Multiplicative parameters and estimators: applications in economics and finance

Ann Oper Res (2016) 238:299–313 DOI 10.1007/s10479-015-2035-x Multiplicative parameters and estimators: applications in economics and finance Helena Jasiulewicz1 · Wojciech Kordecki2 Published online: 19 October 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract In this paper, we pay our attention to multiplicative parameters of random variables and their estimators. We study multiplicative properties of the multiplicative expectation and multiplicative variation as well as their estimators. For distributions having applications in finance and insurance we provide their multiplicative parameters and their properties. We consider, among others, heavy-tailed distributions such as lognormal and Pareto distributions, applied to the modelling of large losses. We discuss multiplicative models, in which the geometric mean and the geometric standard deviation are more natural than their arithmetic counterparts. We provide two examples from the Warsaw Stock Exchange in 1995–2009 and from a bid of 52-week treasury bills in 1992–2009 in Poland as an illustrative example. Keywords Geometric mean · Geometric variance · Lognormal distribution · Pareto distribution · Multiplicative estimators 1 Introduction Two measures frequently used in descriptive statistics are the arithmetic mean and the standard deviation. The geometric mean is used less often, while the geometric standard deviation connected with the geometric mean is used even more rarely. This work was supported by National Science Centre, Poland. B Helena Jasiulewicz Wojciech Kordecki 1 Institute of Economics and Social Sciences, Wrocław University of Environmental and Life Sciences, Wrocław, Poland 2 Faculty of Technical and Economic Science, The Witelon State University of Applied Sciences in Legnica, Legnica, Poland 123 300 Ann Oper Res (2016) 238:299–313 When is it better to use arithmetic (additive) parameters and when geometric (multiplicative) ones? A lot of attention has been paid to these problems in the economic and finance literature. One of the firsts papers on this topic was the article by Latané (1959), who introduced the geometric-mean investment strategy into the finance and economics literature. Weide, Peterson and Maier wrote in their paper (1977). Most of this work has been devoted to the investigation of various properties of the geometric mean strategy. Among the properties of optimal geometric-mean portfolios recently discovered are (i) they maximize the probability of exceeding a given wealth level in a fixed amount of time, (ii) they minimize the long-run probability of ruin, and (iii) they maximize the expected growth rate of wealth. In the paper (Weide et al. 1977), they consider either the computational problem of finding the optimal geometric mean portfolio or the question of the existence of such a portfolio. They analysed both of these problems under various assumptions about the investor’s opportunity set and the form of his/her subjective probability distribution of holding period returns. Let us assume that the gross return R in a single period has a lognormal distribution. The 2 unknown parameter is a = E (R) = em+σ /2 . To estimate this parameter one can use the arithmetic mean of gross returns: 1  Ri . N N R= i=1 It is an unbiased estimator of the parameter a. Another unknown parameter considered in (Cooper 1996) is the geometric mean of the gross return b = EG (R) = em . The parameter b can be estimated as the geometric mean  R G = eln R = exp  N 1  ln Ri . N i=1   In (Cooper 1996; Jacquier et al. 2003, 2005), the expected value E R G is calculated. This   value is an asymptotically unbiased estimator of b. Moreover, the variance D2 R G , which tends to zero, is determined. In our paper, we point out that the quality of the geometric estimator should be examined by the geometric mean and variance, not by their arithmetic counterparts as in (Cooper 1996; Jacquier et al. 2003, 2005). In the paper (Hughson et al. 2006), the authors point out that forecasting a typical future cumulative return should be more focused on estimating the median of the future cumulative return than on the median of the expected cumulative return. Expectation of the cumulative return is always higher than the median of the cumulative return. The probability distribution of returns from risky ventures is positively skewed. It is frequently assumed that returns have lognormal distributions. For a lognormal distribution, the median and the geometrical expectation are equal. Another distribution frequently used in finance and insurance is the Pareto distribution, in which the geometric mean is close to the median and far from the arithmetic mean. Arithmetic and geometric means are somewhat controversial measurements of the past and future investment returns. Critical remarks on this topic are given in the paper (Missiakoulis et al. 2007). A review of basic equalities and inequalities in the context of a gross income from the investment in a discrete time can be found in the article (Cate 2009). Properties of various kinds of means can be found in the review paper (Ostasiewicz and Ostasiewicz 2000). 123 Ann Oper Res (2016) 238:299–313 301 In this paper, unlike in the results discussed above, the issue concerning multiplicative parameters, including a geometric mean, is also extended with interpretations and applications of multiplicative variance as a measure of dispersion. Such a measure, as we justify in more detail in the next sections, is a better and more natural measure of deviation between random variables and their geometric mean. The geometric variance is invariant with respect to multiplication by a constant. From this property it follows that the variance of an economic quantity given in different monetary units is constant, independent of the choice of the unit. For example, if the monetary unit is $1 or one monetary unit is $100 then the variance is the same. Moreover, the geometric variance is a dimensionless measure of variability. For example, it allows to compare the variability of exchange rates between different currencies. In Sect. 2.1 we give definitions and properties of multiplicative parameters. We discuss multiplicative models, in which the geometric mean and the geometric standard deviation are more natural than their arithmetic counterparts. In Sect. 2.2 we introduce typical distributions for which the multiplicative parameters are more natural than the additive ones. In Sect. 2.3 we provide estimators of the multiplicative parameters considered in Sect. 2.1 and their properties. In Sect. 3 we give real examples of applications. These examples indicate the real benefits of applying the geometric parameters instead of arithmetic ones in real situations in economics and finance. 2 Parameters and models 2.1 Multiplicative parameters and models Let us define the multiplicative (geometric) mean by EG (X ) = eE(ln (...truncated)