On the inversion of moving averages

Abstract

1. Let an infinite sequence of real numbers be given by (1) {η_t } =[…, η _t-2, η _t-1, η _t , η _t+1, η _t+2, . . . ] and let (b) = (b ₀, b ₁,.., b_h ) represent real constants. The sequence(2) {ζ _t } = […, ζ _t-2, ζ _t-1, ζ_t , ζ _t+1, ζ _t+2,…] defined for every t by the relation(3) ζ _t = b ₀ · η _t + b ₁ · η _t-1 — . . . + b _h-1 · η _t-h+1 + b _h · η _t-h is said to be a moving average of {η _t } with weights (b _i ) The variable t, which is restricted to integral values 0, ± 1, ± 2 etc., will in the sequel be spoken of as representing time.     

On a rank-order test for the equality of probability of an event

Abstract

The following situation is considered. A fixed number (= n) or sequence of independent trials T ₁ T ₂,…, T _n is given, and in each of these an event E mayor may not occur, It is further observed that the event E occurs a total of k times amongst the n trials T _i , (i = l,…, n). It is then required to test the hypothesis H ₀ that the probability of the occurrence of E is constant from trial to trial, i.e. H ₀ is the hypothesis: p ₁ = p ₂ = ? = p _n = p, if p _n (i = 1, …, n) represents the probability that E occurs on the ith trial.     

BOOK REVIEWS

Book reviewed in this article: Aggregate Theory and Policy: Modern Capital Theory. By Donald Dewey . Aggregate Theory and Policy: Readings in Macroeconomics. Edited by M. G. Mueller . Aggregate Theory and Policy: National Economic Policy. By James Tobin . Business Finance and Investments: Business Finance: Theory and Management. By Stephen H. Archer and Charles A. D'Ambrosio . Business Finance and Investments: The Economics of Corporate Finance. By Seymour Friedland . Business Finance and Investments: Efficiency of Investment in a Socialist Economy. Edited by Mieczystaw Rakowski . Financial Institutions and Markets: Money and Finance: Readings in Theory, Policy, & Institutions. Edited by Deane Carson . Financial Institutions and Markets: Commercial Bank Behavior and Economic Activity: A Structural Study of Monetary Policy in the Postwar United States. By Stephen M. Goldfeld . International Finance: Financing African Development. Edited by Tom J. Farer . International Finance: Central Banking in the Commonwealth. By J. Obasanmi Olakanpo . International Finance: International Monetary Relations: Theory, History, and Policy. By Leland B. Yeager . Public Finance: Federal Budget Projections. By Gerhard Colm and Peter Wagner .     

On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

Risk‐based explanation for the book‐to‐market effect

This paper proposes a risk‐based explanation for the book‐to‐market (B/M) effect. I decompose B/M into net operating asset‐to‐market (NOA/M) and net financing asset‐to‐market (NFA/M) components. Portfolio analysis shows that (i) positive B/M, NOA/M and NFA/M are positively related to future returns and (ii) negative B/M, NOA/M and NFA/M are negatively related to future returns. To the extent that positive B/M, NOA/M and NFA/M act as measures of asset risk and negative B/M, NOA/M and NFA/M act as inverse measures of borrowing risk, the nonlinear relations between B/M, NOA/M and NFA/M and future returns provide some evidence to support the risk‐based explanation for the book‐to‐market effect in stock returns.     

An Estimation of the Coastal Premium for Residential Housing Prices in San Diego County

The authors attempt to estimate the “coastal premium”—additional value conferred on a residence from being located near the coast—of single family homes in San Diego County, while controlling for other locational and structural characteristics. A previous investigation published in 2001 for south Orange County found that moving away from the coast by one mile was associated with a 42,000 lower housing price. Intrigued by this finding, we investigate whether (a) a similar coastal premium exists for all of San Diego County and (b) the premium varies by incremental distance from the coast (e.g., for 500-feet increments). Using data from 9,755 San Diego County home sales in 2006, results presented here suggest that for a median-priced home (42,000 lower housing price. Intrigued by this finding, we investigate whether (a) a similar coastal premium exists for all of San Diego County and (b) the premium varies by incremental distance from the coast (e.g., for 500-feet increments). Using data from 9,755 San Diego County home sales in 2006, results presented here suggest that for a median-priced home (540,000) at the mean distance from the coast (approximately 9 miles—and considerably farther than the Orange County estimate) a one-mile increase in distance from the coast would reduce the sale price by approximately $8,680. Specifying by specific distance increments, we find that the coastal premium is approximately 101.9% for houses within 500 feet of the coast (i.e., their value is 101.9% higher than similar homes located beyond six miles of the coast), falling to 62.8% for homes between 500 and 1,000 feet, declining to approximately 3.3% for homes located between five and six miles of the coast, disappearing entirely beyond around six miles. Since average comparisons of the sort initially considered in this analysis can be very misleading, researchers should consider the nonlinear incremental distance effects in model specifications.     

