A 65-year old female client is considering either purchasing a single-premium insurance policy or investing in a tax-free municipal bond. The fundamental feature of this insurance policy is that the premium is paid all at once when the policy is written, and this one-payment feature does not run afoul of any tax codes. In her case, each dollar of premium generates 3.6914 dollars of insurance to be paid at her death. On the other hand "buy now and hold until death" investment strategy in tax-free municipal bonds gives return at annual rate of 7.1% (and all of the dividends associated with these bonds can be reinvested at this rate). Thus, in two years the bonds would be worth (1.071)2 = 1.147 dollars per each dollar invested, and, e.g., in 12.5 years they would be worth (1.071)12.5 = 2.357 dollars per each dollar invested. The worksheet Residual Life describes the random variable R representing the residual life of 65-year old females (residual life is the number of years till her death), according to the official government statistical data. To simplify for the purposes of this question, all deaths are assumed to occur half way through the year. That is, if she dies between today and her 66th birthday, R = 0.5. If she dies between her 68th and 69th birthday, R = 3.5. 2.

2. (Financial advice, 3 points) What is the expected value of the random variable R?

3. (Financial advice, 4 points) If one million dollars is invested in the tax-free municipal bond with "buy now and hold until death" investment strategy, as described above, what is the probability that the value of investment at time of death will be less than two million dollars? (Hint: You may want to consider a new random variable with the same outcomes and their probabilities as the random variable R, but with numerical values associated with each outcome transformed to represent the value of investment.)

4. (Financial advice, 4 points) If one million dollars is invested in the tax-free municipal bond with "buy now and hold until death" investment strategy, as described above, what is the expected amount of money on hand at the time of death? (You should ignore any potential tax effects and assume that the probability of default is zero.)

2.31 million dollars

3.55 million dollars

3.81 million dollars

4.17 million dollars

8.1 million dollars

R	Probability p		R= residual life. For example, the probability of R=3.5 is equal to 0.0163119990
0.5	0.0131706012
1.5	0.0140897509		3.6914	= life insurance premium
2.5	0.0151368835
3.5	0.0163119990		7.1%	= annual rate of return for tax-free municipal bond
4.5	0.0176034625
5.5	0.0189763697		For example, the value of $1 bond investment after
6.5	0.0203841813		12.5	years of compounding will be	$ 2.357
7.5	0.0218268973
8.5	0.0232696134
9.5	0.0247355990
10.5	0.0262481239
11.5	0.0278537272
12.5	0.0295524090
13.5	0.0313441692
14.5	0.0332522775
15.5	0.0352301947
16.5	0.0372779206
17.5	0.0393954554
18.5	0.0415478947
19.5	0.0437468731
20.5	0.0494353977
21.5	0.0466188040
22.5	0.0480950989
23.5	0.0447640140
24.5	0.0415459861
25.5	0.0394323119
26.5	0.0355408575
27.5	0.0332125389
28.5	0.0286689587
29.5	0.0242073039
30.5	0.0201421678
31.5	0.0159983833
32.5	0.0121708307
33.5	0.0093312979
34.5	0.0070488575
35.5	0.0047205390
36.5	0.0030820319
37.5	0.0020383028
38.5	0.0012469039
39.5	0.0007619058
40.5	0.0004456739
41.5	0.0002375835
42.5	0.0001556582
43.5	0.0000770098
44.5	0.0000262161
45.5	0.0000196621
46.5	0.0000081925
47.5	0.0000049155
48.5	0.0000016385
49.5	0.0000016385
50.5	0.0000000000
51.5	0.0000000000
52.5	0.0000016385
53.5	0.0000016385
54.5	0.0000000000
55.5	0.0000000000
56.5	0.0000000000
57.5	0.0000000000
58.5	0.0000000000
59.5	0.0000000000
60.5	0.0000016385
61.5	0.0000000000