Calculating annuities and monthly payments

To save up for his driver's license, Carlos decides to deposit $\$60$ per month into a bank account at the end of each period. The bank offers him $2.4 \%$ annual compound interest with monthly compounding. Calculate how much money he will have accumulated after 2 years (24 months) of making these deposits.

Note: Remember to convert the annual interest rate into a monthly interest rate by dividing by 12.

Elena is 18 years old and plans to save to pay for the deposit and move-in expenses of a shared apartment when she finishes her vocational training program in $3$ years ($36$ months). She is going to deposit $150$ dollars at the end of each month into an interest-bearing youth account with a nominal interest rate of $3\%$ per year ($i = \frac{0.03}{12} = 0.0025$ monthly).

1. a) Make a reasoned prior estimate before applying complex formulas: calculate how much she would accumulate if the interest rate were $0\%$, and justify whether, after adding the $3\%$ per year, the final accumulated capital will manage to exceed the $6,000$ dollars threshold or not.
2. b) Apply the future value formula for an ordinary constant savings annuity to calculate the exact amount she will have accumulated after the $36$ months have elapsed.
3. c) Check the exact result against your prior estimate and calculate what percentage of the final accumulated capital comes from the interest paid by the bank.

To cover the expenses of an end-of-year trip in two years' time, a group of students decides to open a savings account. They commit to depositing $80$ dollars at the end of each month. The bank offers them an interest rate of $3.6\%$ APR with monthly compounding.

1. a) Calculate the equivalent monthly nominal interest rate.
2. b) How much capital will the group have at the end of the $24$ months?
3. c) What is the total interest generated by the investment?

A family applies for a loan of $\$5,000$ to buy a used motorcycle. The bank grants them the loan at an annual interest rate of $5 \%$, which they must repay in 4 equal annual installments payable at the end of each year. Determine the amount of this annual loan payment.

Calculate the monthly installment required to repay a debt of $\$2,400$ incurred from buying a professional camera, if the loan is repaid over 2 years at a nominal annual interest rate of $4.8 \%$ with monthly payments.

A 1st Baccalaureate student wants to buy a gaming computer costing $\$1,200$. A computer shop allows him to pay for it in 12 equal monthly instalments at a nominal annual interest rate of $6 \%$. Knowing that the monthly interest rate is calculated as $i = \frac{0.06}{12}$, use the loan repayment formula to calculate the monthly instalment he must pay at the end of each month.

Mateo wants to finance the purchase of a $1,800$ dollars laptop through a $2$-year personal loan ($24$ months) with fixed monthly installments payable at the end of each period. The bank applies an annual nominal interest rate (APR) of $6\%$. To calculate his monthly installment $a$, Mateo writes in his notebook:

and gets $a \approx 873.79$ dollars per month.

1. a) Without needing to calculate the actual installment, multiply Mateo's installment by the $24$ monthly payments in your head and explain with reasoning why the result is an obvious financial absurdity.
2. b) Identify and explain the two serious conceptual errors made by Mateo when formulating the equation.
3. c) Set up the correct equation that determines the ordinary monthly installment that Mateo must pay, calculate its exact amount rounded to the nearest cent, and find the total amount he will end up paying solely in interest.

A family applies for a loan of $15,000$ dollars to renovate their kitchen and bathroom. The bank grants them the loan at an annual interest rate of $5\%$ to be repaid over $4$ years in fixed monthly installments due at the end of each month.

1. a) Determine the monthly installment amount they will have to pay.
2. b) Calculate the total amount they will end up paying back to the bank after $4$ years.
3. c) What percentage of the total amount paid corresponds to interest?

Emma bought an electric scooter by financing the $\$450$ it costs with a loan. The bank charges her fixed monthly installments of $\$40$ at the end of each month for one year. If the total interest paid is the difference between the sum of all installments and the cash price, how much money has Emma paid solely in interest by the end of the year?

Emma is planning to attend a music festival when she finishes high school in 3 years. To cover the expenses, she decides to open a youth savings account offering $3 \%$ annual compound interest. If she deposits $\$250$ at the beginning of each year, what total capital will she have accumulated for her trip at the end of the three years?

A school esports club decides to save up to replace its chairs. They deposit $\$100$ at the beginning of each quarter into an account yielding $4 \%$ annual compound interest. If compounding is quarterly ($n=4$ periods per year), calculate the capital they will have saved after one full year (4 deposits).

A young saver decides to deposit $\$500$ at the start of each year into an investment fund guaranteeing $3.5 \%$ annual compound interest. Use the formula for the future value of an annuity to find how much money they will have after 5 years.

A family wishes to build up a savings fund for their child's college education. They plan to deposit $1,500$ dollars at the start of each year for the next $10$ years. The bank guarantees them a compound interest rate of $4.25\%$ per year.

1. a) What amount will they have accumulated right after making the tenth deposit?
2. b) What amount will they have one year after the last deposit (at the end of the tenth year)?

Note: Take into account whether the annuity is an annuity due (payments at the start of each period) or an ordinary annuity (payments at the end of each period).