Financial Mathematics

The One Thing

Financial mathematics evaluates saving, borrowing, investing, and related decisions by applying percentages, equations, ratios, and sequences to the time value of money. Accurate calculations require the correct formula, matching rate and time units, and attention to compounding, fees, taxes, repayment periods, and risk.

Definitions and Results

• Principal: The original amount of money invested, deposited, or borrowed.
• Rate of Interest: The percentage charged on a loan or earned on an investment during a specified period.
• Simple Interest: Interest calculated only on the original principal throughout the entire period.
• Compound Interest: Interest calculated on the principal and the interest accumulated during earlier periods.
• Amount: The total value obtained by adding interest to the principal.
• Compounding Frequency: The number of times interest is added to an account in one year, such as annually, half-yearly, quarterly, or monthly.
• Present Value: The current value of a sum of money that will be received or paid in the future.
• Future Value: The value to which a present amount will grow after earning interest for a specified time.
• Annuity: A series of equal payments made or received at regular intervals.
• Loan Amortisation: The gradual repayment of a loan through regular instalments containing both interest and principal components.
• Equated Monthly Instalment: A fixed monthly payment made by a borrower to repay a loan over a specified period.
• Depreciation: The decrease in the value of an asset over time because of use, age, or obsolescence.
• Appreciation: The increase in the value of an asset over time.
• Inflation: A general rise in prices that reduces the purchasing power of money.
• Effective Rate of Interest: The actual annual rate earned or paid after considering the effect of compounding.
• Nominal Rate: The stated annual interest rate that does not itself include the effect of intra-year compounding.
• Discount: A reduction from the marked or listed price of a product or service.
• Tax: A compulsory payment made to the government, usually calculated as a percentage of a taxable amount.
• Profit and Loss: Profit occurs when selling price exceeds cost price, while loss occurs when selling price is below cost price.
• Simple interest:

$$SI=\frac{P\times R\times T}{100}$$ where $P$ is the principal, $R$ is the annual rate in percent, and $T$ is time in years. The amount is $$A=P+SI.$$

• Compound interest compounded annually:

$$A=P{\left(1+\frac{R}{100}\right)}^n,$$ where $n$ is the number of years, and $$CI=A-P.$$

• Compound interest compounded $k$ times per year:

$$A=P{\left(1+\frac{r}{k}\right)}^{kn},$$ where $r$ is the annual rate written as a decimal and $n$ is time in years.

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

$$\text{Effective rate}={\left(1+\frac{r}{k}\right)}^k-1.$$

• Present and future value:

$$FV=PV(1+i{)}^n,PV=\frac{FV}{(1+i{)}^n},$$ where $i$ is the interest rate per period.

• Ordinary annuity: For payment $C$ made at the end of each period,

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

• Loan repayment: For a loan of principal $P$, periodic rate $i$, and $n$ instalments,

$$EMI=\frac{Pi(1+i{)}^n}{(1+i{)}^n-1}.$$ The interest portion is calculated on the outstanding loan balance, and the principal portion equals EMI minus that interest.

• Profit and loss:

$$\text{Profit}=\text{Selling Price}-\text{Cost Price},$$ $$\text{Loss}=\text{Cost Price}-\text{Selling Price}.$$ Profit and loss percentages are calculated using cost price as the base: $$\text{Profit percentage}=\frac{\text{Profit}}{\text{Cost Price}}\times 100,$$ $$\text{Loss percentage}=\frac{\text{Loss}}{\text{Cost Price}}\times 100.$$

• Discount:

$$\text{Discount}=\text{Marked Price}-\text{Selling Price},$$ $$\text{Discount percentage}=\frac{\text{Discount}}{\text{Marked Price}}\times 100.$$

• Depreciation: At a fixed annual percentage $r$,

$$\text{Value after }n\text{ years}=P{\left(1-\frac{r}{100}\right)}^n.$$

• Appreciation: At a fixed annual percentage $r$,

$$\text{Future value}=P{\left(1+\frac{r}{100}\right)}^n.$$

• Time conversion: For annual interest calculations, $T=\text{number of months}/12$. When converting days into years, the convention stated in the question must be followed.
• Rate and time units: A monthly rate must be used with the number of months, and an annual rate with the number of years.
• Inflation and real return: When rates are small, real return is approximately related to nominal return minus inflation.
• Financial comparison: Money received today is generally more valuable than the same amount received later because it can be invested and earn returns. A sound comparison must also consider compounding frequency, fees, taxes, repayment period, and risk.
• Model assumptions: Financial models commonly assume constant rates, regular payments, and timely repayment; actual outcomes may differ.

Worked Methods

Calculating simple interest and amount

• Identify the principal $P$, annual rate $R$, and time $T$ in years.
• If the time is given in months, convert it using

$$T=\frac{\text{number of months}}{12}.$$

• Substitute into

$$SI=\frac{P\times R\times T}{100}.$$

• Calculate the amount using

$$A=P+SI.$$

Simple interest remains based only on the original principal throughout the entire period.

