Miyahara, Yoshio

Kyoto University Research Information Repository

Title Evaluation of the Scale Risk (Financial Modeling andAnalysis)Author(s)Miyahara, YoshioCitation数理解析研究所講究録 (2014), 1886: 181-188Issue Date2014-04URL http://hdl.handle.net/2433/195713RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityEvaluation of the Scale RiskYoshio MiyaharaNagoya City University1 IntroductionWe study the evaluation problem of the scle risk. The method we have adopted is the risksensitive value measure (RSVM) method, which have been introduced in [8].This method is developed originally for the project evaluation. Even so this method canbe applied to many evaluation problems in finance. For example we can apply this methodto the scale risk evaluation problems.In this paper we overview the idea of the scale risk evaluation problem. For the details,see [8] and etc.2 Risk-Sensitive Value Measure(RSVM)We give the definition of the Risk-Sensitive Value Measure(RSVM) and summarize the prop-erties of this measure.2.1 Definition of the Risk-Sensitive Value MeasureDefinition 1 (Risk sensitive value measure(RSVM)) Let X be a linear space of ran-dom variables, then the risk sensitive value measure(RSVM) on X is the following functionaldefined on X$U^{(\alpha)}(X)=- \frac{1}{\alpha}\log E[e^{-\alpha X}], (\alpha>0)$ , (2.1)where $\alpha$ is the risk avertion parameter.Remark 1 In the above definition, $X$ is supposed to be the random present value of a cashfllow or a return of some asset.2.2 Properties of the Risk-Sensitive Value MeasureWe first remark the following facts.Proposition 1 (i) The following approximation formula holds true:$U^{(\alpha)}(X)=E[X]- \frac{1}{2}\alpha V[X]+\cdots$ (2.2)(ii) If $X$ is Gaussian, then it holds that$U^{(\alpha)}(X)=E[X]- \frac{1}{2}\alpha V[X]$ . (2.3)1886 2014 181-188 1812.2.1 Concave Monetary Value MeasureDefinition 2 (concave monetary value measure) $A$ function $v(\cdot)$ defined on a linearspace X of random variables is called a concave monetary value measure (or concave monetaryutility function) on X if it satisfies the following conditions:(i) (Normalization): $v(O)=0,$(ii) (Monetary property) : $v(X+m)=v(X)+m$, where $m$ is non-random,(Remark: (i) $+($ ii$)arrow v(m)=m$),(iii) (Monotonicity) : If $X\geq Y$ , then $v(X)\geq v(Y)$ ,(iv) (Concavity) : $v(\lambda X+(1-\lambda)Y)\geq\lambda v(X)+(1-\lambda)v(Y)$ for $0\leq\lambda\leq 1,$(v) (Law invariance) : $v(X)=v(Y)$ whenever law(X) $=$ law$(Y)$ ,Remark 2 We don’t require the following positive homogeneity property:(vi) (Positive Homogeneity): $\forall\lambda\in R^{+},$ $v(\lambda X)=\lambda v(X)$ .We next notice an important property of a concave monetary value measure.Proposition 2 (global concavity) $A$ concave monetary value measure $v(\cdot)$ satisfies thefollowing global concavity condition.(iv) ‘ (global concavity) :$v(\lambda X+(1-\lambda)Y)\leq\lambda v(X)+(1-\lambda)v(Y)$ for $\lambda\leq 0$ or $\lambda\geq 1$Proposition 3 Let $v(\cdot)$ be a concave monetary value measure. Then, for a fixed pair $(X, Y)$ ,$\psi_{X,Y}(\lambda)=v(\lambda X+(1-\lambda)Y)$ is a concave function of $\lambda.$Setting $Y=0$ in this proposition, we obtain the following result:Corollary 1 Let $v(\cdot)$ be a concave monetary value measure. Then $\psi_{X}(\lambda)=v(\lambda X)$ is aconcave function of $\lambda$ and $\psi_{X}(0)=0.$From this corollary we obtain the following concept of “Optimal Scale.”[Optimal Scale]Let $v(\cdot)$ be a concave monetary value measure, and assume that $v(X_{0})>0$ for some fixedrandom variable $X_{0}$ . If $v(\lambda X_{0}),$ $\lambda>0$ , is an upper bounded function of $\lambda$ , then we can findthe maximum point $\overline{\lambda}$ . This value $\overline{\lambda}$ is the optimal scale of $X_{0}.