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).

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.

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.

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).

Value at Risk Models in the Indian Stock Market

This paper provides empirical tests of different risk management models in the Value at Risk (VaR) framework in the Indian stock market. It is found that the GARCH-GED (Generalised Auto-Regressive Conditional Heteroscedasticity with Generalised Error Distribution residuals) performs exceedingly well at all common risk levels (ranging from 0.25% to 10%). The EWMA (Exponentially Weighted Moving Average) model used in J. P. Morgan’s RiskMetrics� methodology does well at the 10% and 5% risk levels but breaks down at the 1% and lower risk levels. The paper then suggests a way of salvaging the EWMA model by using a larger number of standard deviations to set the VaR limit. For example, the paper suggests using 3 standard deviations for a 1% VaR while the normal distribution indicates 2.58 standard deviations and the GED indicates 2.85 standard deviations. With this modification the EWMA model is shown to work quite well. Given its greater simplicity and ease of interpretation, it may be more convenient in practice to use this model than the more accurate GARCH-GED specification. The paper also provides evidence suggesting that it may be possible to improve the performance of the VaR models by taking into account the price movements in foreign stock markets.

When index dissemination goes wrong: How fast can traders add and multiply?

This paper studies an episode of dissemination of wrong stock index values in real time due to a software bug in the Indian Nifty index futures market on the morning of January 18, 2006. The episode provides an opportunity to test various models of cognitive biases and bounded rationality highlighted in behavioural finance. The paper provides strong evidence against cognitive biases like “anchoring and adjustment” (Tversky and Kahneman, 1974) that one might expect under such situations even though the cognitive task involved is quite simple. The futures market tracked the true Nifty index which it could not see while completely ignoring the wrong Nifty index that it could see. However, the paper demonstrates that market efficiency failed in more subtle ways. There is evidence of a partial breakdown of price discovery in the futures markets and a weakening of the bonds linking futures and cash markets. This evidence is consistent with the centrality of “market devices” as argued in “actor network theory” in economic sociology (Muniesa, Millo and Callon, 2007 and Preda, 2006). Well functioning markets today depend critically on a whole set of information and communication technologies. Any failures in these material, socio-technical aspects of markets can make markets quite fragile even if behavioural biases are largely absent.

Regulatory Implications of Monopolies in the Securities Industry

Since the mid-1990s, investors and regulators have benefited from a high degree of competition in the Indian securities industry. Even more than all the policy changes that have taken place, it is technology and competition that have transformed the Indian capital market in the last 7-8 years. This paper shows that there is now considerable evidence that critical elements of the Indian securities industry are becoming significantly less competitive than in the past. Reduced competition would remove the single most important driver of capital market modernisation in this country and would create several serious regulatory problems. The paper argues that rather than applying the traditional solution of “regulated monopolies”, regulators need to adopt strong measures to stimulate competition. The regulator must also ruthlessly discard those elements of the regulatory regime that are anti-competitive in nature.

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

Detection and Implications of a Time-reversal breaking state in underdoped Cuprates

We present general symmetry considerations on how a Time-reversal breaking state may be detected by angle-resolved photoemission using circularly polarized photons as has been proposed earlier. Results of recent experiments utilizing the proposal in underdoped cuprates are analysed and found to be consistent in their symmetry and magnitude with a theory of the Copper-Oxides. These togather with evidence for a quantum critical point and marginal Fermi-liquid properties near optimum doping suggest that a valid microscopic theory of the phenomena in the cuprates has been found.Comment: A statement on detecting the Anyon state is added and some typos are subtracte