W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425--442, September 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... 57 1 Introduction Financial theory has long recognized the interaction of risk and reward. The seminal work of Markowitz [Mar52] made explicit the trade o# of risk and reward in the context of a portfolio of financial assets. Others such as Sharpe [Sha64], Lintner [Lin65] and Ross [Ros76] have used equilibrium arguments to develop asset pricing models such as the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) relating the expected return of an asset to other risk factors. A common theme of these models is the ....

....a way consistent with Markowitz theory, then under additional assumptions, one will be able to learn something about the trade o# between risk and return in a market in equilibrium . This is what the CAPM does. The capital asset pricing model (CAPM) is an equilibrium pricing model (see Sharpe [Sha64] and Lintner [Lin65] which relates the expected return of an asset to the risk free return, to the market s expected return and to the covariance between the market and the asset. In addition to assuming that market participants use the mean variance framework, the model makes two additional ....

W.F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19:425--442, 1964.

....systematic risk. This intuition concerning naive diversification is based simply on portfolio size and correspondingly draws on a version of the classical law of large numbers. It is therefore important to note that it differs from that of the capital asset pricing model (CAPM) of Sharpe [40] and Lintner [28] where the distinction between non diversifiable and diversifiable risks is based on mean variance efficiency, and thereby on the efficient diversification of a portfolio. However, despite several attempts, 2 the intuitive notion of a well diversified portfolio has resisted a ....

....1See [33, pp. 173 197] for a discussion of naive and efficient diversification. We note here that there is no uniform terminology in the literature. For example, the terms non diversifiable risk and diversifiable risk used here for the CAPM are also called systematic risk and unsystematic risk in [40]. On the other hand, systematic risk and unsystematic risk used here for the APT model are also referred to as non diversifiable and diversifiable risk in, for example, 38, pp. 116 120] and [37] 2See [27] we comment further on these attempts in the sequel. 3After presenting the intuitive ....

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W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, J. Finance 33 (1964), 885-901.

.... sstandardpracticeofthecomputationofunidirectionallyprojectedempirical betas, i.e. theirrelativereturnvolatilities,withrespecttoaparticularmarket index,ortheirrelativeattributionsoftheir systematic variation,whichwere proposedbythe1990(joint)NobelPrizewinnerWilliamSharpe[71]. Again,thisseeminglyinnocentpracticeisnotwithoutseriousconsequences. Thereiscurrentlyanalarming,andmisdirected,regulatoryinterestinasinglerisk measuretoclassifymutualfunds[32] Sharpe sbetahasbeenproposedbymany analystsassuchameasure.Thisocialinterestinasingleriskmeasureisjustas ....

....fundsbytheirfundsbytheirrisk returnpro. le.Theriskismeasuredbytherelative rateofreturnvolatility,i.e. asmeasuredrelativetothatofabenchmarkmarket index,andthereturnbysomeaveragereturnoveraappropriateperiod[15] This relativeriskmeasureiscalledSharpe s beta [70] [71] .Whenafund sbeta, isbelowunity,thefundiscategorizedas defensive, becausethevolatilityofits investmentreturnsislowerthanthatofthemarketasawhole.Withagreater thanunity,afundiscategorizedas aggressive. Finally,withaequaltounity, ....

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W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19:425 -- 442, 1964.

....based on integer programming, which is bettered by the implicit enumeration algorithm of Blog et al. 6] Cooper and Farhangian [11] develop a dynamic programming approach for an extension of this problem that incorporates fixed costs of transaction. Assuming that the capital asset pricing model [21, 24, 29] holds, Brennan [8] presents an algorithm for determining the optimal number of securities under fixed transaction costs. However, the validity of that assumption in the presence of fixed transaction costs has been questioned [25] Patel and Subrahmanyam [25] develop an efficient algorithm for ....

W.F. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance, 19 (1964), pp 425--442.

....of data quality is an emerging topic, where approaches from other research fields (as we did in this paper) are investigated [12] 6] to provide better information for data consumers. In the context of investments there are some other models that require fewer input parameters (e.g. Index Model [8]: n securities 3n 2 parameters) but they are not directly applicable to data warehouse environments. Furthermore they require additional information, which can be provided by finance markets but not by data warehouses (at this point in time, because of the lack of appropriate methods) ....

