Capital asset pricing model

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An estimation of the CAPM and the security market line (purple) for the Dow Jones Industrial Average over 3 years for monthly data.

In finance, the capital asset pricing model (CAPM) is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. CAPM “suggests that an investor’s cost of equity capital is determined by beta.”[1]:2

The CAPM was introduced by Jack Treynor (1961, 1962),[2] William Sharpe (1964), John Lintner (1965a,b) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. Sharpe, Markowitz and Merton Miller jointly received the 1990 Nobel Memorial Prize in Economics for this contribution to the field of financial economics. Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version was more robust against empirical testing and was influential in the widespread adoption of the CAPM.

Despite its empirical flaws[3] and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton's portfolio problem), the CAPM still remains popular due to its simplicity and utility in a variety of situations.

Formula[edit]

The security market line, seen here in a graph, describes a relation between the beta and the asset's expected rate of return.

The CAPM is a model for pricing an individual security or portfolio. For individual securities, we make use of the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class. The SML enables us to calculate the reward-to-risk ratio for any security in relation to that of the overall market. Therefore, when the expected rate of return for any security is deflated by its beta coefficient, the reward-to-risk ratio for any individual security in the market is equal to the market reward-to-risk ratio, thus:

\frac {E(R_i)- R_f}{\beta_{i}}  = E(R_m) - R_f

The market reward-to-risk ratio is effectively the market risk premium and by rearranging the above equation and solving for E(Ri), we obtain the capital asset pricing model (CAPM).

E(R_i) = R_f + \beta_{i}(E(R_m) - R_f)\,

where:

  • E(R_i)~~ is the expected return on the capital asset
  • R_f~ is the risk-free rate of interest such as interest arising from government bonds
  • \beta_{i}~~ (the beta) is the sensitivity of the expected excess asset returns to the expected excess market returns, or also \beta_{i} = \frac {\mathrm{Cov}(R_i,R_m)}{\mathrm{Var}(R_m)},
  • E(R_m)~ is the expected return of the market
  • E(R_m)-R_f~ is sometimes known as the market premium (the difference between the expected market rate of return and the risk-free rate of return).
  • E(R_i)-R_f~ is also known as the risk premium

Restated, in terms of risk premium, we find that:

E(R_i) - R_f = \beta_{i}(E(R_m) - R_f)\,

which states that the individual risk premium equals the market premium times β.

Note 1: the expected market rate of return is usually estimated by measuring the Geometric Average of the historical returns on a market portfolio (e.g. S&P 500).

Note 2: the risk free rate of return used for determining the risk premium is usually the arithmetic average of historical risk free rates of return and not the current risk free rate of return.

For the full derivation see Modern portfolio theory.

Modified formula[edit]

CAPM can be modified to include size premium and specific risk. This is important for investors in privately held companies who often do not hold a well-diversified portfolio. The equation is similar to the traditional CAPM equation “with the market risk premium replaced by the product of beta times the market risk premium:”[1]:5

E( R_i ) = R_f + \beta( RP_m ) + RP_s + RP_u

"where:

E(R_i) is required return on security i
R_f is risk-free rate
RP_m is general market risk premium
RP_s is risk premium for small size
RP_u is risk premium due to company-specific risk factor"[1]:4

Security market line[edit]

The SML essentially graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML.

The relationship between β and required return is plotted on the securities market line (SML), which shows expected return as a function of β. The intercept is the nominal risk-free rate available for the market, while the slope is the market premium, E(Rm)− Rf. The securities market line can be regarded as representing a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The equation of the SML is thus:

 \mathrm{SML}: E(R_i)= R_f+\beta_i (E(R_M) - R_f).~

It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for risk. Individual securities are plotted on the SML graph. If the security's expected return versus risk is plotted above the SML, it is undervalued since the investor can expect a greater return for the inherent risk. And a security plotted below the SML is overvalued since the investor would be accepting less return for the amount of risk assumed.

Asset pricing[edit]

Once the expected/required rate of return E(R_i) is calculated using CAPM, we can compare this required rate of return to the asset's estimated rate of return over a specific investment horizon to determine whether it would be an appropriate investment. To make this comparison, you need an independent estimate of the return outlook for the security based on either fundamental or technical analysis techniques, including P/E, M/B etc.

