A Generalization of the MeanVariance Analysis


 Tobias Johns
 1 years ago
 Views:
Transcription
1 A Generalization of the MeanVariance Analysis Valeri Zakamouline and Steen Koekebakker This revision: May 30, 2008 Abstract In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above the reference point. suppose that the decision maker generally distorts the objective probabilities. We also First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This meanpartial moments utility generalizes not only the meanvariance utility of Tobin and Markowitz, but also the meansemivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present the solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure. This performance measure generalizes the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker s skewness preferences have firstorder impact on risk measurement even if the risk is small. Key words: meanvariance utility, quadratic utility, meansemivariance utility, risk aversion, loss aversion, risk measure, probability distortion, partial moments of distribution, risk premium, optimal capital allocation, portfolio performance evaluation, Sharpe ratio. JEL classification: D81, G11. Corresponding author, University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) , University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) , 1
2 1 Introduction Expected Utility Theory (EUT) of von Neumann and Morgenstern has long been the main workhorse of modern financial theory. A von Neumann and Morgenstern s utility function is defined over the decision maker s wealth. The properties of a von Neumann and Morgenstern s utility function have been studied in every detail. The concept of risk aversion was analyzed by Friedman and Savage (1948) and Markowitz (1952). They showed that the realistic assumption of diminishing marginal utility of wealth explains why people are risk averse. Measurement of risk aversion was developed by Pratt (1964) and Arrow (1971). These authors analyzed the risk premium for small risks and introduced a measure which is widely known now as the ArrowPratt measure of risk aversion. The celebrated modern portfolio theory of Markowitz and the use of the meanvariance utility function can be justified by approximating a von Neumann and Morgenstern s utility function by a function of mean and variance, see, for example, Samuelson (1970), Tsiang (1972), and Levy and Markowitz (1979). In this sense, the use of the Sharpe ratio (see Sharpe (1966)) as a measure of performance evaluation of risky portfolios is also well justified. However, not very long ago after EUT was formulated by von Neumann and Morgenstern, questions were raised about its value as a descriptive model of choice under uncertainty. Influential experimental studies have shown the inability of EUT to explain many observed phenomena and reinforced the need to rethink much of the theory. An enormous amount of theoretical effort has been devoted towards developing alternatives to EUT. Many of the alternative models of choice under uncertainty can predict correctly individual choices, even in the cases in which EUT is violated, by assuming that the decision maker s utility has a reference point and/or the decision maker distorts the objective probabilities. Most prominent examples of these types of models are (for a brief description, see the online supplement): the utility function of Markowitz (1952); the meanlower partial moment model of Fishburn (1977) and Bawa (1978); Prospect Theory/Cumulative Prospect Theory (PT/CPT) of Kahneman and Tversky (1979) and Tversky and Kahneman (1992); Disappointment Theory (DT) of Bell (1985) and Loomes and Sugden (1986); Anticipated Utility Theory/Rank Dependent Expected Utility (AUT/RDEU) of Quiggin (1982); etc. Even though the models where the utility function has a reference point and/or the objective probabilities are distorted have been 2
3 known for quite a while, very little is known about general implications of these models for decision making except for the prescription that the decision maker should maximize expected utility. The goal of this paper is to provide some general insights into this broad class of models of choice under uncertainty. In this paper we consider a decision maker with a generalized behavioral utility function that has a reference point. We assume that the decision maker regards the outcomes below the reference points as losses, while the outcomes above the reference point as gains. We suppose that the behavioral utility generally has a kink at the reference point and different functions below and above the reference point. We require only that the behavioral utility function is continuous and increasing in wealth and has at least the first and the second onesided derivatives at the reference point. Moreover, we also suppose that the decision maker generally distorts the objective probability distribution. Note that a utility function in EUT is a particular case of our generalized behavioral utility function, so that our generalized framework encompasses EUT as well. The first contribution of this paper is to provide an approximation analysis of the expected generalized behavioral utility function. Our analysis shows that the expected generalized behavioral utility function can be approximated by a function of mean and partial moments of distribution. The meanpartial moments utility generalizes not only the meanvariance utility of Tobin and Markowitz, but also the meansemivariance utility mentioned by Markowitz (1959) and discussed further by many others. The meanpartial moments utility appears to have reasonable computational possibilities (in, for example, the optimal portfolio choice problem) as well as a great degree of flexibility in modelling different risk preferences of a decision maker. The second contribution of this paper is, in the spirit of Pratt (1964) and Arrow (1971), to derive an expression for a risk premium when risk is small. We show that the meanpartial moments utility allows for a much richer and detailed characterization of a risk premium. It is well known that for a decision maker with the meanvariance utility the only source of risk is variance. Moreover, the risk attitude of this decision maker is completely described by a single measure widely known as the ArrowPratt measure of risk aversion (which is, in fact, the aversion to uncertainty 1 ). In contrast, our analysis shows that a decision maker 1 Here by uncertainty we mean actually a deviation measure of uncertainty which assesses the dispersion of 3
