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Before making comparison between the CAPM and APT, we should first see what they are about. The CAPM is a theory about the way how assets are priced in relation to their risk. The CAPM was brought about to answer the question which came from Markowitzs mean-variance portfolio model. The question was how to identify tangency portfolio. Since then, the CAPM has developed into much, much more. CAPM shows that equilibrium rates of return on all risky assets are a function of their covariance with market portfolio.
APT is another equilibrium pricing model. The return on any risky asset is seen to be a linear combination of various common factors that affect asset returns. These two models in fact are similar to each other in some way. CAPM Assumptions: Investors are risk averse individuals and they maximise their expected utility of their end of period wealth.
They have the same one period of time horizon. Investors are price takers (no single investor can affect the price of a stock) and have homogenous expectation about asset returns that have a joint normal distribution. Investors can borrow or lend money at the risk-free rate of return. The quantities of assets are fixed.
All assets are marketable and perfectly divisible. Asset markets are frictionless and information is costless and simultaneously available to all investors. There are no market imperfections such as taxes, no transaction costs or no restrictions on short selling. As we can see, many of these assumptions behind the CAPM are not realistic. Although these assumptions do not hold in the real world, they are used to make the model simpler for us to use for financial decision making. Most of these assumptions can be relaxed.
The CAPM requires that in equilibrium the market portfolio must be an efficient portfolio. As long as all assets are marketable, divisible and investors have homogenous expectations, all individuals will perceive the same efficient set and all assets will be hold in equilibrium. If every individual holds a percentage of their wealth in efficient portfolios, and all assets are held, then the market portfolio must be also efficient because the market is simply the sum of all individual holdings and all individual holdings are efficient. Without the efficiency of the market portfolio the capital asset pricing model is untestable. The efficiency of market portfolio and the CAPM are inseparable joint hypothesis. EI = EF + (EM EF) &# 946; i &# 946; i = Covim / Vm = &# 963; im / &# 963; 2 m &# 946; is quantity of risk; it is the covariance between returns on the risky asset, I, and the market portfolio, M, divided by the variance of the market portfolio.
If we show how to derive the CAPM equation in a simple way: M: Market portfolio, EF: Risk free rate, I: Risky asset The straight line connecting the risk-free asset and market portfolio is the capital market line. In equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights. The equilibrium proportion of each asset in the market portfolio must be; wi = Market value of individual asset / Market value of all assets A portfolio consisting of a % invested in risky asset I and (1 -a) invested in the market portfolio will have the following mean and standard deviation; EP = aEI + (1 -a) EM, SP = {a 2 VI+ (1 -a) 2 VM+ 2 a (1 -a) Cov IM} 1 / 2 VI: The variance of the risky asset I and Cov IM: The covariance between asset I and the market portfolio. The opportunity set provided by various combinations of the risky asset and the market portfolio is the line IMI in figure 1. To determine the equilibrium price for risk at point M in figure 1: Evaluating dEP/da at where a = 0 gives us = EI-EM dSP/da where a = 0 gives us (Cov IM-VM) /SM In equilibrium the market portfolio already has the value weight, wi percent, invested in the risky asset I.
The percentage a is the excess demand for an individual risky asset. In equilibrium excess demand for any asset must be zero. dEp / da at a = 0 is equal to (EI-EM) dSp / da (Cov IM-VM) /SM This is the slope of the efficiency frontier. At equilibrium the slope of the opportunity set at point M, is equal to capital market line; (EM-EF) /SM. At equilibrium (EM-EF) /SM = (EI-EM) (Cov IM-VM) /SM We solve this for EI EI = EF+ (EM-EF) Cov IM/VM or EI = EF+ (EM-EF) &# 946; (See Copeland/Weston and Elton/Gruber for detailed proof how to derive the CAPM). Equation above is known as the capital asset pricing model.
It is shown in the figure 2 where it is also called security market line. Security market line depicts the trade-off between risk and expected return for individual securities. The CAPM equation above describes the expected return for all assets and portfolios of assets in the economy. Em (market return) and Ef (return on restless asset) are not functions of the assets we examine. Thus, the relationship between the expected return on any two assets can be related simply to their difference in &# 946; . The higher &# 946; is for any security, the higher must be its equilibrium return.
Furthermore the relationship between &# 946; and expected return is linear. This equation tells us something important that &# 946; (systematic risk) is the only important element in determining expected returns and non-systematic risk plays no role. Thus, the CAPM verify what we learned from portfolio theory that an investor can diversify all the risk except the covariance of the risk with market portfolio. Consequently, the only risk which investors will pay a premium to avoid is covariance risk.
We made many assumptions for the CAPM. Are all these assumptions realistic? The CAPM may describe equilibrium returns on macro level, but from individual investors perspective, we all hold different portfolios. Therefore it can not be exactly true. Alternative versions of the CAPM have been derived to take into account these problems which violate its assumptions. Modifying some of its assumptions leaves the general model unchanged, whereas changing other assumptions leads to the appearance of the new terms in the equilibrium relationship or, in some cases, to the modification of old terms.