$2.2.2 Utility Indifference ValueFor a utility indifference value we obtain the following result:Proposition 4 Let $u(x)$ be a utility function defined on $(-\infty, \infty)$ and satisfy the usual prop-erties of a utility function. Then the indifference value $v(X)$ determined by the followingequation$E[u(-v(X)+X)]=u(O)=0$ (2.4)is a concave monetary value measure.182Remark 3 An indifference value does not satisfy the following positive homogeneity condi-tion in general.(Positive Homogeneity): $\forall\lambda\in R^{+},$ $v(\lambda X)=\lambda v(X)$ .Proposition 5 $U^{(\alpha)}(X)$ is the indifference value of the exponential utility function:$u_{\alpha}(x)= \frac{1}{\alpha}(1-e^{-\alpha x}) , -\infty<x<\infty (\alpha>0)$ . (2.5)Corollary 2 $U^{(\alpha)}(X)$ is a concave monetary value measure.Corollary 3 $U^{(\alpha)}(\lambda X)$ is a concave function of $\lambda.$2.2.3 Optimal ScaleFrom the fact that $U^{(\alpha)}(\lambda X)$ is a concave function of $\lambda$ , we can discuss the optimal scale ofthe investment, and we obtain the following result:Proposition 6 Assume that the moment generation function of $X$ converges and that thefollowing conditions satisfied,$E[X]>0, P(X<0)>0$ . (2.6)Then it holds that(i) When $\lambda(>0)$ is small, $U^{(\alpha)}(\lambda X)>0$ , and$\lim_{\lambdaarrow\infty}U^{(\alpha)}(\lambda X)=-\infty$ . (2.7)(ii) The optimal scale $\lambda_{opt}$ is$\lambda_{opt}=\frac{C_{X}}{\alpha}, \alpha>0$ , (2.8)where $C_{X}$ is a solution of $E[Xe^{-C_{X}X}]=0,$2.2.4 Independence-Additivity PropertyDefinition 3 (Independence-Additivity) If a value measure $v(\cdot)$ satisfiese $)$ (independence-additivity): $v(X+Y)=v(X)+v(Y)$ if $X$ and $Y$ are independent,then $v(\cdot)$ is said to have the independence-additivity property.We can suppose that this property is desirable for the project evaluation functional, andthe following proposition is easily proved.Proposition 7 An indifference value determined from an exponential utility function hasthe independence-additivity property.The converse of this proposition is known.183Proposition 8 Let $v(x)$ be an indifference value determined by a utility function $u(x)$ whichis of $C^{(2)}$ -class, increasing, concave, and normalized such as $u(O)=0,$ $u’(O)=1$ , and $u”(0)=$$\alpha$ . Then, if $v(x)$ has the independence-additivity property, $u(x)$ is of the following form$u(x)=u_{\alpha}(x)= \frac{1}{\alpha}(1-e^{-\alpha x})$ . (2.9)2.3 Good Points of Risk Sensitive Value Measure(1) The RSVM is a concave monetary value measure.(2) The RSVM is the utility indifference value of the exponential utility function, and it hasa risk aversion parameter $\alpha.$(3) The optimal scale of a project can be discussed.(4) The RSVM has the independence-additivity property, and the RSVM is the almost onlyone which has this property in the set of all utility indifference values.(5) The dynamic RSVM has the time-consistency property, and the RSVM is the almost onlyone which has this property in the set of all utility indifference values.3 Evaluation of the Scale Risk3.1 What is the Scale RiskLet $X$ be a return for an investment of $I$ . We suppose that the return for the investment $\lambda I$is $\lambda X$ . Assume that $E[X]>0$ and $P(X<0)>0$ . If $\lambda(>0)$ is small then the investment $\lambda I$may be positively valued. But if $\lambda$ is very large, then a very big loss may happen and so theinvestment $\lambda I$ may be negatively valued. This is the “scale risk.”3.2 