Eksamen i Forsikringsvidenskab og Statistik ved Københavns Universitet

Abstract

Let X ₁, X ₂,... be a sequence of independent, identically distributed random variables with P(X?0)=0, and such that pκ = ?₀ ^∞ x ^κ dP(x)<∞, k= 1, 2, 3, 4. Assume that {N(t), t?0} is a Poission stochastic process, independent of the X ₁ with E(N(t))=t. For λ ? 0, let Z _T= max {Σ _t?1 ^N(t) X _t ?t(p ₁+λ)}. Expressions 0 ?t?T for E(Z _T ), E(Z _T ²), and P(Z _T =0) are derived. These results are used to construct an approximation for the finite-time ruin function Ψ(u, T) = P(Z _T >u) for u?0. An alternate method of approximating Ψ(u, T) was presented in [10] by Olof Thorin and exemplified in [11] by Nils Wikstad. One of the purposes of this paper is to compare the two methods for two distributions of claims where the number of claims is a Poisson variate. The paper will also discuss the advantages and disadvantages of the two methods. We will also present a comparison of our approximate figures with the exact figures for the claim distribution      

BOOK REVIEWS

Book reviewed in this article: Aggregate Theory and Policy: Capital and Growth. By John R. Hicks . Aggregate Theory and Policy: Essays in Monetary Policy in Honor of Elmer Wood. Edited by Pinkney C. Walker . Business Finance and Investments: Financial Accounting Theory. By Harold Bierman , Jr . Business Finance and Investments: Investment Decisions and Capital Costs. By James T. S. Porterfield . Financial Institutions and Markets: The Financial Sector and Economic Development: The Mexican Case. By Robert L. Bennett . Financial Institutions and Markets: Valuations of Securities as of December 31, 1965. By Committee on Valuation of National Association of Insurance Commissioners . Financial Institutions and Markets: Analytics and Institutions of Money and Banking. By William J. Frazer , Jr . and William P. Yohe . Financial Institutions and Markets: How to Analyze a Bank Statement. By F. L. Garcia . Financial Institutions and Markets: Cycles in Government Securities: Determinants of Changes in Ownership. By Michael E. Levy . Financial Institutions and Markets: California Banking in a Growing Economy: 1946–1975. Edited by Hyman P. Minsky . Financial Institutions and Markets: Government Securities Market. By Ira O. Scott , Jr . Financial Institutions and Markets: Readings in Money, National Income, and Stabilization Policy. Edited by Warren L. Smith and Ronald L. Teigen . International Finance: Foreign Exchange, Capital Flows, and Monetary Policy. By E. Ray Canterbery . International Finance: International Finance. By Richard Ward . International Finance: Report of the Study Group on the Creation of Reserve Assets: Report to the Deputies of the Group of Ten. Public Finance: Development Finance: Planning and Control. By Ursula K. Hicks . Public Finance: Federal Budget Policy. By David J. Ott and Attiat F. Ott . Public Finance: Pros and Cons of the Property Tax. By Dick Netzer . Public Finance: Fiscal Neutrality Toward Economic Growth: Analysis of a Tax Principle. By Edmund S. Phelps .     

On a class of law invariant convex risk measures

We consider the class of law invariant convex risk measures with robust representation r_h,p(X)=sup_fò₀¹ [AV@R_s(X)f(s)-f^p(s)h(s)] ds\rho_{h,p}(X)=\sup_{f}\int_{0}^{1} [AV@R_{s}(X)f(s)-f^{p}(s)h(s)]\,ds, where 1≤p<∞ and h is a positive and strictly decreasing function. The supremum is taken over the set of all Radon–Nikodym derivatives corresponding to the set of all probability measures on (0,1] which are absolutely continuous with respect to Lebesgue measure. We provide necessary and sufficient conditions for the position X such that ρ _h,p (X) is real-valued and the supremum is attained. Using variational methods, an explicit formula for the maximizer is given. We exhibit two examples of such risk measures and compare them to the average value at risk.     

Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V _T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k _n =k ₀ n ^−α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility [^(s)]_n\widehat{\sigma}_{n} in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value Vⁿ_TV^{n}_{T} to the pay-off V _T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V _T =G(S _T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n ^−1/2 in L ² and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_iⁿ=g(i/n)t_{i}^{n}=g(i/n), where the strictly increasing scale function g:[0,1]→[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1−(1−t) ^β , β≥1. We show that the sequence n^1/2(V_Tⁿ-V_T)n^{1/2}(V_{T}^{n}-V_{T}) converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.     

Le XIImecongrès international d'actuaires

Abstract

We study the following inverse thinning problem for renewal processes: for which completely monotone functions f is f/(p+qf), 0?p?1, q=1-p, completely monotone? A characterisation of such f's is given. We also study the case when f comes from a gamma distribution, and present some ideas for more general results.

The intention of this note is to add some information to a paper by Yannaros (1985), in which thinned renewal processes are considered. Let X_n , n?1, be i.i.d. non-negative random variables, distributed according to a probability measure µ, and let S_n = X ₁+...+X_n (with S ₀=0) be the corresponding renewal process. Replacing µ by the probability measure ν=∑_n?1 pq^n-1 µ^n* (µ^n* =µ* ... µ*, n times) we get a new renewal process, obtained from the original one by independently at each stage preserving the process with probability p. Here and below q= 1-p, and to avoid trivialities we assume that 0 Let µ^(s) = ∫[0,∞) exp (-sx)µ(dx) , s?0, denote the Laplace transform of µ. Then ν^=pµ/(1-µ^). We will study the inverse problem: given a completely monotone function ψ, when does ψ(p+qψ) define a completely monotone function. A complete characterisation, and some of its consequences, is given in §§ 1–3 below. In §§ 4–5 we study the gamma distribution. It is proved that the inverse problem has a negative solution when the parameter a > 1, i.e. 1/(p + q(1 + s) ^a ) is not completely monotone then. In Yannaros (1985) this was proved for a=2, 3, ... with entirely different methods. (That 1/(p+q(1+s)^a is completely monotone for 0?a?1 is easily seen; cf. Yannaros (1985). Finally, in § 6 we give some suggestions to more general results related to thinning. Perhaps the most interesting problem is to find sufficiently general conditions for an absolutely monotone function to have a Bernstein function as its inverse.     

A characterization of the exponential distribution and of the geometric distribution

Abstract

Cook (1978) has proved that n positive random variables X ₁ ..., X _n are independent and follow the same exponential distribution iff the random vectors (X ₁ ..., X _s ) and (X _s+1, ..., X _n ) are independent for some s ∈ {1, ..., n-l} and E(Π} _j=1 ⁿ max {X _j -a _j , 0}) is a function of Σ _j=1 ⁿ a _j for a ₁, ..., a _n ∈ dR ⁺. In this paper a generalization of this characterization of the exponential distribution and an analogous characterization of the geometric distribution are given.     

Consistency and asymptotic normality of maximum likelyhood estimators

Abstract

Let the random variable X denote the time taken in completion of a process. For a fixed a, if the observed value of X is less than a, the X is observable, but if X is greater than a, the process is tampered with and is accelerated or decelerated at time a by some unknown factor α, and Y=a+α(X-a) is observed. If the experimenter has only partial control over the experiment, it may be difficult to get several observations on Y corresponding to the same a value. Thus we have a set of independent but not identically distributed observations. The large sample behavior of m.l.e. of the unknown parameters based on tampered random variables Y ^b1 , ..., Y ^bn is studied. If X follows an exponential distribution with mean (1/--), ... the consistency and asymptotic normality of the m.l.e. of α and -- is established under mild conditions on a ^b1, a ^b2, ... The conditions needed for establishing the consistency of m.l.e. of lX are given when X follows a uniform distribution U(O, --) or when X has any known distributional form     

A probabilistic analysis of the trading the line strategy

We provide analytic models for which the appropriate statistics of the trading the line strategy, N _h , can be derived in closed form. In particular, we provide closed-form expressions concerning the average duration of the open position, E(N _h ), the variance of the open duration, Var(N _h ), the average of the stopped log price, E(S _{N h} ), the variance of the stopped log price, Var(S _{N h} ), the correlation, Corr(N _h , S _{N h} ), and the Laplace transform, E(e^{?s N _h} ). These results are obtained, in discrete time settings, for binomial and other price scenarios. Furthermore, when analytic results are not possible, such as the case of a normal distribution for log returns, we show by simulation that our general conclusions still hold. Using these statistics we point out some of the subtle features of the trailing stops strategy.     

