Repayments & Cost of Borrowing (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Borrowing Money

What do I need to know about borrowing money?

• Individuals and businesses will need to borrow money in many different scenarios

• These could be for the short term

  • E.g. An urgent property repair

• Or for the long term

  • E.g. A mortgage to pay for a house

• Whenever money is borrowed, interest is charged at regular intervals

  • This increases the total amount owed

• Interest is effectively the fee that a borrower pays to the lender for the service of providing them money

• This means that in total, the borrower must pay back more than the amount borrowed

  • This is usually justified by a need to have a large lump sum immediately, rather than having time to save up for it

  • E.g. A loan for a car purchase, or a mortgage for a house purchase

Types of borrowing

• There are numerous methods and systems to borrow money, and many different companies and organisations who provide these

  • Loans are a wide-ranging category, and lenders will advertise a selection of different loan products, each aimed at different types of borrower and use-case

    • For example, a loan targeted at someone purchasing a new car

    • This could be provided by a bank or finance company, a car dealership, or even a car brand themselves

    • Hire purchase is a common form of loan for a car, where the car is not owned by the borrower until the full amount is paid off

    • Loans for home improvements are also common, e.g. for kitchens and bathrooms

  • Student loans are a specific type of long-term loan designed to help students pay for their education

    • Different countries have differing rules and terms for the lending and repayment of these loans

    • For example in the UK, student loans are provided by a state-owned provider, and borrowers will not make repayments until they meet a minimum income threshold

  • Credit cards are issued by banks, finance companies, and even supermarkets

    • They provide short-term lending up to an agreed credit limit for a stated interest rate

    • These may have specific purposes or incentives, e.g. 0% interest for a certain time, or earning cashback when spending in a specific store

  • Payday loans are another type of short-term loan

    • They tend to involve fewer checks but can have extremely high interest rates

  • Overdrafts are a feature of some bank accounts that let the account holder spend more money than is in the account (known as going overdrawn)

    • They are short-term loans

    • There will be an agreed limit for this, as well as a fee and interest rate

  • Mortgages are long-term loans specifically for the purchase of property

    • They are secured against the property itself

    • This means that if the borrower does not keep up monthly payments, the lender has the right to take back the property

Student Loans

What are student loans?

• Student loans are a specific type of long-term loan designed to help students pay for their education

  • This could include tuition fees, educational supplies, and living expenses

• Different countries have differing rules and terms for the lending and repayment of these loans, and they are usually provided or backed by the government

• The terms of student loans are usually quite different to loans provided by banks and other private companies

• In the UK, student loans are provided by the Student Loans Company which is backed by the government

How are student loans repaid?

• Repayments are not taken until after the student has completed their studies, and earns over a certain threshold

  • This threshold has changed over time, and therefore depends on which year the loan was taken out

  • For loans taken out in 2024 ("Plan 5"), students only start repaying the loan when they earn over £25 000 per year

• Interest is also charged on the remaining balance

  • The rate of interest varies depending on when the loan was taken out

  • For Plan 5 the interest rate is currently 7.7% (March 2024)

• Once earning over the threshold, repayments are calculated as a percentage of the amount over the threshold they earn

  • E.g. Someone on Plan 5 must pay back 9% of their income over £25 000 per year (or £2083 per month)

  • Consider someone on Plan 5 earning £30 000 per year

    • This would be £2500 per month

  • Find the monthly income over the threshold

    • £2500 - £2083 = £417

  • Find 9% of this amount

    • 9% of £417 = £37.53

    • This is rounded to the nearest pound

    • So £38 is repaid per month

  • Optionally, extra repayments can be made on top of this

• If the borrower is an employee, repayments are taken out of their salary at the same time as income tax and national insurance

  • This will be shown on the borrower's payslip

  • This is a key difference compared to loans offered by private companies

• Due to the repayment structure, many borrowers will never pay off the full amount of the loan, especially once interest is added on

  • Each plan has a time after which the remaining balance is written off

    • This means any remaining debt is cancelled

  • For Plan 5, loans are written off 40 years after the April repayments were first made

Mortgages

What is a mortgage?

• A mortgage is a long-term loan for buying a property

• Most people will not have enough money to buy a house in cash, so banks offer mortgages, which are then paid off over a number of years

• Mortgages are secured against the property itself

  • This means if the borrower does not keep up monthly payments, the lender has the right to take back the property

• Interest is charged, which increases the total amount owed

  • When the mortgage has been fully paid off, a greater amount than initially borrowed will have been paid to the lender

  • Interest rates may be lower than other types of loan, but the long duration, and large amount borrowed, means the total interest paid will be high

• It is rare for banks to lend the full purchase price of a property

  • A cash deposit is paid by the buyer, usually at least 10% of the purchase price

  • A larger deposit means less money needs to be borrowed in total, reducing the overall costs of the mortgage

  • A larger deposit, relative to the value of the property, often reduces the interest rate offered by the lender

    • LTV (loan to value) is the ratio of the amount borrowed to the value of the property, it is usually expressed as a percentage

• Mortgages can have different term lengths, usually between 20 and 30 years

  • Due to increasing house prices, there has been a shift towards longer terms, up to 40 years

How much can be borrowed?

