A method for determining risk aversion functions from uncertain market prices of risk

Insurance: Mathematics and Economics 47 (2010) 84–89

Contents lists available at ScienceDirect

Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

A method for determining risk aversion functions from uncertain market prices of risk Henryk Gzyl a,∗ , Silvia Mayoral b a b

Centro de Finanzas, IESA, Caracas, Venezuela Department Business Administration, Universidad Carlos III de Madrid, 28093 Getafe (Madrid), Spain

article

info

Article history: Received October 2008 Received in revised form January 2010 Accepted 29 March 2010 Keywords: Distortion function Spectral measures Risk aversion function Maximum entropy in the mean Inverse problems for noisy data

abstract In Gzyl and Mayoral (2008) we developed a technique to solve the following type of problems: How to determine a risk aversion function equivalent to pricing a risk with a load, or equivalent to pricing different risks by means of the same risk distortion function. The information on which the procedure is based consists of the market prices of the risk. Here we extend that method to cover the case in which there may be uncertainties in the market prices of the risks. © 2010 Elsevier B.V. All rights reserved.

1. Introductory remarks In a previous paper, see Gzyl and Mayoral (2008) we solved the inverse problem consisting of determining a risk aversion function φ from the knowledge of the market prices of some risks, that is of solving for φ in

πi =

1

Z

qXi (u)φ(u)du

(1.1)

0

where Xi for i = 1, . . . , M is some given collection of risks, and qXi denotes the left quantile of Xi . Also φ has to be in the class of continuous increasing functions on [0, 1]. For that we made use of a maxentropic technique introduced in Gamboa and Gzyl (1997). That technique is well suited to solved ill posed problems like (1.1). Pricing risks by means of risk distortion functions, or equivalently, by risk aversion functionals indicated in the right hand side of (1.1), is a well established technique as described in the many references quoted in Gzyl and Mayoral (2008). We were able to solve (1.1) when all prices were obtained by the same (but unknown) distorted risk function, or when we were presented with a price obtained by loading a mean. A natural question arises: How to proceed when there are uncertainties in the prices. A few typical situations are the

∗

Corresponding author. E-mail addresses: [email protected] (H. Gzyl), [email protected] (S. Mayoral). 0167-6687/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2010.03.011

following. Consider for example one given risk which is priced by different firms with different risk aversion functions. That is R1 we are given πi = 0 qX (u)φi (u)du for say i = 1, . . . , M. Or consider a given risk, say X which different agents price like πj = E [X ](1 +`i ) with `i being positive numbers which may be different for different market agents. Or assume that the pricing agent uses empirical data to compute the price, making whatever pricing method sample dependent. Regardless of the method used, there may be some overpricing or underpricing in the quoted market prices πj . A possible way to restate our problem in this case goes as follows: we are provided with a list {π1 , . . . , πM } of prices of some reference liabilities {X1 , . . . , XM }. We want to determine a risk aversion function φ ∗ and a vector ∗ such that

πi =

1

Z

qXi (u)φ ∗ (u)du + i∗ . 0

With the determined risk aversion function φ ∗ , we can compute R1 the ‘‘true’’ price π (X ) = 0 qX (u)φ ∗ (u)du of the given or any other liability X . By true we clearly mean the price computed with the reconstructed distortion function φ ∗ , which we interpret as the true distortion function. Not only that, if we interpret R1 q (u)φ ∗ (u)du as the ‘‘true’’ price of Xi , then it makes sense to 0 Xi interpret the excess price i∗ as an underpricing or overpricing of Xi with respect to its true price. φ is required to be increasing and continuous, then the related risk measure, defined by the right hand side of (1.1) is coherent. In our previous work we provided a list of basic references on