4 with a generalized behavioral utility distinguishes between three sources of risk: expected loss, uncertainty in losses, and uncertainty in gains. Consequently, a decision maker in our framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Loss aversion leads to different weights of losses and gains in the expression for the risk premium. The third contribution of this paper is to generalize the famous Arrow s solution (see Arrow (1971), page 102) to the optimal capital allocation problem of an investor. In this setting the investor s objective is to allocate optimally the wealth between a riskfree and a risky asset. It is widely known that a meanvariance utility maximizer will always want to allocate some wealth to the risky asset if the risk premium is nonzero, no matter how small it might be. By contrast, we show that an investor with the generalized behavioral utility function will want to allocate some wealth to the risky asset only when the perceived risk premium is sufficiently high 2 (how high depends on the level of loss aversion and the degree of probability distortion). Otherwise, if the perceived risk premium is small, the investor avoids the risky asset and invests all wealth in the riskfree asset. This result may help explain why many investors do not invest in equities 3. This result may also help explain the equity premium puzzle discovered by Mehra and Prescott (1985). One possible explanation is presented by Bernartzi and Thaler (1995) who showed that loss aversion can explain why stock returns are too high relative to bond returns. Another explanation suggested by our analysis is that some particular types of probability distortion (for example, pessimistic distortion) may cause the observed equity premium puzzle. Our fourth contribution is to derive an expression for the portfolio performance measure of an investor with the generalized behavioral utility. This measure generalizes the Sharpe ratio (see Sharpe (1966)) and the Sortino 4 ratio (see Sortino and Price (1994)). As compared to either the Sharpe or Sortino ratio where the investor s risk preferences seemingly disappear, the computation of the generalized performance measure usually requires the knowledge of distribution. 2 Previously, this was briefly mentioned by Segal and Spivak (1990). Using a simple oneperiod binomial model with no probability distortion, Gomes (2005) has also shown that loss aversion causes avoidance of risky asset when the risk premium is small. 3 See, for example, Agnew, Balduzzi, and Annika (2003) who report that about 48% of participants of retirement accounts do not invest in equities. This behavior clearly contradicts EUT. 4 The Sortino ratio is a rewardtosemivariability type ratio that has been used by many researchers. Some examples are Klemkosky (1973), Ang and Chua (1979), and Ziemba (2005). 4
5 the investor s risk preferences. This means that this performance measure is not unique for all investors, but rather an individual performance measure. The explanation for this is the fact that in our generalized framework an investor distinguishes between several sources of risk. Since each investor may exhibit different preferences to each source of risk, investors with different preferences might rank differently the same set of risky assets. The fifth contribution of this paper is to provide some new insights on risk measurement. In modern financial theory most often one uses variance as a risk measure. This is because in the EUT framework variance has the firstorder impact on risk measurement, at least when the risk is small. We demonstrate that in many alternative theories variance is generally not a proper risk measure even if the risk is small. In these theories it is the decision maker s skewness preferences that have firstorder impact on risk measurement. This is the consequence of the presence of loss aversion and the fact that the decision maker s degrees of risk aversions below and above the reference point might be substantially different. In the latter case if probability distributions are not symmetrical, then, depending on the signs and the values of skewness preferences, either the downside or the upside part of variance is a more proper risk measure than variance. The rest of the paper is organized as follows. In Section 2 we briefly review the justification of the meanvariance analysis and present the results we want to generalize. In Section 3 we present assumptions, definitions, and notation that we will use in our generalized framework. In Section 4 we perform the approximation analysis and generalize the meanvariance utility and the quadratic utility. In Section 5 we generalize the ArrowPratt risk premium. In Section 6 we analyze the optimal capital allocation problem. In Section 7 we derive the expression for a portfolio performance measure. In Section 8 we discuss briefly the impact of the decision maker s skewness preferences on risk measurement in our generalized framework. Section 9 concludes the paper. As a supporting document this paper has an extensive online supplement with lots of illustrations and additional information. 2 Expected Utility Theory and the MeanVariance Analysis In this section we present a brief justification of the meanvariance analysis as well as some of its most important results. Throughout the paper we consider a decision maker with (random) 5
6 wealth W. In this section we assume that the decision maker has a von NeumannMorgenstern utility function which is defined over wealth as a single function U(W ). The decision maker s objective is to maximize the expected utility of wealth, E[U(W )], where E[.] is the expectation operator. We suppose that the utility function U is increasing in wealth and is a differentiable function. Then, taking a Taylor series expansion around some (deterministic) W 0, the expected utility of the decision maker can be written as E[U(W )] = n=0 1 n! U (n) (W 0 )E[(W W 0 ) n ]. Our intension now is to keep the terms up to the second derivative of U and disregard 5 the terms with higher derivatives of U. This gives us E[U(W )] U(W 0 ) + U (1) (W 0 )E[(W W 0 )] U (2) (W 0 )E[(W W 0 ) 2 ]. Since a utility function is unique up to a positive linear transformation and U (1) > 0, then a convenient form of equivalent expected utility is [ ] U(W ) E[Ũ(W )] = E U(W0 ) U (1) = E[(W W 0 )] 1 (W 0 ) 2 γe[(w W 0) 2 ], (1) where γ = U (2) (W 0 ) U (1) (W 0 ) (2) is the ArrowPratt measure of risk aversion. The equivalent expected utility (1) can be interpreted as the meanvariance utility since the term E[(W W 0 ) 2 ] is proportional to variance. Observe that we can arrive at the same expression for the expected utility as (1) if we assume that the utility function of the decision maker is quadratic U(W ) = (W W 0 ) 1 2 γ(w W 0) 2. (3) 5 This can be justified when at least one of the following conditions is satisfied: the investor utility function is quadratic (see Tobin (1969)); the probability distribution is normal (see Tobin (1969)); the probability distribution belongs to the family of compact or small risk distributions (see Samuelson (1970)); the aggregate risk is small compared with the wealth (see Tsiang (1972)). 6
