Why geometric models appear in finance

A geometric sequence is a sequence where each term comes from multiplying the previous term by a constant ratio. Financial balances follow this pattern when the same percentage change is applied at regular intervals. That links compound growth, depreciation and inflation directly to exponential models and functions.

Simple and compound interest

Simple interest is calculated only on the original amount invested or borrowed. The accumulated amount is

The same amount of interest is added each year, so the growth is linear rather than geometric. Simple interest provides a useful introduction, although most longer-term financial products use compounding.

Compound interest is calculated on the current balance. As a result, earlier interest earns interest too. When compounding takes place several times per year,

where $k$ is the number of compounding periods per year ($\text{year}^{-1}$). Yearly compounding has $k=1$, half-yearly has $k=2$, quarterly has $k=4$ and monthly has $k=12$. The periodic geometric ratio is $1+\frac{r}{k}$.

Make sure the rate and time use the same period. Divide the annual rate by the number of compounding periods per year, then multiply the number of years by that same number. Built-in financial applications on a graphic display calculator or spreadsheet can handle the calculations, but you still have to identify the principal, annual rate, frequency and time correctly. Formula derivations are not required here.

The graph shows why compound interest eventually pulls away from simple interest, even though both start with the same principal and quoted annual rate.

Annual depreciation

Annual depreciation is a repeated percentage reduction in an asset's value at the end of each year. It is modelled by

The geometric ratio is $1-d$, not $d$.

Reducing a value by a fixed percentage does not reduce it by a fixed monetary amount. Since each year's depreciation is based on a smaller balance, the amount lost generally decreases over time. Other exponential-decay models use the same structure.

Inflation and real value

Inflation is a sustained percentage increase in the general price level, reducing the purchasing power of a fixed amount of money. An investment can rise in nominal currency terms yet gain very little purchasing power.

If inflation is compounded annually, the investment's real value in today's money is

When interest is compounded annually, the equivalent annual real growth factor is $\frac{1+r}{1+i}$. Subtracting the two percentage rates is only an approximation; dividing the growth factors gives the exact result.

Loans and financial decisions

Compound growth helps explain why unpaid loan or credit balances may rise quickly. This connects financial mathematics with economics and business management because the quoted rate, compounding frequency, fees and repayment schedule all affect the true cost of lending and borrowing.

These calculations are not ethically neutral. Clear mathematical information can help borrowers compare products, whereas confusing rates or aggressive high-cost lending can hide risk. Views on interest, investment and profit-sharing also vary across cultural and religious traditions, so societies don't all treat financial arrangements in the same way.

Technology has made lengthy financial calculations and comparisons almost immediate. Judgement is still needed. Software simply returns the consequences of the entries supplied, even if a rate, period or cash-flow sign was entered incorrectly.

For enrichment, increasingly frequent compounding leads to the constant $e$ through

This motivates continuous compounding, but continuous-compounding calculations are not assessed in this topic.

Annuities and end-of-period payments

An annuity is a financial arrangement in which equal payments are made at regular intervals. Regular deposits into a retirement account are annuities, as are regular loan repayments, though the cash flows move in opposite directions.

An ordinary annuity has payments at the end of each payment period. That end-of-period convention is used here. A cash-flow timeline helps separate the initial balance at time zero from the first payment, which occurs one full period later.

The present value of equal end-of-period payments can be written as

The formula helps explain the model. However, you aren't required to memorize, derive or apply it manually; the assessed skill is the correct use of technology.

Amortized loans

Amortization is a repayment process in which regular payments gradually reduce a debt through interest payment and principal repayment. In each period, interest is calculated first on the outstanding balance. Whatever remains from the payment reduces the principal.

The balance follows the recurrence

The repeated multiplication by $1+j$ links this recurrence to geometric sequences and exponential models.

At the start of a typical repayment schedule, the balance is large, so more of each payment goes towards interest. Later, the interest charge is smaller and more of the same payment reduces the principal. An amortization table shows how this split changes.

Fixed-payment amortization schedule at 2% per period.

Using a financial package

A graphic display calculator's finance application normally uses these entries:

• $PV$ is the present value or initial balance (currency units).
• $PMT$ is the regular payment (currency units per payment period).
• $FV$ is the value remaining or accumulated after the final payment (currency units).
• $I\%$ is the quoted annual interest rate (percent per year).
• $P/Y$ is the number of payments per year ($\text{year}^{-1}$).
• $C/Y$ is the number of compounding periods per year ($\text{year}^{-1}$).

Enter $N$ as the total number of payments, not just the number of years. Set the payment mode to the end of the period. Then enter the known quantities and solve for the one unknown: the repayment, loan amount, final balance, interest rate or repayment duration.

Financial packages follow a cash-flow sign convention. Money entering an account and money leaving it must have opposite signs, so a borrowed amount and its repayments can't normally be entered with the same sign. Either direction may be chosen as positive, provided the choice is used consistently.

When payment and compounding frequencies match, the periodic rate is $j=\frac{r}{k}$. If the frequencies differ, enter $P/Y$ and $C/Y$ carefully, then let the package perform the required conversion.

Using a spreadsheet

A spreadsheet can construct an annuity or amortization model period by period. For a loan, set up columns for the opening balance, interest charged, payment, principal repaid and closing balance. Each row should refer to the previous row's closing balance. Fixed rate or payment cells need absolute references when the formula is filled down.

This approach is particularly useful for investigating credit-card debt, student loans or alternative repayment plans because every period is displayed, rather than only the final answer. Built-in spreadsheet financial functions can also calculate present values, future values and payments directly.

Interpreting financial results

For a savings annuity or retirement plan, the future value given by the package is nominal. Apply the inflation adjustment from section 1.4 to assess its purchasing power. If the money will be spent in another currency, an exchange rate can then convert the value, but this conversion doesn't remove inflation risk.

For a loan, don't compare only the regular repayment. The total repaid, total interest, term, fees and effect of missed payments also matter. Mathematics can protect people by revealing the long-term cost of short-term high-interest lending. Poor presentation of the same mathematics can make exploitation easier instead.

Societies have different attitudes towards debt, interest and investment, so their financial products may be structured differently. The underlying calculations give a common basis for comparing consequences. On their own, though, they don't determine whether a borrowing or lending arrangement is socially acceptable.