A lotto to raise funds for a hospital is advertising a $240,000 prize. The winner will recieve $1000 every month for 20 years, starting a year from now. If......... the lotto were able to...

A lotto to raise funds for a hospital is advertising a $240,000 prize. The winner will recieve $1000 every month for 20 years, starting a year from now. If....

..... the lotto were able to negotiate an interest rate of 9.3% per annum, compounded annualy, how much would be invested now?

Asked on by islnds

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lfryerda | High School Teacher | (Level 2) Educator

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Since the lotto is paying a fixed amount every month for 20 years, this is an annuity.  To determine the amount in order to pay for the annuity, we are trying to find the present value of the annuity.  Annuity calculations must have the payment (or rent) and the interest calculations over the same period.  Since the interest rate is compounded annually, then the rent must also be per year.  Therefore the rent is `R=12 times 1000=12000` . The interest rate is `i=9.3%=0.093` .  Finally, the number of payments is 20.  

The amount required to be invested is calculated by the present value formula:

`P=R/i(1-(1+i)^{-n})`  sub in the values

`=12000/0.093(1-(1+0.093)^{-20})`  evaluate

`=107240.36`

The lotto needs to invest $107240.36 now to pay for the annuity over the next 20 years.

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