7 Consider now a risk averse decision maker with deterministic wealth W 0. According to Arrow (1971) (page 90) a risk averter is defined as one who, starting from a position of certainty, is unwilling to take a bet which is actuarially fair. More formally, if x is a fair gamble such that E[x] = 0, then E[U(W 0 + x)] < U(W 0 ). (4) Beginning from the landmark papers of Arrow and Pratt, it became a standard in economics to characterize the risk aversion in terms of either a certainty equivalent, C(x), or a risk premium, π(x), which are defined by the following indifference condition E[U(W 0 + x)] = U(W 0 + C(x)) = U(W 0 + E[x] π(x)). (5) This says that the decision maker is indifferent between receiving x and receiving a nonrandom amount of C(x) = E[x] π(x). Observe that for a fair gamble C(x) = π(x). If the risk of a fair gamble is small, then the decision maker s preferences can be approximated by quadratic utility. The use of quadratic utility (3) in the indifference condition (5) for W = W 0 + x combined with the assumption of risk aversion (4) gives the following expression 6 for the risk premium and certainty equivalent π(x) = C(x) = 1 γvar[x]. (6) 2 This is the famous result of Pratt (1964). Now consider the optimal capital allocation problem of an investor with quadratic utility function. The investor wants to allocate the wealth between a riskfree and a risky asset. The return on the risky asset over a small time interval t is x = µ t + σ t ε, (7) where µ and σ are, respectively, the mean and volatility of the risky asset return per unit of time, and ε is some (normalized) stochastic variable such that E[ε] = 0 and Var[ε] = 1. The 6 We need also to disregard the term with C 2 (x). This is valid since we assume that the risk is small. 7
8 return on the riskfree asset over the same time interval equals r = ρ t, (8) where ρ is the riskfree interest rate per unit of time. We assume that the risky asset can be either bought or sold short without any limitations and the riskfree rate of return is the same for both borrowing and lending. We further assume that the investor s initial wealth is W I and he invests a in the risky asset and, consequently, W I a in the riskfree asset. Thus, the investor s wealth after t is W = a(x r) + W I (1 + r). (9) The investor s objective is to choose a to maximize the expected utility E[U (W )] = max E[U(W )]. (10) a Observe that there is some ambiguity in the choice of the level of wealth W 0 around which we perform the Taylor series expansion. A reasonable choice is W 0 = W I (1 + r). With this choice W 0 does not depend on a and the resulting risk measure (for example, a E[(x r) 2 ]) exhibits the homogeneity property 7 in a. If t is rather small, then the risk is small and the use of quadratic utility is well justified. Consequently, using quadratic utility (3) in the investor s objective function (10) we arrive to the following maximization problem E[U (W )] = max a ae[x r] 1 2 γa2 E[(x r) 2 ]. The firstorder condition of optimality of a gives a = 1 γ E[x r] E[(x r) 2 ], (11) which is the famous Arrow s solution (see Arrow (1971), page 102). Finally, using the expression 7 In the landmark paper of Artzner, Delbaen, Eber, and Heath (1999), the authors argue that a sensible risk measure should satisfy several properties. One of these properties is the homogeneity property. This says that if one invests the amount a in the risky asset, the measure that assesses the investment risk should be a homogeneous function in a. 8
9 for the optimal value of a, we obtain that the maximum expected utility of the investor is given by E[U (W )] = 1 E[x r] 2 2γ E[(x r) 2 ]. Note that for any investor the higher the value of utility irrespective of the value of γ. Thus, the value of E[x r] 2, the higher the maximum expected E[(x r) 2 ] E[x r] SR = E[(x r) 2 ], (12) which is nothing else than the absolute value of the Sharpe ratio 8 (see Sharpe (1966)), can be used as the ranking statistics in the performance measurement of risky assets. Note that the Sharpe ratio is usually presented based on the assumption that either E[x r] > 0 or short sales are restricted. Here we also allow for profitable short selling strategies as well. Therefore we compute the absolute value of the Sharpe ratio in order to properly measure the performance. 3 Assumptions, Definitions, and Notation The purpose of this paper is to generalize the meanvariance analysis and some of its important results presented in the preceding section. Before proceeding to the analysis, in this section we would like to present the assumptions, definitions, and notation. Assumption 1. We suppose that the decision maker s utility function is continuous and increasing in wealth. Assumption 2. We suppose that the decision maker s utility function generally has a kink at the reference point W 0 and U + (W ) if W W 0, U(W ) = U (W ) if W < W 0. This means that above the reference point the utility function is given by U +, whereas below 8 Observe that E[(x r) 2 ] = Var[x] + (E[x] r) 2 Var[x] for a small t. 9