• Lenders will use their own criteria to test affordability and decide a maximum amount to lend to an individual (or often two people together)

  • Affordability is the capacity for the borrower to make repayments each month

  • It depends on the borrower's income, expenses, existing debts, and credit history

• Lenders will take varying levels of risk when offering mortgages

  • This means different lenders will be able to offer different amounts to individuals

  • Generally "higher risk" mortgages (in the eyes of the lender) will be subject to higher interest rates

• Most lenders will lend around 4.5 times the individual's annual salary

  • This is only a guide however, and lenders will use many different factors to determine how much to lend

How are mortgages repaid?

• Mortgages usually start with a fixed term, where a pre-agreed interest rate is charged

  • Fixed terms are usually for either 2, 5, or 10 years

  • Fixed terms offer stability for households, as the monthly payment is fixed

• After the fixed term, the mortgage switches to a variable rate of interest which is linked to the Bank of England base rate (external link) (opens in a new tab)

  • This means the interest rate can fluctuate depending on economic conditions

  • Alternatively, another fixed term can be agreed, known as remortgaging

• As the borrower pays off a portion of the mortgage each month, this builds up equity in the property

  • Equity is how much of the property the borrower owns

  • If a 10% cash deposit is paid at the start, the homeowner's initial equity will be 10%

  • At the end of the mortgage agreement, the homeowner's equity will be 100%

  • This process is known as amortisation

• Consider a mortgage for £180 000 borrowed at a rate of 4.5% annually, for 30 years

  • The lender calculates this will require a monthly payment of £912

• The table below shows how the monthly payments and interest added affect the overall balance owed

  • As the interest is 4.5% annually, the monthly interest is 4.5 ÷ 12 = 0.375%

• Whilst £912 is being paid each month by the borrower, their equity in the property only increases by around £240 each month

• As the total balance owed reduces each month,

  • the interest added on each month reduces,

  • so the amount of equity gained each month will increase

• This means that the mortgage is paid off the slowest at the start, and the fastest at the end

• The graph below shows the outstanding balance of the mortgage over the 30 year term

  • It assumes a fixed interest rate and fixed monthly payment

• The graph has a steeper gradient at the end than at the start

  • This shows that the equity gained (and debt paid off) each year, increases every year

• A mortgage can be paid off even sooner if overpayments are made

  • Overpayments are where the borrower pays more than the agreed monthly amount

  • This could be regular, small payments every month

  • Or larger lump sums at irregular intervals

  • There is usually a limit for the value of overpayments per year before incurring a fee

APR (Annual Percentage Rate)

Why is APR useful?

• Different loans can have very different terms, fees, and repayment plans

  • This makes it harder to compare loans directly to find the best deal

• APR (annual percentage rate) is a figure which takes the total borrowed, interest rate, and repayment amounts and periods all into account to produce an overall figure for comparison

• In the UK, it is a legal requirement that lenders disclose the APR for a loan before an agreement is made

Example of why APR is used

• It is highly unlikely you will be asked to find the APR in this way in an exam

  • It is shown here to explain why APR is important

• Consider a loan that is advertised:

  • Borrow £5000 with 2 payments of £3000

    • One payment of £3000 at the end of year 1

    • One payment of £3000 at the end of year 2

• This means £6000 is paid back in total, so the interest paid is £1000

  • A borrower may assume this means paying back interest of £500 per year

  • £500 ÷ £5000 = 0.10 = 10%

  • 10% interest is probably the assumption a borrower would make

• However it is not this simple, due to the way repayments are made

• Let the "true" annual interest (APR) be denoted by

  • is a decimal, so 12% would be 0.12

• For year 1:

• This remaining balance is then carried forward to year 2, and interest is charged on it

• As this loan only requires two payments to pay off the full amount, the expression for the remaining balance owed at the end of year 2 must be equal to zero

  • 

  • This can be simplified to and solved

  • The positive solution to this quadratic equation is

  • So the APR for this scenario is 13.07% to 2 decimal places

  • This is more than the previously assumed 10%

• This is why the government legislation that lenders must disclose APR is important and a positive for borrowers

What is the formula for APR?

• The formula for APR is given on the formula sheet in your exam

• 

  • is the amount of the loan

  • is the number of repayments

  • is the APR expressed as a decimal

  • is the amount of the ^th repayment

  • is the interval in years between the start of the loan and the ^th repayment

• Writing the sum in full:

  • 

• This formula assumes there are no arrangement or exit fees involved

Example of using the formula for APR

• Consider the same example used earlier, where we want to find the APR

  • Borrow £5000 with 2 payments of £3000

    • One payment of £3000 at the end of year 1

    • One payment of £3000 at the end of year 2

• Identify the values of the variables for the formula

  • 

  • 

  • (APR is being found for this example)

  • and as £3000 is paid at the end of year 1

  • and as £3000 is paid at the end of year 2

• Write the sum using these values

  • 

• This equation can be solved in several ways

  • Algebraically

  • Trial and improvement, usually with a spreadsheet

  • Using an equation solver on a calculator or computer

• When solved algebraically, multiplying both sides by , expanding and simplifying leads to the same equation formed earlier

  • 

  • This solves to give a positive solution of

• So the APR is approximately 13.07%