Social Arithmetic

Telephone, gas and electricity bills usually include a standing charge plus a cost for each unit used. VAT is added to the whole bill.

\[ \text{Units used} = \text{Present reading} - \text{Previous reading} \] \[ \text{Charge} = \text{Units used} \times \text{Cost per unit} \]

VAT is payable on the whole bill.

Secured Loans – Your house is at risk if you default.
Unsecured Loans – Goods are yours; house is safe.
Hire Purchase – Goods become yours after final payment.
Mortgages – Secured loans for houses, ships, etc.

Example

Joe Bloggs wishes to borrow £10,000. Which option is cheapest?

EasyLoan costs £17,343 over 15 years.
Loans R Us costs £11,122.56 over 3 years (unprotected).

Fred’s Finance:

Cheapest: Loans R Us (3 years, unprotected).

HP requires a deposit followed by fixed monthly payments. Goods become yours only after the final payment.

Example

A TV costs £600 cash.
HP: 10% deposit + 36 × £15.75.
How much cheaper is cash?

Cash is £27 cheaper.

Options when buying a car:

Cash: Buy outright.
Car Loan: Spread cost over time.
HP: Own after final payment.
PCP: Pay for depreciation; optional final payment.
PCH / Leasing: Long‑term rental; never own.

Key Features

Contracts up to 5 years.
Flexible deposit.
You own the car at the end.
No mileage limits.

Risks

You do not own the car until the final payment.
Missed payments may lead to repossession.

Key Features

Lower monthly payments.
Options at end: return, upgrade, or buy.

Risks

Mileage limits apply.
Excess wear charges possible.
You only own the car if you pay the final amount.

Key Features

24–48 month contracts.
Flexible initial rental.

Risks

You never own the car.
Charges for excess mileage or damage.

Insurance premiums depend on the probability of an event happening. Higher risk → higher premium.

Example

Bodgit Insurance charges £3.50 per £1000 of house value.
Cost to insure a £189,000 house?

Interest is a percentage of capital charged on loans or paid on savings.

Simple interest is calculated only on the original capital.

Example

Calculate simple interest on £500 for 3 years at 6%.

Compound interest uses interest earned to increase capital.

Example

Calculate compound interest on £500 for 3 years at 6%.

Using the CR^y mnemonic:

Example

For monthly rates, convert years → months.

Example

CR^y can be chained:

Capital × rate₁^term₁ × rate₂^term₂ × …

Example

£5,000 placed in savings on 1 April 2022.

Balance = £5167.33

Effective interest accounts for compounding.

EAR (Effective Annual Rate) is used for loans.
AER (Annual Equivalent Rate) is used for savings.

Example

A loan has a nominal rate of 5%. Find EAR if compounded:

Semi‑annually
Monthly
Daily

Worked Example (EAR)

\[ EAR = \left(1 + \frac{i}{n}\right)^{n} - 1 \] \[ = \left(1 + \frac{0.05}{2}\right)^{2} - 1 \] \[ = (1.025)^{2} - 1 \] \[ = 1.050625 - 1 \] \[ = 0.050625 \] \[ = 5.06\% \]

Explanation

1. Start with the EAR formula:
EAR tells you the “real” interest earned after compounding.

2. Substitute the values:
Interest rate $i = 0.05$ (5%) and compounding $n = 2$ (twice a year).

3. Work out the inside:
$0.05 \div 2 = 0.025$, so the bracket becomes $1.025$.

4. Square it:
$1.025^2 = 1.050625$.

5. Subtract 1:
This removes the original amount, leaving only the interest.

6. Convert to a percentage:
$0.050625 = 5.06\%$.

So the effective annual rate is 5.06%.

Worked Example (EAR)

\[ EAR = \left(1 + \frac{i}{n}\right)^{n} - 1 \] \[ = \left(1 + \frac{0.05}{12}\right)^{12} - 1 \] \[ = (1.004166667)^{12} - 1 \] \[ = 1.051161898 - 1 \] \[ = 0.051161898 \] \[ = 5.12\% \]

Explanation

1. Start with the EAR formula:
EAR shows the true annual return once compounding is included.

2. Substitute the values:
Interest rate $i = 0.05$ (5%) and compounding $n = 12$ (monthly).

3. Work out the monthly rate:
$0.05 \div 12 = 0.004166667$.

4. Add 1:
The bracket becomes $1.004166667$.

5. Raise to the power of 12:
This applies monthly compounding for a full year.

6. Subtract 1:
Removing the original amount leaves the interest earned.

7. Convert to a percentage:
$0.051161898 = 5.12\%$.

So the effective annual rate is 5.12%.

