## Wednesday, April 13, 2016

### Comprehensive Note on Capital Asset Pricing Model and Arbitrage Pricing Theory

Capital Asset Pricing Model (CAPM)
Capital market theory is an extension of the Portfolio theory of Markowitz. The portfolio theory explains how rational investors should build efficient portfolio based on their risk-return preferences. Capital Market Asset Pricing Model (CAPM) incorporates a relationship, explaining how assets should be prices in the capital market. The capital market theory uses the results of capital market theory to derive the relationship between the expected returns and systematic risk of individual securities and portfolios.
Capital asset pricing model is a tool used by investors to determine the risk associated with a potential investment and also gives an idea as to what can be the expected return on the investment. It was developed by William Sharpe along with a formula for working out the risk as  who states that with an investment comes two types of risks:
1) Systematic Risk: These are risks that cannot be diversified away such as interest rates and recessions. As the market moves and changes occur which affect the market, each individual asset is affected to some degree and therefore they are sensitive to change causing a high level of risk.
2) Unsystematic (Specific): These risk can be diversified through increasing the size of an investment portfolio as this risk is specific to individual stocks and effectively represents no correlation between stocks and market movements.
CAPM states that investors are compensated for taking systematic risk however not for taking specific risk as an investor can diversify this risk away. Systematic risk cannot be eliminated of course even by holding all the shares in a stock market, therefore CAPM has introduced a method of calculating that risk.

Mathematical expression of CAPM: It can be expressed mathematically with the help of following equation:
E(rA) = rf + βA(E(rM) - rf)
where:
E(rA) is the expected return of the asset
rf is the risk-free rate
E(rM) is the expected return of the market portfolio
The general idea of CAPM is that investors should be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other part of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is done by taking an estimate of risk, (βA), and multiplying by the MRP, (E(Rm) - rf).
Graphical Presentation of CAMP
An asset is expected to earn the risk-free rate plus a reward for bearing risk as measured by that asset’s beta. The chart below demonstrates this predicted relationship between beta and expected return – this line is called the Security Market Line.
For example, a stock with a beta of 1.5 would be expected to have an excess return of 15% in a time period where the overall market beat the risk-free asset by 10%. The CAPM model is used for pricing an individual security or a portfolio. For individual securities, the security market line (SML) and its relation to expected return and systematic risk (beta) shows how the market must price individual securities in relation to their security risk class.
As the CAPM predicts expected returns of assets or portfolios relative to risk and market return, the CAPM can also be used to evaluate the performance of active fund managers. The difference is “excess return”, which is often referred to as alpha (α). If α is greater than zero, the portfolio lies above the Security Market Line.

Assumptions of CAPM
The capital asset pricing model is based on certain explicit assumptions regarding the behavior of investors. The assumptions are listed below:
1.       Investor make their investment decisions on the basis of risk-return assessments measured in terms of expected returns and standard deviation of return.
2.       The purchase or sale of a security can be undertaken in infinitely divisible unit.
3.       Purchase and sale by a single investor cannot affect prices. This means that there is perfect competition where investors in total determine prices by their action.
4.       There are no transaction costs. Given the fact that transaction costs are small, they are probably of minor importance in investment decision-making, and hence they are ignored.
5.       There are no personal income taxes. Alternatively, the tax rate on dividend income and capital gains are the same, thereby making the investor indifferent to the form in which the return on the investment is received (dividends or capital gains).
6.       The investor can lend or borrow any amount of fund desired at a rate of interest equal to the rate of risk less securities.
7.       The investor can sell short any amount of any shares.

