và không đúng. Đúng tiệm cận I () 1 thường được sử dụng như là một phương sai gần đúng của một ước tính, kể cả trường hợp kích thước mẫu là không lớn. Một vấn đề trong thực tế là, nói chung, tôi () là một chức năng của các tham số không rõ. Để đặt một giá trị gần đúng vào sự khác biệt của, chúng tôi sử dụng ước tính ở vị trí của. | 36 MODELING LONG-TERM STOCK RETURNS stock price at t 10 years given So using both the TSE and S P parameters. In both cases over this long term the left tail is substantially fatter for the RSLN model than for the LN model. This difference has important implications for longer-term actuarial applications. The probability function for the sojourn times can also be used to find unconditional moments of the stock price at any time n. E Sn k E E Sn k Rn k 2 2 E exp k RnR n - R m2 y Rn i n - Rn i r2i T L . k2 2 2 ii . k2 2 E exp IR lk 1 - 2 2 1 - exp Ik n 2 2 no 2 . . k n 2 2 na-ị exp kn 2 k2 n 2 exp r k 1 - 2 y ơ2 - 22 jjpn r THE EMPIRICAL MODEL Under the empirical model of stock returns we use the historic returns as the sample space for future returns each being equally likely sampling with replacement. That is assume we have n observations of the total stock return Return on stocks in t 1 t it t 1 2 3 . n Then we may simulate future values for stock returns for any period r - 1 r as Ir where 1 Pr Ir it for t 1 2 . n n The empirical model assumes returns in successive periods are independent and identically distributed. It provides a simple method for simulation though obviously analytical development is not possible. This distribution is useful as a simple quick method to obtain simulated returns. It suffers from the same problems in representing the data as the LN model which it closely resembles in distribution . Although we are sampling from the historical returns by assuming independence we lose the autocorrelation in the data. The autocorrelation means that low returns The Stable Distribution Family 37 tend to be bunched together giving a larger probability of very poor returns than we get from random sampling of individual historical returns. The autocorreleation is the source of fatter left tails in the accumulation factor distribution. Similarly high returns also tend to be bunched together giving fatter right tails. So the empirical model tends to be .