H. Gzyl, S. Mayoral / Insurance: Mathematics and Economics 47 (2010) 84–89

the subject. Here we only recall that the choice of a risk aversion function φ establishes a pricing procedure. But, there are no rules to decide how one must define the distortion function. We only know that it amounts to a re-weighting of the initial distribution of the liabilities. Sometimes, the choice of the risk aversion function depends on the generic properties that we want the risk measure to satisfy. It is thus interesting to have methods for construction of risk aversion functions from the observed prices of risk. That was the task undertaken in Gzyl and Mayoral (2008), in which references to mathematically related problems were provided. Here we extend that line of work to accommodate the case in which, for whatever reasons, there may be uncertainties in the prices. The import of the paper is not the extension of the method of maximum entropy in the mean to accommodate for the estimation of the error term, but in the development of a method to estimate those errors, which in the context of the present problem are interpreted as underpricing or overpricing. This paper is organized as follows: In Section 2 we recall the concept of a risk aversion function measure and then restate problem (1.1) and establish its discretized version as well. In Section 3 we present the extended version method of maximum entropy in the mean (MEM), which consists of a technique for transforming an ill-posed linear problem with convex constraints into a simpler (possibly unconstrained), but non-linear minimization problem. We then show how to obtain the generic form of the solutions to the discretized problem. In Section 4 we present numerical examples. Finally, Section 5 concludes the paper. 2. Preliminaries We consider a one period market model (Ω , F , P). In the simplest setup, the information about the market, that is the σ algebra F , is generated by a finite collection of random variables, F = σ (X0 , X1 , . . . , XN ), where the {Xj | j = 0, . . . , N } are the basic liabilities traded in the market. We shall model the present worth of our position by X ∈ L2 (P ), that is, essentially all random variables with finite variance. Again, we refer the reader to Gzyl and Mayoral (2008) for a review of the well known relationship between coherent and distorted risk measures, as well as the relationship between distorted risk measures and risk aversion functions. For the purpose of this paper it will suffice to recall that spectral risk measures were proposed in Acerbi (2002), and that they can be expressed as a general convex combination of the quantiles function of the risk. For actuarial applications it is convenient to change slightly the conventions used for financial risk measurement. Definition 2.1. An element φ ∈ C ((0, 1)) is called an admissible risk spectrum if 1. φ ≥ 0. 2. φ is increasing. 3. kφk =

R1 0

φ(t )dt = 1.

Comment 2.1. Actually, for the general definition it would suffice to consider φ ∈ L1 ((0, 1)), but since we want our φ ’s to be associated to coherent risk measures, we need them to be continuous and increasing. Definition 2.2. Let φ ∈ L1 ([0, 1]) be an admissible risk spectrum. The risk pricing measure

ρφ (X ) =

1

Z

qX (u)φ(u)du 0

is called the spectral risk measure generated by φ .

85

The function φ is called the risk aversion function and assigns, in fact, different weights to different p-confidence levels of the left tail. Any rational investor can express her subjective risk aversion by drawing a different profile for the weight function φ . The spectral risk measures are a subset of the coherent risk measures, as Acerbi proves. Specifically, a spectral measure can be associated with a coherent risk measures that has two additional properties, law invariance and comonotone additivity. Below we present a short list of risk aversion functions that we shall use to generate risk prices for Section 4. The properties of these risk aversion functions are discussed in Wang (1996) and Wang (2000). 1. Dual-power risk aversion function:

φ(u) = ν uν−1 ;

ν > 1.

(2.1)

2. Proportional Hazard risk aversion function:

φ(u) =

1

γ

1

( 1 − u) γ − 1 ;

γ > 1.

(2.2)

3. Wang’s risk aversion function:

φα (u) = e−αΦ

−1 (u)−α 2 /2

.

(2.3)

It is also of interest to note that Conditional Value at Risk can be thought of as a spectral risk measure defined by the risk aversion function:

φ(p) =

1 1−α

1{α≤p≤1} .

