Where Utility Functions Do Not Exist - A Note on Lexicographic Orders

There seems to be some amount of confusion in the finance text books regarding the conditions under which an individual’s preferences can be represented by a utility function. Fama and Miller, for example, assert that two axioms (comparability and transitivity) are sufficient to establish the existence of a utility function (when the set of alternatives is Rn). This is totally false: a real valued utility function need not exist even in the single good case (R1). One might hope that a vector valued utility might exist (with lexicographic ordering of the utility vector); but this is not the case. Indeed we cannot salvage the situation even by allowing the utility to be a vector in Rn) (i.e. to have an (countably) infinite number of components); only an uncountable number of real components can do the job. None of these results are new, but they do not seem to be sufficiently well known to researchers in finance. This may be because the original papers are mathematically forbidding or because they are scattered in sources somewhat removed form the mainstream finance literature. If that be so, this note should be of some help; some of our proofs and examples are new and hopefully more elementary (for example we avoid taking recourse to Sierpinski’s lemma).

Risk Management Lessons from the Global Financial Crisis for Derivative Exchanges

During the global financial turmoil of 2007 and 2008, no major derivative clearing house in the world encountered distress while many banks were pushed to the brink and beyond. An important reason for this is that derivative exchanges have avoided using value at risk, normal distributions and linear correlations. This is an important lesson. The global financial crisis has also taught us that in risk management, robustness is more important than sophistication and that it is dangerous to use models that are over calibrated to short time series of market prices. The paper applies these lessons to the important exchange traded derivatives in India and recommends major changes to the current margining systems to improve their robustness. It also discusses directions in which global best practices in exchange risk management could be improved to take advantage of recent advances in computing power and finance theory. The paper argues that risk management should evolve towards explicit models based on coherent risk measures (like expected shortfall), fat tailed distributions and non linear dependence structures (copulas).

A Valuation Model for Indeterminate Convertibles

Many issues of convertible debentures in India in recent years provide for a mandatory conversion of the debentures into an unspecified number of shares at an unspecified time; the conversion ratio (i.e., the number of shares per debenture) is to be determined by the Controller of Capital Issues (CCI). There are serious problems in arriving at a rational value for these "indeterminate convertibles". Even if the investor can make some estimate of the likely conversion terms, there is no valuation model available to arrive at a price. This paper applies the general theory of derivative securities (Cox, Ingersoll and Ross, 1985) to obtain a valuation model for these instruments. The model shows that the naive valuation model which sets the value of the debenture equal to the current stock price times the expected conversion ratio is likely to be a significant overestimate of the price. It also shows that changes in the stock price lead to less than proportionate changes in the debenture price unlike in the case of pre-specified conversion terms. Similarly, the CAPM beta of the debenture would be significantly lower than that of the share. While the model does not obviate the need for obtaining estimates of unobservable parameters related to the market expectations about the likely conversion ratio, the qualitative insights given by the model are quite useful. The model is successful in explaining some of the empirical patterns and anomalies that have been observed in ongoing empirical research into the market prices of these debentures.

Mastershares: Enigmatic Performance

In this paper we have examined the performance of Mastershares, the first all equity close ended growth fund established by the Unit Trust of India (UTI) in the country, using the various portfolio performance measures that have been suggested in the literature. We found that while in terms of return on the Net Asset Value (NAV) the fund has out-performed the market, in terms of returns based on Market Prices it has shown a mixed performance. On further investigation, we inferred that the excellent performance in terms of NAV could neither be ascribed to selectivity nor to timing of decisions. The explanation possibly lies in UTI’s acquisition of stocks in the primary market as well below prevailing market prices and in the manner of allocation of stocks to various funds managed by the Trust. Our analysis also revealed that the market quite irrationally inflates the volatility of the market price of Mastershares as compared to the volatility observed in the NAV. This observation which implies market inefficiency is in line with the recent researches done in the developed capital markets.

Some Experimental Signatures to look for Time-reversal Violating superconductors

We discuss some experimental signatures associated with the topological structures of unconventional superconductor order parameters of form $d_{x^2-y^2}+ix$, where $x=s,p_x \pm p_y$, or $d_{xy}$. Specifically, we study the topological surface states on the $(110)$ and equivalent surfaces of such superconductors which are observable in Andreev tunneling experiments, as well as evaluate the magnetic flux trapped in superconducting rings of such superconductors with multiple grain-boundary Josephson junctions. Previous experiments are examined and several new experiments suggested.Comment: 11 pages, 3 figure

Rupee-Dollar Option Pricing and Risk Measurement: Jump Processes, Changing Volatility and Kurtosis Shifts

Exchange rate movements in the Indian rupee (and many other emerging market currencies) are characterised by long periods of placidity punctuated by abrupt and sharp changes. Many, but by no means all, of these sharp changes are currency depreciations. This paper shows that econometric models of changing volatility like Generalised AutoRegressive Conditional Heteroscedasticity (GARCH) with non normal residuals which perform quite well in other financial markets fail quite miserably in the case of the INR-USD process because they do not allow for such jumps in the exchange rate. The empirical results very convincingly demonstrate the need to model the exchange rate process as a mixed jump-diffusion (or normal mixture) process. Equally importantly, the empirical results provide strong evidence that the jump probabilities are not constant over time. From a statistical point of view, changes in the jump probabilities induce large shifts in the kurtosis of the process. The failure of GARCH processes arises because they allow for changes in volatility but not for changes in kurtosis. The time varying mixture models are able to accommodate regime shifts by allowing both volatility and kurtosis (not to mention skewness) to change. This also shows that the periods of calm in the exchange rate are extremely deceptive; in these periods, the variance of rate changes is quite low, but the kurtosis is so high (in the triple digit range) that the probability of large rate changes is non trivial. The empirical results also show that the Black-Scholes-Garman-Kohlhagen model for valuation of currency options is quite inappropriate for valuing rupee-dollar options and that the Merton jump-diffusion model is the model of choice for this purpose.