**Balancing the books: understanding financial maths in IB® Applications and Interpretations**

Written by Aidan; Aidan is available for private tutoring.

The financial maths component of the IB® A & I course is something that you really will be able to use in everyday life as you grow older. It’s also a prime example of a place where you need to put all of your energy into interpreting the situation before sitting back and letting your calculator do the hard work.

Within the IB®, the good thing about topics like this – i.e. those where you can rely entirely on using your calculator – is that there are no complicated formulae that you need to learn to use. Conversely, the drawback of relying so heavily on your calculator is that it can be hard to see whether or not you’ve made an error – wrong answers might look plausible, and there isn’t really any alternative method you can use to check your working. For that reason, you need to be completely sure about the numbers that you are putting into your calculator’s financial maths application

## Financial products

The problems which involve compound interest may be as simple as conventional savings accounts, where money is paid in at the start and then left for a period of time, or they may involve regular payments into or out of the account either to build up savings, or to repay a loan or mortgage. Interest may be compounded annually, as is most common in the real world, or at other intervals such as monthly or quarterly.

Each of these products has a few key features and these are the things that you need to identify from the question:

- An annual interest rate which is the same for the life of the product (I%)
- A compounding frequency which will be expressed as a number of compounding periods per year (C/Y) – so annually (1), quarterly (4), monthly (12).
- A regular payment (PMT) which will happen a fixed number of times per year (P/Y) (again, usually 1, 4 or 12)
- The total number of payment periods (
*n*) - An initial value (PV)
- A final value (FV)

Some problems may not contain regular payments, but since the idea of a regular payment is a core part of the financial maths application on your calculator, you need to interpret this as a payment value of 0. You can decide whether that’s 0 per month, or 0 per year, or 0 per quarter – and it might seem unimportant (because after all, no money every month is the same as no money every year) but it does make a difference, as you’ll see in a moment.

Many of these values could be positive or negative – getting the signs correct is often the hardest part of these questions.

In order to get the signs right, the most important items to watch for are as follows:

- You need to view each financial product as “closed”. This means that at the start of the product you have zero money in the bank, and at the end of the product, you have zero money in the bank.
- You need to imagine money going into, or out of your pocket. If money leaves your pocket, it’s a negative quantity. If money enters your pocket, it’s a positive quantity.

For example, suppose you keep savings of $5000 in an account and are asked how much it is worth at the end of a certain amount of time. To achieve this, we imagine that your initial bank account value is $0; and so you pay $5000 out of your pocket into the bank. This will be entered in your calculator as an initial value of -5000. To find its value at the end of the account’s life, we imagine that the account will be closed – and so all the money will be paid back to you. This will be a final transaction that gives money to you, and so the final value generated by your calculator will be positive.

Similarly, if you had a bank account starting with $10000 and made a series of withdrawals each of $500, the initial value would be -10000 (because you need to pay $10000 out of your pocket into the bank to have a balance of $10000) and each payment would be +500, because each withdrawal brings money back into your pocket. Suppose we check the “final value” of the account after a certain amount of time. If the final value is positive, that means that when you close the account there is money left and it would be paid into your pocket. In contrast, if the final value is negative, this means that in order to close the account, more money needs to leave your pocket to satisfy your debt to the bank – you are overdrawn.

Getting the signs of these amounts correct every time is really important, so make sure that when you are setting the values of principal value (PV), final value (FV) and payment value (PMT) you simply consider whether you have given money to the bank (-ve) or the bank will give money to you (+ve).

## The right amount of time

Consider the following problem:

*You borrow $10000 at 3% interest per annum compounded monthly. You make quarterly payments of $300 until there is a final balance owing of $3023.65, which you pay off in one additional payment. Determine the amount of time for which you borrowed the money.*

So to answer this: you borrow $10000. That means the bank puts money in your pocket, so PV = +10000. The interest is 3% compounded monthly, so C/Y = 12. You make quarterly payments, so P/Y = 4, and since payments involve money leaving your pocket, PMT = -300. Finally, because you finish by paying extra money back to the bank, FV = -3023.65.

Enter this information into your calculator’s financial maths application and instruct it to solve for *n*. You should get *n = *28. But 28 what? 28 years? 28 months? 28 days?

The answer is that *n* always represents the number of payment periods. In this case, payments were quarterly. So if you have made 28 quarterly payments, the amount of time for which the money was borrowed was 7 years.

If you always remember these key points, you can’t go wrong:

- All products are closed – the bank balance starts at 0 and ends at 0.
- All transactions are from the point of view of your pocket.
- The value of
*n*is the number of payment periods, not the number of years, or the number of compounding periods.

At IB® Higher Level, you might also encounter problems where there’s more than one step – perhaps the interest rate changes, or there are monthly payments which are completely regular except for an extra payment after a certain amount of time. To deal with these, you need to treat the problem as if it is two separate problems. Use the result of the first part to help you answer the second.

## Some practice problems:

As with all maths, the only way to be good at it is to practice. Why not have a go at the questions below? The link at the bottom takes you to the answers. If you get a question wrong, go back and check your working. If you can’t figure out where you’ve gone wrong, the second link takes you to a set of worked solutions.

- You put $15000 in a savings account offering 4% interest compounded annually. Determine the value of your savings after 8 years.
- You borrow $20000 at 3% interest per annum, compounded quarterly. Determine the amount to be repaid if the loan is repaid by a single payment after 4 years.
- You borrow $500000 for a mortgage at a rate of 2.4% per annum compounded annually.

Determine the appropriate monthly payments if the whole amount is to be repaid in 25 years. - An annuity is a form of investment which receives interest whilst simultaneously paying out a regular payment. People often use them in retirement to generate a pension.

You purchase an annuity worth $100000 which receives 2.1% interest compounded annually. The annuity makes monthly payments. Determine how long the annuity will last if monthly payments of $1200 are taken. - A savings account offers interest of 2% per annum, compounded annually.

Anna invests $1000 initially and then $100 every month for five years.

Bhavna invests $1000 initially and then $1200 every year for five years.

Determine the difference between the value of their savings after five years.

The following problems involve multiple stages and are the sort you might encounter at IB® Higher Level.

- You have $15000 in a savings account which pays 1.5% interest p.a. compounded annually. For the first 5 years, you pay in $100 per month. For the next five years, you pay in $200 per month. Find the value of your savings after 10 years.
- A mortgage for $400000 has an initial interest rate of 1.6% per annum over 20 years.

The initial monthly payments are calculated on the assumption that this interest rate will continue for the full life of the mortgage.

After three years, the rate goes up to 2.8% and new payments are calculated.- Find the initial monthly payments.
- Find the monthly payments after three years.
- Find the total cost of the mortgage (i.e. the amount of interest that is paid in total).

Worked solutions to problems, here

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