Assuming that the CAPM is correct, an asset is correctly priced when its estimated price is the same as the present value of future cash flows of the asset, discounted at the rate suggested by CAPM. If the estimated price is higher than the CAPM valuation, then the asset is undervalued (and overvalued when the estimated price is below the CAPM valuation).[4] When the asset does not lie on the SML, this could also suggest mis-pricing. Since the expected return of the asset at time t is E(R_t)=\frac{E(P_{t+1})-P_t}{P_t}, a higher expected return than what CAPM suggests indicates that P_t is too low (the asset is currently undervalued), assuming that at time t+1 the asset returns to the CAPM suggested price.[5]

The asset price P_0 using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship given by

P_0 = \frac{1}{1 + R_f} \left[E(P_T) - \frac{\mathrm{Cov}(P_T,R_M)(E(R_M) - R_f)}{\mathrm{Var}(R_M)}\right]

where P_T is the payoff of the asset or portfolio.[4]

Asset-specific required return[edit]

The CAPM returns the asset-appropriate required return or discount rate—i.e. the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. Betas exceeding one signify more than average "riskiness"; betas below one indicate lower than average. Thus, a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. Given the accepted concave utility function, the CAPM is consistent with intuition—investors (should) require a higher return for holding a more risky asset.

Since beta reflects asset-specific sensitivity to non-diversifiable, i.e. market risk, the market as a whole, by definition, has a beta of one. Stock market indices are frequently used as local proxies for the market—and in that case (by definition) have a beta of one. An investor in a large, diversified portfolio (such as a mutual fund), therefore, expects performance in line with the market.

Risk and diversification[edit]

The risk of a portfolio comprises systematic risk, also known as undiversifiable risk, and unsystematic risk which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all securities—i.e. market risk. Unsystematic risk is the risk associated with individual assets. Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio (specific risks "average out"). The same is not possible for systematic risk within one market. Depending on the market, a portfolio of approximately 30–40 securities in developed markets such as the UK or US will render the portfolio sufficiently diversified such that risk exposure is limited to systematic risk only. In developing markets a larger number is required, due to the higher asset volatilities.

A rational investor should not take on any diversifiable risk, as only non-diversifiable risks are rewarded within the scope of this model. Therefore, the required return on an asset, that is, the return that compensates for risk taken, must be linked to its riskiness in a portfolio context—i.e. its contribution to overall portfolio riskiness—as opposed to its "stand alone risk." In the CAPM context, portfolio risk is represented by higher variance i.e. less predictability. In other words the beta of the portfolio is the defining factor in rewarding the systematic exposure taken by an investor.

Efficient frontier[edit]

Main article: Efficient frontier
The (Markowitz) efficient frontier. CAL stands for the capital allocation line.

The CAPM assumes that the risk-return profile of a portfolio can be optimized—an optimal portfolio displays the lowest possible level of risk for its level of return. Additionally, since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset, (assuming no trading costs) with each asset value-weighted to achieve the above (assuming that any asset is infinitely divisible). All such optimal portfolios, i.e., one for each level of return, comprise the efficient frontier.

Because the unsystematic risk is diversifiable, the total risk of a portfolio can be viewed as beta.

Market portfolio[edit]

An investor might choose to invest a proportion of his or her wealth in a portfolio of risky assets with the remainder in cash—earning interest at the risk-free rate (or indeed may borrow money to fund his or her purchase of risky assets in which case there is a negative cash weighting). Here, the ratio of risky assets to risk-free asset does not determine overall return—this relationship is clearly linear. It is thus possible to achieve a particular return in one of two ways:

  1. By investing all of one's wealth in a risky portfolio,
  2. or by investing a proportion in a risky portfolio and the remainder in cash (either borrowed or invested).

For a given level of return, however, only one of these portfolios will be optimal (in the sense of lowest risk). Since the risk-free asset is, by definition, uncorrelated with any other asset, option 2 will generally have the lower variance and hence be the more efficient of the two.

This relationship also holds for portfolios along the efficient frontier: a higher return portfolio plus cash is more efficient than a lower return portfolio alone for that lower level of return. For a given risk-free rate, there is only one optimal portfolio which can be combined with cash to achieve the lowest level of risk for any possible return. This is the market portfolio.

Assumptions of CAPM[edit]

All investors:[6]

  1. Aim to maximize economic utilities (Asset quantities are given and fixed).
  2. Are rational and risk-averse.
  3. Are broadly diversified across a range of investments.
  4. Are price takers, i.e., they cannot influence prices.
  5. Can lend and borrow unlimited amounts under the risk free rate of interest.
  6. Trade without transaction or taxation costs.
  7. Deal with securities that are all highly divisible into small parcels (All assets are perfectly divisible and liquid).
  8. Have homogeneous expectations.
  9. Assume all information is available at the same time to all investors.