10 the reference point the utility function is given by U. The continuity assumption gives U (W 0 ) = U + (W 0 ) = U(W 0 ). (13) Assumption 3. The decision maker regards the outcomes below the reference points as losses, while the outcomes above the reference point as gains. Consequently, we will refer to U as the utility function for losses, whereas to U + as the utility function for gains. Assumption 4. We suppose that the left and right derivatives of U at the reference point exist and finite. We denote the firstorder leftsided derivative U (1) (W 0) = lim h 0 U (W 0 ) U (W 0 + h). h Similarly, we denote the firstorder rightsided derivative 9 U (1) + (W 0) = lim h 0+ U + (W 0 + h) U + (W 0 ). h The higherorder onesided derivatives of U at the reference point are denoted in the similar manner by U (2) (W 0), U (2) + (W 0), etc. Assumption 5. We suppose that the decision maker generally distorts the objective probability distribution. More formally, suppose that the cumulative objective probability distribution of a gamble X is given by F X. Then the expected payoff of the gamble in the objective world is E[X] = xdf X (x). Observe that the integral above is a LebesgueStieltjes integral which is defined for both discrete and continuous distributions. We denote by Q X the cumulative distorted probability distribution function of X. Under the distortion of probabilities, the expected payoff of the gamble for the decision maker is given by E D [X] = xdq X (x). 9 To be more precise, we need to denote the left derivative as U (1) (1) (W0 ) and the right derivative as U + (W0+), but this would enlarge the notation. 10
11 Note that D does not denote some particular probability measure, but denotes some particular probability distortion rule. Since the probability distortion may depend either on the cumulative objective distribution function or on whether the outcomes of X are interpreted as losses or gains, we need to distinguish between the probability distortion of X and X. For this purpose we denote by Q X the cumulative distorted probability distribution function of the complementary 10 gamble X = X. Observe that generally E D [X] E[X], This says that the expected value of X under distortion of probabilities is generally different from the expected value of X under the objective probabilities. This holds true for all other moments of distribution of X. Moreover, E D [ X] E D [X]. This says that under the same rule of probability distortion the expected value of a complementary gamble X does not generally equal the expected value of a gamble X with the opposite sign. Some illustrations are provided in the online supplement. Assumption 6. If W = W 0 + X where X is some gamble, then we suppose that U(W 0 )dq X = U(W 0 ). (14) This is true when either the sum of all distorted probabilities is equal to 1 (that is, dq X = 1) or the utility function is zero at the reference point (that is, as in PT/CPT U(W 0 ) = 0). Definition 1 (The measure of loss aversion). The measure of loss aversion is given by λ = U (1) (W 0) U (1) + (W 0). (15) This measure of loss aversion was proposed by Bernartzi and Thaler (1995) and formalized by Köbberling and Wakker (2005). Observe that if the decision maker does not exhibit loss 10 By a complementary gamble X = X we mean a gamble which is obtained by changing the signs of the outcomes of gamble X. 11
12 aversion, then λ = 1. Loss aversion implies λ > 1. Conversely, loss seeking behavior implies λ < 1. Finally note that since the firstorder derivatives of U are positive (follows from Assumption 1), the value of λ is also positive, that is, λ > 0. Definition 2 (Two measures of risk aversion). Recall the measure of risk aversion (2) that was introduced by Arrow and Pratt. Since the utility function of the decision maker generally has different functions for losses and gains, we introduce: the measure of risk aversion in the domain of gains γ + = U (2) + (W 0) U (1) + (W 0), and the measure of risk aversion in the domain of losses γ = U (2) (W 0) U (1) (W 0). If, for example, γ + > 0, then the utility function for gains is concave which means that the decision maker is risk averse in the domain of gains. By contrast, if γ + < 0, then the utility function for gains is convex which means that the decision maker is risk seeking in the domain of gains. Finally, if γ + = 0, then the decision maker is risk neutral in the domain of gains. Definition 3 (Lower and Upper Partial Moments). If the decision maker s random wealth is W and the reference point is W 0, a lower partial moment of order n under the distortion of probability is given by W0 LP Mn D (W, W 0 ) = ( 1) n (w W 0 ) n dq W (w). The coefficient ( 1) n is chosen to bring our definition of a lower partial moment in correspondence with the definition of Fishburn (1977). An upper partial moment of order n under the distortion of probability is given by UP M D n (W, W 0 ) = W 0 (w W 0 ) n dq W (w). The lower and upper partial moments of order n computed in the objective world are denoted by LP M n (W, W 0 ) and UP M n (W, W 0 ) respectively. See the online supplement that illustrates the computation of partial moments with and without the probability distortion. 12
13 Definition 4 (Performance Measure). By a performance measure we mean a score (number/value) attached to each financial asset. A performance measure is related to the level of expected utility provided by the asset. That is, the higher the performance measure of an asset, the higher level of expected utility the asset provides. 4 MeanPartial Moments Utility and Generalized Quadratic Utility The purpose of this section is to generalize the meanvariance utility (1) and quadratic utility (3). We consider a decision maker with random wealth W and reference point W 0. The decision maker generally distorts the objective probability distribution of W such that the cumulative distorted probability distribution function is Q W. The decision maker s expected generalized behavioral utility is, therefore, given by E D [U(W )] = W0 U (w)dq W (w) + U + (w)dq W (w). W 0 We apply Taylor series expansions for U (w) and U + (w) around W 0 which yields ( W0 ) E D 1 [U(W )] = n! U (n) (W 0 )(w W 0 ) n dq W (w) + n=0 W 0 1 W0 = U(W 0 )dq W (w) + n! U (n) (W 0 ) (w W 0 ) n dq W (w) + = U(W 0 ) + n=1 n=1 1 n! U (n) (W 0 )( 1) n LP M D n (W, W 0 ) + n=1 ( n=0 ) 1 n! U (n) + (W 0 )(w W 0 ) n dq W (w) 1 n! U (n) + (W 0 ) (w W 0 ) n dq W (w) W 0 n=1 1 n! U (n) + (W 0 )UP M D n (W, W 0 ), (16) supposing that the Taylor series converge and the integrals exist (also recall Assumption 6). Theorem 1. If either utility functions U and U + are at most quadratic in wealth or the probability distribution of W belongs to the family of small risk distributions, then the equivalent expected utility can be written as the following meanpartial moments utility E D [Ũ(W )] = ED [(W W 0 )] (λ 1)LP M D 1 (W, W 0 ) 1 2 ( λγ LP M D 2 (W, W 0 ) + γ + UP M D 2 (W, W 0 ) ), (17) 13