Worked Example (EAR)

\[ EAR = \left(1 + \frac{i}{n}\right)^{n} - 1 \] \[ = \left(1 + \frac{0.05}{365}\right)^{365} - 1 \] \[ = (1.000136986)^{365} - 1 \] \[ = 1.051267496 - 1 \] \[ = 0.051267496 \] \[ = 5.13\% \]

Student‑Friendly Explanation

1. Start with the EAR formula:
EAR shows the true annual return after compounding is included.

2. Substitute the values:
Interest rate $i = 0.05$ (5%) and compounding $n = 365$ (daily).

3. Work out the daily rate:
$0.05 \div 365 = 0.000136986$.

4. Add 1:
The bracket becomes $1.000136986$.

5. Raise to the power of 365:
This applies daily compounding for a full year.

6. Subtract 1:
This removes the original amount, leaving only the interest earned.

7. Convert to a percentage:
$0.051267496 = 5.13\%$.

So the effective annual rate is 5.13%.

Interest Rate Conversion Formula

\[ i_{\text{new}} = \left(1 + i_{\text{old}}\right)^{\frac{C_{f,\text{old}}}{C_{f,\text{new}}}} - 1 \]

Student‑Friendly Explanation

1. What this formula does:
It converts an interest rate from one compounding frequency to another.

2. $i_{\text{old}}$:
The interest rate you already have.

3. $i_{\text{new}}$:
The equivalent rate using a different compounding frequency.

4. $C_{f,\text{old}}$:
How many times per year the old rate compounds.

5. $C_{f,\text{new}}$:
How many times per year the new rate compounds.

6. Why the exponent is a fraction:
It adjusts the growth factor so both rates represent the same annual effect.

7. Subtract 1 at the end:
This removes the principal, leaving only the interest portion.

This formula ensures both rates produce the same effective annual growth.

Source: LibreTexts Mathematics

To convert annual effective → monthly effective:

Annual → Monthly Rate Conversion

\[ i_{\text{month}} = \left(1 + i_{\text{year}}\right)^{\frac{1}{12}} - 1 \]

Student‑Friendly Explanation

1. What this formula does:
It converts a yearly interest rate into the equivalent monthly rate.

2. $i_{\text{year}}$:
The annual interest rate you start with.

3. Why we use the power of $1/12$:
Because we want the monthly rate that, when compounded 12 times, gives the same annual growth.

4. Add 1 first:
This turns the interest rate into a growth factor.

5. Raise to the power of $1/12$:
This finds the “one‑month” version of the annual growth.

6. Subtract 1:
This removes the principal, leaving only the monthly interest.

The result is the true monthly rate that matches the annual rate exactly.

Do NOT divide by 12!

Example

The monthly effective interest rate is 0.31% (1 d.p.).

Example

A loan of £10,000 is borrowed over 3 years. The annual effective interest rate is 7.1%.

Enter the formula =(1+C5)^(1/12)-1 into cell C6 to calculate the monthly effective interest rate.

The Excel function PMT calculates payments based on a constant interest rate and number of periods.

Here:

Rate = effective monthly rate (C6)
NPER = number of payments (C7)
PV = loan value (C4)

This gives a fixed monthly payment of £308.22.

Set up a loan schedule with these headings:

Enter =$C$4 into F12 (type =C4 then press F4).

Extend the table down to 36 months.

Repayment is C8, so enter =$C$8 in C13.

Interest content: =ROUND($C$6*F12,2)

Capital repaid: =C13-D13

New balance: =F12-E13

Copy down to the final row.

The final payment differs. Enter =F47 + D48 into C48.

Summary:

This can be represented by the recurrence relation:
\[ U_{n+1} = 1.0057U_n - 308.22 \]

From the graph, after 15 months the outstanding loan is approximately £6,000.

The effective monthly rate can be used to compare loan offers quickly.

Example

A TV advert offers short‑term loans with a typical annual interest rate of 1271%.

Create a loan schedule for £100 over 6 months.
Calculate the cost of the loan.
Express the cost as a percentage of the original loan.
Comment on your findings.

1. Total repaid = £200.41
2. Cost = £200.41 − £100 = £100.41
3. Percentage = (£100.41 ÷ £100) × 100% = 100.41%
4. The cost of borrowing exceeds the original loan!