AG RISK ANALYDEC MAK
AG RISK ANALYDEC MAK AEB 6182
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Weldon Rau I
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This 21 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 6182 at University of Florida taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/206598/aeb-6182-university-of-florida in Agricultural Economics And Business at University of Florida.
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Date Created: 09/18/15
Savage StateDependent Expected Utility Lecture IV Savage s state dependent expected utility A Savage s subjective expected utility theory takes as the object of choice state dependent outcomes x13 3 x13 Security 1 Security 2 x11 vudxn x21 7116521 x12 712 x12 x22 2x22 x23 3 x23 x14 4 x14 x24 4 x24 x15 715695 x25 auswzs B Writing the expected utility of the gamble out we have N E gmagtzpltigtultxgt What is changing The probability or the states Savage proved a similar result to the von NeumannMorgenstem modeliif agents preferences obey certain axioms they have an expected utlity representation through both the probabilities of each outcome and the utility function Speci cally the probabilities in the expected value framework may be endogenous II Axiomatization of StateiDependent Expected Utility A Start with some notation 1 N E In the stateipreference format we speak of the utility function mapping from statespace into preferences u RS gt R Next we want to introduce the notation of replacement u Cay is the utility of security when state s is replaced with y Specifically we want to compare u cisy 2 u disy III a Note that c d 6 1R3 or are two vectors of outcomes or payoffs while y w 6 R are two scalar incomes b For example take an investment x11 x11 x12 x12 x1 x13 uc4y gtu x13 x14 y x15 x14 B The independence axiom for statedependent expected utility requires that u Cay 2 u disy if and only if u 075w 2 u disw 1 Intuitively the ordering remains the same if we replace on of the states with a common payoff C Theorem 851 Assume that there are at least three states S 2 3 Utility function u has a statedependent expected utility representation iff it obeys the independence axiom Cardinal Utility Systems A A second article Friedman and L Savage 1952 The ExpectedUtility Hypothesis and the Measurability of Utility Journal of Polictical Economy 606 463774 followed the Friedman and Savage article we analyzed last class This article responded to a comment by William Baumol that raised questions about the cardinality of utility measures B Most of our current microeconomic theory is based on ordinal utility measurement where the actual levels of the utility are not important only the order or relative rank C Cardinality assigns importance to the actual numbers D The Cardinal Coordinate Independence axiom is a stronger version of the independence axiom discussed above Proposition 861 Cardional coordinate independence implies independence 2 Theorem 862 Utility function u has a stateindependent expected utility representation iff it obeys the cardinal coordinate independence axiom Expected Utility MeanVariance and Risk Aversion Lecture VII MeanVariance and Expected Utility A Under certain assumptions the MeanVariance solution and the Expected Utility solution are the same 1 N E 4 If the utility function is quadratic any distribution will yield a Mean Variance equivalence Taking the distribution of the utility function that only has two moments such as a quadratic distribution function Any distribution function can be characterized using its moment generating function The moment of a random variable is defined as Exk kafxdx The moment generating function is defined as M I E e If X has mgf M AU then E X l M E 0 where we define dn M E 0 M X t dt 0 First note that 6 can be approximated around zero using a Taylor series expansion 1 MXIEe Ee tewx 0Etzewx 02 t3ewx 03 MXt exp yt aztzj 1ExfEx2Ex3u Note for any moment n My 3 MXt Exquot Exquot1rExquot r2 Thus as t gt0 M o Exquot The moment generating function for the normal distribution can be defined as 2 1 J expx u 2 ta w o 027239 exp yt aztzj B Since the normal distribution is completely de ned by its rst two moments the expectation of any distribution function is a function of the mean and variance 0 However a more speci c solution involves the use of the normal distribution function with the negative exponential utility function Under these assumptions the expected utility has a speci c form that relates the expected utility to the mean variance and risk aversion l N Starting with the negative exponential utility function Ux exp