On a Generalization of certain inequalities by Tchebychef and Jensen

Abstract

The inequalities dealt with in this paper were published by me in Danish in ?Matematisk Tidsskrift? B. 1925 p. 49¹. I was not at the time aware, that an inequality of similar form had already been given by JENSEN; but as the conditions under which his theorem is valid differ from mine and are, in fact, included in them, it appears that my investigation is not superfluous. I am reverting to the subject partly in order to clear up the relationship between JENSEN'S and my own results, partly to show, by means of a few examples, how these inequalities may with advantage be applied to actuarial problems.     

On the probability function in the collective theory of risk

Abstract

Upon reading Dr. LUNDBERG'S paper ?Über die Wahrscheinlichkeitsfunktion einer Risikenmaase?¹ and trying to penetrate it along my own lines of thought, I found another way of deducing some of his formulas, giving the results in a form that directly invites a fairly simple approximation of the probability function. Though time has not permitted my going deeper into the problem, I propose here to give a brief account of the method.     

River stories: Genealogies of a threatened inland river system

Most people now living in Australia's “bread basket”, the much-degraded Murray Darling Basin, are like my family, descendants of convicts or free settlers who came to the inland in the 19th or early 20th centuries. Our legacy includes the dispossession of indigenous peoples, species extinction and the ongoing degradation of the ecological communities which now sustain us. My own family's river stories which “begin” with a pair of impoverished Gaels who migrated with their offspring from the Scottish Highlands, can be considered paradigmatic. I re-narrate it in this essay in response to philosopher Alasdair MacIntyre's challenge—I can only answer the question ‘What am I to do?’ if I can answer the prior question ‘Of what story or stories do I find myself a part?’Some of these family stories I find myself part of, especially those that have been enacted within the catchment of the now-threatened Lachlan River, are very discomforting, but where do they “truly” begin? In seeking to understand my relationship with the river and its catchment, and with the indigenous peoples “my mob” displaced, I explore several possible “beginnings” and ask a further question: what stories do I want to be part of as co-author, co-narrator and protagonist. I then offer my own yet-to-be enacted “truth and reconciliation” stories about the future of the inland plains I love.     

Correlation once more

Abstract

§ 1. Correlation Generally.

In my thesis for Doctorship of 1919, ¹ This Journal, 1919, p. 1. I have made some critical remarks on the theory of correlation, trying especially to exhibit the rather unaccomplished state of this theory, and the danger of releasing its formulae for general use. In a supplementary note, ² Ibidem, p. 204. I have pushed my criticism a little further. Since that time, more than ten years have passed away, but I find my point of view still sustainable. One reproach, however, I have never been able to reject, viz, of having been exclusively negative. In fact, I find this position rather natural, from a philosophical standpoint. The information value of correlation calculations is indeed, as a rule, very small. And this seems the more regrettable, as the importance of the Στχαστι?? Τ?χυη, even for the most difficult questions of knowledge, ought to be very great. In a paper of 1924, “Quelques questions concernant les principes de la théorie des probabilités” ³ Ibidem, 1924, p. 107. I have tried to explain, how I imagine the development of the theory of correlation in order to be more apt to set about such philosophical questions.     

Remarks on the foundations of actuarial science

Abstract

In a paper entjtled ?Critical Thoughts on Actuarial Science? read before the Swedish Actuarial Society and published in this part of the Journal K. ENGLUND has expressed some views which gave rise to objections also from my part in the verbal discussion at the meeting of the Society. In the following lines I have tried to develop my views about the most important questions touched upon in his paper.     

Notes on some present day Questions regarding Mortality

Abstract

In the absence of any definite development of actuarial science in Great Britain which would be a suitable subject for this paper I considered that the most useful course would be to discuss some of the problems which are at present exercising my own mind, and no doubt the minds of many of my colleagues. Probably, many of these questions are akin to others which arise in other countries, so that their discussion at a meeting such as this may prove of mutual interest, while even if they are peculiar to ourselves, a disclosure of our doubts and difficulties will indicate the directions in which future developments may be expected. I must, however, emphasize that any views expressed are purely personal, and do not in any way pretend to represent a consensus of British actuarial opinion. Throughout this paper only Ordinary (as opposed to Industrial) business in considered.     