Problems of CAPM[edit]

A 2014 paper by Harris,[7] argues that there is a math error in the derivation of the CAPM, Arbitrage Pricing Theory and the Black-Scholes Option Pricing Model. It is argued that the error, though subtle, is catastrophic to the models and is the source of most of the observed empirical contradictions in the three models.

Harris’s paper follows from papers by Gurland,[8] Rubin,[9] White[10] and Rao[11] that were overlooked by the economics profession. The argument by Harris reflects the empirical observations by Mandelbrot in his paper The Variation of Certain Speculative Prices.[12]

The paper approaches the problem from both the Bayesian and Frequentist perspectives and follows theorems primarily from White and Rubin. Rubin notes that the maximum likelihood estimator for R for the difference equation

w_{t+1}=Rw_t+\epsilon_{t+1} is the ordinary least squares estimator, regardless of the value of R.

The slope coefficient in the least squares estimator can be thought of as the average slope, a form of the mean. White derives the test statistic using a Frequentist method that also has a Bayesian interpretation.

Because individuals wish to profit in the future, it follows that R>1. In the Frequentist interpretation, the test statistic is the Cauchy distribution when R>1.

As the Cauchy distribution has no defined mean, the test statistic has a precision of zero. Therefore, assuming the CAPM is true, then it follows that it is impossible to measure it. Harris argues therefore it is not a scientific model.

In the Bayesian framework, R is itself a random variable. This view is compatible with White’s proof, however, as is well known, the expectation of the Cauchy distribution does not exist. Therefore in a Bayesian framework E(R) does not exist. Since the expectation is assumed to exist, this gives rise to logical contradiction. Since the expectation does not exist then it follows that \hat{\beta} does not measure systematic risk. As \hat{\beta} is a mean calculated on data drawn from a Cauchy distribution, it should be uncorrelated with the real risk. A note by the National Institute of Standards and Technology on the Cauchy distribution is instructive here.

NIST notes that:

"The mean and standard deviation of the Cauchy distribution are undefined. The practical meaning of this is that collecting 1,000 data points gives no more accurate an estimate of the mean and standard deviation than does a single point."[13]

Harris further argues that the true distribution of returns is not the Cauchy distribution due to the existence of the budget constraint and bankruptcy, but in the simplified world of mean-variance finance, the distribution of returns must be the Cauchy distribution.

In their 2004 review, Fama and French argue that "the failure of the CAPM in empirical tests implies that most applications of the model are invalid".[3]

