Kernel Density Estimation of White Noise for Non-diversifiable Risk in Decision Making

Abstract

Many businesses make profit yearly and tend to invest some of the profit so that they can cushion their organizations against any future unknown events that can affect their current profit making. Since future happenings in businesses cannot be predicted accurately, estimates are made using experience or past data which are not exact. The probability element (which is normally determined by experience or past data) is important in investment decision making process since it helps address the problem of uncertainty. Many of the investment decision making methods have incorporated the expectation and risk of an event in making investment decisions. Most of those that use risk account for diversifiable risk (non-systematic risk) only thus limiting the predictability element of these investment methods since total risk are not properly accounted for. A few of these methods include the certainty (probability) element. These include value at risk method which uses covariance matrices as total risk and the binning system which always assumes normal distribution and thus does not take care of discrete cases. Moreover comparison among various entities lacks since the probabilities derived are for individual entities and are just quantile values. Finite investment decision making using real market risk (non-diversifiable risk) was undertaken in this study. Non-diversifiable risk (systematic risk) estimates of a portfolio of stocks determined by a real risk weighted pricing model are used as initial data. The variance of non-diversifiable risk is estimated as a random variable referred to as random error (white noise). The estimator is used to calculate estimates of white noise (wn). A curve estimation of the wn is made using Kernel Density Estimation (KDE). KDE is a non-parametric way to estimate the probability density function of a random variable. KDE is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. This is used to derive probability estimates of the non-diversifiable risks of the various stocks. This enables determination of total risk with given probabilities of its occurrence thus facilitating decision making under risky and uncertain situations as well as accentuating comparison among the portfolio of stocks.