(2.4)

We devote the remainder of this section to restating our problem as a problem consisting of solving a Fredholm equation, and explain how to discretize that problem. 2.1. Problem statement As mentioned above, our basic problem may be stated as: Given the market prices πi of a finite collection of risk positions Xi for i = 1, . . . , M, find a function spectral risk aversion function φ and estimate mispricing (or noise) components i such that

πi = ρφ (Xi ) =

1

Z

qXi (u)φ(u)du + i ,

i = 1, . . . , M

(2.5)

0

R1

where to accommodate the condition 0 φ(u)du = 1 we choose XM such that qXM (u) = 1 and πM = 1. In general, how to model noise depends on the application at hand. In most engineering or natural science applications it makes sense to assume that the additive noise may be of any sign and of any size, and it is related to the measurement process, in actuarial sciences the sensible interpretation is that the uncertainty in the price (or the noise) is produced by the pricing agent, and that it is a fraction of the actual price. Thus we propose to model the noise by a quantity taking values in a bounded interval, that is, we assume that  ∈ Kn = [e1 , e2 ]M , where the subscript n stands for noise. Below we shall explain how to determine e1 , e2 numerically, and in the last section we offer further remarks on related issues. The maximum entropy method proposed in Gamboa and Gzyl (1997) to solve Fredholm equations like the one at the beginning of the section can be easily modified to tackle (2.5). See Gzyl (2003) for a generic description of the method of maximum entropy in the mean and its application for solving linear inverse problems. We shall explain the extension in the context of the discretization procedure carried out to actually solve this problem in practice. For that we consider a partition of [0, 1] at points uj = j/N. The choice of N depends on the known variability of qX (u) in [0, 1]. Since we are assuming that we know qX and we are reconstructing a continuous function, it is more convenient to take a uniform grid.

86

H. Gzyl, S. Mayoral / Insurance: Mathematics and Economics 47 (2010) 84–89

In case qX were known empirically, and constant on intervals, it would make sense to adapt the partition to the specification of qX . Let us define the M × N matrix B by setting Bi,j = qXi (uj )/N, for i = 1, . . . , M and j = 1, . . . , N. Set φ(aj ) = φj , where aj = 21 (uj + uj−1 ) and u0 = 0. With all this, the problem (2.5) can be restated as: Solve

π = Bφ + ;

φ ∈ K0

(2.6) N

where the constraint set K0 ⊂ R is a convex set defined in this case by K0 = {(φ1 , . . . , φN ) | φ1 < · · · < φj < φj+1 < · · · < φN }. To simplify the description of the constraints, we set φ1 = ψ1 , φ2 = ψ1 + ψ2 , . . . and φN = ψN + · · · + ψ1 , or φ = C ψ where C is the obvious lower diagonal matrix describing the change of coordinates. Setting A = BC we can restate our discretized problem as

π = Aψ + ;

ψ ∈ Ks ;  ∈ Kn

(2.7)

where now the constraint set Ks (s for signal) is a closed convex (bounded or unbounded) subset of RN++ , i.e., the positive orthant in RN , and Kn is as explained below (2.5). Clearly, once the vector ψ is at hand, the φ is easily recovered. In principle there are no theoretical limitations on N and M. In practice this depends on the nature of the matrix A. Actually, since we shall end up minimizing a convex function of M variables, a numerical difficulty may arise from the flatness of the function near its minimum. 2.2. The extension procedure Consider 9 to be the N + M-dimensional vector obtained by ˆ = [A I ] be the M × (N + M )-matrix transposing (ψ, ) and let A obtained by adjoining a M × M-identity matrix to A. The restated problem now consists of solving

