ACTSC 232 Fall 2023 Assignment
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ACTSC 232 Fall 2023 Assignment
Due Monday November 27th, 11:59pm
47 marks Total
Report numerical answers to 6 decimal places for probabilities and 2 decimal places for dollar values. However, you should keep as many decimal places as possible for intermediate steps to avoid rounding errors.
You will be submitting written (or typed) solutions for the questions below and will be graded based on those submissions and not work in Excel. So all written solutions should give details and explanations of what calculations were performed in Excel in order to get full marks.
The Excel file you create to do the calculations will also be submitted in Crowdmark, but mainly for verification purposes if needed. Please complete all calculations in a single Excel file, but on separate tabs as needed.
Solutions to each question must be submitted to the appropriate place in Crowmdark. Please start a new page for each of the following so that they can be uploaded separately: 1a-b, 1c-d, Q2, 3a-b, 3c-d, 4a-b, 4c, 4d.
Submissions with questions that are not properly dragged and dropped to the appropriate places in Crowdmark will receive a 5% penalty. Late submissions receive a penalty of 1% per minute late.
1. [12 marks] Consider two insurance contracts given by the following present value random vari- ables (PVRVs):
(a) [4] Describe in words the insurances given by the two PVRVs.
(b) [3] Derive expressions in terms of standard actuarial notation (big ‘A’s’) for the expected present values for each insurance.
(c) [4] Derive an expression in terms of standard actuarial notation (big ‘A’s’) for the covariance of Z1 and Z2. (Note: it doesn’t need to be simplified).
(d) [1] Without doing any calculations, would the covariance be positive, negative, or 0? Explain why logically.
2. [9 marks] Makeham mortality is given by µx = A + Bcx for all x ≥ 0.
(a) [3] Derive an expression for tpx in terms of A, B, and c. Hint: it is quite similar to the derivation for the Gompertz model we did in class.
(b) [2] Using A = 0.0002, B = 0.00006, and c = 1.12, construct a life table in Excel with columns for x, px and lx for ages x = 20 to x = 109. You should set up your table so that you can change the parameter values for the Makeham model and the values will update.
(c) [4] To your Excel table, add columns for insurance and annuity functions Ax and ¨(a)x assum- ing interest at 6% annual effective per year and using limiting age ω = 110. (i.e. Assume q109 = 1 if ω = 110. Also, you should set it up so that you can change the interest rate and the values will update.) Use backwards recursion from the insurance and annuity values at age 109 to calculate ¨(a)x and Ax at all ages in the table. Show your work/general recursion to explain the calculations done in Excel.
3. [12 marks] Consider a $100,000 20-year discrete endowment insurance issued to a life aged 40. Use the life table constructed in Question 2 and interest of i = 6% in answering this question.
(a) [2] Write an expression for and calculate the EPV of the endowment insurance.
(b) [3] Write an expression for and calculate the standard deviation of the endowment insurance PV.
(c) [4] Calculate the probability that the present value of the payment is less than or equal to $50,000.
(d) [3] Find the 80th percentile of the present value of the benefit.
4. [13 marks] A 30-year term insurance issued to a life aged 30 pays $100,000 immediately on death. Premiums are paid at the beginning of each month. There is a $250 initial expense at time 0, renewal expenses of 0.25% of premium each month, including the first, and a death claim expense of $100.
(a) [3] Write down the loss at issue random variable L0 as a function of the monthly gross premium amount P.
(b) [2] Provide an expression using actuarial EPV notation that would be used to solve for the monthly gross premium amount P under the equivalence principle.
(c) [6] Using your table from Question 2, calculate the monthly gross premium amount at i = 6%. Use three-term Woolhouse to approximate the continuous and monthly payment timings. (Note: A(¯)x:n 
= 1 − δ¯(a)x:n). Do all calculations in your spreadsheet so that P will
recalculate if the interest rate parameter is changed.
(d) [2] The insurer wishes to change the interest assumption from 6% to 5% for the premium calculation. Calculate the updated premium when i = 5% and compare it to the previous
premium value. How has it changed and why?
5. [1 mark] Submit your excel file to Crowdmark.
2023-11-23