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.     

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.     

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.     

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     

Über normale stabile statistische Reihen

Abstract

Es sei X eine zufällige Variable, welche k verschiedene Werte ξ₁, ξ₂, ... ξ _k annehmen kann, wofür die Wahrscheinlichkeiten p ₁, p ₂, ... p _k bestehen, wo p ₁ + p ₂ + ... + p _k = 1; die Gesamtheilt der Werte ξ₁, ξ₂, ... ξ _k und der ihnen zugeordneten Wahrscheinlichkeiten p ₁, p ₂, ... p _k wird das Verteilungsgesetz von X genannt. Werden an der Variablen X Versuche in der Weise vorgenommen, dass das Verteilungsgesetz stets dasselbe bleibt und die einzelne Versuche alle von einander unabhängig sind, so wird die Reihe der unter solchen Bedingungen zu stande kommenden empirisch-zufälligen Werte von X als normal stabil bezeichnet.     

BOOK REVIEWS

Book reviewed in this article: Aggregate Theory and Policy: Unemployment, Money Wage Rates, and Inflation. By George L. Perry . Monetary and Fiscal Policy. By Harold A. Wolf . Business Finance and Investments: The Corporate Merger. Edited by William W. Alberts and Joel E. Segall . Corporate Financial Reporting in a Competitive Economy. By Herman W. The Elusive Art of Accounting: A Brash Commentary on Financial Statements. By Howard Ross . Capital Budgeting: Current Practices and Their Efficiency. By George A. Christy . Investment Efficiency in a Socialist Economy. By Henryk Fiszel . The Management of Business Finance. By Paul G. Hastings . Financial Analysis of Investment Alternatives. By Raymond R. Mayer . Readings in Finance. Edited by Harold A. Wolf and Lee Richardson . Financial Institutions and Markets: Pension Funds: Measuring Investment Performance. By Peter O. Dietz . International Finance: Exchange-Rate Devaluation in a Semi-Industrialized Country: The Experience of Argentina 1955–1961. By Carlos F. Díaz Alejandro . International Liquidity: A Study in the Economic Functions of Gold. By Ian Shannon . Public Finance: A Capital Budget Statement for the U.S. Government. By Manard S. Comiez . Federal Tax Policy. By Joseph A. Pechman .     

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 .     

Osculatory mechanical quadrature

Abstract

The literature contains many formulas of mechanical quadrature¹, most of which are expressible in the form where the A's are constants, f(a _v) represents the functional value of f(x) at each of the n + 1 points x=a _v (v=0,1,2,..., n), and R is the remainder term. Two general and important types of the above formula are the Newton-Cotes ² formula in which the points a _v are equally spaced from c to d, and the Gaussian ³ quadrature formula in which the a's are chosen so as to obtain the greatest accuracy. The Euler-Maclaurin ⁴ formula of summation and quadrature uses the functional values f(a _v ), and the odd ordered derivatives of f(x) at the end points of the interval of integration. Steffensen ⁵ developed a formula for approximate integration employing not only the functional values but the first derivatives, f'(a _v ).     

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     

A characterization of geometric distributions by distributional properties of order statistics

Abstract

Let X ₁, X ₂ be independent identically distributed positive integer valued random variables. H the X _i 's have a geometric distribution, then the conditional distribution of R = max(X ₁, X ₂)-min(X ₁, X ₂), given R > 0, is the same as the distribution of X ₁. This property is shown to characterize the geometric distribution.     

On the convergence rate of fixed-width sequential confidence intervals

Abstract

Let X _f1, X _f2, ... be a sequence of i.i.d. random variables with mean µ and variance σ²∈ (0, ∞). Define the stopping times N(d)=min {n:n ^?1 Σ ⁿ _i=1} (X _i&#x2212;X _n)²+n ^?1?nd ²/a ²}, d>0, where X n =n ^?1 Σ ⁿ _i=1} Xi and (2π)^?½ ∫ ^a _?a exp (?u ²/2) du=α ∈(0,1). Chow and Robbins (1965) showed that the sequence I_n,d =[X_n ?d, X _n + d], n=1,2, ... is an asymptotic level -α fixed-width confidence sequence for the mean, i.e. lim_d→0 P(µ∈I_N(d),d )=α. In this note we establish the convergence rate P(µ∈I_N(d),d )=α + O(d½?δ) under the condition E|X₁|^3+?+5/(28) < ∞ for some δ ∈ (0, ½) and ??0. The main tool in the proof is a result of Landers and Rogge (1976) on the convergence rate of randomly selected partial sums.     

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.     

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.     

BOOK REVIEWS

Book reviewed in this article: Aggregate Theory and Policy; Monetary Economics—Readings on Current Issues. By William Gibson and George Kaufman . Monetary Theory: Inflation, Interest, and Growth in the World Economy. By Robert A. Mundell . Business Finance and Investments; Utility Regulation During Inflation. Edited by Joseph E. Haring and Joseph F. Humphrey . Financial Management of Foreign Exchange. By Bernard A. Lietaer . The Corporate Economy. Robin Marris and Adrian Wood , Eds. Cash Management and the Demand for Money. By Daniel Orr . Risk, Return, and Equilibrium: A General Single-Period Theory of Asset Selection and Capital-Market Equilibrium. By Bernell K. Stone . Financial Institutions and Markets; Credit Rationing and the Commercial Loan Market. By Dwight M. Jaffee . A History of Financial Intermediaries. By Herman E. Krooss and Martin R. Blyn . International Finance; Canada's Experience with Fixed and Flexible Exchange Rates in a North American Capital Market. By Robert M. Dunn . Jr . Capital Transfers and Economic Policy: Canada, 1951–1962. By Richard E. Caves and Grant L. Reuber with Robert W. Baguley , John M. Curtis , Raymond Lubitz . Public Finance; Indirect Taxation in Developing Countries. By John F. Due . Tax Incentives and Capital Spending. Edited with commentary by Gary Fromm . Modern Public Finance: The Study of Public Sector Economics. Second Edition. By Bernard P. Herber . State and Local Finance. Edited by William E. Mitchell and Ingo Walter .