Question

You recently purchased a 20-year investment that pays you $100 at t = 1, $500 at t = 2, $750 at t = 3, and some fixed cash flow, X, at the end of each of the remaining 17 years. You purchased the investment for $5,544.87. Alternative investments of equal risk have a required return of 9%. What is the annual cash flow received at the end of each of the final 17 years, that is, what is X?

Answer 1

The vale of X = $276.11 (to 2 decimal places)

Step-by-step explanation:

First of all, let us lay out the particulars for clarity.

pay at t 1 = $100

pay at t 2 = $500

pay at t 3 = $750

Amount investment was purchased for = $5,544.87

percentage return on investment = 9% = 0.09.

Next let us calculate amount earned as return on investment from percentage return;

amount earned from percentage;

= 9% of $5544.87 = 0.09 × 5544.87 = $499.0383

Next let us calculate the total expected return on investment after 20 years

= Amount invested + amount earned from percentage

= 5544.87 + 499.0383 = $6043.9083

Therefore, after 20 years, the invested is expected to yield $6043.9083.

Next let us add the total amount gotten after the first 3 years;

t1 + t2 + t3 = 100 + 500 +750 = $1,350

To get the total amount to be earned in the remaining 17 years, we will subtract the amount gotten after the first 3 years from the total amount expected;

= 6043.9083 - 1350 = $4693.9083

Hence in the next 17 years, the amount to be earned is $4693.9083.

X is the fixed cash flow at the end of each year for the remaining 17 years, so to calculate this, we divide the total amount earned in the 17 year period by 17.

∴ X = 4693.9083 ÷ 17 = $276.1123 = $276.11 ( to 2 decimal places)

Answer 2

The amount given for the subsequent 17 years assuming the invesmtent yield 9% return is $521.23

Step-by-step explanation:

We have to solve for the installment of a 17 years annuity discounted at 9% annual considering the first 3 years cash flow and the purchase price:

`(Maturity)/((1 + rate)^(time) ) = PV`

discount rate 0.09

# Cashflow Discounted

0 -5544.87 -5544.87

1 100 91.74

2 500 420.84

3 750 579.14

NPV -4453.15

We solve for the PMT that give that amount:

`PV / (1-(1+r)^(-time) )/(rate) = C\\`

PV $4,453.1500

time 17

rate 0.09

`4453.15 / (1-(1+0.09)^(-17) )/(0.09) = C\\`

C $ 521.225