However we should be careful, when we change assumptions simultaneously, the departure from standard CAPM may be serious (see Elton/Gruber). There has been many empirical testing of the CAPM model (There are some problems inherent in the test of CAPM). To test the CAPM we must transform it from exact form to the export form that uses observed data. When CAPM is tested, it is generally written in this form: R&# 61449; I = &# 948; 0 + &# 948; 1 &# 946; I+&# 949; I RI = RF+ (RM-RF) &# 946; I+&# 949; I (RI-RF) = (RM-RF) &# 946; I+&# 949; I, &# 948; 1 = RM-RF R&# 61449; I = the excess return; (RI-RF) Conclusions from empirical tests; Estimated &# 948; 0 is not equal zero, estimated &# 948; 1
It is linear also in &# 946; . They also found out that factors other than &# 946; are successful in explaining that part of security returns not captured by &# 946; . They also found out that price / earning ratios, size of the firm, management of the firm and other factors have effect in explaining returns. These showed there are other factors other than &# 946; explaining returns. We should mention Rolls critique quickly; Roll pointed out that There is problem With testing efficient portfolio (Remember the Joint hypothesis).
Roll said that there is nothing unique about the market portfolio. You can choose any efficient portfolio or an index (if performance is measured relative to an index), then find the minimum variance portfolio that is uncorrelated with the selected efficient index. If index turns out to be ex-post efficient, then every asset will exactly fall on the security market line. There will be no abnormal returns. If there are systematic abnormal returns, it simply means that the index that has been chosen is not ex-post efficient.
Roll argues that tests performed with any portfolio other than the true market portfolio are not tests of the CAPM. They are simply tests of whether the portfolio chosen as a proxy for the market is efficient or not. Since over an interval of time efficient portfolios exist, a market proxy may be chosen that satisfies all the implications of the CAPM model, even when the market portfolio is inefficient. On the other hand, an inefficient portfolio may be chosen as proxy for the market and the CAPM rejected when the market itself is efficient.
What Roll says, that we do not know the true market portfolio. Most tests use some portfolio of common stocks as the market, but the true market contains all risky assets (marketable and non marketable, human capital, coins, buildings, land etc). APT offers a testable alternative to the CAPM. The CAPM predicts that security rates of return will be linearly related to a single common factor; the rate of return on the market portfolio. APT has similar assumptions as CAPM has, like perfectly competitive markets, frictionless capital markets, and assumption of homogenous expectations.
APT replaces Capm's assumption which is based on mean variance framework by assumption of the process generating security returns. Returns on any stock linearly related to asset of indices as shown below: Ri = &# 945; i+bi 1 I 1 +bi 2 I 2 ++bin Im+&# 949; i, &# 945; i = E (Ri) Im = the value of the index that affect the return to asset i; macro economic factors, size of firm, inflation, etc). bin = the sensitivity of the ith asset to the nth factor. &# 949; i = a random error term with mean equal to zero and variance equal to &# 963; 2 ei In APT, We assume that covariances that exist between security returns can be attributed to the fact that the securities respond to one degree or another pull of one or more factors. We assume that the relationship between the security returns and the factors in linear. According to APT, in equilibrium all portfolios that can be selected from among the set of assets under consideration and that satisfy the conditions of (a) using no wealth and (b) having no risk must earn no return on average. These portfolios are called arbitrage portfolios.
Lets see a simple proof; wi = change in wealth in invested in asset i, as a percentage of an individuals total wealth, &# 931; wi = 0 Rp = &# 931; wi Ri = &# 931; wi (&# 945; i+bi 1 I 1 +bi 2 I 2 ++bin Im+&# 949; i) To eliminate risk (diversi able and undiversiable) and get a restless arbitrage portfolio; (1) choose small wi; percentage changes in investment ratios (2) diversify in a large number of asset in portfolio. &# 931; wi bik = 0, k = 1, 2. n. wi must be small, i must be large. wi&# 8776; 1 /n (n chosen to be a large number), &# 931; wi bik = 0 (this eliminates all systematic risk). Consequently, the return on our arbitrage portfolio becomes a constant.
Correct choice of the weights has eliminated all uncertainty, so that RP is not a random variable. RP becomes; Rp = &# 931; wi &# 945; i If the individual arbitrageur is in equilibrium, then return on any arbitrage portfolio must be zero. Rp = &# 931; wi &# 945; i = 0 &# 931; wi = 0, &# 931; wi bik = 0, Rp = &# 931; wi &# 945; i = 0 = &# 931; wiE (Ri) These equations are statements in linear algebra. w e = 0, w b = 0, w &# 945; = 0, underlined w, b, &# 945; are vectors. e is the constant vector. &# 945; must be linear combination of e and b, &# 945; must be orthonogol to e and b as well. &# 945; or E (Ri) must be linear combination of the constant vector and the coefficient vector.
There must exist a set of k+ 1 coefficients, &# 955; 0, &# 955; 1, ... &# 955; k. &# 945; i = E (Ri) = &# 955; 0 +&# 955; 1 bi 1 ++ &# 955; kit, (remember bik are the factor loadings; sensitivities of the returns on the ith security to the kth factor). If there is a restless asset with a restless rates of returns; RF then, bik = 0 and RF = &# 955; 0 Rewrite the...
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