Financial Mathematics
Financial Mathematics
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Financial Mathematics
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Learning goal
I can calculate and compare simple interest, compound interest, savings growth, and loan costs using the correct financial mathematics formulas.
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What words come to mind when you hear financial mathematics?
Use this to surface prior ideas such as interest, loans, savings, banks, debt, investing, and repayments. Pick a few entries to link back to the learning goal.
Compound Interest
Compound interest means interest is added to the balance, and then future interest is calculated on the new total.
This is why compound growth is faster than simple interest growth over time.
Open with the key contrast: simple interest grows by equal amounts each period, compound interest grows by increasing amounts because interest earns interest.
Single-sum deposits
Start with one principal amount, called Pv
No regular payment, so Pmt = 0
The balance grows to Fv
Interest earned = Fv - Pv
Emphasise that this model is for one lump sum invested at the start, not for repeated deposits.
Regular payments: annuities
An annuity uses equal regular payments
Pmt represents the payment each period
Payments are usually set to End, meaning each payment is made at the end of the period
Total paid in by deposits = Pmt × N
Interest earned = Fv - [Pv + (Pmt × N)]
Keep the focus on interpretation of variables rather than calculator steps. Students should see why total contributions must be separated from interest earned.
With and without an initial investment
With initial investment
Pv is the starting amount
Regular deposits may also be added
Without initial investment
Pv = 0
Growth comes only from the regular payments
Interest earned = Fv - (Pmt × N)
Students often miss that an annuity can begin with or without a starting lump sum. This changes how interest earned is calculated.
Key setup values
N = n × K
n = number of years
K = payments or compounding periods each year
Match P/Y and C/Y to the situation
Clarify that N is the total number of periods. For example, monthly for 5 years gives N = 5 × 12 = 60.
Why does compound interest grow faster over time?
Interest adds to balance
Rate changes yearly
Payments stop early
Principal becomes zero
Checks the key contrast from slides 2–6: compound interest earns interest on previous interest, unlike simple interest.
Savings setup
For a single-sum deposit, the starting amount is Pv and there is no regular payment, so Pmt = 0. In an annuity, equal regular payments are added each period, and the total number of periods is N.
All three blanks are load-bearing variables taught in this first subtopic.
How would you explain N = n × K in your own words?
Expected elements: N is the total number of periods; n is the number of years; K is the number of periods each year; example such as 5 years monthly gives 60 periods.
Simple Interest
Simple interest is calculated only on the original principal.
Formula: SI = P × i × n
Remind students that i is written as a decimal in the formula, for example 6% = 0.06.
What the formula means
P = principal
i = interest rate per year as a decimal
n = time in years
Interest increases by the same amount each year
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Rearranging the formula
To find principal: P = SI / (i × n)
To find rate: i = SI / (P × n)
To find time: n = SI / (P × i)
Use the same formula flexibly, depending on the missing value
This is the algebra link in the lesson. Students should recognise that finance formulas can be rearranged just like other equations.
Simple vs compound interest
Simple interest: interest is based only on the original principal
Compound interest: interest is based on the growing balance
Over longer times, compound interest usually gives a much larger final value
This comparison resolves why two investments with the same rate can end with different totals, depending on the interest model.
In simple interest, what does i represent?
Final value
Annual rate
Time in months
Regular payment
Distractors come from nearby finance variables: Fv, n, and Pmt.
Simple and compound interest
Simple interest uses the formula SI = P × i × n, so interest is calculated only on the original principal. With compound interest, future interest is worked out on the growing balance, so the final value is usually larger over time.
This anchors the meanings of i, principal, and balance while reinforcing the comparison in slide 10.
Superannuation and Loans
The same financial mathematics ideas can model both saving and borrowing.
For superannuation, money grows over time. For loans, repayments reduce what is owed while interest increases the cost.
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Superannuation: saving phase
Regular contributions are added over many periods
Usually Pv = 0 if there is no starting balance
Pmt is the regular contribution
The final balance is Fv
Interest earned = Fv - (Pmt × N)
Frame superannuation as a long-term annuity. The mathematical structure is the same as repeated saving.
