensions and modifications of the CAPM, a sound understanding of the original version is necessary in order to understand them. Accordingly, this lesson presents the original version of the CAPM.
To ensure that you have a solid understanding of the capital asset pricing model, the following topics will be covered in this lesson:
* Assumptions
* Capital Market Line (CML)
* Security Market Line (SML)
* Market Model
Upon completion of this lesson, you should be able to:
* List the assumptions behind the CAPM and state their implications,
* Describe the theory of capital market line,
* Explain the separation theorem,
* Define market portfolio,
* Explain the security market line,
* Distinguish between CML and SML, and
* Detail the relationship between the market model and CAPM.
Describe the Capital Market Line
The capital market line (CML) represents the linear efficient set in the world of CAPM. All investors will hold a portfolio lying on the CML. It is the efficient frontier when borrowing and lending at the risk-free rate are permitted.
It can be described as the most desirable asset allocation line. It denotes the set of most desirable risky portfolios that can be generated by borrowing and lending at the risk-free rate of interest. Assuming homogeneous expectations and perfect markets, the CML, therefore, represents the efficient set.
The slope of the CML is equal to the difference between the expected return of the market portfolio and that of the risk-free security,
(r⎯⎯M−rf)
Divided by the difference in their risks,
(σM−0)
Or,
(rM−rf)/σM
As the vertical intercept of the CML is rf, the straight line characterizing the CML has the following equation:
rP=rf+[rM−rfσM]σp
where r⎯⎯p and σp refer to the expected return and standard deviation of an efficient portfolio. This formula represents the expected return of the portfolio equals the risk-free rate plus the risk premium for the asset.
Two key numbers characterize equilibrium in the securities market:
* The first is the vertical intercept of the CML, that is, the risk-free rate. It is often referred to as the reward for waiting.
* The second is the slope of the CML, which is often referred to as the reward per unit of risk borne.
In essence, the security market provides a place where time and risk can be traded, with their prices determined by the forces of supply and demand. Thus, the intercept and slope of the CML can be thought of as the price of time and the price of risk, respectively. In the example, they are equal to 4% and 1.21, respectively.
Exam Tip: The distinguishing feature of the capital market line is that the denominator is the standard deviation of the market. This will help you recognize the capital market line equation on the CFP® exam.
CML Calculation Example:
The market portfolio associated with a risk-free rate of 4% consisted of Able, Baker, and Charlie in the proportions of 0.12, 0.19, and 0.69, respectively. This is under the assumption that these stocks are the only ones that exist. The expected returns for the portfolio and standard deviation for the market portfolio with these proportions are 22.4% and 15.2%, respectively.
* What is the equation for the resulting CML?
The equation for the resulting CML is:
r⎯⎯p=4+[22.4−415.2]σp=4+1.21σp
AUDIO:
Exam Tip: The distinguishing feature of the capital market line is that the denominator is the standard deviation of the market.
This will help you recognize the capital market line equation on the CFP® exam.
- Not likely to have to calculate on exam. If so, drop in variables, and play order of operations
- Conceptually could be exam question
- SML is looking at a particular security and expected return given a risk free rate
- CML takes it a bit further. In the world of CAPM, all risky assets, can be plotted along a line and represent a new efficient frontier
- In this calculation, standard deviation is used as the risk factor
- Everything on that line represents the most efficient portfolios for a return for a given amount of risk (measured by standard deviation)
Exam: Highly-testable calculation Capital Asset Pricing Model (CAPM)
Exam Tip:
* The Capital Asset Pricing Model (CAPM) is a highly-testable formula that is** included on your CFP® Board-provided formula sheet**.
* Check out exam tip to learn about the variables & additional need-to-know facts about CAPM.
- Highly-testable calculation: The Capital Asset Pricing Model (CAPM)
- Security Market Line (SML) is simply CAPM expressed in a graphic form
- Likely will have to calculate
- One of the provided formulas – be able to recognize it by sight and recognize that’s what they’re asking for in the question
- Order of operations:
- Rm – Rf: market return minus the risk-free return
- Multiply by beta
- Add to risk free return
- And that would be CAMP – risk free return
Section 1 – Capital Asset Pricing Model (CAPM) Summary
The capital asset pricing model is widely used by analysts to value securities. It is a theory about equilibrium prices in the markets for risky assets. CAPM focuses on the relationship between systematic risk and returns.
In this lesson, we have covered the following
* The assumptions behind the CAPM are mainly that investors are risk-averse and never satiated. They lend or borrow at a common risk-free interest rate. Investors evaluate portfolios by analyzing expected returns and standard deviations over the same one-period horizon. They have homogeneous expectations regarding expected returns and risks of securities. The implications of these assumptions are that all investors will hold the same efficient portfolio of risky assets, differing only in the amounts of risk-free borrowing or lending they undertake.
* The capital market line is the linear efficient set of the CAPM. The CML represents the equilibrium relationship between the expected return and standard deviation of efficient portfolios. The separation theorem states that an investor’s optimal risky portfolio can be determined without reference to the investor’s risk-return preferences. The market portfolio is the risky portfolio held by all investors consisting of all securities, each weighted in proportion to its market value relative to the market value of all securities.
