An investor purchases an annuity that will pay a constant payment starting today of $100. Beginning with the payment 12 years from today each payment will be 5% greater than the previous payment. The last payment will be received 25 years from today. The annuity is bought to yield an effective interest rate of i such that (vi)^4 = 0.85.
A. What is the purchase price of the annuity today?

Question

contestada

Respuesta :

DeniceSandidge DeniceSandidge · Answer 1 · 2020-02-11T08:15:18+01:00

Answer:

present value = $1921.62

Explanation:

given data

payment = $100

rate g = 5 %

time = 12 year

solution

we get here interest rate that is

v = [tex]\frac{1}{1+i}[/tex] .............1

[tex]v^4 = \frac{1}{(1+i)^{-4}}[/tex]

0.85 = [tex]\frac{1}{(1+i)^{-4}}[/tex]

i = [tex]0.85^{4}[/tex] - 1

i = 4.14 %

now we get here present value of annuity that is

present value of annuity = payment × [tex]\frac{1-(1+i)^{-t}}{i}[/tex] × (1+i) .................2

put here value

present value of annuity = 100 × [tex]\frac{1-(1+0.0414)^{-12}}{0.0414}[/tex] × ( 1 + 4.14 % )

present value of annuity = $969.15

and here present value of growing annuity that is

present value of growing annuity = [tex]\frac{payment}{(i-g)} * ( 1- (\frac{1+g}{1+i})^t )[/tex] ............3

present value of growing annuity = [tex]\frac{100}{(0.0414-0.05)} * ( 1- (\frac{1+0.05}{1+0.0414})^{14} )[/tex]

present value of growing annuity = $1489.16

so now we get

present value that is

present value = present value of annuity + present value of growing annuity .............4

present value = 969.15 + [tex]\frac{1489.16}{(1+0.0414)^{11}}[/tex]

present value = $1921.62