Viewing Risk Measures as Information

In this paper we present the thesis that the reporting of a risk measure is the revelation of a piece of information about the financial health of an institution. We show that there are multiple possibilities for loss distributions when only a VaR is reported. We then introduce two other measures, the Expected Shortfall and the Maximum Loss. We show that reporting any one of these measures can lead to multiple possible loss distributions.

Following the old adage that more information is always better, we consider the situation of reporting any two risk measures. Again, we show that there is ambiguity about the loss distribution, no matter which two measures we choose. We show the same result for three measures and for five measures. We then conclude with some suggestions for regulators and with some suggestions for further research.

There is little wonder that people are confused by what the term “risk” ought to mean. For instance, the website businessdictionary.com defines risk in seventeen different “general categories”. Now, each of the seventeen definitions are relevant to the situations of banks and securities firms. The unifying theme for each of the definitions is that risk requires both uncertainty and exposure. If a company already knows that a loan will default, there is no uncertainty and thus no risk. And if the bank decides not to loan to a business that is considered likely to default, there is also no risk for that bank as the bank has no exposure to the possibility of loss.

If we desire to measure risk in some way, one goal could be to have a single risk measure that will account for all the risk that a bank or securities firm might encounter. Some have objected to the risk measure being a single number, but there is some support for this idea. Investing is always a binary decision-either one invests or one chooses not to invest. Thus the argument is that, given a single number, one should have enough information to decide whether to invest or not. There have been some general agreements about the kinds of properties that such a risk measure ought to possess.

Virtually everyone would agree that if the payout for an investment is always positive (after accounting for the risk-free interest rate), then there is truly no risk of loss in the investment. Many people would say that there is ten times as much risk in investing $10,000 as there is in investing $1000 simply because there is ten times as much money at stake. And most would acknowledge that having cash on hand makes them feel safer about making an investment. It was exactly these thoughts that were, respectively, codified into the definition of risk measure.

Let X be a random variable. Then ρ is a risk measure if it satisfies the following properties:

The idea behind this definition is that a positive number implies that one is at risk for losing capital and should have that positive number of a cash balance on hand to offset this potential loss. A negative number would say that the company has enough capital to take on more risk or to return some of its cash to other operations or to its shareholders.

The risk measure that is most used is the Value at Risk (VaR). Essentially the α-Value at Risk is that number L so that we can expect the losses to be worse than L exactly 1 -α of the time. For instance if the 95%-Value at Risk of our position is $100, we would expect to lose more than $100 only 5% of the time. A more formal definition can be stated as follows:

The α-Value at Risk of a position X , VaR α (X ) = -in f {x : P(X > x) ≤ 1 -α} Since the 1951 paper of Markowitz, many have also favored the usefulness of diversification. VaR does not account for this preference, and a simple example shows this. Consider the case where a bank has made two $1 million loans and one $2 million loan, each with a 0.04 probability of default and all pairwise independent. Then the 95%-VaR for each loan is $0. Thus, if we construct a portfolio consisting solely of the $2 million loan, we must have a 95%-VaR of $0. If we instead choose diversification and make our portfolio out of the two independent $1 million loans, something paradoxical happens. The probability of both loans defaulting is 0.0016, but the probability of exactly one loan defaulting is 0.0768. This implies that the 95%-VaR of our diversified portfolio is $1 million. Thus, VaR does not favor diversification. This leads Artzner, Delbaen, Eber, and He

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