Evaluation of the Scale Risk
Yoshio Miyahara
Nagoya City University

1Introduction

We study the evaluation problem of the scale risk. The method we have adopted is the risk
sensitive 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 method
to 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 properties 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 random variables, then the risk sensitive value measure(RSVM) on X is the following functionaldeﬁned on X
1
U (α) (X) = − log E[e−αX ], (α > 0),(2.1)α
where α is the risk avertion parameter.
Remark 1 In the above deﬁnition, X is supposed to be the random present value of a cash
ﬂlow or a return of some asset.

2.2

Properties of the Risk-Sensitive Value Measure

We ﬁrst remark the following facts.
Proposition 1 (i) The following approximation formula holds true:1
U (α) (X) = E[X] − αV [X] + · · · .
2(2.2)(ii) If X is Gaussian, then it holds that1
U (α) (X) = E[X] − αV [X].
21(2.3)

2.2.1

Concave Monetary Value MeasureDeﬁnition 2 (concave monetary value measure) A function v(·) deﬁned on a linear
space 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(0) = 0,(ii) (Monetary property) : v(X + m) = v(X) + m, where m is non-random,(Remark: (i) + (ii) → v(m) = m),(iii) (Monotonicity) : If X ≥ Y , then v(X) ≥ v(Y ),(iv) (Concavity) : v(λX + (1 − λ)Y ) ≥ λv(X) + (1 − λ)v(Y ) for 0 ≤ λ ≤ 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): ∀λ ∈ R+ , v(λX) = λv(X).
We next notice an important property of a concave monetary value measure.
Proposition 2 (global concavity) A concave monetary value measure v(·) satisﬁes thefollowing global concavity condition.
(iv) ’ (global concavity) :v(λX + (1 − λ)Y ) ≤ λv(X) + (1 − λ)v(Y ) for λ ≤ 0 or λ ≥ 1Proposition 3 Let v(·) be a concave monetary value measure. Then, for a ﬁxed pair (X, Y ),
ψX,Y (λ) = v(λX + (1 − λ)Y ) is a concave function of λ.
Setting Y = 0 in this proposition, we obtain the following result:Corollary 1 Let v(·) be a concave monetary value measure. Then ψX (λ) = v(λX) is aconcave function of λ and ψX (0) = 0.
From this corollary we obtain the following concept of “Optimal Scale.”[Optimal Scale]
Let v(·) be a concave monetary value measure, and assume that v(X0 ) > 0 for some ﬁxed
random variable X0 . If v(λX0 ), λ > 0, is an upper bounded function of λ, then we can ﬁnd
the maximum point λ̄. This value λ̄ is the optimal scale of X0 .
2.2.2

Utility Indiﬀerence Value

For a utility indiﬀerence value we obtain the following result:
Proposition 4 Let u(x) be a utility function deﬁned on (−∞, ∞) and satisfy the usual properties of a utility function. Then the indiﬀerence value v(X) determined by the followingequation
E[u(−v(X) + X)] = u(0) = 0(2.4)
is a concave monetary value measure.
2Remark 3 An indiﬀerence value does not satisfy the following positive homogeneity condition in general.
(Positive Homogeneity): ∀λ ∈ R+ , v(λX) = λv(X).
Proposition 5 U (α) (X) is the indiﬀerence value of the exponential utility function:uα (x) =)1(1 − e−αx ,α−∞ < x < ∞(α > 0).

(2.5)

Corollary 2 U (α) (X) is a concave monetary value measure.
Corollary 3 U (α) (λX) is a concave function of λ.
2.2.3

Optimal Scale

From the fact that U (α) (λX) is a concave function of λ, 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 λ (> 0) is small, U (α) (λX) > 0, andlim U (α) (λX) = −∞.

λ→∞(2.7)(ii) The optimal scale λopt isCXλopt =,α[]where CX is a solution of E Xe−CX X = 0.
2.2.4

α > 0,(2.8)

Independence-Additivity Property

Deﬁnition 3 (Independence-Additivity) If a value measure v(·) satisﬁese) (independence-additivity): υ(X + Y ) = υ(X) + υ(Y ) if X and Y are independent,
then v(·) 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.
3Proposition 8 Let v(x) be an indiﬀerence value determined by a utility function u(x) which
is of C (2) -class, increasing, concave, and normalized such as u(0) = 0, u0 (0) = 1, and u00 (0) =α. Then, if v(x) has the independence-additivity property, u(x) is of the following formu(x) = uα (x) =2.3)
1(1 − e−αx .
α(2.9)

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 α.
(3) The optimal scale of a project can be discussed.
(4) The RSVM has the independence-additivity property, and the RSVM is the almost only
one 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 only
one which has this property in the set of all utility indiﬀerence values.

3

Evaluation of the Scale Risk

3.1

What is the Scale Risk

Let X be a return for an investment of I. We suppose that the return for the investment λIis λX. Assume that E[X] > 0 and P (X < 0) > 0. If λ(> 0) is small then the investment λI
may be positively valued. But if λ is very large, then a very big loss may happen and so theinvestment λI may be negatively valued. This is the “scale risk.”