14 which means that we can assume the following quadratic form for the utility function 11 U (W W 0 ) 1 2 U(W ) = γ +(W W 0 ) 2 if W W 0, λ ( (W W 0 ) 1 2 γ (W W 0 ) 2) if W < W 0. (18) Proof. If utility functions U and U + are at most quadratic, then U (n) (W 0) = 0 and U (n) + (W 0) = 0 for n 3. If the probability distribution belongs to the family of small risk distributions, then we can assume that all the terms in (16) with LP M D n (W, W 0 ) and UP M D n (W, W 0 ), n 3, are of smaller order than the second partial moments. To demonstrate this, suppose that W = W 0 + x and x = tε where ε is some random variable, and let t be sufficiently small. Hence, if we keep only the first and the second partial moments of distribution, the expected utility is E D [U(W )] = U(W 0 )+ 2 n=1 1 n! U (n) (W 0)( 1) n LP M D n (W, W 0 )+ 2 n=1 1 n! U (n) + (W 0)UP M D n (W, W 0 ). Since the utility function is unique up to a positive linear transformation and U (1) + (W 0) > 0, an equivalent expected utility can be given by [ ] E[Ũ(W )] = E U(W ) U(W 0 ) U (1) + (W 0) ( (2) U (W 0) U (1) (W 0) U (1) (W 0) U (1) = U (1) (W 0) U (1) + (W 0) LP M 1 D (W, W 0 ) + UP M1 D (W, W 0 ) (2) + (W ) 0) U (1) + (W 0) UP M 2 D (W, W 0 ). + (W 0) LP M D 2 (W, W 0 ) + U To arrive at (17) recall Definitions 1 and 2 and note that UP M D 1 (W, W 0) LP M D 1 (W, W 0) = E D [W W 0 ]. Corollary 2. In the EUT framework, the meanpartial moments utility (17) reduces to the meanvariance utility (1), and the generalized quadratic utility (18) reduces to the quadratic utility (3). Proof. For a von NeumannMorgenstern utility function the left and right derivatives of U at point W 0 are equal. Hence, for a von NeumannMorgenstern utility function λ = 1 and γ = γ + = γ. Finally note that LP M 2 (W, W 0 ) + UP M 2 (W, W 0 ) = E[(W W 0 ) 2 ]. 11 This utility function can be denoted as piecewisequadratic utility. 14
15 Corollary 3. If λ = 1, γ + = 0, and there is no probability distortion, then the meanpartial moments utility (17) reduces to the meansemivariance utility of Markowitz E[Ũ(W )] = E[(W W 0)] 1 2 γ LP M 2 (W, W 0 ). Remark 1. Observe that the meansemivariance utility is a particular case of the meanpartial moments utility when the decision maker exhibits no loss aversion, risk aversion in the domain of losses, and risk neutrality in the domain of gains. Note that for a decision maker with the meanvariance utility there is only one source of risk, namely, the variance which measures the total uncertainty (or dispersion). In contrast, a decision maker with the meanpartial moments utility generally distinguishes between three sources of risk: the lower partial moment of order 1 which is related to the expected loss, the lower partial moment of order 2 which is related to the uncertainty in losses, and the upper partial moment of order 2 which is related to the uncertainty in gains. Note also that in the meanpartial moments utility the measure of risk aversion in the domain of losses is scaled up by the measure of loss aversion. This allows us to say that a decision maker with loss aversion puts more weight on the uncertainty in losses than on the uncertainty in gains. Observe that the generalized quadratic utility function (18) is very flexible with regard to the possibility of modelling different preferences of a decision maker. In this function λ controls the loss aversion, γ + controls the concavity/convexity of utility for gains, whereas γ controls the concavity/convexity of utility for losses. Some possible shapes of this utility function are presented in the online supplement. 5 Generalization of the ArrowPratt Risk Premium In this section we consider a risk averse decision maker equipped with the generalized behavioral utility function and who generally distorts the objective probability distribution. Recall that we say that the decision maker is risk averse if a fair (in the objective world) gamble decreases the utility of the decision maker. More formally, for a fair gamble x such that E[x] = 0 E D [U(W 0 + x)] < U(W 0 ). 15
16 We want to derive the expressions for the certainty equivalent, C D (x), and the risk premium, π D (x), which are now defined by the following indifference condition E D [U(W 0 + x)] = U ( W 0 + C D (x) ) = U ( W 0 + E D [x] π D (x) ). (19) Theorem 4. If either utility functions U and U + are at most quadratic in wealth or the risk of a fair gamble is small, then for a risk averse decision maker the certainty equivalent and the risk premium of the fair gamble are given by C D (x) = 1 λ ( UP M1 D (x, 0) 1 ) ( 2 γ +UP M2 D (x, 0) LP M1 D (x, 0) + 1 ) 2 γ LP M2 D (x, 0), (20) π D (x) = λ 1 λ ( E D [x] + LP M D 1 (x, 0) ) ( γ+ λ UP M D 2 (x, 0) + γ LP M D 2 (x, 0)). (21) Proof. If either utility functions U and U + are at most quadratic in wealth or the risk of a fair gamble is small, then, according to Theorem 1, the decision maker s preferences can be represented by the generalized quadratic utility (18). Consequently, the expected utility of W = W 0 + x is given by E D [U(W 0 + x)] = UP M1 D (x, 0) 1 ( 2 γ +UP M2 D (x, 0) λ LP M1 D (x, 0) + 1 ) 2 γ LP M2 D (x, 0). For a risk averse decision maker the certainty equivalent of a fair (in the objective world) gamble is negative. Consequently, (22) U ( W 0 + C D (x) ) ( = λ C D (x) 1 2 γ ( C D (x) ) ) 2. (23) Since the risk is small, ( C D (x) ) 2 C D (x) which means that the term with ( C D (x) ) 2 can be disregarded. Thus, the use of (22) and (23) in the indifference condition (19) gives the expression for the certainty equivalent (20). To derive the expression for the risk premium, we use π D (x) = E D [x] C D (x). Corollary 5. In the EUT framework, the risk premium (21) reduces to the ArrowPratt risk premium (6). 16
17 Proof. For a von NeumannMorgenstern utility function λ = 1 and γ = γ + = γ. This gives π(x) = 1 2 (γlp M 2(x, 0) + γup M 2 (x, 0)) = 1 γvar[x]. (24) 2 Observe that since a decision maker with a von NeumannMorgenstern utility exhibits no loss aversion, both (infinitesimal) losses and gains are treated similarly (as it is clearly seen from equation (24)). Consequently, the ArrowPratt measure of risk aversion is really a measure of aversion to variance, or a measure of aversion to uncertainty (or dispersion) of the probability distribution. In other words, when risk is small, a decision maker with a von NeumannMorgenstern utility exhibits only aversion to uncertainty. In addition, the risk premium is fully characterized by a measure of uncertainty aversion and the variance, which is a measure of uncertainty. A utility function with loss aversion and different functions for losses and gains allows a much richer and detailed characterization of risk aversion. According to equation (20) a decision maker exhibits three different types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. The