^{1}

^{*}

^{1}

This paper fits into the research stream started by Aumann and Serrano (2008) with their index
*R*
_{AS}, and introduces a new index of riskiness called
*I*
_{θ}. In particular, our intuition moves from the observation that the
*R*
_{AS} index is defined over the set of gambles whose expected value of losses is lower than the one of gains. We then restrict the attention on investment opportunities in a proper relation with a benchmark value
and we build the index
*I*
_{θ}. The mathematical features of the new index are discussed in deepest detail, providing evidence that
*I*
_{θ} can be used by the investor in comparing risks:
*θ* plays the role of a threshold value such that once fixed in the range (0,1] ,
*I _{θ}* suggests to take into consideration a gamble (an investment, an asset)

*g*only if the ratio between the expected value of its losses and the expected value of its gains is lower than such

*θ*. Moreover,

*I*

*nests*

_{θ}*R*

_{AS}as special case when

*θ*= 1, and it satisfies both homogeneity and duality properties. In the light of these features,

*I*

_{θ}seems to satisfy the common need among practitioners for flexible index of riskiness.

Riskiness is a key issue in a broad number of situations, spanning from finance to insurance and economics. The first attempt to quantify risk is contained in [_{1} and X_{2}: while X_{1} equally assures either a gain of $600 or a loss of $500, X_{2} gives the investor equal probability either to gain $700 or to lose $490. Clearly X_{2} gives lower losses and higher gains than X_{1}. Nevertheless, X_{2} is the gamble with higher standard deviation:

A new path is now coming from the research vein started in 2008 by Aumann and Serrano [^{1} measure of riskiness that a decision maker can use to compare gambles. Suppose a gamble g to be a random variable with positive expected value and positive probability of negative values; [_{AS} which associates to every gamble g the value _{AS} exhibits a number of very appealing features, such as monotonicity with respect to first order stochastic dominance, and positive homogeneity; moreover it makes possible to reach a complete ordering of all considered gambles. However, the most important feature of the R_{AS} index mainly relates to the duality axiom, which essentially requires that if a gamble is accepted by an investor, less risk averse investors accept less risky gambles. In this light, this is an economically motivated axiom, and therefore rightly Aumann and Serrano claimed their R_{AS} index to be an economic index of riskiness. With this background, the work of Aumann and Serrano may be straightforwardly considered a very inspiring contribution that has shown many lines towards which address future research efforts. A first element of innovation, for instance has been kept by [_{FH}: whereas R_{AS} orders gambles (investments) looking at the utility function regardless of wealth, R_{FH} examines the investments looking for the critical wealth regardless of utility. Extensions towards this direction include the contributions of [_{FH} to continuous random variables and to dynamic environments; [_{AS} concept, studied an extension of the stochastic dominance order, which is equivalent to the one introduced by the index of Aumann and Serrano. Moreover, [_{AS} index. More recently [_{AS} as a special case, and allows riskiness to be examined relative to benchmark levels of (net) return other than zero: it is also provided empirical evidence that the ordering of two financial assets in terms of their relative riskiness may vary even with only small changes in the benchmark level of return. Another interesting contribution is that of [_{AS} and R_{FH} were created with a risky world in mind, but not an uncertain one; starting from this point, decision under uncertainty is modeled and a complete and objective rankings of sets of gambles is obtained, which extends the previous riskiness measures.

Moving to practical applications, [_{AS} were also provided independently one to each other in [_{AS} is the reciprocal of the adjusting coefficient that is used in the insurance risk literature as a proxy to monitor the risk of going bankruptcy. Moreover, [_{AS} index was proposed for gambles in terms of absolute outcomes, it can be applied to excess returns, since the latter can be regarded as the outcome of a zero investment strategy that consists in borrowing $1 and investing it in a risky asset for a given time span. Furthermore, [_{AS} on different bunches of mutual and hedge funds, comparing the results to those obtained with more standard risk measures. Finally [_{AS} to the lower tail of investment returns thus developing a new measure of pertinent risk for risk-averse investors.

Our paper contributes to the related literature by proposing an index of riskiness in the route of that of Aumann and Serrano: we consider gambles _{AS} index is defined over the set of gambles _{AS} ranks the gambles whose expected value of losses

Thereafter, what remains of the paper is organized as follows. Section 2 introduces and describes the basic features of the Aumann and Serrano model. Sections 3 presents some preliminary concepts useful from a mathematical viewpoint to introduce our index of riskiness. This latter is derived in Section 4. Section 5 concludes.