Further remarks on Laplace' and Gauss' summation formulas

1. The practical form of the remainder terms of Laplace' and Gauss' summation formulas which I have employed in earlier papers and in my book on Interpolation suffers from the inconvenience that it is inapplicable in the case of infinite limits of summation and integration. The purpose of the present paper is to show, how this inconvenience may be avoided.     

Remarks on risk theory

§ 1. The Scheme U(n).

Following remarks are the result of some deliberations which I have made in order to find a simple starting point for rough estimates of the risk reserves necessary under given conditions. They do not appear with any pretensions. The idea, upon which my thoughts were based, is far from original: I have hoped to be able to simplify the theory by simplifying in a high degree the hypotheses and by confining myself to small numbers and arithmetical methods.     

LEVERAGED ESOPS AND CORPORATE RESTRUCTURING

The ESOP is treated as simply an employee benefit, but it is really a device to save the human race. —Louis T. Kelso, founder of ESOP¹

The American public is not too well-informed….these plans have been heralded as the basic solution for many of our economic ills. Specifically, one of our chief proponents… has said that widespread adoption of ESOPs will accomplish the following objectives: The restoration and acceleration of economic growth to unprecedented levels; create legitimate full employment for two or three decades; and lay the foundation for arresting inflation. I must confess that these are some claims. Certainly no one since I have been chairing this committee has come before us with program that promises that much. —Sen. Hubert Humphrey, Joint Economic Committee Hearings, 1975

(Kelsonomics is) an amateurish and crankish fad. —Paul Samuelson, San Juan Star, April 27, 1972

The ESOP is first and foremost an employee benefit plan. —Rosen, Klein, Young, 1986

     

Some results on the compound Markov binomial model

Abstract

Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula¹ which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed² that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders¹, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions.     

Samuelson Hypothesis & Indian Commodity Derivatives Market

Samuelson (1965) devised that futures price volatility increases as the futures contract approaches its expiration. The relation amid the volatility and time to maturity has significant inference for hedging strategies. Interestingly, so far the empirical evidence in favor of the Samuelson Hypothesis (maturity effect) is mixed in various markets. Considering no significant work to examine the relationship is so far carried out in commodity derivative markets of India, this paper ordeal the Samuelson Hypothesis on 8 commodities traded on Multi-Commodity Exchange (MCX), India. We have examined the issue by applying different regression techniques to test the hypothesis for 8 commodities (Aluminium, Nickel, Copper, Gold, Silver, Natural Gas, Crude Oil and Wheat) using inter-day data on MCX India. In order to test the Samuelson’s hypothesis, tests have been conducted using a series of GARCH, EGARCH and TGARCH models by including trading volume, open interest and time-to-maturity in the conditional variance equation. From our results, it is concluded that Samuelson’s hypothesis does not hold true for majority of commodity contracts considered. Our results also find that volatility series depend on the trading volume, compared to the time-to-maturity or open interest. As Samuelson hypothesis does not hold true for majority of commodity contracts, traders in Indian commodity derivative markets should not bias their decisions solely based on the time-to-maturity, but should also consider trading volume and open interest as they are an important determinant of price volatility. They should also consider the possibility of leverage effect while predicting future price volatilities, and the associated margin requirements.     

Least squares,and a generalisation of the »Student»-Fisher theorem

Abstract

1. Following the methods employed by Steffensen ¹ a simple analytical proof of a generalisation of the »Student»-Fisher theorem is obtained. Geometrical proof of this generalisation has been given by Sterne.²     

Positive accounting theory: A review

Watts and Zimmerman's Positive Accounting Theory provides a refreshing, controversial and important contribution to accounting thought. It is important because of its vigorous emphasis on the entity's actual choice of financial accounting technique (or, more broadly, financial reporting activity). It is controversial because the theory and empirical techniques it conveys are not fully developed. It is refreshing because it challenges us to expand our thinking about the nature of accounting institutions.In this essay, I try to document these claims about the content and significance of Positive Accounting Theory. The first section presents my interpretation of the work. The next section lists some concerns which I feel are important in consuming the work and in providing directions for continued labor. The third section discusses some unusual strengths in the work, while the final one provides some speculation about where accounting thought might next venture.Before proceeding, though, a caveat emptor warning must be issued. This essay presents both aspects which I learned about and also those which I worried about while consuming Positive Accounting Theory. It is an experiential report.     