Numerical ExampleLet $X,$ $Y,$ $Z$ be random variables whose distributions are$P(X=-10)=0.02, P(X=4)=0.5, P(X=8)=0.48$ (3.1)$E[X]=5.64, V[X]=8.9104$ , (3.2)$P(Y=-2)=0.15, P(Y=4)=0.7, P(Y=10)=0.15$ (3.3)$E[Y]=4.00, V[Y]=10.8000$ , (3.4)$P(Z=-1)=0.3, P(Z=4)=0.6, P(Z=16)=0.1$ (3.5)$E[Z]=3.70, V[Z]=21.8100$ . (3.6)From the scale risk point of view, $X$ has a big scale risk, $Z$ has a less scale risk and $Y$ isbetween $X$ and $Z$ . Remak here also that$E[X]>E[Y]>E[Z]$ (3.7)$V[X]<V[Y]<V[Z]$ (3.8)184We calculate the values of $\lambda X,$ $\lambda Y$ and $\lambda Z$ . In the following table,$MV_{X}( \lambda)=E[\lambda X]-\frac{1}{2}\alpha V[\lambda X], RSVM_{X}(\lambda)=U^{(\alpha)}(\lambda X)$, (3.9)$MV_{Y}( \lambda)=E[\lambda Y]-\frac{1}{2}\alpha V[\lambda Y], RSVM_{Y}(\lambda)=U^{(\alpha)}(\lambda Y)$ , (3.10)$MV_{Z}( \lambda)=E[\lambda Z]-\frac{1}{2}\alpha V[\lambda Z], RSVM_{Z}(\lambda)=U^{(\alpha)}(\lambda Z)$ , (3.11)where $MV_{X}$ is the mean variance value of $X.$$\alpha=0.05$$\lambda$ $MV_{X}$ $RSVM_{X}$ $MV_{Y}$ $RSVM_{Y}$ $MV_{Z}$ RSVMZ1 5.417240 5.381304 3.730000 3.729802 3.154750 3.213878210.388960 10.043808 6.920000 6.917064 5.219000 5.649037314.915160 13.521364 9.570000 9.556959 6.192750 7.511068418.995840 15.127878 11.680000 11.646280 6.076000 8.9227915 22.631000 14.163164 13.250000 13,188966 4.868750 9.9592796 25.820640 10.355244 14.280000 14.200910 2.571000 10.6717507 28.564760 4.096341 14.770000 14.712311 -0.817250 11.1008378 30.863360 -3.853309 14.720000 14.766822 -5.296000 11.2827189 32.716440 -12.796062 14.130000 14.417932-10.865250 11.25123510 34.124000 -22.26819413.000000 13.723921 -17.525000 11.0381231135.086040 -32.00848211.330000 12.742895 -25.275250 10.67257712 35. $602560$ $-41.881393$ 9.120000 11.528915 -34.116000 10.18077213 35. $673560$ $-51.819265$ 6.370000 10.129651 -44.047250 9.58558814 35. $299040$ $-61.788863$ 3.080000 8.585410 -55.069000 8.90658815 34.479000 -71.773961 -0.750000 6.929195 -67.181250 8.16018116 33.213440 -81.766643 -5.120000 5.187368 -80.384000 7.35992217 31.502360 $-91.763044-10.030000$ 3.380592 -94.677250 6.51687018 29. $345760-101.761270-15.480000$ 1.524834-110.061000 5.63995919 26. $743640-111.760395-21.470000$ $-0.367698-126.535250$ 4.73634820 23. $696000-121.759963-28.000000$ $-2.287744-144.100000$ 3.811738From the above table we can see that the RSVM is a desirable value measure whichcontains the evaluation of scale risk.4 Hedging of the Scale RiskA numerical example185Let $X$ and $W$ be given as follows,$P(\{\omega_{1}\})=0.02, P(\{\omega_{2}\})=0.5, P(\{\omega_{3}\})=0.48$ , (4.1)$X(\omega_{1})=-10,$ $X(\omega_{2})=4,$ $X(\omega_{3})=8$ ; $E[X]=5.64,$ $V[X]=8.9104$, (4.2)$W(\omega_{1})=10,$ $W(\omega_{2})=-1,$ $W(\omega 3)=-1$ ; $E[W]=-O.7800,$ $V[W]=2.3716$ . (4.3)(The distribution of $X$ is same as before. )Then we obtain$U^{(0.05)}(X)=5.381304>0, U^{(0.05)}(10X)=-22.268194<0$ (4.4)$U^{(0.05)}(W)=-0.8301<0, U^{(005)}(10W)=-9.5976<0$ . (4.5)So, $X$ may be carried out but $10X,$ $W$ and $10W$ are not carried out.On the other hand, we obtain the following results,$U^{(0.05)}(X+W)=4.7498>0, U^{(0.05)}(10X+10W)=38.4748>0$ . (4.6)Therefore, both $X+W$ and $10X+10W$ may be carried out. This means that $W$ or $10W$are valueless, but we can hedge the scale risk of $10X$ by the use of $10W.