3.2

Numerical Example

Let X, Y, Z be random variables whose distributions areP (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 is
between X and Z. Remak here also that
E[X] > E[Y ] > E[Z](3.7)

V [X] < V [Y ] < V [Z]

(3.8)

4

We calculate the values of λX, λY and λZ. In the following table,
M VX (λ) = E[λX] − 12 αV [λX], RSV MX (λ) = U (α) (λX),(3.9)

M VY (λ) = E[λY ] − 21 αV [λY ],RSV MY (λ) = U (α) (λY ),(3.10)M VZ (λ) = E[λZ] − 21 αV [λZ],

RSV MZ (λ) = U (α) (λZ),(3.11)

where M VX is the mean variance value of X.
α = 0.05
λM VX
RSV MX
M VY
RSV MYM VZ RSV MZ
1 5.417240
5.381304 3.730000 3.729802
3.154750 3.213878
2 10.388960 10.043808 6.920000 6.917064
5.219000 5.649037
3 14.915160 13.521364 9.570000 9.556959
6.192750 7.511068
4 18.995840 15.127878 11.680000 11.646280 6.076000 8.922791
5 22.631000 14.163164 13.250000 13.188966 4.868750 9.959279
6 25.820640 10.355244 14.280000 14.200910 2.571000 10.671750
7 28.564760
4.096341 14.770000 14.712311 -0.817250 11.100837
8 30.863360 -3.853309 14.720000 14.766822 -5.296000 11.282718
9 32.716440 -12.796062 14.130000 14.417932 -10.865250 11.251235
10 34.124000 -22.268194 13.000000 13.723921 -17.525000 11.038123
11 35.086040 -32.008482 11.330000 12.742895 -25.275250 10.672577
12 35.602560 -41.881393 9.120000 11.528915 -34.116000 10.180772
13 35.673560 -51.819265 6.370000 10.129651 -44.047250 9.585588
14 35.299040 -61.788863 3.080000 8.585410 -55.069000 8.906588
15 34.479000 -71.773961 -0.750000 6.929195 -67.181250 8.160181
16 33.213440 -81.766643 -5.120000 5.187368 -80.384000 7.359922
17 31.502360 -91.763044 -10.030000 3.380592 -94.677250 6.516870
18 29.345760 -101.761270 -15.480000 1.524834 -110.061000 5.639959
19 26.743640 -111.760395 -21.470000 -0.367698 -126.535250 4.736348
20 23.696000 -121.759963 -28.000000 -2.287744 -144.100000 3.811738

From 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 Risk

A numerical example

5Let X and W be given as follows,
P ({ω1 }) = 0.02, P ({ω2 }) = 0.5, P ({ω3 }) = 0.48,
X(ω1 ) = −10, X(ω2 ) = 4, X(ω3 ) = 8;
W (ω1 ) = 10, W (ω2 ) = −1, W (ω3 ) = −1;(4.1)

E[X] = 5.64, V [X] = 8.9104,(4.2)

E[W ] = −0.7800, 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 (0.05) (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 α of the following equation
U (α) (X) = 0(5.1)

is called the inner rate of risk avertion (IRRA) of X, and denoted by α0 (X) .
Remark 4 The larger α0 (X) is, the smaller the risk of X is. So the IRRA can be a rating
index of assets.

5.2Existence of the IRRA

For the existence of IRRA, we obtain the following result:
Proposition 9 Assume that the moment generation function of a random variable X converges, and the following conditions satisﬁed,
E[X] > 0andP (X < 0) > 0.

Then the IRRA α0 (X) of X exists and is unique.

6(5.2)

6Concluding Remarks

The 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.

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Series in Financial Engineering.
[2] Cheridito, P., Delbaen, F. and Kupper, M. (2006), Dynamic Monetary Risk Measures
for Bounded Discrete-Time Processes, Electronic J. Probab. 11, 57-106.
[3] Föllmer, 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 Thermal
Power Plant Based on RNPV Probit Model Considering Real Option,’ Journal of Real
Options and Strategy, Vol.5, No.1, 1-18. (in Japanese)[5] Kupper, M. and Schachermayer, W. (2009), ‘Representation Results for Law Invariant
Time Consistent Functions, Mathematics and Financial Economics, Vol.2, No.3, 189210.
[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,’ Discussion
Papers 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.
7

[9] Miyahara, Y. (2012), Option pricing in Incomplete Markets: Modeling Based on Geometric 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 Investment by Probit Model Simpliﬁed UNPV Method,’ (submitted).
[12] Miyahara, Y. and Tsujii, Y. (2011),‘ Applications of Risk-Sensitive Value Measure
Method to Portfolio Evaluation Problems,’ Discussion Papers in Economics, Nagoya
City University, No.542, 1-12.
[13] Miwa, M. and Miyahara, Y. (2010), ‘Real Option Approach to Evaluation of Plants
Maintenance,’ 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 for
Insurance and Finance, Wiley.
Nagoya City University, Japan
E-mail address: yoshio m@zm.commufa.jp
名古屋市立大学

8宮原孝夫