Sharpe, William F.: Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, Vol. 19(3), September 1964, 425-442

....to accept, the optimal portfolio can be obtained by solving a convex quadratic programming problem. This mean variance model has had a profound impact on the economic modeling of nancial markets and the pricing of assets the Capital Asset Pricing Model (CAPM) developed primarily by Sharpe [26], Lintner [17] and Mossin [23] was an immediate logical consequence of the Markowitz mean variance portfolio theory. In 1990, Sharpe and Markowitz shared the Nobel Memorial Prize in Economic Sciences for their work on portfolio allocation and asset pricing. In spite of the theoretical success of ....

....factor model and uncertainty sets for the mean return vector, the factor loading matrix, and the covariance 3 matrix of the residual return. We also formulate robust counterparts of the mean variance optimal portfolio selection problem [19] the maximum Sharpe ratio portfolio selection problem [26] and the value at risk (VaR) portfolio selection problem. The uncertainty sets introduced in this section are ellipsoidal (intervals in the one dimensional case) and may appear quite arbitrary. Before demonstrating that these sets are, indeed, natural, we rst establish in Section 3 that the ....

W. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425-442, 1964.

....it components. NPV (A B) NPV (A) NPV (B) 23 0.18 Risky cash ows returns Reading; C W 1 3,6 7 We haven t yet said where discount rates come from so this section tells us. It results from work on Portfolio Theory by Markowitz (1959) 31] and the Capital Asset Pricing Model (CAPM) of Sharpe [50], Lintner [28] 1963) et al. 0.19 Market returns risk (Log) Market returns are risky, normally distributed with mean variance statistics return r 1 = P 1 D 1 P 0 P 0 = P 1 D 1 P 0 1 = capital gain dividend yield log return lr 1 = log e P 1 D 1 P 0 = ln (1 r 1 ) ....

W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425-442, 1964.

....mentioned it in Assumption 4.2 but have placed no discussion on it so far. A related issue is found in the option pricing theory which allows the underlying stock to have jumps in its price process [16] 18] 39] 40] 11 . They resorted to the conventional CAPM (Capital Asset Pricing Model) [42] by assuming that jump processes describe nonsystematic or idiosyncratic risks, which implies that risks such as firms defaults have a too wide variety of backgrounds with no good reason to be pre distributed to appear in a global risk premium. This is an extreme assumption and there are a lot of ....

William F. Sharpe. "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk". The Journal of Finance, Vol. 19, No. 3, pp. 425--442, September 1964.

....a risk premium, on average, of 3.60 percent per year. Our results suggest that the momentum effect is related to systematic skewness. The low expected return momentum portfolios have higher skewness than high expected return portfolios. THE SINGLE FACTOR CAPITAL ASSET PRICING MODEL CAPM of Sharpe 1964 and Lintner 1965 has come under recent scrutiny. Tests indicate that the crossasset variation in expected returns cannot be explained by the market beta alone. For example, a growing number of studies show that fundamental variables such as size, book to market value, and price to earnings ....

Sharpe, William, 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425--442.

....fund s alpha. Henceforth we use returns to denote rates of return in excess of a riskless interest rate or payos on zero investment spread positions. The choice of benchmarks is often guided by a pricing model, as in Jensen s (1969) pioneering use of the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) to investigate mutual fund alphas relative to a single market index benchmark. Other studies, beginning with Lehmann and Modest (1987) examine fund alphas with respect to a set of multiple benchmarks viewed as the relevant factors for pricing in a multifactor model, such as ....

Sharpe, William F., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442.

....NSW, 2007, Australia 1 Introduction to Benchmark Pricing Various alternative methodologies for the modelling of asset prices and #nancial markets have been proposed in the literature. The Capital Asset Pricing Model #CAPM#, which is a mean variance one period equilibrium model of exchange, see Sharpe #1964#, Lintner #1965# and Mossin #1966#, has been designed to model asset price dynamics. This model has been crucial for the understanding of the relationship between mean and variance of returns in equilibrium. Merton #1973# developed an intertemporal CAPM from portfolio selection behaviour of ....

Sharpe, W. F. #1964#. Capital asset prices: A theory of market equilibrium under conditions of risk. J. Finance 19, 425#442.