Note on the collective theory of risk

Abstract

What follows has grown out of a discussion with Carl Philipson following a lecture [1] on the collective theory of risk. Although I give here nothing else but a refined interpretation of Paul Lévy's form (see, e.g., [2], p. 322) of identically distributed random variables the result still seems of interest for all those working in the field of collective risk theory. I thank Carl Philipson for stimulating my interest in this matter.     

Formulae of recurrence for the semiinvariants at a class of frequency functions of more variables

Abstract

In a paper in this journal¹ Mr. Paul Qvale has proved following theorem:     

On certain inequalities by Steffensen

Abstract

Extract

In this note we are concerned with the inequalities published by Steffensen in this Journal under the title: ?On a generalization of certain inequalities by Tchebycheff and Jensen?. I will show how these inequalities are represented in one form by the integrals of Stieltjes. These integrals have the advantage of containing both sums and the integrals of Riemann. The proof of this generalized theorem is also more simple, which shows that this theorem appears only in this generalization in its most natural form.     

Samuelson's Fallacy of Large Numbers and Optional Stopping

Accepting a sequence of independent positive mean bets that are individually unacceptable is what Samuelson called a fallacy of large numbers. Recently, utility functions were characterized where this occurs rationally, and examples were given of utility functions where any finite number of good bets should never be accepted. ¹ 1 Ross, 1999 .
Here the author shows how things change if you are allowed the option to quit early: Subject to some mild conditions, you should essentially always accept a sufficiently long finite sequence of good bets. Interestingly, the strategy of quitting when you get ahead does not perform well, but quitting when you get behind does. This sheds some light on more possible behavioral reasons for Samuelson's fallacy, as well as strategies for handling a series of sequentially observed good investments.     

An introduction into Lundberg's theory of risk

Abstract

In this number of the journal a paper of Dr. Filip Lundberg is published, in which he thoroughly deals with certain problems of the theory of risk. As all the former works of Dr. Lundberg about the theory of risk with only one exception (the paper “Über die Theorie der Rückversicherung” in the transactions of the Congress of Actuaries in Wien 1909) are published in the Swedish language and consequently inaccessible to the international insurance world, a simultaneous report of some of the fundamental ideas in this former production perhaps will offer some interest. Though the latest paper follows a special line, the starting points and the manner of consideration are unaltered, and hence an aquaintance with the simpler problems which here will be dealt with will be rather illustrative.     

On the correlation of acting probabilities

Abstract

In a paper printed in this journal n:r 4–5 1916, I have treated the problem of correlation in homograde statistics. This led in the case of constant acting probabilities to the discussion of an extension of the theorem of Bernoulli to embrace the occurrences of two attributes.     

The perception of time,risk and return during periods of speculation

Abstract

What return should you expect when you take on a given amount of risk? How should that return depend upon other people's behaviour? What principles can you use to answer these questions? In this paper, I approach these topics by exploring the consequences of two simple hypotheses about risk.

The first is a common-sense invariance principle: assets with the same perceived risk must have the same expected return. It leads directly to the well known Sharpe ratio and the classic risk-return relationships of arbitrage pricing theory and the capital asset pricing model.

The second hypothesis concerns the perception of time. I conjecture that in times of speculative excitement, short-term investors may instinctively imagine stock prices to be evolving in a time measure different from that of calendar time. They may perceive and experience the risk and return of a stock in intrinsic time, a dimensionless time scale that counts the number of trading opportunities that occur, but pays no attention to the calendar time that passes between them.

Applying the first hypothesis in the intrinsic time measure suggested by the second, I derive an alternative set of relationships between risk and return. Its most noteworthy feature is that, in the short-term, a stock's trading frequency affects its expected return. I show that short-term stock speculators will expect returns proportional to the temperature of a stock, where temperature is defined as the product of the stock's traditional volatility and the square root of its trading frequency. Furthermore, I derive a modified version of the capital asset pricing model in which a stock's excess return relative to the market is proportional to its traditional beta multiplied by the square root of its trading frequency.

I also present a model for the joint interaction of long-term calendar-time investors and short-term intrinsic-time speculators that leads to market bubbles characterized by stock prices that grow super-exponentially with time.

Finally, I show that the same short-term approach to options speculation can lead to an implied volatility skew.

I hope that this model will have some relevance to the behaviour of investors expecting inordinate returns in highly speculative markets.