Financial Mathematics

The One Thing

Financial mathematics evaluates money across time using interest, discounting, annuities, depreciation, and investment measures. The central principle is that amounts received at different times must be converted to a common date using an appropriate rate and timing convention.

Definitions and Results

• Principal: The original amount of money invested, borrowed, or deposited.
• Interest: Additional money earned on an investment or paid for borrowing money.
• Rate of interest: The percentage of the principal charged or earned during a specified period.
• Simple interest: Interest calculated only on the original principal throughout the investment period.
• Compound interest: Interest calculated on the principal together with interest accumulated in earlier periods.
• Compounding frequency: The number of times interest is added in one year, such as annually, half-yearly, quarterly, or monthly.
• Present value: The value today of a future amount after allowing for the applicable interest rate.
• Future value: The value at a future date of money invested or deposited today.
• Annuity: A sequence of equal payments or receipts made at equal time intervals.
• Ordinary annuity: An annuity in which payments are made at the end of each period.
• Annuity due: An annuity in which payments are made at the beginning of each period.
• Loan amortisation: Gradual repayment of a loan through regular instalments containing both interest and principal.
• EMI: Equated Monthly Instalment, a fixed monthly payment used to repay a loan over a specified period.
• Nominal rate: The stated annual interest rate before considering the effect of compounding during the year.
• Effective annual rate: The actual annual rate earned or paid after considering compounding frequency.
• Depreciation: Decrease in an asset’s value because of use, age, or obsolescence.
• Appreciation: Increase in an asset’s value over time.
• Return on investment: Gain or loss from an investment relative to the amount invested.
• Risk: The possibility that the actual financial outcome differs from the expected outcome.
• Simple-interest result:

$$I=PRT$$ where $P$ is principal, $R$ is the annual rate as a decimal, and $T$ is time in years. The amount is $$A=P+I=P(1+RT).$$

• Compound-amount result:

$$A=P{\left(1+\frac{r}{n}\right)}^{nt},$$ where $r$ is the annual rate in decimal form, $n$ is the number of compounding periods per year, and $t$ is time in years.

• Compound interest:

$$CI=A-P.$$

• Annual compounding:

$$A=P(1+r{)}^t.$$

• Half-yearly compounding: Use rate $r/2$ and $2t$ periods.
• Quarterly compounding: Use rate $r/4$ and $4t$ periods.
• Monthly compounding: Use rate $r/12$ and $12t$ periods.
• Present value:

$$PV=\frac{FV}{(1+r{)}^t},$$ where $FV$ is future value, $r$ is the periodic interest rate, and $t$ is the number of periods.

• Future value:

$$FV=PV(1+r{)}^t.$$

• Effective annual rate: For nominal rate $r$ compounded $n$ times per year,

$$i_{\text{eff}}={\left(1+\frac{r}{n}\right)}^n-1.$$

• Future value of an ordinary annuity: For payment $R$ each period,

$$FV=R\left[\frac{(1+i{)}^n-1}{i}\right],$$ where $i$ is the interest rate per period and $n$ is the number of payments.

• Present value of an ordinary annuity:

$$PV=R\left[\frac{1-(1+i{)}^{-n}}{i}\right].$$

• Future value of an annuity due: Multiply the future value of the corresponding ordinary annuity by $(1+i)$.
• Present value of an annuity due: Multiply the present value of the corresponding ordinary annuity by $(1+i)$.
• Loan instalment or EMI: For a loan of principal $P$, repaid by $n$ equal instalments at periodic rate $i$,

$$E=\frac{Pi(1+i{)}^n}{(1+i{)}^n-1}.$$

• Amortisation schedule: Each payment is divided into interest for the period and repayment of principal. As the outstanding balance decreases, the interest component generally decreases and the principal component increases.
• Outstanding loan balance: After $k$ payments, it can be found using the present value of the remaining instalments or by subtracting the accumulated value of payments already made from the accumulated loan amount.
• Straight-line depreciation:

$$\text{Annual depreciation}=\frac{\text{Cost}-\text{Salvage value}}{\text{Useful life}}.$$

• Straight-line book value: After $t$ years,

$$V=C-\frac{t(C-S)}{n},$$ where $C$ is original cost, $S$ is salvage value, and $n$ is useful life.

• Reducing-balance depreciation:

$$V=C(1-r{)}^t,$$ where $r$ is the depreciation rate per period.

• Percentage return:

$$\text{Return percentage}=\left(\frac{\text{Gain or Loss}}{\text{Initial investment}}\right)\times 100.$$

• Investment result: Profit or loss should consider purchase price, selling price, income received, transaction costs, and the time period.
• Rate and time condition: Rates and time units must match. A monthly rate must be used with the number of months, and an annual rate must be converted before monthly calculations.
• Timing condition: A date or payment timeline is essential because payments at the beginning and end of periods have different present and future values.
• Practical limitations: Taxes, inflation, fees, changing interest rates, and credit risk may affect the result even when the mathematical model is correct.

Worked Methods

1. Calculating simple interest

• Identify the principal $P$, annual rate $R$, and time $T$ in years.
• Express the annual rate as a decimal.
• Apply

$$I=PRT.$$

• Calculate the total amount using

$$A=P+I=P(1+RT).$$

Simple interest remains based only on the original principal.