px The expected utility can then be written as EUx 1 exp pxlfxo392dx no 1 2 Lo exp px a exp xU Z Lodx Combining the exponential terms and taking the constants outside the integral yields 1 J 0 exp l j px dx 01 27239 2 039 Next we propose the following transformation of variables FM 039 In general terms the distribution of a transformation of a random variable can be derived given that the transformation is a onetoone mapping Assume that z is a function of x such that z gx EUxl If the mapping is onetoone the inverse function can be de ned x g391z Given this inverse mapping we know what x leads to each 2 The only requird modi cation is the Jacobian or the relative change in the mapping 6912 62 Putting the pieces together assume that we have a distribution function x and a transformation zgx The distribution of 2 can be written as f 2 f 912 In this particular case the onetoone functional mapping is xz 652 62 and the Jacobian is dxzadz 3 The transformed expectation can then be expressed as x z u 039 320x uxuzo EUx 1 J 0 O39eXp lZZ pZO39 pJdz 039 27239 2 Next we want to complete the square 2 2 2 zz p039z pu lz2 p039z 1p20392 pZO39Z pJ II MeanVariance Versus Direct Utility Maximization A Due to various nancial economic models such as the Capital Asset Pricing Model that we will discuss in our discussion of market models the nance literature relies on the use of meanvariance decision rules rather than direct utility maximization In addition there is a practical aspect for stockbrokers who may want to give clients alternatives between ef cient portfolios rather than attempting to directly elicit each individual s utility function Along the later tack the study by Kroll Levy and Markowitz examines the acceptability of the MeanVariance procedure Speci cally they attempt to determine whether the expected utility maximizing choice is contained in the MeanVariance ef cient set 03 1 Looking forward to next week s lectures we assume that the decision maker is faced with allocating a stock portfolio between various investments Two approaches for making this problem are to choose between the set of investments to maximize expected utility maxE U x st 211 x1 1 x Z 0 The second alternative is to map out the ef cient MeanVariance space by solving max 039 x st x Qx St q 2 0 A better formulation of the problem is max 039 x x Qx st x1 2 0 And where p is the Arrow Pratt absolute risk aversion coef cient Table III Optimal Investment Strategies with a Direct Utility Maximization Utility California Carpenter Chrysler Conelco Texas Average Standard Function Gulf Return Deviation e 443 347 02 55 153 224 XO1 332 360 136 172 233 323 X 5 422 344 234 259 494 LnX 379 348 111 162 231 294 111 Table V Optimal EV Portfolios for Various Utility Functions Utility California Carpenter Chrysler Conelco Texas Average Standard Function Gulf Return Deviation e39x 394 386 50 170 225 270 XO1 285 434 86 86 231 300 XO5 418 321 261 257 473 LnX 329 418 74 187 229 289 Utility Functions Risk Aversion Coef cients and Transformations Lecture VI An examination of the ArrowPratt Coefficients for particular functions A Quadratic Utility Function To specify the appropriate shape of the utility function the quadratic function becomes Uw aw bw2 U39w a 2bw Uquotw 2b ArrowPratt absolute risk aversion coefficient 2b 2b RAw a wa RAW a wa dRA w gt 0 C i 1 M 61W 61 2bw2 dx fx c2 ArrowPratt relative risk aversion coefficient RRW2bw2b a 2bw 2 2b w dRR W 213 213 2 2b gt 0 dw a2bw Cl wa B Power Utility Function w 7r U w 1 r U39w wquot Uquotw rwquot391 ArrowPratt absolute risk aversion coefficient 471 rw r R w Alt gt w dRA wLlt0 dw w2 ArrowPratt relative risk aversion coefficient rw39Hw RR w r 7r w d R w R 0 dw Constant relative risk aversion C Negative Exponential Utility Function U w exp pw U39w pexp pw Uquotw p2 exp pw ArrowPratt absolute risk aversion coef cient p2 exp pw RA w Z l PeXPEPWl 2 p dRA w 0 dw Constant absolute risk aversion ArrowPratt relative risk aversion coef cient 2 ex w RRw p p p w pw p exp pw d RR w dw D HARAiHyperbolic Absolute Risk Aversion V uw1Tv bgt0 pgt0 lv ArrowPratt absolute risk aversion coef cient a 611 v awb1 y awb1 y 1 v 1 As 7 gt lit becomes risk neutral 2 y 2 is a quadratic 3 y gt oo and b l is the negative exponential 4 b 0 and y lt l is the power utility function Interpretations and Transformations of Scale for the PrattArrow Absolute Risk Aversion coefficient Implications for Generalized Stochastic Dominance A To this point we have discussed technical manifestations of risk aversion such 03 as where the risk aversion coefficient comes from and how the utility of income