# A Probability Metrics Approach to Financial Risk Measures

A Probability Metrics Approach to Financial Risk Measures

Note that the preference order is defined by the economic agent; various agents may have different preference orders. In the equivalent numerical representation, it is the utility function u x which characterizes U and, therefore, determines the preference order. As we explained, lotteries may be discrete, continuous, or mixed. If the lottery is discrete, then the payoff is a discrete random variable and equation 3. The formula for the fair value in the St Petersburg Paradox given by Daniel Bernoulli has the form of equation 3. Thus, the St Petersburg Paradox is resolved by calculating the fair value through the expected utility of the lottery.

If the lottery is such that it has only one possible outcome i. For example, concerning certain prospects, all investors who prefer more to less are called non-satiable. If there are two prospects, one with a 48 3. The outcomes x and y can be interpreted as the payoffs of two opportunities without an element of uncertainty, i.

Suppose that the investor gains a lower utility from a venture with some expected payoff and a prospect with a certain payoff, equal to the expected payoff of the venture: that is, the investor is risk averse. By definition, if a utility function satisfies 3. Some common examples of utility functions are listed below. For example, all non-satiable investors have non-decreasing utility functions and all risk-averse investors have concave utility functions. Thus, different classes of investors can be defined through the general unifying properties of their utility functions.

Suppose that there are two portfolios X and Y, such that all investors from a given class do not prefer Y to X. This means that the probability distributions of the two portfolios differ in a special way that, no matter the particular expression of the utility function, if an investor belongs to the given class, then Y is not preferred by that investor. In this case, we say that portfolio X dominates portfolio Y with respect to the class of investors.

Such a relation is often called a stochastic dominance relation or a stochastic ordering. Since it is only a relationship between the probability distributions of X and Y which determines whether X dominates Y for a given class of investors, it appears possible to obtain a criterion characterizing the stochastic dominance, involving only the cumulative distribution functions c.

Thus, we are able to identify by only looking at distribution functions of X and Y if either of the two portfolios is preferred by an investor from the class. This section discusses such criteria for three important classes of investors.

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Denote by U1 the set of all utility functions representing nonsatiable investors: that is, the set contains all non-decreasing utility functions. The condition in terms of the c.

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Figure 3. If X and Y describe the payoff of two portfolios with distribution functions such as the ones plotted in Figure 3. Consequently, if X is preferred by all non-satiable investors, then it is preferred by the investor with 52 3. In general, the converse statement does not hold. If the expected payoff of a portfolio exceeds the expected payoff of another portfolio, it does not follow that any non-satiable investor would necessarily choose the portfolio with the larger expected payoff.

This is because the inequality between the c. In effect, there will be non-satiable investors who would choose the portfolio with the larger expected payoff and other non-satiable investors who would choose the portfolio with the smaller expected payoff. It depends on the particular expression of the utility function: for example, whether it is a logarithmic or a power utility function. This conclusion also holds for the subcategory of the non-satiable investors who are also risk-averse. Therefore, the condition in 3. This is demonstrated in the following example.

The distribution functions of X and Y do not satisfy 3. Nevertheless, a non-satiable, risk-averse investor would never prefer Y. Denote by U2 the set of all utility functions which are nondecreasing and concave. In contrast to the FSD, the condition in 3. It 54 3. Such an illustration is provided in Figure 3. Rothschild and Stiglitz introduce a slightly different order by dropping the requirement that the investors are non-satiable.

The class of risk-averse investors is represented by the set of all concave utility functions, which contains the set U2. Thus, the condition in 3. This conclusion holds for the non-satiable risk-averters as well and, therefore, the relation in 3. The converse relation is not true. This can be demonstrated with the help of the example developed in section 3. Those who are non-satiable would certainly prefer the larger sum but this is not universally true for all risk-averse investors because we do not assume that u x is nondecreasing. Generally, its values vary for different payoffs depending on the corresponding derivatives of the utility function. Larger values of rA x correspond to a more pronounced risk-aversion effect.

For smaller values of x, the graph is more curved while for larger values of x, the graph is closer to a straight line and, thus, to risk neutrality. In section 3. The logarithmic utility function is an example of a utility function exhibiting decreasing absolute risk aversion.

