Maggie wins the lottery and is awarded 500,000 at the beginning of each quarter

Question

Maggie wins the lottery and is awarded 500,000 at the beginning of each quarter for the next 20 years. The government takes 40% of her winnings and she spends 90% of what is left. The remainder is put into a special account with an insurance company that guarantees 7% compounded monthly for the first 10 years and 4% compounded 6 times a year for the following 10 years. At the end of 20 years the accumulated value in the account is used to purchase a perpetuity-immediate with semi-annual payments of X based on a force of interest of 6% per year. Find X.

Explanation / Answer

Lottery awared at the begining of each quarter =$500,000

amount left to Maggie after payment to government =$500,000*60%

=$300,000-------(A)

Amount left for savings after maggie spending 90% of A =$300,000*10% =$30,000.

Amount deposited at the begining of the quarter for each of next 20 years.

For the first 10 years, the amount is compounded monthly at 7%.

So,the future value of $30,000 deposited each quarter with a monthly compounding interest rate of 7% is given by the formula :

A = R * [(1+i)n -1]/i

A= maturity amount at the end of year 10, R is the quarterly deposit =$30,000

i = interest rate =7%/12 =0.005833

n=number of times the deposit is made =4*10 =40

Substituting the values in the formula A=$30,000[(1+0.005833)40-1]/0.005833

=$1347144.654.

This amount is compounded 6 times a year for the next 10 years.

A=P(1+r/100)n is the formula for calculation of compounded interest.

Here P =$1347144.654.

n=6*10 = 60

r=4% =(4%/12)*2

=0.0066667

Substituting the values in the formula A =$1347144.654. * (1+0.0066667)60

=$2007037.681-------------(A)

For the next 10 years quarterly deposit is $30,000 and the interest earned is 4% compounded 6 times a years, the maturity amount is arrived by using the formula :

A = R * [(1+i)n -1]/i

A= maturity amount at the end of year 10, R is the quarterly deposit =$30,000

i = interest rate =(4%/12)*2 =0.0066667

n=number of times the deposit is made =4*10 =40

Substituting the vlaues in the formula A=$30,000*[(1+0.0066667)40 -1]/0.0066667

=$1370026.249----(B)

Total amount received at the end of 20 years =A+B

=$3,377,063.93

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