is derived However I want to start turning to the question How do we apply the concept of risk aversion Several procedures exist for integrating risk into the decision making process such as direct application of expected utility mathematical programming using the expected valuevariance approximation or the use of stochastic dominance All of these approaches however require some notion of the relative size of risk aversion 1 Risk aversion directly uses a risk aversion coefficient to parameterize the negative exponential or power utility functions 2 Mathematical programming uses the concept of the tradeoff between variance and expected income 3 Stochastic dominance uses measures of risk aversion to bound the utility function The current study gives some guidance on using previously published risk aversion coefficients Specifically the article looks at the effect of location and scale on the risk aversion coefficient 1 As a starting place we develop an interpretation of the PrattArrow coefficient in terms of marginal utility u x u x du39lt u39 x d Elnu x d l u xu x d x This algebraic manipulation develops the absolute risk aversion coefficient as the percent change in marginal utility at any level of income a Therefore r is associated with a unit of change in outcome space If the risk aversion coefficient was elicited in outcomes of dollars then the risk aversion coefficient is 0001 b This result indicates that the decisionmaker s marginal utility is falling at a rate of 01 per dollar change in income This association between the risk aversion and the level of income then raises the question of the change in outcome scale a For example what if the original utility function was elicited on a per acre basis and you want to use the results for a whole farm exercise rx N b Theorem 1 Let rxu xu x Define a transformation of scale on x such that wxc where c is a constant Then rwcrx l The proof lies in the change in variables Given Ux Uwc dUdUdx U dw dx dwc x d2U d dU d U 2U dw2dw dw dwc Jo C x 2 In other words if the scale of the outcome changes by c the scale of the risk aversion coefficient must be changed by the same amount c Theorem 2 If vx c where c is a constant then rvrx Therefore the magnitude of the risk aversion coefficient is unaffected by the use of incremental rather absolute returns D Example Suppose that a study of Us farmers gives a risk aversion coefficient of r0001 US Application to the Australian farmers whose dollar is worth 667 of the Us dollar is r0000667 Australian Lecture XXI Crop Insurance Valuing Crop Yield Insurance A The general concept of insurance is the construction of an instrument or gamble that pays the purchaser in the event of some adverse occurrence N E Frequently purchased insurance contracts include life insurance that pays in the event of the holders death car insurance that pays in the case of an accident catastrophic health insurance that pays in the event of a major medical event such as cancer etc Each of these contracts speci es a payable event an indemnity the amount to be paid on the event and a premium the amount paid for insurance contract Under commercial insurance arrangements the premium charged for the insurance is generally considered to be actuarially sound Speci cally the expected indemnity payments are exactly equal to premiums charged a If the premiums exceeded the expected indemnity payments then insurance rms would earn abnormal pro ts These abnormal pro ts would be bid out of the market by new rms entering the insurance arena If premiums fell short of the expected indemnity the insurance rm would loose money and ultimately exit the industry Fquot B The actuarial value of an insurance contract can then be written as Vijfydy where a Vis the value of crop yield insurance b P is the price of the crop c y is the variable of integration d y is the probability density function for crop yields and e y is the minimum insured yield trigger yield in crop insurance C Current debates in the area of crop yield insurance involve 1 Estimation of the probability density function for yields y a Most common statistical applications assume that the probability density function is normal or asymptotically normal This assumption may have serious shortcomings in the valuation of crop insurance From an agronomic perspective yields are bounded by zero on the downside and limiting nutrients such as nitrogen on the up side Hence at the least the x eoooo of the normal would appear to be violated However the truncated normal distribution may be appropriate for crop yields The debate of potential normality of crop yields