Life Contingencies II
Project
Colin Sproat
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Table of Contents
Page 1.........................................................................................................................Title Page
Page 2...........................................................................................................Table of Contents
Page 3.................................................................Mathematical Description of the Problem
Page 4......................................................................................Description of Methods Used
Page 5-7..............................................................................................................................Plots
Pages 8.....................................................................................................................Conclusion
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ID: 1100503
Mathematical Description of the Problem
Before stating the problem, some background information is needed.
Consider the follow set of equations:
The first two equations are the crude survival functions for our decrements for cause 1 and
cause 2. These are defined as: S’( j ) = exp(-exp(-m/s)*(exp(x/s)-1)). The following g
functions are defined as the derivative of the jth decrement’s survival function divided by the
jth value given by Frank’s copula. The equations for the derivatives of the survival functions
(along with their weights) and the jth Frank’s copula are given below.
For the project, I will be using the following values:
p = 0.40
m1=90 σ1 = 10
m2=70 σ2 = 20
α = -20, -0.0001, 20
Now we can state the problem. Consider the following ordinary differential equation
(ODE):
.
This ODE is what I am intending to solve.
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Description of Methods Used
I will be solving this system of ODEs by using the bivariate Newton-Raphson
approximation technique. Hence, this derivative can be approximated by the following
equation:
.
Substitution and algebra on the previous two equations yields:
.
As you can see, the value of the next net survival function due to cause j is simply the value
of the previous net function plus our time (h) multiplied by our increment : g(t, y(t)). So,
using our knowledge that at time zero the survival function is equal to one (by definition) we
obtain the initial condition to our ODE as:
.
This will give us a recursive formula to approximate the value of the net survival function at
time h. In order to obtain a more accurate approximation I am going to choose h = 0.10, in
other words, I am going to compute the value of the survival function at times t = 0.1, 0.2,
0,3 … 114.9, 115. Ideally, I would choose h as close to infinity as possible, however this
would take increasingly longer to compute, so I will choose h = 1/10. Hence, I have
essentially discretized time into intervals of one-tenth of a year. Please note that I only take
each tenth value (hence each year) that is computed, thus generating the jth net survival
function for the years: 0,1,2....114,115.
Other values of interest are:
The Crude Survival Functions (which is just a modified Gompertz model) are:
S’( j )(t) = exp(-exp(-m/s)*(exp(x/s)-1))
The PDF is found by taking the derivative of the death function:
d/dt {F’( j )(t)} = d/dt {1 - S’( j )(t)}
d/dt {F’( j )(t)} = -exp(-exp(-m/s)*(exp(x/s)-1))*(exp((x-m)/s)/s)
And then adding the weights w1 = 0.40, w2 = 0.60 to the pdf of cause 1 and 2 respectively.
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Plots
Crude Survival Functions:
Probability Density Functions (PDF):
Net Survival Functions @ α = - 20:
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Net Survival Functions @ α = - 0.00001:
Net Survival Functions @ α = 20:
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Solutions:
Note: The preceding plots as well as the data associated with the plots and the g-functions
have all been sent to you in an e-mail to save paper. In addition the code, written in R, has
also been sent to your e-mail address.
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Conclusion
The results of the following analysis support my findings:
Proper Convergence of Algorithm:
By examining the g-functions (this is included in my e-mail in a text file) I was able to
determine that they decreased in a manner which was very well-behaved in terms of
convergence since there were no serious deviations from the decreasing pattern.
Values Are Correctly Bounded and Realistic:
All survival functions lie in the interval (0,1) and the area under the curves of the PDFs seem
likely to be one. In addition, the probabilities given by the pdfs and the survival functions
appear very realistic (having a high value initially then a low probability of survival as age
approaches 115).
The Net Survival Functions and Graphs Match Those on the Website:
From the website http://gompertz.math.ualberta.ca/ :
“Here are the values at age 70 for the first net survival function. They are .6484(-20),
.9299(0), .9496(20). Here are the values at age 70 for the second net survival function.
They are .5916(-20), .6213(0), .6280(20).”
The calculated and plotted values presented earlier (included in the excel file entitled Net
Survival Functions in the e-mail) concur with those cited precisely.
In light of the preceding evidence, I conclude with certainty that I have found the required
Net Survival Functions.
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