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Some Applications in Economy for Utility Functions Involving Risk Theory

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Some Applications in Economy for Utility Functions Involving Risk Theory Available online at www.sciencedirect.com ScienceDirect Procedia Economics and Finance 22 ( 2015 ) 595 – 600 2nd International Conference ‘Economic Scientific Research - Theoretical, Empirical and Practical Approaches’, ESPERA 2014, 13-14 November 2014, Bucharest, Romania Some Applications in Economy for Utility Functions Involving Risk Theory Corina Cipua, Carmen Gheorgheb* aUniversity Politehnica of Bucharest, Romania, Str. Polizu, no. 1-7, Sector 1, Bucharest, 011061, Romania bNational Institute for Economic Research “Costin C. Kiriţescu”, Casa Academiei Române, Calea 13 Septembrie, no. 13, Sector 5, Bucharest, Romania Abstract We present in the first part of the article types of utility functions that can describe the behavior of the investor and their applications to optimize portfolio. The second part of the paper refers to applications in calculating insurance premiums aggregated risk in zero utility principle. © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (©http://creativecommons.org/licenses/by-nc-nd/4.0/ 2015 The Authors. Published by Elsevier B.V.). Selection and/orand/or peer peer-review-review under under responsibility responsibility of ofthe the Scientific Scientific Committee Committee of ESPERA of ESPERA 2014 2014. Keywords: utility function, risk theory, zero utility principle 1. Maximize the value of the business for Risk Averse Investor There are three important principles of investments in order to maximize the value business. These are: the investment decision, the financing decision and the dividend decision. The investment decision can be taken by investing in assets that gain a higher return than the smallest acceptable hurdle rate. The hurdle rate should reflect risk degree of the investment and the mixture of debt and equity employed to finance it, and the return should reflect the magnitude and phasing of cash payments. * Corresponding author. E-mail address: [email protected], [email protected]; 2212-5671 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of the Scientific Committee of ESPERA 2014 doi: 10.1016/S2212-5671(15)00268-3 596 Corina Cipu and Carmen Gheorghe / Procedia Economics and Finance 22 ( 2015 ) 595 – 600 In order to find the most suitable type of debt and the correct blend of debt and equity to finance our business operations, we have to apply the financing decision. It is very important to match the correct type of tenor for your assets to maximize the firm value. With the dividend decision can be stated the sum to return to the business investors in cash or buybacks. The utility functions allow the measurement of the preferences of an investor in the desire to increase wealth in view of its risk aversion. We present here several criteria for the selection of a utility function that evolved over time. Among them are: [1] u() is an increasing function for xf(0, ) . This is true when the first derivative (marginal utility) is strictly positive; [2] u() is concave; m. It is important to isolate the large values thatsuch that ux() ܴא݉ u() is bounded: there exists [3] occur rarely from main preferences; [4] as wealth increases, the absolute risk aversion A() decreases. A fifth criterion is occasionally advanced: [5] utility is a constant function for negative values of wealth xux0, '( ) 0 . The utility functions we use in this paper are continuous and differentiable at zero, and they have the properties of i. normalization uu'(0) 1, (0) 0 (empty financial budget has no utility); ݕ൑ݖ՜ݑሺݕሻ൑ݑሺݖሻandܴאii. monotonicity ݕǡݖ ǡͲ<m>1, u(my+(1-m)z) ൒ ݉ݑሺݕሻ ൅ሺͳെ݉ሻݑሺݖሻ (relative increase of the utility getsܴאconcavity ݕǡ ݖ smaller when y grows). Functions that do not meet the above criteria are ux() x (not concave); ux() 1 eOx ,O ! 0 (not bounded); ux() xO ,0O 1 (fails [4]). Utility functions obtained from Weibull and Pareto distribution functions that do a meet the above criteria for proper parameters ux() 1 eOx , a 1,O ! 0, ux() 1 (OO x 1),,a a ! 0. Remark: The normalization conditions could always be fulfilled for utility functions with u '0! if we §·u 0 consider a change function xuxo . ¨¸ ©¹u '0 If u is twice differentiable we can write the [1]-[3] properties uu'0,''0,(0)0td u and u'(0) 1 . 