.... all questions in an area even the pathbreaking contributions to capital structure theory made by Modigliani and Miller (1958, 1963, 1966) to portfolio theory made by Markowitz (1952, 1959) to the efficient market literature made by Harry Roberts (1959) and to asset pricing theory made by Sharpe (1964), Lintner (1965) and Mossin (1966) It is interesting that we spend so little time as scientists thinking about the research process as a whole, and this is reflected in the fact that we do not have very good models of the process. As a journal editor, I ve become more impressed over time with ....

Sharpe, William F. "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk", Journal of Finance, Vol. XIX (September, 1964), pp. 425-42.

....unconditional beliefs regarding assets and that they can change on the introduction of 1 Markowitz, H. Portfolio selection, Journal of Finance, Vol. 7(1) March 1952. 2 Tobin, J. Liquidity preference as behaviour towards risk, Review of Economic studies, No.61, February 1958, p.65 86. 3 Sharpe, W.E. Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, Vol. 19(3) September 1964. 4 Merton, R.C. An Intertemporal Capital Asset Pricing Equation, Econometrica, 1973, 41, 867 887. 5 Ross, S, The arbitrage theory of capital pricing, Journal of ....

Sharpe, W.E., Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, Vol. 19(3), September 1964, p.425 -- 442.

....premium is charged to satisfy risk management and return on risk capital requirements. The profit and loss distributions are priced based on a combination of Value at Risk and return on capital approach. Its existing counterpart is the equilibrium capital asset pricing model (CAPM) developed by Sharpe (1964), Lintner (1965) and Mossin (1966) In the derivatives markets, there are at least two stylized facts that puzzle the profession. One is that implied volatility is on average larger than realized volatility; the other is that implied volatility curves exhibit smile or smirk effects for ....

Sharpe, W., (1964), "Capital Asset Prices, A Theory of Market Equilibrium", Journal of Finance, September.

....true probabilities and termed risk neutral probability, with the property that prices of traded assets are expected discounted cash flows under this revised probability. An important special case of these theories is the equilibrium capital asset pricing model (CAPM) of Sharpe, Lintner, and Mossin [18], 13] 15] The CAPM broadly asserts that the risk neutral probability may be defined as a function of the return on the market portfolio. Much empirical research has been concerned with testing this theory and the Ross [17] extensions. The basic conclusion has been that although there may be ....

Sharpe, W.F. (1964), "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, 19, 425-442.

....have continued to add foreign stocks to their portfolios, but the share of foreign stocks has leveled off, reflecting gains in U.S. stock prices. I. Introduction The international version of the classical capital asset pricing model (ICAPM) based on traditional portfolio theory developed by Sharpe (1964) and Linter (1965) predicts that to maximize risk adjusted returns investors should hold the world market portfolio of risky assets, irrespective of their country of residence. In practice, however, the proportion of foreign assets in investors portfolios tends to be very small. In the case of ....

Sharpe, W., 1964. Capital asset prices: A theory of market equilibrium under the condition of risk. Journal of Finance, 19: 425-442.

....account the differences between observable and expected returns for risky assets and for the market portfolio of all traded assets, as well as and the effects of excluded variables. Using this approach, we provide 1. evidence that the CAPM is alive and well. The Capital Asset Pricing Model of Sharpe (1964), The views expressed are those the authors, not necessarily those of the Board of Governors or staff of the Federal Reserve System, the Comptroller of the Currency, or the Department of the Treasury. The authors thank James Barth, Peter von Thomas Larry Mote whose comments considerably improved ....

Sharpe, William F., 1964, Capital asset prices: a theory market equilibrium under conditions risk, Journal of Finance 19, 425- 442.

....however, in light of massive empirical evidence, this assumption has finally begun to lose credibility. For reasons as yet to be explored, markets are not always efficient nor are investors always rational. Such discoveries effectively cause such prominent and well respected theories as the CAPM (Sharpe, 1964; Lintner, 1965; Mossin, 1966) and Arbitrage Pricing Theory (APT) Ross, 1976) to be subjugated, rather unceremoniously, to special case status. They are appropriate models if investors behave rationally. The irrational behavior at the root of this controversy would seem to necessitate the ....

....lie on the efficient frontier is no longer respected as it once was. In fact, investors rarely behave as the CAPM demands. Empirical evidence has shown that each of the nine standard assumptions used to prove the CAPM (from riskless borrowing and investor homogeneity to risk aversion and perfect 4 Sharpe, W. F. Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. Journal of Finance 19 (1964) 425. Sharpe attributes ignoring quasirationality to the formidable computational problems. inherent in calculating and using semi variances] This may indicate that deviations from ....