- The security market line is the linear relationship between market covariance and expected return. The slope of the SML indicates the level of aggregate investor risk aversion. Like the CML, the SML is an equilibrium risk-return relationship. In SML the relevant measure of risk for individual securities is the contribution that they make to the standard deviation of the market portfolio as measured by their respective covariances with the market portfolio or their betas. The beta is a measure of the covariance between the security and the market portfolio relative to the market portfolio’s variance.
- The market model is not an equilibrium model of security prices as is the CAPM. However, the beta from the CAPM is similar in concept to the beta from the market model. The market model differs from the CAPM in that it is a factor model while the CAPM is an equilibrium model. Further, the market model uses a market index while the CAPM uses the market portfolio. The market index is a subset of the CAPM’s market portfolio. The total risk of a security can be separated into market risk and non-market risk, under the CAPM. Each security’s non-market risk is unique to that security and hence is also termed its “unique risk.”
Identify the variables that represent the ‘market risk premium’ within the CAPM formula.
ri = rf + (rm – rf)ßi
* rf + (rm – rf)
* (rm – rf)Bi
* (rm – rf)
* rf
(rm – rf)
* The ‘market risk premium’ is the difference between the market return and the risk-free rate [i.e., (rm – rf)].
* This is the premium given to investors for taking on systematic risk.
* When multiplied by ß, it becomes the ‘stock risk premium.’
Which one of the following is not a key assumption underlying the CAPM?
* Investors prefer portfolios with lower standard deviations.
* Assets are infinitely divisible.
* Investors may borrow or lend at a single risk-free interest rate.
* Taxes and transaction costs reduce market liquidity.
Taxes and transaction costs reduce market liquidity.
* The assumption regarding taxes and transactions costs under CAPM is that they are irrelevant. It is not assumed that they reduce market liquidity.
* The CAPM also assumes that investors are risk averse and therefore prefer portfolios with lower standard deviations. Other assumptions are that assets are infinitely divisible and that investors may borrow or lend at a single risk-free rate.
If the risk-free rate is 4%, the beta on Intel is 1.1, and the rate of return of the market portfolio is 12.0, what is the expected return on Intel?
* 12.8%
* 11.2%
* 12%
* 13.1%
12.8%
* The expected rate of return
* = 4% + (12.0 – 4) (1.1)
* = 12.8%.
In the equilibrium world of the CAPM, a security that is not part of the market portfolio: (Select all that apply)
* Is not owned by investors
* Has an equilibrium price of zero
* Is attractive to the very risk-averse investor
* Has a market value of zero
Is not owned by investors
Has an equilibrium price of zero
Has a market value of zero
* The market portfolio is the risky portfolio held by all investors. Therefore, a security that is not part of the market portfolio is not attractive to the risk-averse investor.
* An important feature of the CAPM is that in equilibrium each security must have a nonzero proportion in the composition of the tangency portfolio.
* That is, no security can, in equilibrium, have a proportion in the portfolio that is zero.
* Hence, the market portfolio is a set of securities that can be freely owned by investors.
Match the term with the correct description:
Market risk
Non-market risk
Risk-free rate of return
Total risk of a security
* It is composed of market risk and non-market risk.
* The portion of a security’s total risk that is related to events specific to the security and not to the movements in the market portfolio.
* In the CAPM world, the expected return of a security with a beta of zero equals this rate of return.
* The portion of a security’s total risk that is related to movements in the market portfolio and hence to the beta of the security.
- Market risk - The portion of a security’s total risk that is related to movements in the market portfolio and hence to the beta of the security.
- Non-market risk - The portion of a security’s total risk that is related to events specific to the security and not to the movements in the market portfolio.
- Risk-free rate of return - In the CAPM world, the expected return of a security with a beta of zero equals this rate of return.
- Total risk of a security - It is composed of market risk and non-market risk.
Section 2 – Arbitrage Pricing Theory (APT)
The capital asset pricing model is an equilibrium model that describes why different securities have different expected returns. In particular, this economic model of asset pricing asserts that securities have different expected returns because they have different betas. However, there exists an alternative model of asset pricing that was developed by Stephen Ross. It is known as arbitrage pricing theory, and in some ways it is less complicated than the CAPM.
One primary APT assumption is that each investor, when given the opportunity to increase the return of his or her portfolio without increasing its risk, will proceed to do so. The mechanism for doing so involves the use of arbitrage portfolios. An arbitrage portfolio is defined by three conditions:
* Self Financing: Does not require additional funds from investor.
* Riskless: There is no sensitivity to any factor; there is zero variance and covariance with other portfolios; and there is negligible nonfactor risk.
* Positive Return: The riskless arbitrage will result in a positive return.
To ensure that you have a solid understanding of arbitrage pricing theory, the following topics will be covered in this lesson:
* Factor Models
* Identifying the Factors
Upon completion of this lesson, you should be able to:
* Explain arbitrage pricing theory,
* Explain arbitrage opportunities arising from disequilibirum, and
* List factors that might affect expected returns in APT.