loss is measured by the expected loss, and the uncertainties in gains and losses are measured by the second upper partial moment of x and the second lower partial moment of x respectively. Observe that losses and gains have different weights in the computation of the risk premium. In particular, for a loss averse decision maker, losses are λ times more important than gains. Finally, a brief comparative static analysis of the expressions for the certainty equivalent and the risk premium is presented by means of the following corollaries. Corollary 6. The certainty equivalent decreases as the decision maker s loss aversion increases. Conversely, the risk premium increases as the decision maker s loss aversion increases. Proof. The firstorder derivatives of the certainty equivalent and risk premium with respect to λ C D (x) λ = πd (x) λ = 1 (UP λ 2 M1 D (x, 0) 1 ) 2 γ +UP M2 D (x, 0). Consider the sign of B = UP M D 1 (x, 0) 1 2 γ +UP M D 2 (x, 0). 17
18 Note that UP M D 1 (x, 0) > 0 and UP M D 2 (x, 0) > 0. Obviously, if γ + 0, then the sign of B is positive. However, even if γ + > 0 the sign of B is positive since in this case we need to impose the upper limit max(x) < 1 γ + outcomes of x. Therefore B = 1 γ + 0 to ensure that the utility function is increasing 12 for all (y 12 γ +y 2 ) dq x (y) > 0, since y 1 2 γ +y 2 > 0 for all y < 1 γ +. Finally we obtain C D (x) λ = πd (x) λ < 0. Corollary 7. The certainty equivalent decreases as the decision maker s risk aversion in the domain of gains increases. Conversely, the risk premium increases as the decision maker s risk aversion in the domain of gains increases. Proof. The firstorder derivatives of the certainty equivalent and risk premium with respect to γ + since UP M D 2 (x, 0) > 0. C D (x) γ + = πd (x) γ + = 1 2λ UP M D 2 (x, 0) < 0, Corollary 8. The certainty equivalent decreases as the decision maker s risk aversion in the domain of losses increases. Conversely, the risk premium increases as the decision maker s risk aversion in the domain of losses increases. Proof. The firstorder derivatives of the certainty equivalent and risk premium with respect to γ since LP M D 2 (x, 0) > 0. C D (x) γ = πd (x) γ = 1 2 LP M D 2 (x, 0) < 0, 12 To motivate for this, consider the generalized quadratic utility function (18). If γ + > 0, then the function U(W ) = (W W 0) 1 2 γ+(w W0)2 is increasing for 0 < W W 0 < 1 γ +. 18
19 6 Optimal Capital Allocation in the Generalized Framework The set up of the problem is the same as that described in Section 2. That is, we consider an investor who wants to allocate the wealth between a riskfree and a risky asset. The returns on the risky and the riskfree assets over a small time interval t are given by equations (7) and (8) respectively. The investor s initial wealth is W I and the investor s objective is to maximize the expected utility of his future wealth. Since the resulting expression for the investor s expected utility depends on whether the investor buys and holds or sells short the risky asset, we need to consider these two cases separately. In particular, if the investor uses the amount a 0 to buy the risky asset, his future wealth is given by equation (9). If the investor sells short the risky asset such that the proceedings are a 0, his future wealth is given by W = a(r x) + W I (1 + r). Note that in both cases the value of a is nonnegative. This is necessary to be able to distinguish between the probability distortion of x r and r x. Before attacking the optimal capital allocation problem, we need to choose a suitable reference point W 0 to which gains and losses are compared. One possible reference point is the status quo, that is, the investor s initial wealth W I. Unfortunately, with this choice it is not possible to arrive at a closedform solution for the optimal capital allocation problem unless W I = 0. However, according to Markowitz (1952), Kahneman and Tversky (1979), Loomes and Sugden (1986), and others, the investor s initial wealth does not need to be the reference point. Following Barberis, Huang, and Santos (2001) we assume that the reference point is W 0 = W I (1 + r). This is the investor s initial wealth scaled up by the riskfree rate. This choice of reference point is sensible due to many reasons: (1) This level of wealth serves as a benchmark wealth. The idea here is that the investor is likely to be disappointed if the risky asset provides a return below the riskfree rate of return; (2) With this reference point the performance measure does not depend on the investor s wealth (see the subsequent section). That is, the investor s utility of returns is independent of wealth; (3) De Giorgi, Hens, and Mayer (2006) show that with this choice of reference point a prospect theory investor satisfies 19
20 the mutual fund separation; (4) This choice is also justified if we require that all partial moments should exhibit the homogeneity property in a (for explanation, see footnote 7). For example, if the investor is equipped with the Fisburn s meanlower partial moment of order 2 utility, then the risk of investing the amount a, as measured by downside deviation, equals a LP M2 D (x r, 0), which seems reasonable. Having decided on the investor s reference wealth, we are ready to state and prove the following theorem. Theorem 9. If either utility functions U and U + are at most quadratic in wealth or the investment risk is small, then the investor s optimal capital allocation problem has the following solution: If E D [x r] (λ 1)LP M1 D (x r, 0) > 0 (25) and λγ LP M D 2 (x r, 0) + γ + UP M D 2 (x r, 0) > 0, (26) then the optimal amount invested in the risky asset is given by a = E D [x r] (λ 1)LP M1 D (x r, 0) λγ LP M2 D(x r, 0) + γ +UP M2 D (27) (x r, 0), and the investor s maximum equivalent expected utility from the buyandhold strategy is given by EBH[U D (W )] = 1 ( E D [x r] (λ 1)LP M1 D(x r, 0)) 2 2 λγ LP M2 D(x r, 0) + γ +UP M2 D (28) (x r, 0). If E D [r x] (λ 1)LP M D 1 (r x, 0) > 0 (29) and λγ LP M D 2 (r x, 0) + γ + UP M D 2 (r x, 0) > 0, (30) then the optimal amount of the risky asset that should be sold short is given by a = E D [r x] (λ 1)LP M1 D (r x, 0) λγ LP M2 D(r x, 0) + γ +UP M2 D (31) (r x, 0), 20
Basic Utility Theory for Portfolio Selection
Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationAnalyzing the Demand for Deductible Insurance
Journal of Risk and Uncertainty, 18:3 3 1999 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State
More informationC2922 Economics Utility Functions
