Expected Discounted Valuation With Gamma and Brownian Hitting-Time Distributions — Extends the Life Span and Reduces the Premium

Abstract

The gamma and hitting-time distributions extend the geometric and exponential distributions allowing for greater flexibility in modelling termination events and their premia. In an example, the termination premia were calculated using prior beliefs and mean-variance approximation in addition to using the gamma and hitting-time distributions; they gave risk premia that were within 4 basis points of each other because skewness, kurtosis and higher order cumulants did not contribute significantly. Since the mean and variance are the significant contributors, the gamma and hitting-time distributions can be used interchangeably if the mean and variance of one distribution correspond with those of the other distribution. If spreadsheets are used, it is not necessary to add a termination premium to the discount rate; instead, the spreadsheet cells can be truncated at some point before the expected life span of the income stream and the income discounted without the termination premium added to the discount rate. Two applications demonstrate the use of the hitting-time and gamma distributions. The hitting-time distribution was used for a problem that involved an expenditure rate, variance of expenditure and time when the funds are fully expensed. The gamma distribution was used in another application where the survivability value contributed to the life span and consequently a lower termination premium. It demonstrated that a fund with a higher rate of negative monthly performance can have a lower termination premium than another with a lower rate of negative performance due to the survivability value. The examples used in the article assumed that the termination premium was not included in the CAPM-based discount rate; if it was, the discount rate would have to be adjusted to remove the termination premium. A regression of returns on the risk-free rate and excess market returns is a possible test to determine whether the termination premium was included.