Destabilizing Properties of a VaR or Probability-of-Ruin Constraint when Variances May Be Infinite

Abstract

Var is perhaps the most commonly used measure of risk by financial institutions. At many of these the head of risk management gets a daily report on firm-wide Var as well disaggregated Var across the firm's businesses. Often chief risk manager (CRM) is a top level officer in many financial institutions with a direct report to the board of directors as well as the chief executive officer (CEO). Though mentioned most often as a risk measure, Var can also be a constraint. For example, it can be used as a constraint on the probability of ruin, which is risk measure that has been of interest to insurers since Condorcet introduced it in 1784. A constraint on the probability of ruin is a special case of a VaR constraint. There may be significant negative consequences for the CEO or CRM on whose watch the firm has a credit downgrade or worse fails. Hence we cannot rule out (until proven otherwise) that managers behave as if they have a VaR constraint or less strongly are influenced in some manner by such a constraint. Two properties of the marginal price of risk with a binding Var constraint for an expected value maximizing manager are: the marginal price of risk is strictly additive and not just subadditive and depending on the financial health of the firm managers can switch from pricing a lottery (risky asset or contract) at a premium over the risk-neutral value to pricing it at a discount or the switch can be the other way. For a class of bivariate distributions for a lottery and the firm's portfolio that include some with infinite variances but have finite expectations and conditional expectations of the lottery given portfolio value, the manager switches from averting risk, as measured by the variance, to preferring it when the firm is in bad financial health. Risk, lacking well-defined variances, is defined in terms of a notion dubbed cotendency. Also given a binding VaR constraint, when in bad financial health, the firm will prefer a homogeneous increase in risk over decreasing homogeneous risk or leaving it constant. The opposite is true when the firm is in good financial health.