Question

Sofia must pay $ 110000 within 2 years. With this purpose she begins to save $ 3000 monthly for 18 consecutive months, earning 1.6% monthly capitalizable each month during the first year and 1.9% monthly capitalizable each month the rest of the time. Sofia wants to know how much she will have to deposit at the end of month 20 earning 1.9% monthly capitalizable each month to be able to obtain a total amount at 2 years equal to the amount of the debt.

Answer 1

Given information:

Interest rate for months 1-12 = 1.6%

Interest rate for months 13-24 = 1.9%

Deposits at the end of each month for 18 months = $3,000

Deposit at the end of month 20 = x (say)

Payment at the end of 2 years = $110,000

From the given information, the equation can be formed as:

Future value of all deposits at the end of 2 years = Payment at the end of 2 years

Future value of all deposits at the end of 2 years

= Future value of 18 monthly deposits of $3,000 each

+ Future value of deposit of $x at the end of month 20

As the interest rate changes at the end of 1 year, i.e. 12 months the future value would be calculated as follows:

Component I: Future value of deposits of $3,000 for months 1-12 after 2 years

Component II: Future value of deposits of $3,000 for months 13-18 after 2 years

Component III: Future value of deposit of $x at the end of 20 months after 2 years

Component I: Future value of deposits of $3,000 for months 1-12 after 2 years can be calculated by calculating future value of annuity at the end of year 1 at 1.6% and then calculate the future value of that amount at the end of year 2 at 1.9%.

Thus,

Component I = $3,000 * Annuity Factor at 1.6% for 12 periods * Future Value Factor at 1.9% for 12 periods

= $3,000 * ((1+1.6/100)^(12)-1)/(1.6/100) * (1+1.9/100)^(12)

= $3,000 * 13.11440041 * 1.25340149

= $49,312.83

Component II: Future value of deposits of $3,000 for months 13-18 after 2 years can be calculated by calculating future value of annuity at the end of 18 months at 1.9% and then calculate the future value of that amount at the end of year 2 at 1.9%.

Thus,

Component II = $3,000 * Annuity Factor at 1.9% for 6 periods * Future Value Factor at 1.9% for 6 periods

= $3,000 * ((1+1.9/100)^(6)-1)/(1.9/100) * (1+1.9/100)^(6)

= $3,000 * 6.29232367 * 1.11955415

= $21,133.79

Component III: Future value of deposit of $x at the end of 20 months after 2 years

Component III = x * (1+1.9/100)^4

= 1.07819357x

Thus, the equation would be:

Component I + Component II + Component III = $110,000

$49,312.83 + $21,133.79 + 1.07819357x = $110,000

1.07819357x = $39,533.38

x = $36,684.86.

Therefore, amount required to be deposited at the end of 20 months would be $36,684.86.