Sharpe, W. Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. Journal of Finance 19 (1964): 425-442.

....we can not reject this model. Recent research has documented evidence of time variation in expected returns, return volatilities, and betas of nancial assets. Meanwhile, researchers have identied a number of anomalies against the unconditional version of the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) The evidence against this constant beta model is so forceful that some argue that the CAPM is dead (see Fama and French (1992, 1996b) However, as Dybvig and Ross (1985) and Hansen and Richard (1987) show theoretically, the conditional version of the CAPM can hold perfectly ....

Sharpe, William F., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442.

....in equilibrium. See also [10] The CAPM predicts that equilibrium prices will be such that the market portfolio (i.e. the aggregate supply of risky securities) is optimal for mean variance preferences. That is, the market portfolio generates maximum mean return for its volatility. See, e.g. [12, 11]. This prediction is independent of agents levels of risk aversion, which is significant, because these cannot readily be measured, and moreover, may change during the course of the experiment. Consequently, to determine whether experimental markets have equilibrated, we compare the rewardto ....

Sharpe, W. (1964): "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance 19, 425-442.

....the convergence process halts, statistical tests reject the CAPM. JEL Classification: G12, C92, D59. Keywords: Capital Asset Pricing Model (CAPM) Experimental Economics, Financial Markets, Equilibrium, Equilibration. 1 Introduction In many respects, the Capital Asset Pricing Model (CAPM) of Sharpe [1964] and Lintner [1965] is the cornerstone of modern finance. Its decision theoretic foundation, mean variance analysis, has become a major guidance to asset al..location. Its equilibrium restriction provides the most important risk correction in the evaluation of portfolio performance. It is widely ....

Sharpe, W. (1964): "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance 19, 425-442.

....hence, weak. So, the experimental setting needs to be specified further in order to generate a second, potentially cardinal, theoretical prediction. The second situation where (1) acquires more precise content is one where the predictions of the Capital Asset Pricing Model (CAPM, developed in Sharpe [1964], Lintner [1965] and Mossin [1966] can be invoked. This requires either normally distributed payo#s or quadratic utility. 3 Quadratic utility is unlikely to provide a general description of subjects global preferences. Still, quadratic utility may be a reasonable local approximation of ....

Sharpe, W. (1964): "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance 19, 425-442.

....Capital Asset Pricing Model (CAPM) Experimental Economics, Financial Markets, Equilibrium, Equilibration. 2 The CAPM In Thin Experimental Financial Markets Peter Bossaerts and Charles Plott 1 Introduction This paper studies the extent to which the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) explains pricing and trading in simple, thin experimental financial markets. The CAPM predicts that equilibrium prices will be set such that expected returns in excess of the riskfree rate will be proportional to the covariance with aggregate risk. Aggregate risk is measured by ....

Sharpe, W. (1964): "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance 19, 425-442.

....in three ways. First, we include the three additional explanatory variables in Carhart s (1997) four factor model of returns. These are the Fama and French (1993) size and book to market factors and Carhart s momentum factor. The additional factors have been shown to capture the major anomalies of Sharpe s (1964) single factor CAPM, and we include them to avoid rewarding managers for simply exploiting these anomalies. Second, we include lagged 6 values of the four factors to capture the effect of infrequent trading of individual stocks on mutual fund returns. Third, we use as the dependent variable the ....

....between the asset in question and the market. Harlow and Rao (1989) test the mean lower partial moment asset pricing model and fail to reject it empirically, whereas they are able to reject the CAPM. Bawa and Lindenberg point out that the graphical representation of their model is identical to Sharpe s (1964) model except that standard deviation is replaced by the square root of ( f r F 2 . This implies a portfolio performance metric that is analogous to the Sharpe ratio. The slope of the line connecting the riskless asset and a risky asset can be interpreted as a reward to risk ratio that ....

Sharpe, W., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442.

....In addition, the size and book to market factors of Fama and French (1993) and the labor income risk factor of Jagannathan and Wang (1996) do not explain the CAPM deviations. 2 1 Introduction A problem arises when testing the conditional version of the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) While the conditional CAPM imposes cross sectional restrictions on moments of asset returns, the model does not describe exactly how these moments vary over time. The conditional CAPM states that conditional expected excess returns on securities at any given point in time are ....

