代写ACTL 30001 Actuarial Modeling I Assignment 2代写C/C++编程

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ACTL 30001

Actuarial Modeling I

Assignment 2

• Due date: By 5:00PM on Friday, May 23 2025

• This is an individual assignment and you have to sign the cover sheet.

• Assignment 2 will be worth 15%.

• Assignments do not have to be typed; nevertheless, handwriting that is very dif-ficult 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 to-gether with your R markdown 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

1. (15 marks) The Qu1-data-Assignment2.xlsx dataset contains data from an ex-perimental study of recidivism of 411 male prisoners, who were observed for a year after being released from prison. The researchers now want to under-stand what factors affect the likelihood of those prisoners being rearrested. The following variables are included in the dataset:

• week: weeks of first arrest after release.

• arrest: the event indicator, equal to 1 for those arrested during the period of the study and 0 for those who were not arrested, or right censored at one year or within one year.

• fin: a dummy variable, equal to 1 if the individual received financial aid after release from prison, and 0 if he did not

• prio: number of prior convictions.

Consider a proportional hazards model under which prisoners with no prior con-victions and did not receive financial aid after release from prison, are subject to the baseline hazard.

(a) Write down a formula for the hazard rate at time t for a prisoner with the two covariates (risk factors): fin and prio. Specify all quantities/variables used in this PH model, and in particular specify the random variable for which the hazard rate is modeled.

(b) Let L(β1, β2 ) be the partial likelihood function, where β1 and β2 are two regression coefficients to be estimated. Plot the natural log of the partial likelihood lnL(β1, β2), for −1 < β1, β2 < 1, using R or Excel.

2. (20 marks) In a mortality study, N independent lives are observed between age x and x + 1. For the i-th life, observation starts at age x + ai and ends at age x + bi, or ealier if this life dies, where 0 ≤ ai < bi ≤ 1.

For 20 lives observed at age 80, the following data were collected:

In Unit 3, x + bi is the age when our observation of life i must cease and it is either x + 1 or the age at the end of the observation period.

‘Withdrawal’ in the table above means that this life withdrew from our obser-vation prior to the end of the observation period or prior to age 81. For such a life with mode of exit ‘withdrawal, the waiting time is Vi = bi − ai and Di = 0.

In what follows, you calculate each estimated probability to 5 decimal places.

(a) Under the binomial model of mortality, use Excel Solver, or Excel Goal Seek, or a numerical method such as Newton method, or otherwise, to find

i. the maximum likelihood estimate of q80, under UDD, CFM, and the Balducci assumptions on the fractional age, respectively.

ii. the moment estimate of q80 under UDD, CFM, and the Balducci as-sumptions on the fractional age, respectively.

(b) In the binomial model, calculate the moment estimate of q80, under UDD assumption, by using

(d) Under the binomial model, calculate the modified moment estimate of q80.

(e) If the age at death for the first death (the one on the fifth row of the left column) is 80.7, the age at death for the second death (the one on the second last row of the left column) is 80.5, and the age at death for the second death (the one on the right column) is 80.3, calculate

• the actuarial estimate of q80;

• The MLE and ME of q80 under the two-state Markov mortality model.

3. (12 marks) In the two-state Markov model, if Balducci assumption is made,

(a) Write down the likelihood function L as a function of qx.

(b) Write down the log-likelihood function lnL(qx).

(d) Calculate ˆqx for the data set in the Table for Question 2 and the death ages for Question 2(e), by using Excel Solver or R.

4. (25 marks) An insurance company sells a disability income insurance policy which also provides a death benefit. If the policyholder dies as a result of an accident, the amount of the death benefit is doubled. In order to price this policy, the insurer constructs the multiple state model shown below. The arrows indicate the possible transitions.

Let Zx be the state of a life at age x and let denote the transition intensity from state i to state j at age x, and let denote the probability that a life currently in state i is in state j in t years’ time.

(a) Let = min{t ≥ 0 : Zx+t = j|Zx = i}, for i = 0, 1 and j = 2, 3. Write down the pdf of and , respectively.

(b) A whole life insurance is issued to a life currently aged x and in state 0: a death benefit of $ 10,000 payable immediately at death if the death is not caused by an accident, while the death benefit will be doubled if death is caused by an accident.

i. Find an integral expression for the expected present value (EPV) of the death benefits. The force of interest for calculating EPV is assumed to be δ > 0.

ii. Find the probability that the insured received a benefit of $20,000.

(c) An insured aged x (currently in state 0 at time 0) receives a continuous disability benefit at annual rate 10, 000 while the insured is in state 1, and pays a premium continuously at an annual rate P while the insured is in state 0.

i. Find an expression for the expected total disability benefit (without accumulation).

ii. Assume that = 0. Find an integral expression for the expected present value of premium payments if the force of interest is assumed to be δ > 0. (The assumption of = 0 only applies to c(ii))

(d) Write down a pair of differential equations satisfied by and .

(e) Suppose that the transition intensities are constant, so that for all x. Find a second order differential equation for .

(f) Suppose that µij = 0.01 for all possible transitions.

i. Find an expression for by solving the 2nd order ODE.

ii. Find an expression for .

iii. Find an expression for and , by using the results in Question 4 of Tutorial 8.

(g) Again suppose that the transition intensities µ ij x = µ ij are constant. Write down expressions for the likelihood associated with two individuals

(i) an individual A who is in state 0 at time 0, remains in state 0 until transferring to state 1 at time t1, and remains in state 1 until trans-ferring to state 3 at time t1 + t2, and

(ii) another individual B who is in state 1 at time 0, remains in state 1 until transferring to state 0 at time t3 and remains there till time t3+t4 and transfers to state 2 instantly at t3 + t4.

5. (18 marks) Let {Mt ;t ≥ 0} be a continuous-time MC with n states labeled as 1, 2, . . . , n. The intensity from i to j (≠ i) is given by

for i = 1, 2, . . . , n − 1.

Define

Let and be the transition probability matrix and intensity matrix, respectively.

(a) Write down the intensity matrix µ.

(b) Write down the Kolmogorov’s forward DE for p 1 j (t), j = 1, 2, . . . , n.

(d) Find an expression for pij(t), for i ≤ j.

(e) Define T(i) = min{t ≥ 0 : Mt ≠ i|M0 = i}, i = 1, 2, . . . , n. Identify the distribution of T (i) .

(f) Define Ti,j = min{t ≥ 0 : Mt = j|M0 = i}, 1 ≤ i < j ≤ n. Identify the distribution of T1,n.

6. (10 marks) Suppose that you are a tutor for ACTL30001 and were asked by the subject coordinator to set one final exam question worth of 10 marks. Please

(a) set the question (may or may not include several small parts) related to content upto Unit 4;

(b) provide detailed solution to your question in (a).