C2922 Economics Utility Functions T.C. Johnson October 30, 2007 1 Introduction Utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. For example,
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino email: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationMidterm Exam:Answer Sheet
Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a riskfree and risky asset, with returns given by r f and E(r p ), respectively. Let y be the
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More information3. The Economics of Insurance
3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationAn Overview of Asset Pricing Models
An Overview of Asset Pricing Models Andreas Krause University of Bath School of Management Phone: +441225323771 Fax: +441225323902 EMail: a.krause@bath.ac.uk Preliminary Version. Crossreferences
More informationComputer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR
Computer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR David N. Nawrocki  Villanova University ABSTRACT Asset allocation has typically used
More informationA Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824
A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More informationLecture 15. Ranking Payoff Distributions: Stochastic Dominance. FirstOrder Stochastic Dominance: higher distribution
Lecture 15 Ranking Payoff Distributions: Stochastic Dominance FirstOrder Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) firstorder stochastically dominates G( ) if
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationChap 3 CAPM, Arbitrage, and Linear Factor Models
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
More informationThe Values of Relative Risk Aversion Degrees
The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des SaintsPeres, F75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article
More informationFE670 Algorithmic Trading Strategies. Stevens Institute of Technology
FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 MeanVariance Analysis: Overview 2 Classical
More information1 Capital Asset Pricing Model (CAPM)
Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available
More informationLecture 1: Asset Allocation
Lecture 1: Asset Allocation Investments FIN460Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating
More informationAFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang
AFM 472 Midterm Examination Monday Oct. 24, 2011 A. Huang Name: Answer Key Student Number: Section (circle one): 10:00am 1:00pm 2:30pm Instructions: 1. Answer all questions in the space provided. If space
More information2. Meanvariance portfolio theory
2. Meanvariance portfolio theory (2.1) Markowitz s meanvariance formulation (2.2) Twofund theorem (2.3) Inclusion of the riskfree asset 1 2.1 Markowitz meanvariance formulation Suppose there are N
More informationCHAPTER 1: INTRODUCTION, BACKGROUND, AND MOTIVATION. Over the last decades, risk analysis and corporate risk management activities have
Chapter 1 INTRODUCTION, BACKGROUND, AND MOTIVATION 1.1 INTRODUCTION Over the last decades, risk analysis and corporate risk management activities have become very important elements for both financial
More informationLife Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans
Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined
More informationOn the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information
Finance 400 A. Penati  G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationThe Effect of Ambiguity Aversion on Insurance and Selfprotection
The Effect of Ambiguity Aversion on Insurance and Selfprotection David Alary Toulouse School of Economics (LERNA) Christian Gollier Toulouse School of Economics (LERNA and IDEI) Nicolas Treich Toulouse
More informationOn characterization of a class of convex operators for pricing insurance risks
On characterization of a class of convex operators for pricing insurance risks Marta Cardin Dept. of Applied Mathematics University of Venice email: mcardin@unive.it Graziella Pacelli Dept. of of Social
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationChoice Under Uncertainty
Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange
More informationThe Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment
THE JOlJKNAL OF FINANCE VOL. XXXVII, NO 5 UECEMREK 1982 The Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment NICHOLAS ECONOMIDES* IN HIS THEORETICAL STUDY on the demand
More informationLecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier
Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The
More informationThe Equity Premium in India
The Equity Premium in India Rajnish Mehra University of California, Santa Barbara and National Bureau of Economic Research January 06 Prepared for the Oxford Companion to Economics in India edited by Kaushik
More informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closedbook exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationCapital Allocation Between The Risky And The Risk Free Asset. Chapter 7
Capital Allocation Between The Risky And The Risk Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in riskfree versus risky assets asset allocation
More informationRISK MANAGEMENT IN THE INVESTMENT PROCESS
I International Symposium Engineering Management And Competitiveness 2011 (EMC2011) June 2425, 2011, Zrenjanin, Serbia RISK MANAGEMENT IN THE INVESTMENT PROCESS Bojana Vuković, MSc* Ekonomski fakultet
More informationAsset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006
Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More information1 Introduction to Option Pricing
ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationCHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS
CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional
More informationTarget Strategy: a practical application to ETFs and ETCs
Target Strategy: a practical application to ETFs and ETCs Abstract During the last 20 years, many asset/fund managers proposed different absolute return strategies to gain a positive return in any financial
More informationInsurance Demand and Prospect Theory. by Ulrich Schmidt
Insurance Demand and Prospect Theory by Ulrich Schmidt No. 1750 January 2012 Kiel Institute for the World Economy, Hindenburgufer 66, 24105 Kiel, Germany Kiel Working Paper No. 1750 January 2012 Insurance
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationThis is the tradeoff between the incremental change in the risk premium and the incremental change in risk.