Sharpe, William F., 1964,\Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk", Journal of Finance 19, 425-442.

....It is therefore important to assess to what extent regression models for stock returns with persistent regressors, as have appeared in numerous recent studies, may be susceptible to spurious regression effects. 2 In asset pricing model applications, such as the Capital Asset Pricing Model [Sharpe (1964)] Intertemporal Asset Pricing model [Merton (1973) Breeden (1979) or Arbitrage Pricing Model [Ross (1976) expected stock returns are determined by the sum of one or more betas, each multiplied by a return index or factor. In some applications the betas are fixed parameters and the factor ....

Sharpe, William. F., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-42.

....a signi#cant impact on the pricing of credit risky instruments in #nancial markets. A parallel may be drawn with the relationship between equity returns and compensation for systematic risk, as established by the Modern Portfolio Theory of Markowitz #1952# and the Capital Asset Pricing Model of Sharpe #1964#. One can also envisage far reaching implications of this development for the capital adequacy regime to which banks are subjected. Since the introduction of the Basle Accord in 1988, see Basle Committee on Bank Supervision #1988#, capital charges are determined for individual assets. These ....

Sharpe, W. #1964#. Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 19, 429#442.

....is accomplished by comparing the return of the particular trading strategy with the riskless interest rate and either the market return or the expected riskadjusted return for the considered time interval. The expected riskadjusted return is calculated using the Capital Asset Pricing Model (CAPM) [15]. The CAPM postulates the following relation beween risk and return of a risky asset: E(r i ) r f (E(rM ) Gamma r f ) COV(i; M ) VAR(rM ) r f (E(rM ) Gamma r f ) fi i Hereby E(r i ) is the expected return of asset i, r f denotes the riskless interest rate, E(rM ) is the expected ....

W. F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk", Journal of Finance, pp. 425-442, Sep. 1964.

....is accomplished by comparing the return of the particular trading strategy with the riskless interest rate and either the market return or the expected riskadjusted return for the considered time interval. The expected riskadjusted return is calculated using the Capital Asset Pricing Model (CAPM) [Sha]. The CAPM postulates the following relation between risk and return of a risky asset: E(r i ) r f (E(r M ) Gamma r f ) COV(r i ; r M ) VAR(rM ) r f (E(r M ) Gamma r f ) fi i Hereby, E(r i ) is the expected return of asset i, r f denotes the riskless interest rate, E(rM ) is the ....

W. F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk", Journal of Finance, pp. 425-442, Sep. 1964.

....in two ways. First, we include the three additional explanatory variables in Carhart s (1997) four factor model of returns. These are the Fama and French (1993) size and book to market factors and Carhart s momentum factor. The additional factors have been shown to capture the major anomalies of Sharpe s (1964) single factor CAPM, and we include them to avoid rewarding managers for simply exploiting these anomalies. Second, we include lagged values of the four factors to capture the effect of infrequent trading of individual stocks on mutual fund returns. Neither TM nor HM detects more than a few ....

....between the asset in question and the market. Harlow and Rao (1989) test the mean lower partial moment asset pricing model and fail to reject it empirically, whereas they are able to reject the CAPM. Bawa and Lindenberg point out that the graphical representation of their model is identical to Sharpe s (1964) model except that standard deviation is replaced by the square root of ( f r F 2 . This implies a portfolio performance metric that is analogous to the Sharpe ratio. The slope of the line connecting the riskless asset and a risky asset can be interpreted as a reward to risk ratio that ....

Sharpe, W., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442.

....of risk, and a mean variance model for portfolio selection based on minimizing variance subject to a given level of mean return. The mean variance approach has been widely used in financial decision making; it has also been the basis of mean variance capital market equilibrium models (e.g. see Sharpe, 1964, 1970, 1991) But arguments have been made that mean variance models are appropriate only if the investor s utility function is quadratic or the joint distribution of returns is normal. However, these conditions are rarely satisfied in practice. In fact, even expected utility theory, used as the ....

