This paper defines the concept of simple strategy and introduces three kinds of simple strategies: wealth-invariant, scale-invariant and "wealthier-accept more". For three commonly used utility function families: CARA, CRRA and DARA equivalent characterizations are obtained in terms of the corresponding simple strategy, in terms of the buying and selling price properties, and in terms of the utility function properties as expressed by Cauchy functional equations. Moreover, an extension of famous Pratt (1964) theorem is proved which involves buying price for a lottery as an alternative measure of comparative risk aversion. Additionally a number of propositions on both selling and buying price for a lottery and CRRA utility class are proved.
In the paper we present robust estimation methods based on bounded innovation propagation filters and quantile regression, applied to measure Value at Risk. To illustrate advantage connected with the robust methods, we compare VaR forecasts of several group of instruments in the period of high uncertainty on the financial markets with the ones modelled using traditional quasi-likelihood estimation. For comparative purpose we use three groups of tests i.e. based on Bernoulli trial models, on decision making aspect, and on the expected shortfall.
We develop a fully Bayesian framework for analysis and comparison of two competing approaches to modelling daily prices on different markets. The first approach, prevailing in financial econometrics, amounts to assuming that logarithms of prices behave like a multivariate random walk; this approach describes logarithmic returns most often by the VAR(1) model with MGARCH (or sometimes MSV) disturbances. In the second approach, considered here, it is assumed that daily price levels are linked together and, thus, the error correction term is added to the usual VAR(1)–MGARCH or VAR(1)–MSV model for logarithmic returns, leading to a reduced rank VAR(2) specification for logarithms of prices. The model proposed in the paper uses a hybrid MSV-MGARCH structure for VAR(2) disturbances. In order to keep cointegration modelling as simple as possible, we restrict to the case of two prices representing two different markets.
The aim of the paper is to show how to check if a long-run relationship between daily prices exists and whether taking it into account influences our inference on volatility and short-run relations between returns on different markets. In the empirical example the daily values of the S&P500 index and the WTI oil price in the period 19.12.2005 – 30.09.2011 are jointly modelled. It is shown that, although the logarithms of the values of S&P500 and WTI oil price seem to be cointegrated, neglecting the error correction term leads to practically the same conclusions on volatility and conditional correlation as keeping it in the model.
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