  • The model assumes that the variance of returns is an adequate measurement of risk. This would be implied by the assumption that returns are normally distributed, or indeed are distributed in any two-parameter way, but for general return distributions other risk measures (like coherent risk measures) will reflect the active and potential shareholders' preferences more adequately. Indeed risk in financial investments is not variance in itself, rather it is the probability of losing: it is asymmetric in nature. Barclays Wealth have published some research on asset allocation with non-normal returns which shows that investors with very low risk tolerances should hold more cash than CAPM suggests.[14]
  • The model assumes that all active and potential shareholders have access to the same information and agree about the risk and expected return of all assets (homogeneous expectations assumption).[citation needed]
  • The model assumes that the probability beliefs of active and potential shareholders match the true distribution of returns. A different possibility is that active and potential shareholders' expectations are biased, causing market prices to be informationally inefficient. This possibility is studied in the field of behavioral finance, which uses psychological assumptions to provide alternatives to the CAPM such as the overconfidence-based asset pricing model of Kent Daniel, David Hirshleifer, and Avanidhar Subrahmanyam (2001).[15]
  • The model does not appear to adequately explain the variation in stock returns. Empirical studies show that low beta stocks may offer higher returns than the model would predict. Some data to this effect was presented as early as a 1969 conference in Buffalo, New York in a paper by Fischer Black, Michael Jensen, and Myron Scholes. Either that fact is itself rational (which saves the efficient-market hypothesis but makes CAPM wrong), or it is irrational (which saves CAPM, but makes the EMH wrong – indeed, this possibility makes volatility arbitrage a strategy for reliably beating the market).[citation needed]
  • The model assumes that given a certain expected return, active and potential shareholders will prefer lower risk (lower variance) to higher risk and conversely given a certain level of risk will prefer higher returns to lower ones. It does not allow for active and potential shareholders who will accept lower returns for higher risk. Casino gamblers pay to take on more risk, and it is possible that some stock traders will pay for risk as well.[citation needed]
  • The model assumes that there are no taxes or transaction costs, although this assumption may be relaxed with more complicated versions of the model.[citation needed]
  • The market portfolio consists of all assets in all markets, where each asset is weighted by its market capitalization. This assumes no preference between markets and assets for individual active and potential shareholders, and that active and potential shareholders choose assets solely as a function of their risk-return profile. It also assumes that all assets are infinitely divisible as to the amount which may be held or transacted.[citation needed]
  • The market portfolio should in theory include all types of assets that are held by anyone as an investment (including works of art, real estate, human capital...) In practice, such a market portfolio is unobservable and people usually substitute a stock index as a proxy for the true market portfolio. Unfortunately, it has been shown that this substitution is not innocuous and can lead to false inferences as to the validity of the CAPM, and it has been said that due to the inobservability of the true market portfolio, the CAPM might not be empirically testable. This was presented in greater depth in a paper by Richard Roll in 1977, and is generally referred to as Roll's critique.[16]
  • The model assumes economic agents optimise over a short-term horizon, and in fact investors with longer-term outlooks would optimally choose long-term inflation-linked bonds instead of short-term rates as this would be more risk-free asset to such an agent.[17][18]
  • The model assumes just two dates, so that there is no opportunity to consume and rebalance portfolios repeatedly over time. The basic insights of the model are extended and generalized in the intertemporal CAPM (ICAPM) of Robert Merton,[19] and the consumption CAPM (CCAPM) of Douglas Breeden and Mark Rubinstein.[20]
  • CAPM assumes that all active and potential shareholders will consider all of their assets and optimize one portfolio. This is in sharp contradiction with portfolios that are held by individual shareholders: humans tend to have fragmented portfolios or, rather, multiple portfolios: for each goal one portfolio — see behavioral portfolio theory[21] and Maslowian portfolio theory.[22]
  • Empirical tests show market anomalies like the size and value effect that cannot be explained by the CAPM.[23] For details see the Fama–French three-factor model.[24]

See also[edit]

References[edit]

  1. ^ a b c James Chong; Yanbo Jin; Michael Phillips (April 29, 2013). "The Entrepreneur's Cost of Capital: Incorporating Downside Risk in the Buildup Method". Retrieved 25 June 2013. 
  2. ^ French, Craig W. (2003). "The Treynor Capital Asset Pricing Model". Journal of Investment Management 1 (2): 60–72. SSRN 447580. 
  3. ^ a b Fama, Eugene F; French, Kenneth R (Summer 2004). "The Capital Asset Pricing Model: Theory and Evidence". Journal of Economic Perspectives 18 (3): 25–46. doi:10.1257/0895330042162430. 
  4. ^ a b Luenberger, David (1997). Investment Science. Oxford University Press. ISBN 978-0-19-510809-5. 
  5. ^ Bodie, Z.; Kane, A.; Marcus, A. J. (2008). Investments (7th International ed.). Boston: McGraw-Hill. p. 303. ISBN 0-07-125916-3. 
  6. ^ Arnold, Glen (2005). Corporate financial management (3. ed. ed.). Harlow [u.a.]: Financial Times/Prentice Hall. p. 354. 
  7. ^ D. Harris, "Why Heavy Tails?," Southwestern Economic Review, Vol. 41 (2014), pp.127-151
  8. ^ J. Gurland, "Inversion formulae for the distribution of ratios," Ann. Math. Stat. Vol. 19 (1948), pp. 228-237
  9. ^ H. Rubin, "Consistency of maximum likelihood estimates in the explosive case," Statistical Inference in Dynamic Economic Models, T. Koopmans (ed.), John Wiley & Sons, 1950
  10. ^ John S. White, "The limiting distribution of the serial correlation coefficient in the explosive case," Ann. Math. Stat., Vol. 29, No. 4 (Dec. 1958), pp. 1188-1197
  11. ^ M.M. Rao, "Consistency and limit distributions of estimators of parameters in explosive stochastic difference equations," Ann Math Stat, Vol. 32 No. 1 (Mar. 1961), pp. 195-218
  12. ^ Benoit Mandelbrot, The variation of certain speculative prices, The Journal of Business, Vol. 36 No. 4 (1963), pp. 394-419
  13. ^ NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
  14. ^ http://www.barclayswealth.com/Images/asset-allocation-february-2013.pdf
  15. ^ Daniel, Kent D.; Hirshleifer, David; Subrahmanyam, Avanidhar (2001). "Overconfidence, Arbitrage, and Equilibrium Asset Pricing". Journal of Finance 56 (3): 921–965. doi:10.1111/0022-1082.00350. 
  16. ^ Roll, R. (1977). "A Critique of the Asset Pricing Theory’s Tests". Journal of Financial Economics 4: 129–176. doi:10.1016/0304-405X(77)90009-5. 
  17. ^ http://ciber.fuqua.duke.edu/~charvey/Teaching/BA453_2006/Campbell_Viceira.pdf
  18. ^ Campbell, J & Vicera, M "Strategic Asset Allocation: Portfolio Choice for Long Term Investors". Clarendon Lectures in Economics, 2002. ISBN 978-0-19-829694-2
  19. ^ Merton, R.C. (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica 41 (5): 867–887. doi:10.2307/1913811. 
  20. ^ Breeden, Douglas (September 1979). "An intertemporal asset pricing model with stochastic consumption and investment opportunities". Journal of Financial Economics 7 (3): 265–296. doi:10.1016/0304-405X(79)90016-3. 
  21. ^ Shefrin, H.; Statman, M. (2000). "Behavioral Portfolio Theory". Journal of Financial and Quantitative Analysis 35 (2): 127–151. doi:10.2307/2676187. 
  22. ^ De Brouwer, Ph. (2009). "Maslowian Portfolio Theory: An alternative formulation of the Behavioural Portfolio Theory". Journal of Asset Management 9 (6): 359–365. doi:10.1057/jam.2008.35. 
  23. ^ Fama, Eugene F.; French, Kenneth R. (1993). "Common Risk Factors in the Returns on Stocks and Bonds". Journal of Financial Economics 33 (1): 3–56. doi:10.1016/0304-405X(93)90023-5. 
  24. ^ Fama, Eugene F.; French, Kenneth R. (1992). "The Cross-Section of Expected Stock Returns". Journal of Finance 47 (2): 427–465. doi:10.2307/2329112. 