$5 Inner Rate of Risk Avertion(IRRA)5.1 Definition of the Inner Rate of Risk Avertion (IRRA)Definition 4 Let $X$ be an asset. Then a solution $\alpha$ of the following equation$U^{(\alpha)}(X)=0$ (5.1)is called the inner rate of risk avertion (IRRA) of $X$ , and denoted by $\alpha_{0}(X)$Remark 4 The larger $\alpha_{0}(X)$ is, the smaller the risk of $X$ is. So the IRRA can be a ratingindex of assets.5.2 Existence of the IRRAFor the existence of IRRA, we obtain the following result:Proposition 9 Assume that the moment generation function of a random variable $X$ con-verges, and the following conditions satisfied,$E[X]>0$ and $P(X<0)>0$ . (5.2)Then the IRRA $\alpha_{0}(X)$ of $X$ exists and is unique.1866 Concluding RemarksThe books and articles relating to this paper are listed in the References. ([1, 2, 3, 5, 6, 7, 8,9, 10, 14, 15] $)$[Problems to which the Risk-Sensitive Value Measure Method can be Applied](1) Project evaluation.(2) Evaluation of financial (or real) assets.(3) Evaluation of big projects (energy or resources exploitation).(4) Evaluation of research projects.(5) Evaluation of the intellectual property.(6) Evaluation of the credit risk.(7) Evaluation of a portfolio.(8) Evaluation of a company.The papers, [4], [11], [12], [13] are relating to those applications.References[1] Carmona, R. (ed.) (2008), Indifference Pricing: Theory and Applications, PrincetonSeries in Financial Engineering.[2] Cheridito, P., Delbaen, F. and Kupper, M. (2006), Dynamic Monetary Risk Measuresfor Bounded Discrete-Time Processes, Electronic J. Probab. 11, 57-106.[3] F\"ollmer, H. and Schied, A. (2004), Stochastic Finance (2nd Edition), Walter de Gruyer,Berlin and New York.[4] Hirose, T., Miyauchi, H. and Misawa, T. (2012), ‘Project Value Assessment of ThermalPower Plant Based on RNPV Probit Model Considering Real Option,’ Journal of RealOptions and Strategy, Vol.5, No.l, 1-18. (in Japanese)[5] Kupper, M. and Schachermayer, W. (2009), ‘Representation Results for Law InvariantTime Consistent Functions, Mathematics and Financial Economics, Vol.2, No.3, 189-210.[6] Misawa, T. (2010), ‘Simplification of Utility Indifference Net Present Value Method’,OIKONOMIKA, Nagoya City University, Vol.46, No.3, 123-135.[7] Miyahara, Y. (2006), (Project Evaluation Based on Utility Expectation,’ DiscussionPapers in Economics, Nagoya City University, No.446. 1-21. (in Japanese)[8] Miyahara, Y. (2010), ‘Risk-Sensitive Value Measure Method for Projects Evaluation,’Journal of Real Options and Strategy, Vol.3, No.2, 185-204.187[9] Miyahara, Y. (2012), Option pricing in Incomplete Markets: Modeling Based on Geo-metric Levy Processes and Minimal Entropy Martingale Measures, ICP.[10] Miyahara, Y. (2013) ‘Scale Risk and its Evaluation,’ OIKONOMIKA, Nagoya CityUniversity, Vol. 49, 45-56. (in Japanese)[11] Miyauchi, H., Hirata, N., and Misawa, T. (2011), ‘Risk Assesment for Generation In-vestment by Probit Model Simphfied UNPV Method,’ (submitted).[12] Miyahara, Y. and Tsujii, Y. $(2011),($ Applications of Risk-Sensitive Value MeasureMethod to Portfolio Evaluation Problems,’ Discussion Papers in Economics, NagoyaCity University, No.542, 1-12.[13] Miwa, M. and Miyahara, Y. (2010), ‘Real Option Approach to Evaluation of PlantsMaintenance,’ Journal of Real Options and Strategy, Vol.3, No.1, 1-23. (in Japanese)[14] Owari, K. (2009), A Note of Utility Maximization with Unbounded Random Endowment,(preprint).[15] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999), Stochastic Processes forInsurance and Finance, Wiley.Nagoya City University, Japan$E$-mail address: yoshio $\lrcorner$n@zm.commufa.jp$8B\ovalbox{\tt\small REJECT}\hslash E\star\not\in g\ovalbox{\tt\small REJECT}\yen\star$188

Evaluation of the Scale Risk (Financial Modeling and Analysis)

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Evaluation of the Scale Risk (Financial Modeling and Analysis)

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