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

Answer:

(a) $850,169.64

(b) $566,277.662

Explanation:

Given that,

Principal amount (P) = $75,000

No. of installments = 20

Period (t) = 19 years

Discount rate (r) = 7%

(a)

$Present value of annuity due=P+P[\frac{1-\frac{1}{(1+r)^{t}} }{r}]$

$Present value of annuity due=75,000+75,000[\frac{1-\frac{1}{(1+0.07)^{19}} }{0.07}]$

$Present value of annuity due=75,000+75,000[\frac{1-\frac{1}{(1.07)^{19}} }{0.07}]$

                                                     = $75,000 + 75,000 × 10.3355952

                                                     = $75,000 + 775,169.64

                                                     = $850,169.64

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

$Present value of annuity due=P+P[\frac{1-\frac{1}{(1+r)^{t}} }{r}]$

$Present value of annuity due=75,000+75,000[\frac{1-\frac{1}{(1+0.14)^{19}} }{0.14}]$

$Present value of annuity due=75,000+75,000[\frac{1-\frac{1}{(1.14)^{19}} }{0.14}]$

                                                     = $75,000 + 75,000 × 6.55036883

                                                     = $75,000 + 491,277.662

                                                     = $566,277.662