Describe the Factor Model of Arbitrage Pricing Theory (APT)
Stephen Ross formulated various arbitrage arguments into a formal Arbitrage Pricing Theory (APT) that uses any number of risk factors. The theory is based on the law of one price, which states that if a security’s price is different in different markets, then a riskless profit exists for investors to buy the security from the market with the lower price and sell it in the market with the higher price.
ri = a + b1F1 + b2F2 + bkFk + ei
Where:
ri = rate of return on security i
ai = the zero factor: the expected return when all factors = zero
Fk = the value of the factor, such as the rate of growth in industrial production
ei = random error term
In this equation, bi is known as the sensitivity of security i to the factor. It is also known as the factor loading for securityi or the attribute of securityi.
The law of one price tells us that assets with equal betas have the same amount of undiversifiable risk and, therefore, should have identical expected rates of return. Furthermore, they should also have identical expected rates of return and intercept terms. The scenario described here is an equilibrium situation in which it will not be profitable to perform arbitrage between assets. When multiple assets are in equilibrium, their Arbitrage Pricing Theory Lines would be identical. The y-intercept and slope are the same for all assets. Every asset that plots above the arbitrage pricing line is underpriced, and every asset that plots below the APT line is overpriced.
In a theoretically ideal market, a smart investor might use the law of one price to earn riskless arbitrage profits. By setting up an imperfect hedge with the imbalanced portfolios, the smart investor can create a profit without investing any money or without taking any risk. For example, if two assets have the same sensitivity to a factor but different expected returns, then the investor would buy the one with the higher return while selling the one with the lower return. The proceeds from the sale will pay for the purchase and the return would be the difference between the two securities.
TEST TIP
The factors are the distinctive characteristic of the arbitrage pricing theory. If a question on the certification exam begins to talk about factors and sensitivity to factors, it is referring to the APT.
Cheri is trying to determine the return for a security that has zero factor of 4%, expected return from economic growth of 8% with sensitivity to the growth of 0.8. If the error term is 0, what is the return of this security?
* 8%
* 12%
* 13.6%
* 10.4%
10.4%
* The one factor model for this security is
* r = .04+(.8)(.08)
* =10.4%
List the 4 Factors
Stephen Ross employed in APT
Stephen Ross and Richard Roll employed factor analytic techniques to analyze 1,260 NYSE-listed stocks divided into 42 groups that contained 30 stocks each. They analyzed one decade of daily stock price returns. They concluded there were four or fewer significant risk factors. These factors represented unanticipated changes in four variables:
* Changes in the rate of inflation
* Changes in the index of industrial production
* Changes in the yield spread between high-grade and low-grade corporate bonds, a measure of investor confidence
* Changes in the slope of the term structure of interest rates, as measured by the difference between the yields on long-term government bonds and T-bills
Different researchers reported other risk factors. The most commonly identified factors that affect expected returns are indicators of aggregate economic activity, such as corporate earnings and dividends, inflation and interest rates.
Section 2 – Arbitrage Pricing Theory (APT) Summary
The APT requires fewer underlying assumptions and includes a wider array of variables into the analysis than the SML. The APT is a more general theory than the SML.
The APT can be shown to be mathematically equivalent to the SML when the market portfolio is the only risk factor in both models. Other similarities show that the two theories do not contradict each other. Moreover, the two theories are similar because both delineate undiversifiable commonalties that form the basis for risk premiums in market prices and returns. Since APT has been in existence for fewer years than the SML, it has not been tested as extensively. However, the results from initial tests look favorable.
In this lesson, we have covered the following:
- Factor Models are the basis of arbitrage pricing theory because APT assumes that security returns are generated by a factor model but does not specify the number or identity of the factors. Arbitrage is the process of earning riskless profits by taking advantage of differential pricing for the same physical asset or security. An arbitrage portfolio includes long and short positions in securities. It is self-financing, riskless, and has a positive expected return. The model states that securities or portfolios with equal factor sensitivities will behave in the same way except for non-factor risk. Therefore, securities or portfolios with the same factor sensitivities must offer the same expected returns.
- Factors: Empirical research has isolated four risk factors that significantly influenced securities returns:
- Unanticipated changes in the rate of inflation
- Unanticipated changes in the index of industrial production
- Unanticipated changes in the yield spread between high-grade and low-grade corporate bonds
- Unanticipated changes in the slope of the yield curve
Which of the following is true about the arbitrage pricing theory? (Select all that apply)
* Investors will take advantage of arbitrage opportunities thus eliminating them.
* Investors will not act on arbitrage opportunities.
* Arbitrage has fewer assumptions than the CAPM.
* Arbitrage opportunities are expensive and risky.
Investors will take advantage of arbitrage opportunities thus eliminating them.
Arbitrage has fewer assumptions than the CAPM.
* The logic behind APT is that investors will observe and take advantage of arbitrage opportunities and eliminate them.
* When all arbitrage possibilities have been eliminated, the equilibrium expected return on a security will be a linear function of its sensitivities to the factors.