2. Calculating compound interest

• Identify $P$, the annual rate $r$, the compounding frequency $n$, and time $t$.
• Express $r$ as a decimal.
• Substitute into

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}.$$

• Find compound interest using

$$CI=A-P.$$

For annual compounding, use $A=P(1+r{)}^t$. For half-yearly, quarterly, and monthly compounding, use $r/2$ with $2t$, $r/4$ with $4t$, and $r/12$ with $12t$, respectively. For the same positive rate and period, compound growth is generally greater than simple interest because accumulated interest earns further interest.

3. Converting between present and future value

To find the future value:

• Identify the present value, periodic rate, and number of periods.
• Apply

$$FV=PV(1+r{)}^t.$$

To find the present value:

• Identify the future value, periodic rate, and number of periods.
• Discount the future amount using

$$PV=\frac{FV}{(1+r{)}^t}.$$

This conversion is required before amounts received at different times can be compared directly.

4. Finding the effective annual rate

• Identify the nominal annual rate $r$ and the number of compounding periods $n$.
• Substitute into

$$i_{\text{eff}}={\left(1+\frac{r}{n}\right)}^n-1.$$

• Express the result as a percentage if required.

For a positive nominal rate, more frequent compounding usually produces a higher effective return.

5. Valuing an ordinary annuity

For a sequence of equal end-of-period payments:

• Identify payment $R$, periodic rate $i$, and number of payments $n$.
• For future value, use

$$FV=R\left[\frac{(1+i{)}^n-1}{i}\right].$$

• For present value, use

$$PV=R\left[\frac{1-(1+i{)}^{-n}}{i}\right].$$

6. Valuing an annuity due

For payments made at the beginning of each period:

• First calculate the corresponding ordinary-annuity value.
• Multiply by $(1+i)$.
• Use this adjustment for both future value and present value:

$$F{V}_{\text{due}}=F{V}_{\text{ordinary}}(1+i),$$ $$P{V}_{\text{due}}=P{V}_{\text{ordinary}}(1+i).$$

7. Calculating an EMI and interpreting amortisation

• Identify the loan principal $P$, periodic interest rate $i$, and number of instalments $n$.
• Apply

$$E=\frac{Pi(1+i{)}^n}{(1+i{)}^n-1}.$$

• For each payment, calculate the interest component from the outstanding balance and the periodic rate.
• Subtract that interest component from the payment to obtain the principal repayment.
• Reduce the outstanding balance by the principal repayment.
• Repeat for the amortisation schedule.

Each instalment contains both interest and principal. As the outstanding balance decreases, the interest component generally decreases while the principal component increases. The balance after $k$ payments may also be found from the present value of the remaining instalments or by comparing accumulated loan and payment values.

8. Calculating depreciation

For straight-line depreciation:

• Identify cost, salvage value, and useful life.
• Calculate annual depreciation:

$$\frac{\text{Cost}-\text{Salvage value}}{\text{Useful life}}.$$

• Calculate book value after $t$ years:

$$V=C-\frac{t(C-S)}{n}.$$

For reducing-balance depreciation:

• Identify original cost $C$, depreciation rate $r$, and number of periods $t$.
• Apply

$$V=C(1-r{)}^t.$$

9. Calculating investment return

• Identify the initial investment, purchase price, selling price, income received, transaction costs, and time period.
• Determine the gain or loss.
• Calculate

$$\text{Return percentage}=\left(\frac{\text{Gain or Loss}}{\text{Initial investment}}\right)\times 100.$$

• Interpret the result alongside risk rather than considering return alone.

Where It Goes Wrong

• Using a percentage rate directly in a formula instead of converting it to decimal form.
• Mismatching rates and time units, such as using an annual rate with monthly periods without conversion.
• Treating compound interest as though it were calculated only on the original principal, thereby omitting interest accumulated in earlier periods.
• Forgetting that an ordinary annuity is paid at the end of each period, whereas an annuity due is paid at the beginning; the annuity-due value requires multiplication by $(1+i)$.
• Treating every loan instalment as principal repayment and omitting the interest component in the amortisation schedule.
• Ignoring payment dates, taxes, inflation, fees, changing interest rates, or credit risk when applying an otherwise correct mathematical model.

What Gets Asked

This material supports questions requiring students to:

• Calculate simple interest, total amount, compound amount, and compound interest.
• Adjust compound-interest calculations for annual, half-yearly, quarterly, and monthly compounding.
• Convert between present value and future value.
• Calculate an effective annual rate from a nominal rate and compounding frequency.
• Find the present or future value of an ordinary annuity or annuity due.
• Calculate an EMI and construct or interpret a loan amortisation schedule.
• Find an outstanding loan balance after a stated number of payments.
• Calculate straight-line annual depreciation and book value.
• Calculate reducing-balance depreciation.
• Determine gain, loss, or percentage return on an investment.
• Compare investments by considering both expected return and risk.
• Check whether the stated rate, time unit, payment timing, and practical assumptions are consistent with the selected formula.