AS 3429/9429 Long Term Actuarial Math II Assignment
Instruction
1. Students could work individually or in a group (of maximum two members) to complete the Assignment using Excel (or other mathematical software).
2. Each student must submit the following two files via Assignments on OWL:
• One solution report with the formulas and methods explained, required graphs, and appendix (if needed, e.g., R codes).
• One Excel spreadsheet showing all the calculations.
*Note: If you work in group, please state clearly the group members at the beginning of your report.
3. The assignment is due on Friday, Dec 6, by 11:59pm. For late submission, a penalty of 50% mark per day will be applied on an hourly basis.
4. Evaluation: The assignment will be graded out of a total mark of 50, with weights
Assignment
1. Construct a special select and ultimate survival model based on the SSSM (Standard Select and Ultimate Survival Model).
Recall that the ultimate part of SSSM assumes Makeham’s Law with: µx = A + B · Cx ,
where A = 0.00022, B = 2.7 × 10−6, C = 1.124. For this new model, you are given:
• The select period is three years.
• Functions for this model are denoted by an asterisk, * . For all values of x,
q*[x] = 0.97q[x]; q*[x]+1 = 0.98q[x]+1; q*[x]+2 = 0.99q[x]+2; qx(*) = qx.
Construct a new table for this special select and ultimate survival model with the values
of p*[x] , p*[x−1]+1, p*[x−2]+2, px(*) , l*[x] , l*[x−1]+1 , l*[x−2]+2 and lx(*) . Also,¨(a)x(*) , Ax(*) , 2 Ax(*) ,¨(a)x(*): 10 , Ax(*): 10 ,
¨(a)x(*): 20i ,Ax(*): 20i , 5Ex(*) , 10 Ex(*), 20Ex(*) , ¨(a)x(*)(: , Ax(*)(: , as well as ¨(a)*[x] , ¨(a)*[x(]4) , A*[x(]4) , A*[x] , 5E] , 10 E] ,
20 E] , at integer ages, with limiting age ω = 130. Assume l2(*)0 = 100, 000 and interest
rate i = 5%. Use UDD for fractional ages where applicable.
Use the special select and ultimate survival model in Question 1 for Questions 2–4.
2. An insurer issues a 20-year term life insurance policy to a select life [35]. The sum insured of $200,000 is payable at the end of the year of death, and premiums are paid annually throughout the term of the contract and calculated using equivalence principle. The basis for calculating premiums and policy values is:
• Interest: 5% per year effective;
• Initial Expenses: $200 plus 25% of the first premium;
• Renewal Expenses: $25 plus 5% of each premium after the first year.
(a) Calculate the gross premium policy values at each time t, for t = 0, 1, . . . , 20, and plot them on a graph. Is there any negative policy value? If so, explain the reason.
(b) Now consider that the insurer earns an actual interest of 6% each year (mortality and expenses are as assumed). Assume that 80% of the profit is distributed as cash dividends to policyholders who are still alive at the end of the year. Calculate the EPV of the profit to the insurer per policy issued. (This is a participating life insurance.)
(c) Now assume that the premiums are now paid quarterly for maximum 5 years, and the death benefit is paid at the end of month of death. Ignoring the expenses, calculate the premium policy values at each year and each premium payment time, and plot them on a graph.
*Note that the policy values have jumps at the premium payment time.
3. Suppose a 20-year endowment insurance with sum insured $20,000 and survival benefit $10, 000 issued to a select life [35]. Assume the death benefit is paid at the end of year of year, while the level premiums are paid annually throughout the term of the contract and calculated using equivalence principle. The basis for calculating premiums and policy values is:
• Interest: 5% per year effective;
• Initial Expenses: $200 plus 25% of the first premium;
• Renewal Expenses: 5% of each premium after the first year.
(a) Calculate the gross premium policy values, net premium policy values and FPT policy values at each year. Plot them on a graph, and briefly discuss your conclusion by comparing their values.
(b) Discuss why the premium basis and policy value basis might be different in practice.
4. Consider a continuous 10-year deferred 10-year term life annuity of $2,000 per year on a selected life aged [35]. A level premium of P is payable continuously each year during the first 10 years. Assume the interest rate is i = 5% and death are uniformly distributed (UDD) within each year of age.
(a) Calculate the exact premium rate P.
(b) Write out the Thiele’s differential equation for t ∈ (0; 20), and give any relevant boundary conditions.
(c) Determine the premium rate P by solving Thiels’s differential equation using Euler’s method, with a time step h = 0:05.
(d) Recalculate Part (c) using a time step h = 0:025, and compare the result with Part (a) and Part (c).
(e) Plot the graph of tV for t ∈ (0; 20).
*You may need the “Goal Seek” function in Excel.
5. Consider the following model for an insurance policy combining disability income insurance benefits and critical illness benefits.
The transition intensities are as follows:
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01 x
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= a1 + b1 exp{c1 x};
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x(02) = a2 + b2 exp{c2 x};
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12 x
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= x(02) ; x(32) = 1:1 x(02) ;
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10 x
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= 0:1 x(01) ; x(03) = 0:05
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x(01) ; x(13) = x(03) ;
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where
a1 = 4 × 10−4; b1 = 3:5 × 10−6; c1 = 0:14; a2 = 5 × 10−4; b2 = 7:6 × 10−5; c2 = 0:09:
Using Euler’s method with a step size of h = 0:05, calculate values of 20p30(ij) for i = 0; 1 and j = 0; 1; 2; 3.