UNSW
ACTL1001 Actuarial Studies and Commerce
Sample Solutions Exercises 9
NOTE: the following questions can be computed either by hand or by
excel. You should ensure that you are able to use both methods.
Exercise 1
Results below
1. To calculate the probability of being on di/erent discount levels for the
&fth year we need to raise the transition probability matrix to the power
of 5. Firstly the probability of no claims (and moving to the next higher
level in a year) is
e
&
0
:
1
= 0
:
904837
and the probability of one or more
claims is 0.095163. The transition matrix will be
NCD level
0
1
2
0 0.095163 0.904837
0
1 0.095163
0 0.904837
In Excel we can multiply matrices using the MMULT(,) function. To
do this, select an area for the results the size of the resulting matrix (
in this case 3x3). Then enter =mmult(array1, array2) and then press
CTRL, SHIFT, ENTER (not just ENTER) and the selected area (3X3)
will have the resulting matrix in it. In this case array1 is just the tran
sition matrix and array2 is the same matrix so we multiply the matrix
by itself to get the square of the matrix and so on to the 5 th power in
this case. The results are given in this case below
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentTransition matrix
Prob of no claims
0.904837
NCD level
0
1
2
0 0.095163 0.904837
0
1 0.095163
0 0.904837
2
0 0.095163 0.904837
Year 2
0.095163 0.086107 0.818731
0.009056 0.172213 0.818731
0.009056 0.086107 0.904837
This is the end of the preview.
Sign up
to
access the rest of the document.
 One '09
 Nicole
 Markov chain, 0 1 2 0 0.095163 0.904837 0 1 0.095163 0 0.904837 2 0 0.095163 0.904837 Year, 2 0.095163 0.086107 0.818731 0.009056 0.172213 0.818731 0.009056 0.086107 0.904837 Year, 3 0.01725 0.164019 0.818731 0.01725 0.086107 0.896643 0.009056 0.094301 0.896643 Year, 4 0.01725 0.093521 0.889229 0.009836 0.100935 0.889229 0.009836 0.093521 0.896643 Year

Click to edit the document details