π = Aˆ 9,

9∈K

(2.8)

where K = Ks × Kn is the full constraint space. Once a pair ψ∗ , ∗ are found, Aψ∗ could be thought of as the true price and ∗ as the overprice or underprice of the market price relative to the true price. If (2.8) were a linear regression problem, ψ ∗ would stand for the estimated parameters whereas ∗ would stand for the residuals. An interpretation for Kn = [e1 , e2 ]M , is that it is a box that we can place about a point Aψ ∗ in the image AK s such that π = Aψ∗ + ∗ for some ∗ in Kn . 3. The method of maximum entropy in the mean In order for this note to be self contained, we borrow some of the material developed in Gzyl and Mayoral (2008). The difference between the two cases lies in the nature of the matrix Aˆ introduced in the previous section, not in the formal development of the method which comes below. 3.1. Basic methodology The method of maximum entropy in the mean (MEM) is a technique for transforming an ill-posed linear problem with convex constraints like (2.8) into a simpler (possibly unconstrained) but non-linear minimization problem. The number of variables in the auxiliary problem being equal to the number of equations in the original problem, M in our case. To carry out the transformation one thinks of the ψj there as the expected value of a random variable Ψj with respect to some measure Q which is to be determined. The basic datum is a sample space (Ω , F ) on which Ψ is to be de-

fined. In our setup the natural choice is to take Ω = K , F = B (K ), the Borel subsets of K , and Ψ = idK as the identity map. To continue we need to select a reference or prior (but not in the Bayesian sense) measure dQ o (ξ ) on (Ω , F ). The only restriction that we impose on it is that the closure of the convex hull of supp(Q ) is K . This prior measure embodies knowledge that we may have about ψ . To get going we define the class

ˆ Q [9] = π} P = {Q | Q  Q o ; AE

(3.1)

and observe now that the algebraic problem (2.7) is transformed into the problem consisting of finding a measure Q ∈ P. Note that dQ for any Q ∈ P having a strictly positive density ρ = dQ o , then EQ [9] ∈ int(K ). This follows since expectation is basically a linear convex combination. The procedure to explicitly produce such Q ’s is known as the method of maximum entropy, exponential tilting or the Esscher transform. A couple of references with applications and further references are Vyncke et al. (2003) and Wang (2006). The first step of the procedure consists of assuming that P 6= ∅, which amounts to say that our problem has a solution and define SQ o : P → [−∞, ∞) by the rule SQ o ( Q ) = −



Z ln Ω



dQ

dQ

dQ o



(3.2)



is Q -integrable and SQ o (Q ) = −∞ otherwise. This entropy functional is concave on the convex whenever the function ln

dQ dQ o

set P. To guess the form of the density of the measure Q ∗ that maximizes SQ o is to consider the class of exponential measures on Ω defined by e−hλ,AΨ i ˆ

dQλ =

Z (λ)

dQ o

(3.3)

where the normalization factor is Z (λ) = EQ o [e−hλ,AΨ i ] = ˆ

Z

e−hλ,AΨ i dQ o . ˆ

Ω

Here λ ∈ RM . If we define the dual entropy function

Σ (λ) : D (Q o ) → (−∞, ∞] by the rule

Σ (λ) = ln Z (λ) + hλ, πi

(3.4)

or Σ (λ) = ∞ whenever λ 6∈ D (Q ) ≡ {µ ∈ R | Z (µ) < ∞}. It is easy to prove that, Σ (λ) ≥ SQ (Q ) for any λ ∈ D (Q o ), and any Q ∈ P. Thus if we were able to find a λ∗ in the interior of D (Q o ) such that Qλ∗ ∈ P, we are done. To find such a λ∗ it suffices to minimize (the convex function) Σ (λ) over (the convex set) D (Q ). Clearly, when the minimum is reached in the interior of D (Q o ), the vanishing of the gradient ∇ Σ (λ∗ ) is equivalent to Qλ∗ ∈ P. o