I. The Capital Asset Pricing Model A. Assumptions and implications 1. Security markets are perfectly competitive. a) Many small investors b) Investors are price takers. Markets are frictionless a) There
More information2. Information Economics
2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More information1 Interest rates, and riskfree investments
Interest rates, and riskfree investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)
More information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationChapter 1 INTRODUCTION. 1.1 Background
Chapter 1 INTRODUCTION 1.1 Background This thesis attempts to enhance the body of knowledge regarding quantitative equity (stocks) portfolio selection. A major step in quantitative management of investment
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationAuke Plantinga and Sebastiaan de Groot 1. November 2001. SOMtheme E: Financial markets and institutions
5,6.$'867('3(5)250$1&(0($685(6 $1',03/,('5,6.$77,78'(6 Auke Plantinga and Sebastiaan de Groot 1 November 2001 SOMtheme E: Financial markets and institutions $EVWUDFW In this article we study the relation
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationOn Compulsory PerClaim Deductibles in Automobile Insurance
The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory PerClaim Deductibles in Automobile Insurance CHUSHIU LI Department of Economics, Feng Chia
More informationHow to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing ShortTerm Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
More informationFund Manager s Portfolio Choice
Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled
More informationOptimal Consumption with Stochastic Income: Deviations from Certainty Equivalence
Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence Zeldes, QJE 1989 Background (Not in Paper) Income Uncertainty dates back to even earlier years, with the seminal work of
More informationECO 317 Economics of Uncertainty Fall Term 2009 Week 5 Precepts October 21 Insurance, Portfolio Choice  Questions
ECO 37 Economics of Uncertainty Fall Term 2009 Week 5 Precepts October 2 Insurance, Portfolio Choice  Questions Important Note: To get the best value out of this precept, come with your calculator or
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationWHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES. A D Wilkie, United Kingdom. Summary and Conclusions
WHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES A D Wilkie, United Kingdom Summary and Conclusions The question of whether a risk averse investor might be the more willing to hold shares rather
More informationFinancial Services [Applications]
Financial Services [Applications] Tomáš Sedliačik Institute o Finance University o Vienna tomas.sedliacik@univie.ac.at 1 Organization Overall there will be 14 units (12 regular units + 2 exams) Course
More informationPDF hosted at the Radboud Repository of the Radboud University Nijmegen
PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is a publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/29354
More informationInsurance Demand under Prospect Theory: A Graphical Analysis. by Ulrich Schmidt
Insurance Demand under Prospect Theory: A Graphical Analysis by Ulrich Schmidt No. 1764 March 2012 Kiel Institute for the World Economy, Hindenburgufer 66, 24105 Kiel, Germany Kiel Working Paper No. 1764
More informationMean Variance Analysis
Finance 400 A. Penati  G. Pennacchi Mean Variance Analysis Consider the utility of an individual who invests her beginningofperiod wealth, W 0,ina particular portfolio of assets. Let r p betherandomrateofreturnonthisportfolio,sothatthe
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationChapter 6 Arbitrage Relationships for Call and Put Options
Chapter 6 Arbitrage Relationships for Call and Put Options Recall that a riskfree arbitrage opportunity arises when an investment is identified that requires no initial outlays yet guarantees nonnegative
More informationInvesco Great Wall Fund Management Co. Shenzhen: June 14, 2008
: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationA Simple Utility Approach to Private Equity Sales
The Journal of Entrepreneurial Finance Volume 8 Issue 1 Spring 2003 Article 7 122003 A Simple Utility Approach to Private Equity Sales Robert Dubil San Jose State University Follow this and additional
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationTaxation of Shareholder Income and the Cost of Capital in a Small Open Economy
Taxation of Shareholder Income and the Cost of Capital in a Small Open Economy Peter Birch Sørensen CESIFO WORKING PAPER NO. 5091 CATEGORY 1: PUBLIC FINANCE NOVEMBER 2014 An electronic version of the paper
More informationOptimal demand management policies with probability weighting
Optimal demand management policies with probability weighting Ana B. Ania February 2006 Preliminary draft. Please, do not circulate. Abstract We review the optimality of partial insurance contracts in
More informationINTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE
INTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE J. DAVID CUMMINS, KRISTIAN R. MILTERSEN, AND SVEINARNE PERSSON Abstract. Interest rate guarantees seem to be included in life insurance
More informationA Coefficient of Variation for Skewed and HeavyTailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University
A Coefficient of Variation for Skewed and HeavyTailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a
More informationCurrent Accounts in Open Economies Obstfeld and Rogoff, Chapter 2
Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2 1 Consumption with many periods 1.1 Finite horizon of T Optimization problem maximize U t = u (c t ) + β (c t+1 ) + β 2 u (c t+2 ) +...
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationRank dependent expected utility theory explains the St. Petersburg paradox
Rank dependent expected utility theory explains the St. Petersburg paradox Ali alnowaihi, University of Leicester Sanjit Dhami, University of Leicester Jia Zhu, University of Leicester Working Paper No.
More informationUniversity of Pennsylvania The Wharton School. FNCE 911: Foundations for Financial Economics
University of Pennsylvania The Wharton School FNCE 911: Foundations for Financial Economics Prof. Jessica A. Wachter Fall 2010 Office: SHDH 2322 Classes: Mon./Wed. 1:303:00 Email: jwachter@wharton.upenn.edu
More informationSAMPLE MIDTERM QUESTIONS
SAMPLE MIDTERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationWe show that prospect theory offers a rich theory of casino gambling, one that captures several features
MANAGEMENT SCIENCE Vol. 58, No. 1, January 2012, pp. 35 51 ISSN 00251909 (print) ISSN 15265501 (online) http://dx.doi.org/10.1287/mnsc.1110.1435 2012 INFORMS A Model of Casino Gambling Nicholas Barberis
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationNOTES ON THE BANK OF ENGLAND OPTIONIMPLIED PROBABILITY DENSITY FUNCTIONS
1 NOTES ON THE BANK OF ENGLAND OPTIONIMPLIED PROBABILITY DENSITY FUNCTIONS Options are contracts used to insure against or speculate/take a view on uncertainty about the future prices of a wide range
More informationCHAPTER 11: ARBITRAGE PRICING THEORY
CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times
More informationTPPE17 Corporate Finance 1(5) SOLUTIONS REEXAMS 2014 II + III
TPPE17 Corporate Finance 1(5) SOLUTIONS REEXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder
More informationSolution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:
Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the riskfree rate is 6%. What is your optimal position
More informationChapter 21 Valuing Options
Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher
More informationThe Rational Gambler
The Rational Gambler Sahand Rabbani Introduction In the United States alone, the gaming industry generates some tens of billions of dollars of gross gambling revenue per year. 1 This money is at the expense
More informationDynamic Asset Allocation Using Stochastic Programming and Stochastic Dynamic Programming Techniques
Dynamic Asset Allocation Using Stochastic Programming and Stochastic Dynamic Programming Techniques Gerd Infanger Stanford University Winter 2011/2012 MS&E348/Infanger 1 Outline Motivation Background and
More information6. Budget Deficits and Fiscal Policy
Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 2012 6. Budget Deficits and Fiscal Policy Introduction Ricardian equivalence Distorting taxes Debt crises Introduction (1) Ricardian equivalence
More informationThis paper is not to be removed from the Examination Halls
~~FN3023 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON FN3023 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
More informationMATHEMATICS OF FINANCE AND INVESTMENT
MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationOptimal proportional reinsurance and dividend payout for insurance companies with switching reserves
Optimal proportional reinsurance and dividend payout for insurance companies with switching reserves Abstract: This paper presents a model for an insurance company that controls its risk and dividend
More information