Question

Time value Personal Finance Problem You have ​$1,800 to invest today at 8​% interest compounded annually.

a. Find how much you will have accumulated in the account at the end of​ (1) 3 ​years, (2) 6 ​years, and​ (3) 9 years.

b. Use your findings in part a to calculate the amount of interest earned in​ (1) the first 3 years​ (years 1 to 3​), ​(2) the second 3 years​ (years 4 to 6​), and​ (3) the third 3 years​ (years 7 to 9​).

c. Compare and contrast your findings in part b. Explain why the amount of interest earned increases in each succeeding 3 dash year period.

Answer 1

a. Find how much you will have accumulated in the account at the end of​–

(1) 3 years

For the 3 years, you can save $1,800 per year; we can use FV of an Annuity due formula (as the deposits are made at the beginning of the year) to calculate the future value of after 3 years savings by you

FV = PMT *(1+i) {(1+i) ^n−1} / i

Where,

Future value of annual savings FV =?

PMT = Annual savings = $1,800

n = N = number of payments = 3 years

i = I/Y = interest rate per year = 8%

Therefore,

FV = $1,800 *(1+8%) {(1+8%) ^3−1} /8%

FV = $6,311.00

Therefore you will accumulate $6,311.00 after 3 years of deposits.

(2) 6 years

FV = PMT *(1+i) {(1+i) ^n−1} / i

Where,

Future value of annual savings FV =?

PMT = Annual savings = $1,800

n = N = number of payments = 6 years

i = I/Y = interest rate per year = 8%

Therefore,

FV = $1,800 *(1+8%) {(1+8%) ^6−1} /8%

FV = $14,261.05

Therefore you will accumulate $14,261.05 after 6 years of deposits.

(3) 9 years

FV = PMT *(1+i) {(1+i) ^n−1} / i

Where,

Future value of annual savings FV =?

PMT = Annual savings = $1,800

n = N = number of payments = 9 years

i = I/Y = interest rate per year = 8%

Therefore,

FV = $1,800 *(1+8%) {(1+8%) ^9−1} /8%

FV = $24,275.81

Therefore you will accumulate $24,275.81 after 9 years of deposits.

b. Use your findings in part a to calculate the amount of interest earned in​–

(1) The first 3 years (years 1 to 3)

The amount of interest earned = Total value of account at the end of year 3 – 3* annual deposits

= $6,311.00 – 3* $1,800

= $6,311.00 – $5,400

= $911.00

(2) The second 3 years (years 4 to 6)

The amount of interest earned = Total value of account at the end of year 6 – Total value of account at the end of year 3 - 3* annual deposits

= $14,261.05 – $6,311.00 - 3* $1,800

= $14,261.05 -$6,311.00 – $5,400

= $2,550.04

(3) The third 3 years (years 7 to 9)

The amount of interest earned = Total value of account at the end of year 9 – Total value of account at the end of year 6 - 3* annual deposits

= $24,275.81 – $14,261.05 - 3* $1,800

= $24,275.81 – $14,261.05 – $5,400

= $4,614.77

c. Compare and contrast your findings in part b. Explain why the amount of interest earned increases in each succeeding 3 dash year period.

Findings in part b.-

The amount of interest earned increases in each succeeding 3 year period because the interest rate is compounding annually and second years on-wards you are getting interest on your eared interest amount. In later years, the accumulation of interest is more therefore interest earned increases in each succeeding 3 year period.