Graduation Mathematics Question Paper

Graduation Mathematics

Institution: Kenyatta University question papers

Exam Year:2009

KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2009/2010
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
ECONOMICS, BACHELOR OF ARTS AND BACHELOR OF EDUCATION
(ARTS)

BMS 408: GRADUATION MATHEMATICS

DATE:
Tuesday 22nd December 2009
TIME: 8.00am – 10.00am

INSTRUCTIONS:
1.
Answer Question 1 (30 marks) and any other two questions (each 20 marks)
2.
Support your answers with relevant illustrations/examples

Q1.
a)
List four factors in respect of which life insurance mortality statistics are
often
subdivided.
[4marks]

b)
Explain why crude mortality rates are graduated before being used
for
financial
calculations.
[4marks]

c)
List two methods of graduating a set of crude mortality rates and state, for
each
method:

i)
Under what circumstances it should be used; and
ii)
How
smoothness
is
ensured.
[8marks]

d)
A life insurance company has a small group policies written on impaired

lives and has conducted an investigation into mortality of these

policy holders. It is proposed that the crude mortality rates be graduated

for use in future premium calculations. Discuss the suitability of two

methods of graduation that the insurance company could use. [4marks]

e)
Define the terms undergraduation and explain three indicators underlying
undergraduation [4marks]
Page 1 of 3

f)
When is it appropriate to use smooth-junction method of interpolation?

[4marks]

g)
Enumerate at least four non-parametric method of graduation. [2marks]

Q2.
An insurance company is investigating the mortality of its annuity policyholders.

It is proposed that the crude mortality rates be graduated for use in future
premium
calculations.

i)
a)
Suggest, with reasons, a suitable method of graduation in this case.

b)
Describe how you would graduate the crude rates.

ii)
Comment on any further considerations that the company should take into

account before using the graduated rates for premium calculations.

Q3.
Use the expression 3v -3300 - 6x +15 2
x = 0 together with successive values of
x

x = ,
1
15
....
3
,
2
to obtain the value v and use the moving average method to
x

smooth the values by averaging four, five and six terms. Comment on the

smoothed values in each column.

Q4.
An investigation was undertaken into the mortality of male term assurance

policyholders for a large life insurance company. The crude mortality rates were

graduated using a formula of the form:
0

yx
q = a + ße
x

Page 2 of 3

An extract of the results is shown below.

Age Exposure

Crude
Graduated
Standardized
(years)
mortality rate
mortality rate
deviation
x Ex qˆ
qˆ
x
x
?
o ?
E qˆx q
x ?
- x ?
?
?
= =
x
0 ?
o ?
E q 1 q
x
x ? -
x ?
?
?

40 11.037
0.0029
0.00348
-1.035
41 12.010
0.00333
0.00358
-0.459
42 11.654
0.003
0.00368
-1.212
43
9.658
0.003
0.00379
-1.264
44
8.457
0.00319
0.00391
-1.061
45 10.541
0.00427
0.00402
0.406
46 7.410
0.00472
0.00415
0.763
47 12.042
0.00399
0.00428
-0.487
48 14.038
0.00406
0.00441
-0.626
49 11.479
0.00375
0.00455
-1.274
50 12.480
0.00409
0.00469
-0.981
51 10.567
0.00407
0.00485
-1.154
52 9.187
0.00512
0.00500
0.163
53 14.027
0.00456
0.00517
-1.007
54 11.581
0.00466
0.00534
-1.004

i)
Test the graduation for goodness of fit using the chi-squared test.
ii)
a)
By inspection of the data, suggest one aspect of the graduated rates where
adherence
to
data seems inadequate.

b)
Explain why this may not be detected by the chi-squared test.

c)
Carry out one other test that may detect this deficiency.

iii)
Suggest how the graduation could be adjusted to correct the deficiency identified.
Page 3 of 3

Graduation Mathematics Question Paper

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