Financial Applications (DP IB Maths: AI HL)

Emily decides to buy a new boat at a cost of $20 000, but cannot afford the full amount. The boat dealership offers two options to finance a loan.

First option:

A 5 year loan at a nominal annual interest rate of 15%, compounded monthly.
No deposit required and repayments are made each month.

Find the repayment made each month.

A 5 year loan at a nominal annual interest rate of % compounded quarterly. Terms of the loan require a 5% deposit and a monthly repayment of $400.

Find the annual interest rate, .

State which option Emily should choose. Justify your answer.

On the first of January 2018 Phillip invests 25 000 US dollars (USD) in the S&P 500. The S&P 500 is a stock market index that tracks the performance of 500 large companies listed on the stock exchange in the United States. Over the past 30 years the S&P 500 has delivered an average annual growth rate of 10.7%, compounded annually.

Calculate the expected value of Phillip’s investment on the first of January 2024. Give your answer to 2 decimal places.

On the first of January 2024 the value of Phillip’s investment 32 762.41 USD.

Calculate the percentage error between the expected value of the investment on the first of January 2024, found in part (a), and the actual value of the investment on the first of January 2024.

Calculate the average annual growth rate of the S&P 500 between 2018 and 2024.

In 2018 the S&P 500 delivered an annual return of 4.38%.

Calculate the average annual growth rate of the S&P 500 between 2019 and 2024.

Ben and Evan are twins and both receive €44 000 on their 21st birthday. Ben deposits his €44 000 into a savings account that pays a nominal annual interest rate of 3.27%, compounded annually. Evan invests his €44 000 into a fixed income fund that returns €1750 per year.

Calculate:

Find the year in which the amount in Ben’s account surpasses the amount in Evan’s account.

Georgia buys a new computer for herself that costs $1099. At the same time, she buys her son, Duncan, a new gaming computer that costs $2749.

It is anticipated that Georgia’s computer will depreciate at a rate of 11% per year, whereas Duncan’s gaming computer will depreciate at 18% per year.

Estimate the value of Georgia’s computer after 6 years.

Georgia and Duncan’s computers will have the same estimated value  years after they were purchased.

Find:

the value of

the estimated value after years.

Sasha takes out a loan of $19 800 to purchase new tools for his gardening business. He agrees to pay the bank $840 at the end of every quarter to amortise the loan. The loan has a nominal annual interest rate of 4.34% per year, compounding semi-annually.

Find out how long it takes to pay back the loan.

Calculate the total amount that Sasha will pay in amortising the loan.

The bank offers Sasha an option to make a higher quarterly repayment of $980.

Calculate how much Sasha will save if he decides to make the higher payment.

John has just retired at the age of 65. He has $445 000 in his savings fund. He “rolls over” the money into an annuity fund which returns a nominal annual interest rate of 4.17%, compounded quarterly.

John decides to withdraw $2200 every month to live on.

Find the number of years and months for the money in the fund to run out.

After 5 years, John wants to make a withdrawal from his annuity to invest in a new online educational platform. Terms of the annuity state John is allowed to withdraw up to 9.5% of his account value without paying a surrender charge.

Calculate the maximum amount John can withdraw so that he does not have to pay the surrender charge.

Joshua wants to invest in Blueprint Construction LTD and so decides to buy 55% of the business for $660 000.

Calculate the total value of Blueprint Construction LTD.

Stephanie, Joshua’s financial advisor, has prepared the following table detailing the 5-year expected annual returns from Joshua’s investment in different economic states (boom or bust) and the probability of each state occurring.

Calculate the 5-year expected return of Joshua’s investment in Blueprint Construction LTD.

In 5 years, Blueprint Construction is valued at $1 800 000.

Find:

Malcolm and Julie have decided to move out of their home to a new apartment in town, so that they can rent it to a young family who agree to pay $3200 a month.

They decide to take out a mortgage to buy the new apartment costing $545 000. Terms of the mortgage are: 

• a minimum deposit of 12%
• a nominal annual interest rate of %, compounded annually
• monthly repayments of $2695

Calculate:

Malcolm and Julie pay off the loan in exactly 30 years.

Find the value of the interest rate, .

Malcolm and Julie decide to invest the difference between their rental income and their monthly repayments into a savings account paying a nominal annual interest rate of 2.91%, compounded annually.

Calculate the total amount in the savings account once they have finished paying off their loan.

Aaron has been renting out his apartment for $1200 per month, however he wants to renovate the apartment to increase the monthly rent to $1750. He notifies his current tenants and they agree to move out. The renovations are expected to take 5 months.

Calculate the amount of rental income Aaron is foregoing by deciding to renovate the apartment.

Aaron takes out a loan of $25 000 to renovate the apartment. Terms of the loan are

• A nominal annual interest rate of 6.2%, compounded monthly
• Monthly repayments of $850.