Über das Theorem von Tchebycheff

Abstract

Eine Grösse X hänge in der Weise vom Zufall ab, dass sie verschiedene Werte x ₁, x ₂, … XII annimmt, je nachdem das Ereignis E ₁, oder das Ereignis E ₂ oder … oder das Ereignis E _n eintritt, wofür die Wahrscheinlichkeiten p ₁, p ₂, … p _n bestehen sollen; p ₁ + p ₂ + … + p _n = 1. Mann nennt X eine von Zufall abhängige Grösse oder X eine variable Grösse mit dem Wertevorrat (x ₁, x ₂, … x _n), wobei jedem einzelnen dieser Werte eine bestimmte Wahrscheinlichkeit zukommt. Zwei Grössen X, Y mit den Individualwerten x ₁, x ₂, … x _n; y ₁, y ₂, … y _m heissen unabhängig von einander, wenn die Wahrscheinlichkeit p _i von x _i dieselbe bleibt, welches auch der Wert von Y sei, und wenn auch die Wahrscheinlichkeit q _p von y _p dieselbe bleibt, welchen Wert auch X annehmen möge.     

Sur une loi de probabilité considérée par J. F. Steffensen

Abstract

Dans ce même périodique, vous avez considéré¹, à la page 7, la loi de probabilité de deux variables aléatoires X, Y,² où la probabilité élémentaire ?(x, y) dx dy pour que X et Y soient respectivement compris entre x et x + dx, y et y + dy, est de la forme où K, a ₁, a ₂, b ₁, b ₂ sont des constantes. Nous nous proposons, dans ce qui suit, d'apporter quelques compléments à votre exposé.     

Testing for Income Smoothing Using the Backing Out Method: A Review of Specification Issues

This study investigates potentially spurious correlation in prior studies of income smoothing which use a research method that we call backing out. If E is reported income, Y a discretionary earnings component hypothesized to smooth income, and T the smoothing target, the backing out method consists of regressing Y on E – Y – T. A negative regression coefficient is interpreted as evidence that Y is being managed to smooth earnings. We argue that the negative regression coefficient may simply reflect the positive correlation between E and T, which may or may not be the result of manipulating Y. If E is not manipulated by Y, the negative regression coefficient reflects measurement errors in E – Y as an estimate of unmanaged earnings.We replicate the backing out method of prior studies and use each of the accrual and cash flow components of earnings in the Statement of Cash Flows for Y. We find that the regression coefficient, hypothesized to be negative because of smoothing with Y, is always negative and its magnitude depends on the variances of Y and E – T (smoothing error) independent of the nature of Y (discretionary or nondiscretionary). We conclude that the negative coefficient on E – Y – T is equally consistent with three possibilities: (1) managers smooth earnings with Y, (2) managers smooth earnings with something other than Y, and (3) managers do not smooth earnings at all.     

BOOK REVIEWS

Book reviewed in this article: Greg Pound, John J. Willingham and D. R. Carmichael, Australian Auditing — Concepts and Methods Carrick Martin, An Introduction to Accounting D. R. Jeffery and M. J. Wallace, Systems Analysis and Design Yuji Ijiri, Accounting Structured in APL F. M. Wilkes, Capital Budgeting Techniques I. G. Wallschutzky and B. T. Colditz, Australian Income Tax Questions Gabriel B. Rothberg, Structured EDP Auditing K. J. Leo and J. R. Hoggett, Company Accounting in Australia Sir Keith Yorston, E. B. Smyth and S. R. Brown, Advanced Accounting Jack Clark Francis, Management of Investments Michael T. Skully, Merchant Banking in ASEAN — A Regional Examination of its Development and Operations Edna Carew, Fast Money - The Money Market in Australia     

Remark on a theorem by Gurland

Abstract

Let X ₁,X ₂,...,X _n be a random sample of size from a distribution with probability density function p(x|θ), where the unknown parameter θ belongs to a non-degenerate interval I. The unknown true value of θ will be denoted by θ₀.