Math 193b. HOMEWORK ASSIGNMENT.

During the course, the HW assignment may be slightly changed. Please, watch the current assignment.

When solving problems marked below by *, you do not have to provide final calculations: it is enough to know how to solve these problems. Certainly, you may nevertheless do calculations (for example, to compare your answer with that in the book), and in any case, you should make sure that you will be able to do it if needed.

Chapter 7 (you do not have to write solutions in detail: just make sure that you can do it): 1-12, 13a, 14 (Advice: Use the memoryless property to avoid redundant calculations), 16а, 16b (Advice: It makes sense to apply Bayes’ formula. Let B₁ be the event that a person belongs to the first group, and B₂ – to the second group. Then, P(B₁) = w₁ and P(B₂) = w₂. By the formula for total probability,

P(X>x) = P(X>x | B₁ ) P(B₁) +P(X>x | B₂ ) P(B₂) ,

and by Bayes’ formula

w₁ (x) = P(B₁ |X>x)=

= P(X>x | B₁ ) P(B₁ )/[ P(X>x | B₁ ) P(B₁) +P(X>x | B₂ ) P(B₂)];

17*, 19, 20, 21ab, 22 (you may use Exercise 16b or just solve this problem independently using the fact that we are dealing with exponential distributions), 24 (Advice: You may use Exercise 9), 25 (it is enough to consider a heuristic approach), 30, 33, 35 (optional), 36 (optional), 39 (optional, just realize that you can do it).

Chapter 8: 1, 2, 3, 4 (to prove (6.1), first show that K(x) has a geometric distribution and compute P(K=k)), 5, 6 (a bit challenging: it looks as the answer depends on two parameters while you have just one equation. Figure out what the answer depends on as a matter of fact.), 9*, 10 (it will be enough to consider x=60, and use the answers in Example 1.1-1), 13 (Hint: The distribution does not have to be uniform) , 14 (since the distribution is uniform, we do not have to solve the part (b) directly: approximation (2.1.6) is precise in this case (show why)), 15, 16, 18a (Think also whether the APV of the former insurance is always larger than that of the latter insurance, or the situation is more complicated), 18b, 19, 20a, 20b*, 21 (Regarding this problem, realize again what is the answer to the following question: If we have two groups of people with given survival functions, and we know the probabilities that a newborn belongs to the first and the second group, respectively, then how to find the same probabilities (of belonging to the first and the second groups) for a person of age x chosen at random), 22*, 24*, 26*, 27, 28, 29, 30, 31, 32 (one may restrict her/himself to a heuristic proof), 33ab, 33c (optional), 34.

An important additional problem. Look carefully at the posted here Excel file for computing APV’s for term insurances. (The symbol c_k there stands for the amount of payment if it occurs at the kth period.) Using this file and the Illustrative Table, find the net single premium and the variance of the present value for the 5-year term insurance for a 65 years old insured, for benefits payable at the end of the year of death, with a benefit of $100,000 if the payment time occurs within the first three years, and $500, 000 otherwise. Do the same for a similar 5-year endowment policy. In this case, if the insured survives the 5-year term, the payment should be of $500, 000.

Optional reading. I recommend to read the text in pp.440-441, starting from “While all of this is true, …” and up to the end of p.441. The reasoning in this text is not necessary but quite useful since it sheds an additional light on the notion of present value.

Chapter 9: Consider thoroughly Example 1.2-1.

Exercises: 1 (a good idea is to use Excel; in particular, you can use the files posted below, but they should be revamped a bit), 3, 4, 5, 6a, 8-10, 13a, 13b (optional), 15, 16, 19, 20, 21, 22, 23 (easy, but absolutely mandatory), 24, 25*, 26*, 27,

28 (Optional, a bit challenging. Computing the expected present value is not very difficult: denote by N the number of visits, by Y -- the present value under consideration, compute, first, E{Y|N}, and after that use the fact that E{Y}=E{{Y|N}}. For the variance, one may use (7.2.1) from Chapter 0),

29, 31, 36.

Additional problems:

1) Look carefully at the posted here Excel files on computing APV and variances with use of the current payment technique (Fig.1a, Ch.9), and the aggregate payment technique (Fig.1b, Ch.9). Using these files and the Illustrative table, find the expected present value and variance for a five-year temporary annuity-due for a 20 years old insured, with the first three payment of $20,000, and the last two payments of $15,000.

2) Suppose that in the situation of Exercise 29, we are dealing with a homogeneous portfolio of 100 polices (i.e., all policies are issued at the same day, and have the same characteristics). The annuity payments are provided from a fund established at the moment of the policies issue. Using normal approximation, find an initial size of the fund such that the probability that the fund will turn out to be enough for providing all payments, is 0.99.

Chapter 10: 1, 2, 3, 4, 5, 6, 7*, 8, 9*, 10*,

Consider thoroughly Example 1.2.1.-3.

11*, 13, 14, 15*, 17*, 18, 19, 20, 22a*.

Consider Table 1 on p. 510, and make sure that you can provide the answers in the right column on your own just proceeding from the standard logic and notation. Do the same regarding Table 2 on p. 512.

27(optional), 29, 30.

Chapter 2: 60 (optional), 63ab (optional).