代做MATH 376 Actuarial Models MOCK 2022 CLASS TEST代做Web开发

MATH 376

MOCK 2022 CLASS TEST

Actuarial Models

Section A

A1. (a) You are given that , 0 ≤ t ≤ 100 − x.

Find: (i) S0(x), µx, [2 marks]

(ii) central rate of mortality Mx, [2 marks]

(iii) the variance of the future lifetime (x). [2 marks]

[6 marks]

(b) The following table gives a data set of survival times (in years) with censored status (1≡ failure, 0≡ censored) and covariates x.

Assume that the Cox’s proportional hazards model is employed with parameter β. Write down the partial likelihood of β. [4 marks]

[10 marks]

A2. You are given the following table of death (failure) times of a certain group of patients.

where by ∗ denotes the censored observations. Find the Kaplan-Meier es-timator.

[10 marks]

Section B

B3. You are given the following table of death (failure) times of a certain group of patients.

where by ∗ denotes the censored observations.

a) Create survival object. [2 marks]

b) Calculate the Kaplan-Meier estimate of the survival function, Sˆ(t). What is the estimated probability of surviving to 230 days? [3 marks]

c) Plot the Kaplan-Meier curve with the (point-wise) 95% confidence inter-val. [2 marks]

d) What is the median survival time? [1 mark]

e) Estimate the cumulative hazard function by the Nelson-Aalen estimator. [2 marks]

[10 marks]

Required packages:

install.packages(”survival”)

library(”survival”)

B4. A life insurance company priced its whole life contracts using a standard mortality table. The company wishes to establish whether recent mortality experience in the portfolio of business is in line with the pricing base. Thus, the following data are employed:

a) Calculate the crude estimates for mortality mx. [1 mark]

b) Estimate the parameters of Gompertz law via nonlinear least squares estimation. [2 marks]

c) By using the estimated Gompertz law parameters, estimate the param-eters of Makeham law. [2 marks]

d) Plot mortality rate and graduated mortality rate obtained from Make-ham law by age together. [2 marks]

[7 marks]

B5. The UK mortality data is obtained from the Human Mortality Database for the period of 1922-2018 for all ages. The maximum age is 100.

(a) Read the data and create the “demogdata” object from data to fit the Lee-Carter model. [2 marks]

(b) Estimate and define the Lee-Carter model parameters. [3 marks]

(c) Plot the figures of estimated parameters and interpret these figures. [5 marks]

(d) Project the mortality time-index for ten years and plot the figure of projected values. [3 marks]

[13 marks]

Required packages:

install.packages(”demography”)

install.packages(”forecast”)

install.packages(”lifecontingencies”)