Calculate the total amount paid for the renovations.

Aaron has some new tenants ready to move in as soon as the renovations are finished.

Find the number of months it takes for the increase in Aaron’s rental income to match the amount paid for the renovations, found in part (b).

Phillip takes out a loan of $47 000. The unpaid balance on the loan has an interest rate of 4.97%, compounded semi-annually. The loan is to be repaid in payments of $2120 at the end of every quarter.

Calculate the number of years it will take to repay the loan.

After 1.5 years, Phillip misses a payment. The penalty for missing a payment is 5.5% of the remaining balance.

Calculate the total amount paid for the loan.

Paul is looking at two homes for his family to move into. The first home is located out of town and is listed at $775 000 and the second home is located in town and is listed at $620 000.

The bank offer Paul an identical loan for both homes. Terms of the loan are:

• 20% deposit
• 5.77% nominal annual interest rate, compounded quarterly
• Repayments of $4800 to be paid at the end of every month

Calculate the loan amount for the

Find the number of months and years to pay off the loan for the

Calculate the total amount Paul will save if he decides to choose the second home. Give your answer to 2 decimal places.

Anna decides she wants to buy a farm and the bank agree to give her a loan provided she makes a 14% deposit of $40 000.

Calculate the value of the farm. Give your answer to the nearest dollar.

She currently has $15 000 saved up and decides to invest it in some high risk high growth shares forecasted to grow at 65% annually.

Calculate the forecasted number of years it will take for her to be able to afford the deposit.

1.5 years later, the shares outperform their forecasted growth rate and Anna is able to afford the deposit on the farm.

Calculate the percentage error between the forecasted annual growth rate and the actual annual growth rate of the shares.

Anna now takes out the loan from the bank. Terms of the loan are:

• 5.04% nominal annual interest rate, compounded monthly
• Repayments of $3900 to be made at the end of every quarter

Calculate the total number of years and months it will take for Anna to go from making the decision to buy the farm to fully paying off the loan.

In this question, give all your answers to the nearest dollar.

Joe has just retired and he has saved $1 500 000. He “rolls over” his savings into an annuity fund which returns a nominal annual interest rate of 3.55%, compounded semi-annually. Joe is allowed to withdraw up to $25 000 every month, however if he wants to withdraw more than this he must pay a penalty of 5% of the withdrawal amount.

Joe decides to withdraw $5000 every month.

Calculate the value of the fund after 6 years.

After 6 years Joe decides to make a 30% deposit on a $1 040 000 home for his daughter’s family to move into. He makes the withdrawal for the deposit at the same time as he makes his usual monthly $5000 withdrawal.

Calculate the penalty amount Joe will have to pay.

Calculate the amount in the fund after he makes the withdrawal and the penalty has been applied.

In this question, give all your answers to the nearest dollar.

Nina’s grandson, Jasper, has just turned 5 and for his birthday she has set up two passive income streams that he will have access to when he turns 18. The first passive income stream is a savings account that pays a special annual interest rate of 5.42% for kids under the age of 12. The second passive income stream is a well-diversified investment portfolio expected to give annual returns of 6.88%. Nina deposits $200 into the savings account and $100 into the investment portfolio every month.

Calculate

When Jasper turns 12, the interest rate on the savings account drops to 3.53%.

Given that Nina continues to deposit the same amount every month, calculate the total amount in the savings account when Jasper turns 18.

When Jasper turns 18 the average annual returns of the investment portfolio is calculated at 12.4%.

Given that Nina continues to deposit the same amount every month, calculate the total amount available to Jasper when he turns 18.

In this question, give all your answers to the nearest dollar

Roy has an outstanding balance of $35 839.75 on a loan with interest that accumulates at a nominal annual interest rate of 9.96%, compounded monthly. Roy has been making repayments of $720 at the end of every month for 2 years.

Calculate the initial loan amount.

After 2 years of repaying back the loan Roy wants to increase his monthly repayments. The bank offers him an option to increase his payments to $980 and pay a nominal annual interest rate of 10.1%, compounded monthly.

Determine whether Roy should accept the option and justify your answer.

Jeff has an outstanding balance of $66 900 on a loan that he has been repaying $1840 every quarter for 4.5 years. The initial loan amount was $80 000 with a nominal annual interest rate of %, compounded semi-annually.

Find the value of .

Jeff wants to change his repayments such that he makes monthly repayments and pays off the loan within a total of 10 years.

Find Jeff’s new monthly repayment amount.

Helen takes out a loan of $105 000 with a nominal annual interest rate of 4.42%, compounded monthly. Helen makes 78 monthly repayments to amortise the loan.

Calculate

4 years into the loan Helen is considering paying off the loan in full, incurring a prepayment penalty of 8.2% of the remaining balance.

Determine whether Helen will save money by paying off the loan early.