M

3.2. A solution scheme As is clear from the statement of (2.7), the actual implementation scheme depends on the assumptions that we place on the constraint set K . Here we shall develop one of the many possible alternatives consisting of assuming K to be bounded. This choice is adequate when we assume the φj are bounded, which is a natural assumption under simple regularity assumptions on φ , like continuity for example. Thus let us assume that for appropriate a and b, we know that a ≤ ψj ≤ b∀j. This amounts to assuming that K = [a, b]N × [e1 , e2 ]M . We should add that the N a0 s, b0 s, and the M e01 s and e2 could be assumed different with no problem at all. Once this aspect of the modeling process is decided, the other

H. Gzyl, S. Mayoral / Insurance: Mathematics and Economics 47 (2010) 84–89

87

degree of freedom that one has corresponds to the choice of the reference measure Q o . Since any point in [a, b] is a convex combination of the end points, a simple assumption consists of putting dQ o (ξ ) =

N Y

(pδa (dξj ) + (1 − p)δb (dξj ))

j =1

×

M Y

(qδe1 (dηj ) + (1 − q)δe2 (dηj )).

(3.5)

j =1

We use the standard notation δa (dx) to denote the unit point mass measure concentrated at a (the Dirac measure at a). The parameters p, q are both 0 < p, q < 1 and they reflect the possible bias of the ψj0 s or the i0 s towards one of the ends of the intervals [a, b] or [e1 , e2 ]. When no bias is assumed, one chooses p = q = 1/2, which is what we will do below. The next step consists of computing the normalization factor Z (λ). Clearly N Y

Z (λ) =

ζs ((At λ)j )

j =1

M Y

Fig. 1. Original and reconstructed φ ’s.

ζn ((I λ)j ),

j=1

where ζs (τ ) and ζn (τ ) denote, respectively, the Laplace transforms of pδa (dx) + (1 − p)δb (dx) and qδe1 (dηj ) + (1 − q)δe2 (dηj ), that is, for example

ζs (τ ) =

Z

e−xτ pδa (dx) + qδb (dx) = pe−aτ + qe−bτ .

The following step has to be carried out numerically. It consists of finding the minimizer λ∗ in (3.4). Once that is accomplished, it is easy to see that the maxentropic reconstruction ψ ∗ and ∗ are given by

ψj∗ = ap∗j + b(1 − p∗j ),

for j = 1, . . . , N

j∗ = e1 q∗j + e2 (1 − q∗j ),

for j = 1, . . . , M ,

(3.6)

where, respectively, for j = 1, . . . , N, and k = 1, . . . , M, we have p∗j =

t ∗ t ∗ e−a(A λ )j + e−b(A λ )j

e−e1 λk

qk =

e−e1 λk + e−e2 λk ∗

,

and

!

∗

∗

!

t ∗ e−a(A λ )j

∗

.

The φj∗ must be recovered from the ψj∗ as described at the end of Section 2.1. Notice that the MEM procedure has shifted the parameters of the distribution. That is the post-data, maximum entropy distribution Q ∗ is different from the prior (reference) measure Q o in two respects. First, the components of ψ are no longer independent (the distribution is not a product of 1-dimensional distributions), but are independent of the  ’s and second, the original bias in the choice of p and q has been modified, as shown in last identity above. Before we describe the numerical examples, let us mention that the reconstruction errors mentioned below are just mean square errors |π − Bφ∗ |. Here is where the effectiveness of the method is apparent, but we shall not forget that we are finding the solution to an inverse problem with infinitely many solutions. The method provides us with a very good solution. Also, there is no interpretative relationship between the reconstruction error and the price indeterminacy error. This error, which may be large or small, is reconstructed very precisely by the method. 4. Numerical examples In this section we examine several examples that illustrate the many possibilities of the method. But first a word about the