Calculating compound interest annually

• Identify $P$, the annual percentage rate $R$, and the number of years $n$.
• Calculate the amount:

$$A=P{\left(1+\frac{R}{100}\right)}^n.$$

• Subtract the principal to obtain compound interest:

$$CI=A-P.$$

Compound interest grows faster than simple interest over long periods because interest from each period can itself earn interest.

Calculating compound interest with intra-year compounding

• Identify the annual rate $r$ as a decimal, the compounding frequency $k$, the principal $P$, and time $n$ in years.
• Use

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

• For half-yearly compounding, use the rate $R/2$ and $2n$ periods.
• For quarterly compounding, use the rate $R/4$ and $4n$ periods.
• Subtract the principal if compound interest rather than amount is required:

$$CI=A-P.$$

A higher compounding frequency usually produces a higher final amount for the same stated annual rate and time.

Finding the effective annual rate

• Express the nominal annual rate $r$ as a decimal.
• Identify the number $k$ of compounding periods per year.
• Apply

$${\left(1+\frac{r}{k}\right)}^k-1.$$

• Convert the result to a percentage if required.

This gives the actual annual rate after accounting for intra-year compounding.

Converting between present value and future value

To calculate future value:

• Identify the present value $PV$, periodic interest rate $i$, and number of periods $n$.
• Use

$$FV=PV(1+i{)}^n.$$

To calculate present value:

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

$$PV=\frac{FV}{(1+i{)}^n}.$$

Money received today can be invested and therefore generally has greater value than the same nominal amount received later.

Calculating the future value of an ordinary annuity

• Identify the equal periodic payment $C$, periodic interest rate $i$, and number of payments $n$.
• Confirm that payments are made at the end of each period.
• Use

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

Calculating the present value of an ordinary annuity

• Identify $C$, $i$, and $n$.
• Confirm that the payments occur at the end of each period.
• Use

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

Calculating an EMI and separating its components

• Identify the loan principal $P$, periodic interest rate $i$, and number of instalments $n$.
• Calculate the fixed periodic payment:

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

• For each instalment, calculate interest on the outstanding loan balance.
• Calculate the principal portion:

$$\text{Principal portion}=EMI-\text{Interest portion}.$$

• Reduce the outstanding balance by the principal portion before calculating the next instalment’s interest.

Loan amortisation therefore involves both interest and principal repayment. Reducing the outstanding principal generally reduces future interest charges.

Calculating profit, loss, and discount

• Identify the cost price, selling price, and, where relevant, marked price.
• If selling price exceeds cost price, calculate:

$$\text{Profit}=\text{Selling Price}-\text{Cost Price}.$$

• If selling price is below cost price, calculate:

$$\text{Loss}=\text{Cost Price}-\text{Selling Price}.$$

• Use cost price as the denominator for profit and loss percentages.
• For a discount, calculate:

$$\text{Discount}=\text{Marked Price}-\text{Selling Price}.$$

• Use marked price as the denominator for discount percentage.

Calculating depreciation and appreciation

For depreciation:

• Identify the initial value $P$, fixed annual rate $r$, and number of years $n$.
• Use

$$\text{Value after }n\text{ years}=P{\left(1-\frac{r}{100}\right)}^n.$$

For appreciation:

• Identify the initial value $P$, fixed annual rate $r$, and number of years $n$.
• Use

$$\text{Future value}=P{\left(1+\frac{r}{100}\right)}^n.$$

In both cases, the repeated percentage change is represented by a geometric sequence.

Managing units and rounding

• Match the interest rate to the period of calculation.
• Convert months into years for annual-interest formulas using months divided by $12$.
• Convert days into years according to the convention specified in the question.
• Keep full precision during intermediate calculations.
• Round generally at the final step, particularly in money-related problems.

Where It Goes Wrong

• Applying simple interest to accumulated interest instead of only to the original principal.
• Using the annual rate with a number of months, or using a monthly rate with a number of years, without converting the units.
• Treating a nominal rate as the effective annual rate without accounting for compounding frequency.
• Using cost price as the base for discount percentage or marked price as the base for profit percentage; the correct bases are cost price for profit and loss, and marked price for discount.
• Calculating every EMI interest portion from the original loan instead of from the outstanding loan balance.
• Rounding intermediate values, which can produce avoidable errors in final money amounts.

What Gets Asked

• Calculate simple interest and the total amount for a given principal, rate, and time.
• Calculate compound interest compounded annually, half-yearly, quarterly, or at another stated frequency.
• Find the amount, compound interest, or effective annual rate from a nominal rate and compounding frequency.
• Convert between present value and future value using a stated periodic interest rate.
• Calculate the future or present value of an ordinary annuity with equal end-of-period payments.
• Determine an EMI and separate each instalment into interest and principal components.
• Calculate profit, loss, profit percentage, loss percentage, discount, and discount percentage.
• Calculate the depreciated or appreciated value of an asset after a stated number of years.
• Convert months or days into the appropriate time unit and identify the consequences of mismatched units.
• Compare financial choices while considering interest rate, compounding frequency, fees, taxes, repayment period, risk, inflation, and the assumptions of constant rates, regular payments, and timely repayment.