FM for Actuaries

314 CHAPTER 10
Applying equations (10.18) and (10.19), we obtain
E ( )
a 5⌉ = 4.54697,
Var ( )
a 5⌉ = 0.72268.
Finally, employing formulas (10.20) to (10.23), we obtain
E ( )
ä 5⌉
= 1+E ( )
a 4⌉ =1+3.69483 = 4.69483,
Var ( )
ä 5⌉
= Var ( )
a 4⌉ =0.40836,
E ( )
s 5⌉
= 1+E (¨s )
4⌉
=1+4.51648 = 5.51648,
Var ( )
s 5⌉
= Var (¨s )
4⌉
=0.64414.
10.4 Autoregressive Model
Let Y t =ln(1+i t ). Under the independent lognormal model, we have
Y t =ln(1+i t ) ∼ N(µ, σ 2 ), (10.24)
where there are no correlations between Y t and Y t−k for all k ≥ 1. In this section
we consider interest rate models that allow some dependence among Y t ’s, say,
Corr(Y t ,Y t−k ) ≠0, for some k ≥ 1, (10.25)
while maintaining the lognormal assumption in (10.24).
In the literature of statistical time series analysis there are many classes of
discrete time series models which may be applied to characterize the dependence
among Y t ’s. 4 As an introductory illustration of the basic idea we consider the class
of first-order autoregressive models, denoted by AR(1). The AR(1) model has the
form
Y t = c + φY t−1 + e t , (10.26)
where c is the intercept and φ is the autoregressive parameter and e t are independently
and identically distributed normal random variates each with mean zero and
variance σ 2 .For|φ| < 1, the correlation structure of the interest-rate process {Y t }
is
Corr(Y t ,Y t−k )=φ k , for k =1, 2, ··· , (10.27)
4 For example, see Wei, W.W.S., Time Series Analysis: Univariate and Multivariate Methods,2 nd
Edition, Pearson Addison Wesley, 2006.