In this topic of IBDP Mathematics, we will be discussing loans and annuities, and solving questions related to these topics.
Loans
In IBDP Mathematics, loans is one type of finanical mathematics that you will need to know. It has a more direct connection with the real world, thus is an important topic not only in IBDP Mathematics, but is also useful in the real world.
Loans are large amounts of money that are borrowed from a financial institution (i.e. a bank) in order to help finance larger purchases. People often take out loans when purchasing houses and cars, as well as for renovations, education expenses etc.
- When you take out a loan, you will be given an amount of money from the bank, and you are expected to repay that amount plus interest over a set amount of time.
- Typically, one will be able to repay the loan with a series of regular payments - this process is called amortisation.
With each payment back to the bank, a part of the payment will be you repaying the amount you initially borrowed - this part is known as the principal. However, the other part of the payment will be you paying the interest, which is based on the interest rate.
- The interest for each repayment is calcuated based on the reducing balance (i.e. the amount you still have to repay).
- Given that you will be repaying your loan, the balance would reduce with each repayment. As a result, the interest will also decrease as the loan is repaid.
- As the amount you repay each time is fixed, and given that the interest decreases as more of the loan is repaid, the principal amount of repayment will increase with each repayment.
- In other words, the proportion of interest in each repayment decreases, whilst the proportion of principal repayment in each repayment increases.
- At the end of the series of repayments, you will have paid back the amount you initially borrowed (this is the sum of the principal component of each repayment), as well as interest.
You will not necessarily be tested on this knowledge, as it is unlikely that you will be asked to calculate the amount of interest paid for each repayment. However, it is useful to understand how these loans work.
Instead, it is likely that you will be asked how much interest is paid in total. This can be calculated using the formula below:
Other questions that are likely to be asked include how much each repayment should be, or what the interest rate is given a particular repayment value. These questions can be answered using the financial solver (i.e. TVM) in your calculator.
Using the financial solver, there should be several pieces of information which you have to input in order to calculate your desired value. Although these might be labelled differently in different calculators, the information required should be similar (Note: the following notation is based on the financial solver in the TI-Nspire calculator from Texas Instruments).
- N = the number of payments made
- I = the annual interest rate (written in percentages)
- PV = present value (i.e. the amount of money one has in the present)
- PMT = the amount for each repayment
- FV = future value (i.e. the amount of money one hopes to have in the future)
- PpY = payments per year
- CpY = compounding periods per year
Most of the information is given in the question - you will only be able to solve for one value if you have all the other values inputted. This is shown in the following example.
Example: Charlie is taking out a personal loan for $12,000 to purchase a car. He will repay it over 5 years at 6% p.a. Calculate the monthly repayments, total repayment and interest charged.
- First, we can deduce what types of information is provided in the question:
- We are given the interest rate (annual interest rate of 6% p.a. - the p.a. means per annum) - this gives us the value for I (I = 6 )
- We are also given the value for N (N=60 ) - Charlie is repaying the loan over 5 years, and making monthly repayments. This gives us a total number of 60 repayments, as he will be making 12 repayments each year, for 5 years.
- We are given the value of PV (PV = 12,000 ) - this is the amount he has borrowed, thus in the present he has $12,000.
- We can also deduce the future value (FV = 0 ) - this is because he wishes to repay all of his loan, thus in the future he hopes to have $0.
- The value of PpY and CpY (in this question) are the same - they are both 12.
- For PpY, it is fairly straight forward. The question states that he will be making monthly repayments, thus Charlie will be making 12 payments per year.
- For CpY, if not stated, then the value is the same as PpY. This is because it is assumed that the interest is compounded each time a repayment is made. Therefore, for monthly repayments, the interest would be compounded 12 times a year, so CpY = 12.
- Sometimes, the question will state the compounding period per year (e.g. 6% p.a. compounded quarterly). In this case, the CpY would be 4, as the interest is compounded 4 times a year.
- As you can see, we have deduced all the information required, except PMT (which is the value we want to find). In order to calculate PMT, input all the other information into the right place, and click enter in the field for PMT.
- You should get PMT = -232.00, thus the monthly repayments is $232.
- In order to calculate the total repayment, multiply the montly repayment by N.
- For this question, you should get total repayment = $232 x 60 = $13,920.
- To determine the interest charged, subtract the total repayment by the amount borrowed.
- For this question, you should get interest paid = $13,920 - $12,000 = $1920.
Many students have trouble with the signs for PV and PMT. For questions related to loans, the value of PV will always be positive, as you are receiving the amount loaned (i.e. this is the amount of money you currently have). The value of PMT (i.e. the value of repayments) will be negative, as you are repaying that amount, so the money is going away from you.
Annuities
The second type of financial mathematics you will need to know in IBDP Mathematics is annuities. Annuities are a type of investment where you deposite a certain amount of money into the bank, and make withdrawls from it over a fixed period of time. The investment earns interest according to the balance of the annuity at each time.
- Essentially, it is the opposite of a loan. With a loan, you are borrowing money from the bank, and are paying it back over time. With annuities, you are depositing money in the bank, and withdrawing money over time.
- Similar to interest on loans, as the balance in the annuity decreases, so does the interest earned. However, the regular repayments remain the same.
Annuities are typically used for people who have retired, and want a regular source of income. They start depositing money at an earlier age to build their annuity. As they retire, they are then able to withdraw money from their annuity regularly.
Questions on annuities typically include asking how long one can withdraw from their annuity given regular withdrawls of the same amount, or the amount one should withdraw each time if they want to be able to withdraw over a fixed amount of time. The following example shows how we can used the financial solver in questions about annuities.
Example: Jo has retired at age 68 with $830,000 in his savings fund. She rolls the money into an annuity fund that earns 4% p.a. compounded monthly. If Jo wants the money to last until she is 85, how much can she withdraw each month?
- Similar to solving questions about loans, the first step is to extract all the information given in the question.
- We can figure out the value of N using the information about Jo's age.
- Given that she is currently 68, and wants to withdraw money until she is 85, she wants the fund to last 17 years. Not only that, but she wants to withdraw money each month, thus N = 17 x 12 = 204.
- The information about the interest rate gives us the value of I - I = 4
- We are also able to figure out the values for PpY and CpY - they are both 12.
- For PpY, Jo wants monthly withdrawls, thus she will withdraw money 12 times a year.
- For CpY, the questions states that the interest is compounded monthly, therefore it is compounded 12 times a year.
- The value of PV is also given - Jo has deposited $830,000 into her annuity fund, therefore PV = -830,000
- The sign for PV is negative, as this is the money that she doesn't currently posses with her - it is deposited into the annuity fund.
- We can deduce the value of FV from the question. At the end of the 17 years, she will not be making anymore withdrawls (i.e. her annuity fund would be empty), thus FV = 0.
- Input all this information to calculate PMT - this is the amount she can withdraw each month.
- You should get PMT = 5614.06, therefore Jo can withdraw $5614.06 each month.
You may have noticed that the signs of PV is negative, and the sign for PMT is positive. This is because with annuities, the person is first depositing money into the bank, thus at the present time, the person will have negative that amount of money. PMT is positive as the person is withdrawing that amount of money each time (the money is going to them).
Helpful Tips:
- When answering questions related to loans and annuities, there are lots of different types of questions that can be asked, thus it is a good idea to do lots of practice.
- If you forget what the signs are for PV and PMT, don't worry. You should get the same values regardless. Just make sure that if the sign for PV is positive, the sign for PMT should be negative (and vice versa).
- Always round up when answering questions about loans, and round down when answering questions about annuities.
This is the end of this topic.