Following [

Consider a probability space^{2}, and with positive expected value and positive probability of negative values [

Denote by

Moreover, an investor is characterized by a utility function in the Von Neumann-Morgenstern sense [^{3} agent, and up to an additive and positive multiplicative constant, his utility function is:

Within this framework, provided that

An agent with utility function u accepts the gamble g at wealth w if:

In order to compare gambles and decide whether a gamble is more risky than another, define an index Q to be a mapping that assigns a positive real number to each gamble g: [_{AS} that satisfies both duality and homogeneity; in particular, for every gamble g,

Thus a CARA agent

Roughly, former properties characterize R_{AS} in an axiomatic way. The duality axiom gives a mathematical interpretation of the following fact: provided two agents, one uniformly no less risk averse than the other, this latter will take a fortiori a position less riskier (according to the newly introduced index) than that one the former agent will consider. In addition, the homogeneity axiom embodies the cardinal nature of riskiness, making possible to quantify it. More precisely, it makes sense to say that if g is a gamble, then 2g is not only riskier than g but it as twice as risky as g.

To conclude, R_{AS} suggests by way of the duality and homogeneity axioms an order for gambles, but from an operational point of view, one might be tempted to ask more, for instance by requiring stronger conditions to either accepting or rejecting a risky asset. Keeping this in mind, in next sections we work on the hypotheses (1) and (2), hence modifying the original model of Aumann and Serrano.

Using the notational conventions we already introduced in Section 2, we now consider the following definition.

Definition 3.1 Let

where:

By construction, in (4) we examine the ratio between the decrement in the expected utility due to all possible outcomes of

In order to discuss some mathematical features of R, we preliminary recall that g first order stochastically dominates on h if and only if:

where F_{g}, F_{h} are the cumulative distribution functions of g and h, respectively. We now introduce a theorem establishing for every utility function u and every wealth w how the measure

Theorem 1 Assume:

Proof. We set:

that gives:

If

and:

and we can conclude.

Another interesting feature of R relates to its behaviour under monotone transformations of the investor’s utility function. We summarize this result in Theorem 2.

Theorem 2 Consider two utility functions

Proof. We have:

iff:

Let:

and:

By Fubini’s theorem, (10) is equivalent to:

As u_{1} is a [strictly] concave and increasing transformation of u_{2}, we have:

with:

Note that for all

Remark 3.1 Given two utility functions u_{1}, u_{2}, corresponding to investors i_{1} and i_{2} respectively, by the Arrow-Pratt theorem we get: _{1} is a concave [strictly concave] and increasing transformation of u_{2}.

Consider now two CARA agents

Now, for every gamble g and for every

Assuming a CARA agent with wealth w and utility function

Then, we can introduce the following theorem.

Theorem 3 For every gamble

and:

Proof. From (14) it follows that

In a similar fashion:

We are now ready to introduce our index of riskiness

Definition 4.1 Consider an agent with utility function u, wealth

otherwise the agent rejects g at the level θ.

Remark 4.1 From (13), if for a gamble g we have:

Let us now illustrate Definition 4.1 with an example.

Example 4.1 Let us set to zero the initial wealth:

Note that by Theorem 2, assuming

Assume now a CARA agent, a gamble

has a unique solution iff:

Now, we define the set:

and for all

As the function

Example 4.2 Let us consider the gambles g and h already employed in the Example 4.1. The equation:

_{0.7} and I_{0.9} are not ordinally equivalent. Finally, we observe that the agent characterized by the function u defined in Example 4.1 and with wealth

We are now going to introduce a number of mathematical features shared by the

Assume

0.5 | 0.6 | 0.7 | 0.8 | 0.85 | 0.9 | 0.95 | 1 | |
---|---|---|---|---|---|---|---|---|

114.776 | 30.622 | 18.492 | 13.580 | 12.068 | 10.897 | 9.960 | 9.193 |

0.7 | 0.8 | 0.85 | 0.9 | 0.95 | 1 | |
---|---|---|---|---|---|---|

25.781 | 13.736 | 11.315 | 9.696 | 8.535 | 7.661 |

Furthermore:

Theorem 4 Consider two gambles g, h, being

・

・

Proof. Assume

Moreover, by (13), given two gambles g, h it is:

Finally, since

However, the most important property of the index _{ }is summarised in the following theorem.

Theorem 5 Let two gambles

then:

Proof. By hypothesis and by Theorem 2 we have:

therefore by Theorem 3 we have:

Theorem 5 posits a duality statement. In fact, it says that given two gambles g and h and two agents such that the first one is uniformly no less risk averse than the other, if the first agent with wealth w accepts g at the threshold θ, and ^{4}.

In this paper we presented a new index of riskiness that we obtained using the index of Aumann and Serrano (2008) as starting point, and having in mind the Gain/Loss ratio of [

In particular, our intuition moved from the observation that the R_{AS} index is defined over the set of gambles whose expected value of losses is lower than the one of gains. In practical applications, however, the investor could be more demanding, restricting the attention on investment opportunities in a proper relation with a benchmark value θ: we assumed that _{AS} as a special case for

Nevertheless, some issues need to be addressed. In particular, the risk framework we discussed in this paper still stays on the theoretical stage. Future research efforts will move basically towards two directions, by deriving efficient computational algorithms to calculate it, and by discussing some simulations studies, in order to compare