Time Dependent Relative Risk Aversion

Risk management has developed in the recent decades to be one of the most fundamental issues in quantitative finance. Various models are being developed and applied by researchers as well as financial institutions. By modeling price fluctuations of assets in a portfolio, the loss can be estimated using statistical methods. Different measures of risk, such as standard deviation of returns or confidence interval Value at Risk, have been suggested. These measures are based on the probability distributions of assets' returns extracted from the data-generating process of the asset. However, an actual one dollar loss is not always valued in practice as a one dollar loss. Purely statistical estimation of loss has the disadvantage of ignoring the circumstances of the loss. Hence the notion of an investor's utility has been introduced. Arrow [2] and [10] were the first to introduce elementary securities to formalize economics of uncertainty. The so-called Arrow-Debreu securities are the starting point of all modern financial asset pricing theories. Arrow—Debreu securities entitle their holder to a payoff of 1$ in one specific state of the world, and 0 in all other states of the world. The price of such a security is determined by the market, on which it is tradable, and is subsequent to a supply and demand equilibrium. Moreover, these prices contain information about investors' preferences due to their dependence on the conditional probabilities of the state of the world at maturity and due to the imposition of market-clearing and general equilibrium conditions. The prices reflect investors' beliefs about the future, and the fact that they are priced differently in different states of the world implies, that a one-dollar gain is not always worth the same, in fact its value is exactly the price of the security.