typically revolves around skewness and kurtosis i Skewness is a measure of nonsymmetry of the distribution The normal distribution is symmetric and hence yields zero Fquot O skewness A signi cant portion of the literature supports skewness in yields but as pointed out by Just and Weninger it does not reach a consensus on the direction of skewness ii Kurtosis measures the relationship between the area in the tails and the area around the means 2 The second area of debate in the area of crop insurance is the moral hazardincentive compatibility dimension of crop insurance a A basic problem in any insurance contract is the determination of the insurable event and the amount of damages i The classic scenario of ling lawsuits for pain and suffering after a car accident due to whiplash is a famous lawyer joke ii In the current case the insurance company must determine when a yield reduction has occurred and the amount of that reduction A second problem is the dif culty of selfselection Speci cally as in health insurance contracts riskier farmers will be willing to pay more money for insurance than safer farmers D Valuing crop yield insurance a Using the data from Ramirez Moss and Boggess we derive the parameters of the normal distribution function for corn as u17303 6871 f 1 y 173032 V P eX yam2n pi 15188 Fquot b Assuming a corn price of 300bushell the value of insurance becomes Insurance Premiums 90 17303 1073 85 17303 63 80 17303 01 II Integrating Price Insurance A In order to integrate price risk the actuarial premium becomes ff V prfypdydp 0 0 B The joint distribution is speci ed using the futures price as an ef cient estimate of the price at harvest time 1 The price at harvest time can be estimated as a function of the futures price at planting pzh a0 a1f2 82 J39 I u u 39 Given an is the J A basis and a1 is equal to one The distribution of price is then a function of the distribution of 8 N Lecture XXI Applications of Stochastic Dominance Moss Charles B Stephen A Ford and Mario Castejon Effect of Debt Position on the Choice of Marketing Strategies for Florida Orange Growers A Risk Ef ciency Approach Southern Journal of Agricultural Economics 1991 103711 A The risk of a rm can be decomposed into business risk and nancial risk 1 Business risk is the risk inherent in the productionbusiness environment 2 Financial risk results from the leverage debt decisions based on that business environment 3 Collins 1985 demonstrated this decomposition using a DuPont expansion r i 6K R 5 1 5 where RE is the rate of return to equity rp is the return to production activities A is the total level of output 139 is the rate of capital gains 5 is the debttoasset position and K is the cost of debt capital a Following this development R A r 139 3 039 b Therefore the expected rate of return on equity and the variance of the rate of return on equity can be derived as HA 6K 0 a 2 1 6 1 5 assuming that the cost of debt is nonstochastic B The linkage between debt and asset returns has been addressed several ways in economic literature Following ModiglianiMiller the value of assets is independent of the capital structure 2 The ModiglianiMiller results assume in nite arbitrage opportunities Debt can be increased by issuing bonds and using the proceeds to buy stock increasing the relative level of debt Alternatively someone could sell stock and buy bonds to increase the relative level of equity 5 3 In agriculture such arbitrage is typically not possible ie there are very few publicly traded rms 4 Literature following Collins 1985 allows for endogeneity but holds the cost of debt constant 5 This study Moss Ford and Castejon allows the cost of capital to be stochastic AEB 61827Agricultural Risk Analysis and Decision Making Fall 2004 Professor Charles B Moss Lecture 21 R Pm V Dr where R is the return on investment 139 in period t Pt is the price of output using marketing instrument 139 in period t K is the yield in period t V is the variable cost D is the level of debt and r is the cost of debt capital in period t C This study then applies the above de nition to derive a series of returns for oranges marketing their output as FCOJ using cash markets futures and participation 1 Applying second degree stochastic dominance we derived the efficient choices of marketing instruments based on the solvency level presented in table 1 Table 1 Second Degree Stochastic Dominance Results Solvency Participation Cash Hedge Ratio Decem ber 0 30 40 50 60 February 0 30 40 50 60 gtltgtltgtltgtltgtlt gtltgtltgtltgtltgtlt April 0 30 40 50 60 gtltgtltgtltgtltgtlt gtltgtltgtltgtltgtlt 2 Given