2. Principle of expected utility maximization For F - feasible investment alternatives, XI() - random variable giving the ending value of the investment for the time period considered, a rational investor acts to select an investment IFopt which maximizes his expected utility function u Eu(m X Iopt ax((() EX I) IF 3. Investment problem We take X as a random variable; x1 and x2 two realizations; x1 represents good outcome and x2 bad outcome. The set of feasible investment alternatives has only two elements: pXx P( 1 ) , 1P()pXx 2 . Question: Which alternative (do nothing or make investment) does the investor choose if he follows the principle of expected utility maximization? Corina Cipu and Carmen Gheorghe / Procedia Economics and Finance 22 ( 2015 ) 595 – 600 597 The Certainty Equivalent (Cert. Eq ) for X is cu 1 MuX(( )i.e. u()c) MuX( (), meaning: if his current wealth is c , he will be indifferent between undertaking the investment and doing nothing. For investment problem Cert.(1) Eq px12 p x . One can measure the level of the risk aversion of an investor, in two ways: ௨ᇲᇲሺ௫ሻ : ሺݔሻ ൌെܣa) measure of the absolute risk aversion by: ܴ ሺݔሻ ൌ ௔ ௨ᇲሺ௫ሻ ᇱሺݔሻ ൐Ͳ฻ݑᇱᇱᇱሺݔሻݑᇱሺݔሻ ൏ݑᇱᇱሺݔሻଶ then as wealthܣ :x For the increasing absolute risk aversion behaviour increases will be hold fewer Euro in risky assets. ᇱሺݔሻ ൌͲ฻ݑᇱᇱᇱሺݔሻݑᇱሺݔሻ ൌݑᇱᇱሺݔሻଶܣ x For the stationary case ᇱሺݔሻ ൏Ͳ then as wealth increases will be hold moreܣ :x For the decreasing absolute risk aversion behaviour Euro in risky assets. ᇲᇲ ᇲᇲᇲ ᇲ ᇲᇲ మ ௫௨ ሺ௫ሻ ᇱ ௨ ሺ௫ሻ௨ ሺ௫ሻି௨ ሺ௫ሻ . ሺݔሻ ൌെ మ ܣ ሺݔሻ ൌെ , withܣb) measure of the relative risk aversion: ܴ௥ሺݔሻ ൌ ݔ ௨ᇲሺ௫ሻ ൫௨ᇲሺ௫ሻ൯ 4. Application in insurances. Utility premiums for linear truncated and quadratic utility Compensations are paid by IC (insurance company) based on insurance policies owned by the insured following the conclusion of a contract. Premiums paid by insurers must cover any damages and other costs: fees, taxes, maintenance costs. The amount of damage or loss associated with a contract for a period of time is a random variable X and represents the risk assumed by IC. The zero utility principle for different scale invariant utility functions ug(())O x ux() ,O ! 0,x , (4.1) O g()O with ݑǣ ܴ ՜ ܴ a classical utility function, will be used in order to obtain the zero utility premium HO,1 that is the implicit solution of the following equation: EHY((PO ))0, (4.2) Y being the risk assured, with unit expectation. For the scaled risk XY P with P expectation the zero utility premium HOP, is also uniquely determined: HHOP,/, P OP1 (4.3) We shall compare the variation of the zero utility premium HOP, for linear truncated and quadratic utility versus O parameter for an Exponential and Pareto distribution of the risk with P expectation. The linear truncated utility uxO () min{,} xO leads to the approximate solution HOP, | P . For X exponential with unit expectation equation (4.1) is written: ()H O Eu[(O H X )] OO e , H t (4.4) and one obtain ­OOOdln , 1 HO,1 ® (4.5) ¯ 1,O t 1. which is generalized for the risk X with P expectation ­ PPO,0dd ° HOP, ® §·P (4.6) °OPtln¨¸ , PO . ¯ ©¹O 598 Corina Cipu and Carmen Gheorghe / Procedia Economics and Finance 22 ( 2015 ) 595 – 600 Also, the mean value of uHYO () for the Pareto risk Y with unit expectation and D parameter is D ­ §·H O °OO¨¸1,H t Eu[(O H Y )] ® ©¹D (4.7) ° ¯ HHd1, O leading to ­ D ­ PPO,0dd OO1,1 d ° D ° HO,1 ® O and HOP, ® P (4.8) ° °OPDPa D ,. PO t ¯ 1,O t 1. ¯ O for the Pareto risk X with P expectation and D parameter. In the same manner, for quadratic utility ­ x2 °xxd, O ° 2O uxO () ® (4.9) O ° ,,x t O ¯° 2 the approximate solution is P 2 H |P (4.10) OP, 2O for any type of risk. For the exponential risk Y with unit expectation the mean of uHYO () is ­ O 2() 2eH O ° , H t O ° 2O Eu[(O H Y )] ® (4.11) (1OO22)((1))H ° , H d O ¯° 2O with unique solution ­OOO1122 t ° H 2 (4.12) O,1 ® O ° OOdln , 2. ¯ 2 and ­OP O22 P,0 dd PO /2 ° 2 (4.13) HOP, ® 2P ° OPtln , PO / 2. ¯ O 2 for the exponential risk X with P expectation. Corina Cipu and Carmen Gheorghe / Procedia Economics and Finance 22 ( 2015 ) 595 – 600 599 Fig 1. Zero utility premiums Fig 2. Quadratic utility and a Pareto risk for exponential and Pareto risks In Fig. 1 zero utility premiums for exponential and Pareto risks with P expectation were represented. One observe that between Pareto cases the Pareto for bigger D parameter is more convenient to the investor and more appropriate with the approximate solution. For quadratic utility and a Pareto risk Y with unit expectation and parameter D , the solution of equation (4.1) becomes ­ DD2 °OD , Od D 1 D 1 ° D 1 O 2 HO,1 ® 2D (4.15) ° OD2 2 ° OOtln , . ¯° 21D and for Pareto scaled risk XY P with P expectation and parameter D , the zero utility premium is ­ 2 21DP§· D °ODPDPD 1 ¨¸, Pt O ° DO12©¹ D HOP, ® (4.16) ° 22DD11 ° OP O P,0dd PO , ¯ DD12 P and was represented in Figure 2 for [0,2] and different values of parameter D {1.8;2;2.5;3}.
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