Question

The state​ lottery's million-dollar payout provides for ​$1.5 million to be paid in 20 installments of ​$75,000 per payment. The first ​$75,000 payment is made​ immediately, and the 19 remaining​$75,000 payments occur at the end of each of the next 19 years. If 7 percent is the discount​ rate, what is the present value of this stream of cash​ flows? If 14 percent is the discount​ rate, what is the present value of the cash​ flows?

Answer 1

(a) $850,169.64

(b) $566,277.662

Step-by-step explanation:

Given that,

Principal amount (P) = $75,000

No. of installments = 20

Period (t) = 19 years

Discount rate (r) = 7%

(a)

`Present value of annuity due=P+P[(1-(1)/((1+r)^(t)) )/(r)]`

`Present value of annuity due=75,000+75,000[(1-(1)/((1+0.07)^(19)) )/(0.07)]`

`Present value of annuity due=75,000+75,000[(1-(1)/((1.07)^(19)) )/(0.07)]`

= $75,000 + 75,000 × 10.3355952

= $75,000 + 775,169.64

= $850,169.64

(b) When discount rate changes to 14%, then

`Present value of annuity due=P+P[(1-(1)/((1+r)^(t)) )/(r)]`

`Present value of annuity due=75,000+75,000[(1-(1)/((1+0.14)^(19)) )/(0.14)]`

`Present value of annuity due=75,000+75,000[(1-(1)/((1.14)^(19)) )/(0.14)]`

= $75,000 + 75,000 × 6.55036883

= $75,000 + 491,277.662

= $566,277.662