The University of Melbourne ACTL 30001
Center for Actuarial Studies Actuarial Modelling I
Assignment 1
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• Due date: By 5:00PM on Friday, 7 April 2023
• This is an individual assignment and you have to sign the cover sheet.
• Assignment 1 will be worth 15%.
• Assignments do not have to be typed; nevertheless, handwriting that is very
difficult to read may not be marked. Please refrain from using a red pen
anywhere in the assignment.
• Submission: Please submit your typed or scanned answers in PDF format
together with your programming file and/or Excel spreadsheet to LMS by
the due date.
• Please name your files using your student ID number, e.g., idnumber.pdf
and idnumber.xlsx
• Solution will be posted on LMS after the submission date
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1. (25 marks) Suppose force of mortality is given by µy = a + by + cy
2 for all
0 ≤ y < 120 where a, b, and c are constants and c > 0. You are also given
that no one can survive to age 120.
(a) Calculate and simplify the expression for tpx.
(b) What constraints should be imposed on the values of a, b, and c?
(c) You are given a = 0.002, b=−0.000254.
i. Find the value of c such that e0 = 59.5 using goal seeker in Excel.
Describe the steps you took to find the value.
ii. Does there exist a value of c such that e0 = 80 and why?
1
(d) Let l0 = 100, 000. Using the approximation
◦
ex ≈ ex + 0.5 and Excel
to construct a mortality table based on this mortality law using the
values of a and b specified in 1c and the value of c determined in 1(c)i.
The constructed mortality table should have columns with heading
lx, x = 0, 1, 2, . . . , 120,
dx, px, qx, µx,
◦
ex, Tx, Lx, x = 0, 1, 2, . . . , 119.
Round lx, Lx and Tx to the nearest integer. Describe how you obtain
Tx and Lx.
(e) Comment on whether the constructed mortality table is suitable for
human mortality.
(f) Based on your constructed Life Table in 1d,
i. compute f50(1.5) to 4 significant digits, if the uniform distribution
of deaths (UDD) assumption applies, where fx(t) is the density of
Tx at time t;
ii. compute 0.5|1.5q50.5 to 4 significant digits, if CFM assumption ap-
plies;
2. (25 marks) Suppose µy = m1 for all y ≤ 30 and µy = m2 for all y > 30.
(m1 < m2 are constants).
(a) Determine the distribution function of Tx, the future lifetime of (x).
(b) Calculate and simplify the expression for
◦
ex and ex.
(c) Let m1 = 0.03 and m2 = 0.04. Compute 10p25,
◦
e25 and e25.
(d) Let Kx = [Tx]. Find the probability mass function of Kx.
(e) Define µ¯y = P(K0 = y|K0 ≥ y) to be the hazard function of K0 at y
for y = 0, 1, 2, . . . . Find µ¯y.
3. (5 marks) You are told that for a fixed x0 > 0 and all t ≥ 0, tpx0 =
atbe−ct
2−dt, where a, b, c, d are constants. Find all the restrictions on a, b, c, d.
(For instance, a > 2, b < 0, etc.)
4. (5 marks) Which of the following functions could be the hazard rate µx of
some positive random variable (not necessarily representing human lifetime),
for all x larger than some x0 ≥ 0? Justify your answers.
(a) C
(x+1)2
, if C is a positive constant.
(b) C1
x+C2
, if C1, C2 are positive constants.
2
(c) C1e
C2x, if C1, C2 are positive constants.
5. (15 marks) You are given l102 = 10, l103 = 5.
(a) Plot the three curves {1−tqU102+t, 1−tqB102+t, 1−tqC102+t} for 0 ≤ t ≤ 1 on
the same graph, if U refers to UDD, B refers to the Balducci assump-
tion, and C refers to the constant force of mortality assumption.
(b) Compute the maximum of the difference |1−tqU102+t − 1−tqB102+t| for 0 ≤
t ≤ 1.
6. (15 marks) Eight lives aged 40 under observation gave the following data:
deaths times: 3.2, 4.5, 6.8, 9.3
withdrawal times: 1.2, 4.5, 5.5, 8.0
(a) Compute the K-M estimate of S(t), for all t;
(b) Plot the K-M estimate of S(t).
(c) Assume that the future lifetime of (x) follows the distribution Fx(t; θ) =
1− e−θt, θ > 0.
i. Write down the likelihood function from the data above.
ii. Find the MLE of θ.