procedure of minimization of the dual entropy function Σ (λ) in (3.4). Since we do not know beforehand whether π ∈ AKs or not, we have to devise a solution method that includes that possibility. Thus, the minimization procedure includes a loop that evolves step-wise from e1 = e2 = 0 to positive values of e2 and to negative values of e1 in small steps. If a solution were obtained in the case e1 = e2 = 0, this would mean that the observed prices are consistent, that is, that they can be thought of as having been valued by the same load or risk aversion function. Apart from this remark, it is worth mentioning that since Σ (λ) usually is very flat near its minimum, we use an adaptive gradient method to obtain its minimizer λ∗ . Example 4.1. In order to double check the first part of the loop, that is if our procedure can detect whether the prices of different risks are calculated by an agent using the same (but unknown to us) risk aversion function, we considered three risks, X1 ∼ U (0, 1), X2 ∼ Gamma(2, 1) and a generalized Pareto X3 ∼ GP(1/3, 1, 1). We assume that each agent uses the same distortion function to value the three risks, but different agents use different distortion functions. In Table 1 we present, both the market price π and the reconstruction error |πi −(Bφ∗ )i | in each case. For example, in the second row in the first column, we display the market value π (X2 ) of a Gamma distributed risk, priced by means of a Wang type risk aversion function, and next to it the absolute value of the difference between the market price and the price computed with the reconstructed φ∗ . The mispricing vector  is 0 in this case, which correctly suggests that agents used the same risk aversion function to price all risks. In Fig. 1 we display both the original and the reconstructed risk aversion function for each agent. Clearly the agreement is rather good. We carried out a numerical experiment consisting of computing the price of a new liability, say a U (0, 2)-distributed risk, computed with the same three risk aversion functions which we used to compute the prices in Table 1. We also computed the price of that risk with the reconstructed φ ∗ . The prices obtained, and absolute vales of the differences in price are: πWang = 0.98 and |πWang − π ∗ | = 0.36 × 10−3 ; πPH = 1.015 and |πPH − π ∗ | = 0.13 × 10−4 ; and to finish πDP = 1.24 and |πDP − π ∗ | = 0.00012. For the coming example, recall that the outcome of the reconstruction process consists of the vector (φ∗ , ∗ ), and that the components of the vector Bφ∗ may to be interpreted as the ‘‘true’’ prices of the risks, whereas the components of the reconstructed error vector ∗ may be interpreted as the overpricing or underpricing of each risk with respect to the ‘‘true’’ price.

88

H. Gzyl, S. Mayoral / Insurance: Mathematics and Economics 47 (2010) 84–89

Table 1 Reconstruction errors. Distortion

Wang

Uniform Gamma Pareto

PH

π

Error

0.51 2.5 2.56

0.14 × 10 0.35 × 10−10 0.22 × 10−10 −10

DP

π

Error

0.55 2.25 2.43

0.22 × 10 0.29 × 10−11 0.31 × 10−11 −11

π

Error

0.64 2.83 2.79

0.71 × 10−10 0.02 × 10−10 0.32 × 10−10

Table 2 Reconstruction errors. Distortion

Wang

Uniform Gamma Pareto

PH Error

π

Error

π

Error

0.52 2.38 2.62

0.001 × 10−10 0.001 × 10−10 0.02 × 10−10

0.56 2.07 2.36

0.001 × 10−10 0.54 × 10−10 0.69 × 10−10

0.65 2.48 2.66

0.77 × 10−10 0.70 × 10−10 0.70 × 10−10

Table 3 Overvaluing-undervaluing amounts in observed market prices. Distortion

Uniform Gamma Pareto

DP

π

Wang

PH

DP



%



%



%

−0.05

−11.73

−0.03 −0.27 −0.19

−5.96 −12.99 −8.03

0.1 0.27 0.17

15.15 10.88 7.12

0.09

4.02

−0.08

−3.28

Fig. 2. Reconstructed φ ∗ ’s from risk aversion functions.

