# financial maths

## Introduction

Here we're going to look at some aspects of the mathematics used in financial situations. We'll look at the concept of present value, and see how it relates to the APR figures that you see on loan advertisements. We'll then look at how mortgage payments and taxes are calculated.

## 1. Present value

Have a think about these questions:

Would you rather have £1000 now or £1100 in a year's time?

Would you rather have £1000 on January 1st or £100 on the last day of each month for the year?

What information do we need to know to answer this question effectively? How do we decide the value of a payment that will happen in the future. Watch the video discussion of the question above, and do these questions (check your answers on the second page).

## 2. Annual Percentage Rate (APR)

Here's a TV advert for a payday loan company, Wonga (which has since gone out of business).

What does the phrase Representative 1509% APR mean? Annual Percentage Rate is the way that customers are able to compare the value of borrowing money from different lenders, with a lower APR representing better value, i.e. less interest being charged over time. Watch the videos below about calculating with APRs, and then answer these questions. The CAS (computer algebra system) used is geogebra - feel free to use it in answering the questions. Solutions here.

## 4. Tax and National Insurance

... in this world nothing can be said to be certain, except death and taxes.

So said Benjamin Franklin in 1789. Governments charge their citizens taxes in order to fund spending, for example on education, health care and so on. There are various modes of taxation in the UK:

Income Tax, where a proportion of what you own is given to central government;

National Insurance, again a proportion of earnings, specifically used to fund certain benefits that payers are entitled to ;

Value Added Tax (VAT) paid on goods at the point of sale. So your earnings are taxed twice: once when you receive your pay, and once when you spend it.

Vehicle Tax charged to car users, TV licence fees, Council Tax (proceeds go to councils, not central government), Congestion Charge, ...

In this short section, we're going to look at how Income Tax and National Insurance are calculated, and how it affects one's net pay (i.e. what you 'take home' each month.

Have a look at the websites here and here, and then look at the tax calculation examples below. Some questions here, and solutions.

Bob earns a salary of £58 000 a year. What is his monthly take home pay?

Income tax: Bob has a tax-free allowance of £12 500, and pays no tax on this amount. As he earns over £50 000 he is classed as a higher rate tax payer, but he only pays the 40% rate on his income over that £50000. His tax is calculated as:

0% of £12 500 = £0

20% of £37 500 = £7 500

40% of £8 000 = £3 200

So his income tax is £10 700 for the year, or £891.67 per month

National Insurance: Bob (like everyone we'll consider) is a letter A national insurance contributor. His salary is £4 833.33 a month. So each month (using the monthly totals on the website), he pays:

0% of £792 = £0

12% of £3 375 = £405

2% of £666.33 = £13.33

So his NI contributions are £418.33 per month.

His monthly take home pay is then £4833.33 - £891.67 - £418.33 = £3523.33.

Jenny earns £122 000 a year. What is her monthly take home pay?

Income tax: Because Jenny earns over £100 000, her tax-free allowance is reduced by half of the amount she earns over this figure (i.e. £11000). She is also a higher rate tax payer. Her tax is calculated as:

0% of £1500 = £0

20% of £48 500 = £9700

40% of £72 000 = £28 800

So her income tax is £38 500 for the year, or £3 208.33 per month

National Insurance: Jenny's salary is £10 166.66 a month. So each month (using the monthly totals on the website), she pays:

0% of £792 = £0

12% of £3375 = £405

2% of £5 999.66 = £119.99

So her NI contributions are £524.99 per month.

Her monthly take home pay is then £6 433.34.