Please show all step and formulas used it would mean a lot 2

Please show all step and formulas used! it would mean a lot!

2. Your best friend has won the Power Ball and he is given the following options of receiving his money, on every January 29th for the next 30 years, starting in year 2019: Option A: 811,000,000 each year. Option B: S1 first time, with each subsequent payment is double that of the last. (a) Set up and specify the present values of the 2 options. (b) Which option should he choose if the risk-free rate is 0%? (c) Which option should he choose if the risk-free rate is 8%? (d) At what risk-free interest rate range should he choose each of the options?

Solution

Option A

He will receive $11,000,000 each year for next 30 years then PW for this option will be

11mn[(1/1+r)+(1/1+r)^2+(1/1+r)^3+(1/1+r)^4+(1/1+r)^5+...+(1/1+r)^30]=11mn[1-(1/1+r)^30]/(1-(1/1+r))

Option B

We get $1 first year and then gets doubled then PW for option 2 is

1/(1+r)+2*(1/1+r)^2+4*(1/1+r)^3+8*(1/1+r)^4+16*(1/1+r)^5+32*(1/1+r)^6+64*(1/1+r)^7+128*(1/1+r)^8+256*(1/1+r)^9+512(1/1+r)*10+1024*(1/1+r)^11+...+2^29*(1/1+r)

Answer for B

If r =0

then PW for Option A is =11mn*30=330mn

then PW for Option B is = (2^29-1)/1=536.87mn

Hence Option B chosen over Option A

Answer for C)

If r=8%

PW for Option A is 11mn [1-(1/1+r)^30]/(1-(1/1+r))=11mn * 12.15=133 mn

PW for Option B is 125.26 mn

Hence in this case Option B chosen over Option A

Answer for d

8.75% is the rate at which PW for both Options A and B is same