8.       Investors share homogeneity of expectations. This implies that investors have identical expectations with regard to the decision period and decision inputs. Investors are presumed to have identical expectations regarding expected returns, variance of expected returns and covariance of all pairs of securities.
CAPM has been a popular model for calculating risk for over 40 years now and is therefore a proven method, some advantages are:
a)      Ease-of-use: CAPM is a simplistic calculation that can be easily stress-tested to derive a range of possible outcomes to provide confidence around the required rates of return.
b)      Systematic Risk: It considers only systematic risk, reflecting a reality in which most investors have diversified portfolios from which unsystematic risk has been essentially eliminated.
c)       Business and Financial Risk Variability: When businesses investigate opportunities, if the business mix and financing differ from the current business, then other required return calculations, like weighted average cost of capital (WACC) cannot be used. However, CAPM can.
d)      Determination of firm’s required return: To develop this overall cost of capital, the manager must have an estimate of the cost of equity capital. To calculate a cost of equity, some managers estimate the firm’s beta (often from historical data) and use the CAPM to determine the firm’s required return on equity.
e)      Public utility: The CAPM can also be used by the regulations of public utilities. Utilities rates can be set so that all costs, including costs of debt and equity capital, are covered by rates charged to consumers. In determining the cost of equity for the public utility, the CAPM can be used to estimate directly the cost of equity for the utility in question. The procedure is like that followed for any other firm. The beta and risk-free and market rates of return are estimated, and the CAPM is used to determine a cost of equity.
f)       Useful tool for investment managers: Investment practitioners have been more enthusiastic and creative in adapting the CAPM for their uses. The CAPM has been used to select securities, construct portfolios, and are forecastle considered under-valued, that is, attractive candidates for purchase.
g)      Most reliable and effective tool: Furthermore, in the opinion of most experts it is a more reliable and effective method of calculating risk than other models such as the Dividend Growth Model as CAPM takes into account a company's level of systematic risk against the stock market as a whole; this is a benefit as it allows for a company to compare itself to the market.
Drawbacks of CAPM
Despite the consistent use of the model over the years there has been some criticism for a few reasons:
1)      Unrealistic assumptions: Capital asset pricing model is based on a number of assumptions that are far from the reality. For example it is very difficult to find a risk free security. A short term highly liquid government security is considered as a risk free security. It is unlikely that the government will default, but inflation causes uncertain about the real rate of return.
2)      Based on future expectations: The CAPM is based on expectations about the future but empirical tests and data for practical use are exclusively based on historical returns.
3)      Risk-free Rate (Rf): The commonly accepted rate used as the Rf is the yield on short-term government securities. The issue with using this input is that the yield changes daily, creating volatility.
4)      Return on the Market (Rm): The return on the market can be described as the sum of the capital gains and dividends for the market. A problem arises when at any given time, the market return can be negative. As a result, a long-term market return is utilized to smooth the return. Another issue is that these returns are backward-looking and may not be representative of future market returns.
5)      Ability to Borrow at a Risk-free Rate: CAPM is built on four major assumptions, including one that reflects an unrealistic real-world picture. This assumption, that investors can borrow and lend at a risk-free rate, is unattainable in reality. Individual investors are unable to borrow (or lend) at the rate the US government can borrow at. Therefore, the minimum required return line might actually be less steep (provide a lower return) than the model calculates.
6)      Determination of Project Proxy Beta: Businesses that use CAPM to assess an investment need to find a beta reflective to the project or investment. Often a proxy beta is necessary. However, accurately determining one to properly assess the project is difficult and can affect the reliability of the outcome.
7)      Single-period investment appraisal: One disadvantage in using the CAPM in investment appraisal is that the assumption of a single-period time horizon is at odds with the multi-period nature of investment appraisal. While CAPM variables can be assumed constant in successive future periods, experience indicates that this is not true in reality.