An illustration is given in Figure 3. Utility functions exhibiting a decreasing absolute risk aversion are important because the investors they represent favor positive to negative skewness. This is a consequence of the decreasing risk aversion — at higher payoff levels such investors are less inclined to avoid risk in comparison to lower payoff levels at which they are much more sensitive to risk taking.

Thus, U3 represents the class of non-satiable, risk-averse investors who prefer positive to negative skewness. Therefore, the condition 3. It measures the variability of X below a target payoff level t.

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Suppose that X and Y have equal means and variances. If X has a positive skewness and Y has a negative skewness, then the variability of X below any target payoff level t will be smaller than the variability of Y below the same target payoff level.

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A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies. A Probability Metrics Approach to Financial Risk Measures, by Svetlozar T. Rachev, Stoyan V. Stoyanov, and Frank J. Fabozzi, , Oxford.

At first sight, 3. In fact, it is only a matter of algebraic manipulations to show that, indeed, if 3. The efficient set of a given class of investors is defined as the set of ventures not dominated 58 3. For example, the efficient set of the non-satiable investors is the set of those ventures which are not dominated with respect to the FSD order. As a result, by construction, any venture which is not in the efficient set will be necessarily discarded by all investors in the class.

The portfolio choice problem of a given investor can be divided into two steps. The first step concerns finding the efficient set of the class of investors which the given investor belongs to. Any portfolio not belonging to the efficient set will not be selected by any of the investors in the class and is, therefore, suboptimal for the investor.

Such a class may be composed of, for example, all non-satiable, riskaverse investors if the utility function of the given investor is nondecreasing and concave. In this case, the efficient set comprises all portfolios not dominated with respect to the SSD order. Note that in this step, we do not take advantage of the particular expression for the utility function of the investor.

Once we have obtained the efficient set, we proceed to the second step in which we calculate the expected utility of the investor for the portfolios in the efficient set. The difficulty of adopting this approach in practice is that it is very hard to obtain explicitly the efficient sets.

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That is why the problem of finding the optimal portfolio for the investor is very often replaced by a simpler one, involving only certain characteristics of the portfolios return distributions, such as the expected return and the risk. In this situation, it is critical that the simpler problem is consistent with the corresponding stochastic dominance relation in order to guarantee that its solution is among the portfolios in the efficient set.

Checking the consistency reduces to choosing a risk measure which is compatible with the stochastic dominance relation. In the following, we examine the FSD and SSD orders concerning logreturn distributions and the connection to the corresponding orders concerning random payoffs. The logarithmic return, or simply the log-return, is a central concept in fundamental theories in finance, such as derivative pricing and modern portfolio theory.

Therefore, it makes sense to consider stochastic orders with respect to log-return distributions rather than payoff. Without loss of generality, we can assume that the stock does not pay dividends. The random variable Pt can be regarded as the random payoff of the common stock at time t, while rt is the corresponding random log-return.

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Even though log-returns and payoffs are directly linked by means of the above formulae, it turns out that, generally, stochastic dominance relations concerning two log-return distributions are not equivalent to the corresponding stochastic dominance relations concerning their payoff distributions. Apparently, v y is not concave.

It turns out that an investor who is non-satiable and risk averse with respect to payoff distributions may not be risk averse with respect to log-return distributions. In fact, v y also has non-positive first derivative but the sign of the second derivative can be arbitrary. Therefore the investor is non-satiable but may not be risk-averse with respect to log-return distributions.

This fact is illustrated in Figure 3. This is true because if v y satisfies the corresponding derivative inequalities, so does u x given by 3. Consequently, it follows that the investors who are nonsatiable and risk averse on the space of log-return distributions are a subclass of those who are non-satiable and risk averse on the space of payoff distributions.

However, such an equivalence does not hold for the SSD order. Such kinds of relations deserve a closer scrutiny as optimal portfolio problems are usually set in terms of returns, and consistency with a stochastic dominance relation implies that the stochastic dominance relation is also set on the space of return distributions, not on the space of payoff distributions. Moreover, in this section we considered only one-period returns. In a multi-period setting, for example in the area of asset-liability management, matters get even more involved.

Otherwise they may be violated.