Example 4.2. Let us now consider identical risks, each of them valued with different risk aversion function by different agents, that is, each of the three agents values the same risk (a Gamma distributed risk) using a different risk aversion function, and in each case the observer sees three different market prices for the same risk. For Table 2 we consider the same liabilities as for Table 1, but this time the table has to be read horizontally. In the second row we list the observed prices as well as the reconstruction error, which this time is |πi − (Bφ∗ )i − i∗ |. The range in which the different  ’s fall is, respectively, (−0.1, 0.1) for the uniform distribution, (−0.27, 0.27) for the gamma, and (−0.19, 0.19) for the Pareto. In practice this would suggest that different agents use different risk aversion functions to price the same risks. Some of them undervalue the risk and some overvalue the risk (if we agree to interpret Bij φ ∗ (j) as the correct price of the i-th risk). In Table 3 we display the following information. In each row, we provide the reconstructed excess prices i ’s, and we provide as well the percent changes (πi − i )/πi . For example, when the risk is Gamma distributed and valued with a dual power risk

aversion function, the last term in the second row asserts that the market price is 0.27 units higher that the true price, (that is, 2 = π2 − (A(φ)∗ )i = 0.27), and that the overprice is 10.88% of the true price ((π2 −2 )/π2 = 10.88%). This results may be related to the critiques made by Wang in Wang (2000), indicating that the dual power and the proportional risk aversions do not satisfy all necessary properties to determine pricing procedures. This is clearly displayed in the fourth and sixth columns of Table 3. In Fig. 2, the panel on the left displays the reconstructed risk aversion function φ(j) when the data consists of the risk prices of the same liability provided by three market agents, each of whom uses one of the three risk aversion functions displayed in the right hand panel. For example, φgamma denotes the risk aversion function that best fits the prices provided by three agents, each of whom respectively uses a risk aversion function of the Wang, proportional hazard or of the dual power types to value a liability of the gamma type. That is the qualitative behavior of φ ∗ follows the qualitative behavior of the (unknown) risk aversion function used to generate the risk prices. Example 4.3. For the following example, the prices that we observe in the market are obtained by loading the mean value of a liability, that is we set π (X ) = E [X ](1 + `), where ` denotes the load. The numerical experiment was conducted for the same list of liabilities, and the loads considered in each case were of three types. ‘‘Small’’ loads: (0.05, 0.01, 0.1); ‘‘medium’’ loads: (0.5, 0.45, 0.4); and ‘‘heavy’’ loads: (1.1, 1, 0.9). That is, each risk price was loaded in three different ways, and then the reconstruction algorithm was carried out. The reconstructed φ ∗ ’s are displayed in Fig. 3. For example, the upper left panel displays the reconstructed risk aversion functions that are equivalent to loading the three different risks with small loads. That is, each curve displays the equivalent risk aversion function that best fits the price of a risk with three small loads. We double checked for consistency that when the loads were very small (` ∼ 0.0001) the φ ∗ ’s were almost equal to 1, or in other words, there was no distortion in the prices. To obtain those pictures, the reconstruction errors were all of the order of 10−10 . To display at least one table with the reconstruction errors, the value of the excess prices and percent excess price, we chose to display the case of the generalized Pareto liability priced with medium loads, just because it has the heaviest tail. We display the numbers in Table 4. The results were similar in all the cases we analyzed. Regarding the present example, we should add that the excess prices obtained in the application of the reconstruction procedure, are consistent with the loads applied to the prices. That is, the higher over-pricing was automatically assigned to the price with the largest load factor, and the smallest under-pricing was assigned to the risk with the smallest load factor. This behavior is obtained regardless of the nature of the risk and the size of the load.

H. Gzyl, S. Mayoral / Insurance: Mathematics and Economics 47 (2010) 84–89

89

Table 4 Results for a Pareto liability with medium loads.

δ = 0.5 δ = 0.45 δ = 0.4

π

Error



%

3.30 3.11 3.14

0.45 × 10−10 0.19 × 10−10 0.86 × 10−10

0.10 −0.003 −0.10

−0.09 −3.39

3.21

Table 5 Market prices, reconstruction errors and excess prices.