Arbitrage Pricing Theory (APT)
Arbitrage Pricing Theory (APT) is an alternate version of Capital Asset Pricing Model (CAPM). This theory, like CAPM provides investors with estimated required rate of return on risky securities. It is a multifactor mathematical model used to describe the relation between the risk and expected return of securities in financial markets. It computes the expected return on a security based on the security’s sensitivity to movements in macroeconomic factors. The resultant expected return can then be used to price the security.
The Arbitrage pricing theory based model aims to do away with the limitations of one factor model (CAPM) that different stocks will have different sensitivities to different market factors which may be totally different from any other stock under observation. In layman terms, one can say that not all stocks can be assumed to react to single and same parameter always and hence the need to take multifactor and their sensitivities. The formula includes a variable for each factor, and then a factor beta for each factor, representing the security’s sensitivity to movements in that factor. A two-factor version of the arbitrage pricing theory would look like as:
r = E(r) + B1F1 + B2F2 + e
r = return on the security
E(r) = expected return on the security
F1 = the first factor
B1 = the security’s sensitivity to movements in the first factor
F2 = the second factor
B2 = the security’s sensitivity to movements in the second factor
e = the idiosyncratic component of the security’s return
As the formula shows, the expected return on the asset/stock is a form of liner regression taking into consideration many factors that can affect the price of the asset and the degree to which it can affect it i.e. the asset’s sensitivity to those factors.
If one is able to identify a single factor which singly affects the price, the CAPM model shall be sufficient. If there are more than one factor affecting the price of the asset/stock, one will have to work with a two factor model or a multi factor model depending on the number of factors that affect the stock price movement for the company.
Basic assumptions of Arbitrage Pricing Theory
a)      It is based on the principle of capital market efficiency and hence assumes all market participants trade with the intention of profit maximisation.
b)      The Investors have homogenous beliefs/expectations.
c)       Risk-return analysis is not the basis.
d)      It assumes no arbitrage exists and if it occurs participants will engage to benefit out of it and bring back the market to equilibrium levels.
e)      Capital markets are perfectly competitive.
f)       The security returns are generated according to a factor model. Several factors affect the return on a security.
g)      There are no transaction costs, no taxes, short selling is possible and infinite number of securities is available.
h)      The relationship between security returns and factors is linear.
Advantages of Arbitrage Pricing Theory
a)      APT model is a multi-factor model. So, the expected return is calculated taking into account various factors and their sensitivities that might affect the stock price movement. Thus it allows selection of factors that affect the stock price largely and specifically.
b)      APT model is based on arbitrage free pricing or market equilibrium assumptions which to a certain extent result in fair expectation of the rate of return on the risky asset.
c)       APT based multi factor model places emphasis on covariance between asset returns and exogenous factors unlike CAPM. CAPM places emphasis on covariance between asset returns and endogenous factors.
d)      APT model works better in multi period cases as against CAPM which is suitable for single period cases only.
e)      APT can be applied to cost of capital and capital budgeting decisions.
f)       The APT model does not require any assumption about the empirical distribution of the asset returns unlike CAPM which assumes that stock returns follow a normal distribution and thus APT a less restrictive model.
Limitations of Arbitrage Pricing Theory:
a)      Problems in listing of various factors: The model requires listing of factors that have impact on the stock under consideration. Finding and listing all factors can be a difficult task and there is a risk that some or the other factor being ignored. Also risk of accidental correlations may exist which may cause a factor to become substantial impact provider or vice versa.
b)      Expected return of various factors: The expected returns for each of these factors will have to be arrived at, which depending on the nature of the factor, may or may not be easily available always.
c)       Difficult to measure Sensitivities of factors: The model requires calculating sensitivities of each factor which again can be a tedious task and may not be practically possible.
d)      Change in factors from time to time: The factors that affect the stock price for a particular stock may change over a period of time. Moreover, the sensitivities associated may also undergo shifts which need to be continuously monitored making it very difficult to calculate and maintain.
e)      Existence of arbitrage is essential: The APT model will prevail only if there is a opportunity of arbitrage. If arbitrage opportunity is not available, then this model does not prevail.
f)       Uncertain size or sign of factors: APT makes no statement about the size of sign of the factors.
g)      Unrealistic assumption: It is based on some assumptions which are not practical.

Difference between APT and CAPM
a)      APT computes the expected return on a security based on the security’s sensitivity to movements in macroeconomic factors. Whereas, CAPM is a tool used by investors to determine the risk associated with a potential investment and also gives an idea as to what can be the expected return on the investment.
b)      The APT can be set up to consider several risk factors, such as the business cycle, interest rates, inflation rates, and energy prices. The model distinguishes between systematic risk and firm-specific risk and incorporates both types of risk into the model for each given factor. Where as CAPM considers only systematic risk, reflecting a reality in which most investors have diversified portfolios from which unsystematic risk has been essentially eliminated.
c)       APT is based on factor model of return and arbitrage Whereas CAPM is based on investors’ portfolio demand and equilibrium.
d)      APT is a multifactor model where as CAPM is a single factor model.
e)      APT places emphasis on covariance between asset returns and exogenous factors whereas CAPM places emphasis on covariance between asset returns and endogenous factors.

f)       APT model works better in multi period cases as against CAPM which is suitable for single period cases only.