Abstract
M.Com.
In this dissertation, we have used a semi-Markovian model to calculate the conditional higher moments of any order of the present value of cash flows generated by an investment, taking into account the state of the market. With the force of interest following a stochastic process, we give an example to illustrate our results. In this work we assume that the state of economy explains the state of a country’s finances. This state is perceptible on the basis of certain indicators such as the index of a stock market. In South Africa it is the All Share Index (ALSI) that we have taken into consideration. We have observed the daily series of the ALSI index for the period from 6 December 1994 to 6 September 2014. We have further subdivided this period into two sub-periods. The first sub-period runs from December 6, 1994 to December 31, 2007 coinciding with the period before the global financial crisis and the second sub-period runs from January 1, 2008 to September 6, 2014, and relates to the post-crisis period. We have found that the daily average of the index for the pre-crisis is much lower than the daily average for the post-crisis period. Consequently, we assumed that each of these two periods put the South African economy in a particular state. Symbolically we denote state1 to represent the pre-crisis state, corresponding to the low level of the index and state0 for which the index has a high average. The consideration of South African data to conduct this study is essentially judged by the importance of its economy on the African continent.
Moreover, this study established a multi-state model based on a semi-Markov approach that calculates the moments of any order of the present value of a given cash-flow.
As results we found explicit formulas for the first two moments for the compound renewal sums with discounted cash flows, for a random interest rate. We observe that the length of stay in a state has a positive influence on the current value of moments in state 0. Instead In the state 1 when the length of stay increases, the present value of moments shows a slight downward trend.
In additional to the above findings, when considering time, the current values of moments of order 1 are growing except in the case where the interest rate is lower in state 1. But when the interest rate increases the present value of moments also increases, even when the length of stays increasing this cause moments to increase. When the interest rate is lower, the moments of order 2 decrease with time regardless of the state in which we stand. The length of stay has the same effect on the moments, it positively influences the moments of order 2 in the state 0,..