that the choice of marketing instrument does not change within the each marketing period what is the role of risk aversion To see this we focus on the February marketing period AEB 61827Agricultural Risk Analysis and Decision Making Fall 2004 Professor Charles B Moss Lecture 21 Table 2 Second Degree Stochastic Dominance with Respect to a Function Results Solvency Participation Cash Hedge Ratio r x 0000002 0000003 0 X X 30 X 40 X 50 X 60 X r x 0000003 0000004 0 X 30 X X 40 X X 50 X 60 X r x 0000004 0000005 0 X 30 X 40 X 50 X X 60 X X 3 The study then nds that the choice of marketing instrument only changes between debttoasset positions only when risk aversion is taken into account D A similar study can be found in Gloy Brent A and Timothy G Baker The Importance of Financial Leverage and Risk Aversion in Risk Management Strategy Selection American Journal of Agricultural Economics 8442002 1130743 II A second paper I want to talk about is Moss Charles B Implementation Of Stochastic Dominance A Nonparametric Kernel Approach A This study focuses on the implementation of the stochastic dominance approach 1 Following the standard First and Second Degree Stochastic Dominance Approach dominance is defined as A Ewen Esme j fx gxUxdx 2 0 31 Gx Fx z 0 Vx E1 inf Gx Fx A51 squx Fx AEB 61827Agricultural Risk Analysis and Decision Making Fall 2004 Professor Charles B Moss Lecture 21 a If A and A have the same sign or 20 one distribution dominates the other A U xj Gx Fxdx l U xj 62 Fzdz dx E 62 Fzdz z 0 Vx R infjGz Fzdz 352 sup J Gz Fzdz b Again if A and A have the same sign or Z 0 one distribution dominates the other in the second degree 2 The dif culty lies in the empirical form of the distribution function Speci cally the empirical distribution function is typically de ned as N s m LV x N which is a step function B My approach is based on nonparametric regression l The general nonparametric regression or kernel regression can be expressed as 91 i W a 9i y 1 kxj A1 J exp 5xj Acl 2 In this application we de ne 9 as the cumulative distribution function de ned like the step function de ned above The empirical kernel can then be written as kxj 1 N A Zkx1xx 11 A k 96px St Petersburg Paradox Savage and StatePreference Lecture III I St Petersburg Paradox A Most of what we know about risk started from games of chance B One of the rst problems was with a bet known as the St Petersburg Paradox 1 To start the question we pose gamble based on a coin toss If the coin comes up heads you earn 1 If two heads in a row come up you earn 2 The expected payoff is then the sum Vi2f 15 2 The expected return of this gamble approaches in nity as N goes to in nity 3 Would you pay in nity for this gamble II Friedman and Savage A Friedman M and LJ Savage 1948 The Utility Analysis of Choices Involving Risk Journal of Political Economy 564 2797304 B Friedman and Savage follow a description of the expected utility hypothesis similar to that presented in the preceding lecture Taking B as the certain alternative andA as a risky alternative we de ne the expected utility of A as UAaUIl1 aUIZ Next we de ne the actuarial value of the risky alternative as 1Aod1 1 a2 C Based on this formulation we have three possibilities 1 U 72U A or the utility of the average income is greater than the utility of the gamble In this case the individual avoids or is averse to risk 2 U 1 U A the individual is indifferent between the gamble and a certain payoff or risk neutral UAU1 NI 3 U 1 S U A the individual prefers the gamble t0 the certain payoff N D Friedman and Savage then consider ve different behaviors Consumer units prefer larger to smaller incomes Lowincome consumer units buy or are willing to buy insurance Lowincome consumer units buy or are willing to buy lottery tickets Many lowincome consumer units buy or are willing to buy both insurance and lottery tickets and Lotteries typically have more than one prices 59 V39 III State Preference Setup Let x j 6 RS be a vector of returns to investment or asset j in states 3 e l S X e M M S is the matrix of all such returns in the economy h is a portfolio of these possible investments or securities The payoff of a portfolio is also a vector hX hXZZhx J M zelRS zhX forsomehelRJ u CG CI is the utility of consumption defined on this statespace 1 c0 is the consumption at t 0 2 c1 is the vector ofpossible consumptions at I l SI 00 S w0 qz 01 S w1 z 2 EM ltzhxgt F State Dependent Expected Utility S uclczcsZucc c iffZ 39SVS 0322723244 51 G State Independent Expected Utility S uclczcs2 uc139cc iffZ SVCSZ Z svcs39 51
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