....from the normative utility preference and make their decisions based on risk value tradeoffs, as in financial decision making. For instance, our risk value models provide extensions with a new theoretical foundation for the traditional mean variance analysis (e.g. Markowitz, 1959, 1987, 1991; Sharpe, 1964, 1970, 1991) so that some more appealing decision models can be developed for financial modeling, such as a mean semivariance model, a disappointment model, and a three moments model. We believe that the potential for contributions of our risk value models in finance is very exciting. And also ....

Sharpe, W.F. (1964), "Capital asset prices: A theory of market equilibrium under conditions of risk," Journal of Finance, 19, 425-442.

....variable resources and their variability in making resource decisions. Third we analyze a pure exchange system in which users exchange their resources in such a way that the cost of their allocations are unchanged. Our analysis borrows heavily ideas from securities models in the finance literature [26, 13, 19, 10] and some of our results are simple versions of the pricing theory there. After some preliminaries in x3, we will look at, in x4, how an equilibrium might be approached, and, in x5, some interesting properties of equilibrium allocations and prices. 3 Assumptions, notations and preliminaries We ....

....says that the return on the variable bandwidth equals the return on the fixed bandwidth plus a risk premium that is proportional to the expected excess return from variable bandwidth over fixed bandwidth. It is a simple version of the equilibrium pricing theory in the capital asset pricing model [26, 10]. Recall that according to Proposition 4 users hold variable bandwidth and variable buffers in multiples of (ae =p 1 ; fl =q 1 ) Define the random variable m as m = ae p 1 R 1 fl q 1 B 1 which costs ae fl = 1. Here m represents the return on the allocation ....

W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19:425--442, 1964.

....theorem to the situation where not both variable bandwidth and buffer are in excess. Indeed if c 0 then the qualification is not necessary (Proposition 6) The theorem has a similar flavor to the well known fact in the context of investment where it is optimal for every investor to diversify [24], 10] 17] 9] The security models there however have an important difference: investors are allowed to hold short positions, i.e. x n ; yn ) can be negative as well as positive. The case that allows negative allocations is treated in [12] The nonnegativity constraint in our model ....

W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19:425--442, 1964.

....the distribution of wealth for instance, are necessary for a characterization of equilibrium prices. This lack of parsimony stands in sharp contrast to a number of alternative valuation models that attain low dimensionality by assuming frictionless environments. Classic examples are the CAPM of Sharpe (1964) and Lintner (1965) and Ross s (1976) APT. More recent examples are factor based approaches such as (among many others) Bossaerts and Green (1989) Chen, Roll, and Ross (1986) Fama and French (1996) Ferson and Harvey (1991) Jagannathan and Wang (1996) and Lehmann and Modest (1988) One ....

Sharpe, W. F., (1964), Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-42.

....that investors evaluate securities. For example, there are no rewards or penalties per se associated with corporate diversification. Of course, diversification could affect value by affecting expected bankruptcy costs and thus net cash flows. 2. 3 Capital Asset Pricing Theory Treynor (1961) Sharpe (1964), and Lintner (1965) apply the normative analysis of Markowitz to create a positive theory of the determination of asset prices. Given investor demands for securities implied by the Markowitz mean variance portfolio selection model and assuming fixed supplies of assets, they solve for equilibrium ....

Sharpe, William (1964): "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance, vol. 19, pp. 425-442.

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W. F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425--442, September 1964.

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Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425--442.

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W. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3):425--442, 1964.

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W. Sharpe, "Capital asset prices: A theory of market equilibrium under conditions of risk," J. Finance, vol. 19, pp. 452--442, 1964.

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W. Sharpe. Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. Journal of Finance, 19(3):425--442, 1964.

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W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, J. Finance 19 (1964) 425--442.

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Sharpe, William F. (1964), `Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,' Journal of Finance Vol. 19, pp. 425-442.

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William Sharpe. Capital asset prices: A theory of market equilibrium. Journal of Finance, September 1964.

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Sharpe, W. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. J.

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Sharpe, William F., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-42.

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W. F. Sharpe, Capital assets prices: a theory of market equilibrium under conditions of risk, Journal of Finance (September 1964), 425-442.

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Sharpe, W. [1964], "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance 19, 425-442 20

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Sharpe, W. 1964. "Capital asset prices: A theory of market equilibrium under conditions of risk." Journal of Finance 19, pp. 425-442.

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Sharpe, William, 1964, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance 19, 425-442. 29

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Sharpe, W.F.(1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19. Pp 425-442.

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