Uniform Gamma Pareto

π

Error



0.50 2.17 0.79

0.02 × 10 0.03 × 10−8 0.61 × 10−8 −8

% 0.003

0.063

−0.20

−9.38

0.22

27.69

Fig. 4.

Fig. 3. Reconstructed φ ∗ ’s for different risks and different loads.

An interesting of our method is the following. If P consequence we interpret Bij φ ∗ (j) as the ‘‘true’’ price of the i-th risk, then j P ∗ j Bij φ (j) − E [Xi ] will be the ‘‘true’’ load. In all experiments we carried out, this was independent of the risk, but varied with the size of the load originally used to compute the price.

valued is known to us, and that it is the same as that used by the pricing agent. After seeing how the method works, some comments regarding the issue of how to model the noise are appropriate. On one hand, it is possible to obtain a ‘‘noiseless’’ solution even in the case of uncertain prices. Thus if we applied the present method to the data used in our previous paper, we would have ended up with the same reconstructions. On the other hand, if we preassign a possible range for the noise, we may obtain a risk aversion function and an estimated noise, even if there was no uncertainty in the prices. All this is due to the nature of the inverse problem (2.5) which admits infinitely many solutions. That is why we chose the procedure outlined at the beginning of Section 4, namely, to start assuming that there is no mispricing and check for solutions. When no solution exists for a zero mispricing, we proceed as described. We hope we have convinced the reader of the potential usefulness of the method, and have provided an interesting enough interpretation of the outputs of the method, that is, to think of Bφ∗ as the vector of ‘‘true’’ risk prices and to think of ∗ as the vector of excess prices. Acknowledgements

Example 4.4. To conclude this list of examples, in Table 5 we present the results of applying the reconstruction method to obtain a risk aversion function when the data consists of the market prices of three risks, the same as in the second example, each of them valued with a different distortion function (the uniform with a Wang, the gamma with a proportional hazard and the Pareto with the dual power). Again, the fact that a non-zero excess price vector is obtained suggests that prices were not obtained from a unique risk aversion function. We list the excess prices obtained and percentage differences with respect to market prices. For example, the entries in the third and last columns of the third row are the overprice (with respect to the reconstructed price) in the market price of a Pareto distributed risk, as well as the percentage by which it differs from the market price. The obtained distortion functions are displayed in Fig. 4.

5. Concluding remarks One important methodological remark has to be kept in mind: to implement the method we assume that the statistical nature (that is, the true distribution function) of the risk that is being

We want to thank the referee’s comments on the first draft of this paper, which helped us to improve our presentation considerably. The research of the second author was partially funded by MEyC Grant CO2009-10796. References Acerbi, C., 2002. Spectral measures of risk: a coherent representation of subjective risk aversion. Journal of Banking and Finance 7, 1505–1518. Gamboa, F., Gzyl, H., 1997. Maxentropic solutions of linear Fredholm equations. Mathematical and Computer Modelling 25, 23–32. Gzyl, H., 2003. Maximum entropy in the mean: a useful tool for constrained linear problems. (A tutorial). In: Bayesian Inference and Maximum Entropy, vol. CP 695. Am. Inst. Phys., pp. 361–385. Gzyl, H., Mayoral, S., 2008. Determination of risk measures from market prices of risk. Insurance: Mathematics and Economics 43, 437–443. Vyncke, D., Goovaerts, M., De Schepper, A., Kaas, R., Dhaene, J., 2003. On the distribution of cash flows using Esscher transforms. Journal of Risk and Insurance 70, 563–575. Wang, S.S., 1996. Premium calculation by transforming the layer premium density. ASTIN Bulletin 26, 71–92. Wang, S.S., 2000. A class of distortion operators for pricing financial and insurance risks. The Journal of Risk and Insurance 67 (1), 15–36. Wang, S.S., 2006. Normalized exponential tilting: pricing and measuring multivariate risks. North American Actuarial Journal 11, 89–99.