Superannuation: withdrawal phase
Money is taken out after the balance has grown
A new present value can be based on the final value after withdrawals
The balance must last through the withdrawal period
Keep this conceptual rather than technical. Students only need to see that the model changes once money starts being withdrawn.
Loans as an annuity
Pv is the amount borrowed
Pmt is the regular repayment
Fv = 0 when the loan is fully repaid
Total paid to the bank = Pmt × N
Interest paid = (Pmt × N) - Pv
Stress that the loan amount is money received at the start, while repayments are money paid out over time.
Split time in a loan
Sometimes loan conditions change after a number of years
First, find the loan value after the first time period
Then use that value as the new Pv for the next stage
This lets one loan be modelled in parts
Examples include a change in repayment size or interest rate. Students do not need a full worked example here unless time allows.
When a loan is fully repaid, what is Fv?
The interest paid
The amount borrowed
The regular repayment
Zero
Tests the loan model from slides 11–15. Students often confuse Fv with Pv or total paid.
Saving and borrowing
In superannuation saving, regular contributions are added over many periods and often Pv = 0 at the start. In a loan, Pv is the amount borrowed and interest paid is total repaid minus Pv.
Echoes the lesson language from slides 12 and 14 without adding a new context.
Why might a loan need to be modelled in parts?
Expected elements: if loan conditions change, find the value after the first stage, then use that amount as the new Pv for the next stage.
Strategies to Minimise Interest and Loan Amount
Loan cost depends on both the amount borrowed and how long the balance stays unpaid.
Changing repayments and timing can reduce the total interest paid.
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Larger and more frequent repayments
Larger repayments reduce the balance faster
More frequent repayments reduce the time interest builds up
Both strategies lower total interest paid
Connect this to the formula idea: reducing the balance sooner means less interest is calculated in later periods.
Reducing the term
A shorter loan term means fewer repayment periods
Fewer periods usually means less total interest
The trade-off is that each repayment may need to be larger
This is a useful opportunity to discuss financial trade-offs without drifting away from the mathematics.
Making lump-sum payments
Extra one-off payments cut the principal quickly
A smaller principal means less future interest
Lump sums can shorten the term or reduce later repayments
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Why these strategies work
Pay more sooner → principal falls faster → less interest is charged later.
In financial mathematics, time and balance size both matter.
Use this as the final causal summary before the glossary.
Which action reduces total loan interest most directly?
Pay later
Borrow more
Pay more sooner
Extend the term
Checks the causal summary across slides 16–20: reducing the balance earlier reduces later interest.
Reducing loan cost
Larger repayments reduce the balance faster. A shorter loan term means fewer repayment periods. Extra lump-sum payments cut the principal quickly and reduce future interest.
Uses the exact strategy language from slides 17–19.
Which loan strategy seems most useful to you, and why?
No single correct answer. Look for reference to the key idea that paying more or earlier reduces the principal sooner, so less interest is charged later.
What is one idea from this lesson you will remember?
Reflection only. Invite brief personal responses about the main takeaway from the lesson.
What helped you understand the maths best today?
Reflection only. Encourage students to name a specific representation, comparison, or formula that helped.
Which part of this lesson still feels hardest?
Reflection only. Use responses to identify where reteaching may be needed.
Glossary
principal - the starting amount invested or borrowed
simple interest - interest calculated only on the original principal
compound interest - interest calculated on the balance after previous interest has been added
annuity - a model with equal regular payments over time
present value (Pv) - the value at the start of the calculation
future value (Fv) - the value at the end of the calculation
payment (Pmt) - the regular amount paid in or out each period
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Glossary
N - the total number of payment or compounding periods
n - the number of years
K - the number of periods each year
interest earned - the amount gained above the total money invested
interest paid - the amount paid above the amount borrowed
loan term - the full length of time allowed to repay a loan
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