Bibliography[edit]

  • Black, Fischer., Michael C. Jensen, and Myron Scholes (1972). The Capital Asset Pricing Model: Some Empirical Tests, pp. 79–121 in M. Jensen ed., Studies in the Theory of Capital Markets. New York: Praeger Publishers.
  • Fama, Eugene F. (1968). Risk, Return and Equilibrium: Some Clarifying Comments. Journal of Finance Vol. 23, No. 1, pp. 29–40.
  • Fama, Eugene F. and Kenneth French (1992). The Cross-Section of Expected Stock Returns. Journal of Finance, June 1992, 427–466.
  • French, Craig W. (2003). The Treynor Capital Asset Pricing Model, Journal of Investment Management, Vol. 1, No. 2, pp. 60–72. Available at http://www.joim.com/
  • French, Craig W. (2002). Jack Treynor's 'Toward a Theory of Market Value of Risky Assets' (December). Available at http://ssrn.com/abstract=628187
  • Lintner, John (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1), 13–37.
  • Markowitz, Harry M. (1999). The early history of portfolio theory: 1600–1960, Financial Analysts Journal, Vol. 55, No. 4
  • Mehrling, Perry (2005). Fischer Black and the Revolutionary Idea of Finance. Hoboken: John Wiley & Sons, Inc.
  • Mossin, Jan. (1966). Equilibrium in a Capital Asset Market, Econometrica, Vol. 34, No. 4, pp. 768–783.
  • Ross, Stephen A. (1977). The Capital Asset Pricing Model (CAPM), Short-sale Restrictions and Related Issues, Journal of Finance, 32 (177)
  • Rubinstein, Mark (2006). A History of the Theory of Investments. Hoboken: John Wiley & Sons, Inc.
  • Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (3), 425–442
  • Stone, Bernell K. (1970) Risk, Return, and Equilibrium: A General Single-Period Theory of Asset Selection and Capital-Market Equilibrium. Cambridge: MIT Press.
  • Tobin, James (1958). Liquidity preference as behavior towards risk, The Review of Economic Studies, 25
  • Treynor, Jack L. (1961). Market Value, Time, and Risk. Unpublished manuscript.
  • Treynor, Jack L. (1962). Toward a Theory of Market Value of Risky Assets. Unpublished manuscript. A final version was published in 1999, in Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics. Robert A. Korajczyk (editor) London: Risk Books, pp. 15–22.
  • Mullins, David W. (1982). Does the capital asset pricing model work?, Harvard Business Review, January–February 1982, 105–113.

External links[edit]