Please help me answering this question?

(Uncertain annuity) Gavin’s grandfather, Mr. Jones, has just turned 90 years old and is applying for a lifetime annuity that will pay $10,0000 per year, starting 1 year from now, until he dies. He asks Gavin to analyze it for him. Gavin finds that according to statistical summaries, the chance (probability) that Mr. Jones will die at any particular age is as follows:

age: 90 91 92 93 94 95 96 97 98 99 10 101

probability:.07 .08 .09 .10 .10 .10 .10 .10 .10 .07 .05 .04

The Gavin (and you) answers the following questions:

(a) What is life expectancy of Mr. Jones?

(b) What is the present value of an annuity at 8% interest that has a lifetime equal to Mr. Jones’s life expectancy? (For an annuity of a nonintegral number of years, use an average method. )

(c) What is the expected present value of the annuity?

(a) Life expectancy = Σ(probability x age) = 90*0.07+91*0.08+92*0.09+…+101*0.04 = 95.13 years.

(Note: check that Σ(probability)=1; this is true here. )

(b) PV = $10,0000/1.08 + $10,0000/1.08^2 + … $10,0000/1.08^5 + $10,0000*0.13/1.08^5.13

= $40,938.

(I assume that in year 6 only fraction 0.13 of the annuity is paid, at the time of death.)

(c) Expected PV of annuity = Σ{probability x PV(n) for Mr Jones dying at end of years 90 to 101 (n=1 to 12)}.

If Mr Jones dies at the end of year …

1 then PV of annuity is PV(1) = $10,0000/1.08 = $9,259.26

2 then PV of annuity is PV(2) = PV(1) + $10,0000/1.08^2 = $17,832.65

3 then PV of annuity is PV(3) = PV(2) + $10,0000/1.08^3 = $25,770.97

…

12 then PV of annuity is PV(12) = PV(11) + $10,0000/1.08^12 = $75,360.78.

Expected PV of annuity = Σ(0.07*PV(1) + 0.08*PV(2) + 0.09*PV(3) + … + 0.04*PV(12))

= Σ(0.07*$9,259.26 + 0.08*$17,832.65 + 0.09*$25,770.97 + … + 0.04*$75,360.78)

= $44,803.