Financial Mathematics

Table of Contents

Interest

  • Interest is essentially the cost of using someone else’s money
  • When you borrow money from a bank or lender, you are charged a percentage of that intial amount extra when it comes time to pay it back
  • In most cases, interest is paid back periodically (over time) rather than as a lump sum (all at once)
  • There are two main kinds of interest: simple and compound

Simple Interest

  • Simple interest is where interest is calculated based on the principle (initial) amount, and the time since the loan started

  • Simple interest $(I)$ is calculated with $\color{orange}I=(P\times R\times n)$

    • $P$ is the principal (initial) sum,
    • $R$ is the rate of interest per unit of time
    • $n$ is the number of time intervals which have passed
  • For example, if you take a $\$ $100 loan at 6% simple interest per year, every year you wait adds $\$ $6 to the amount you have to pay back $\color{orange}(\$100\times 6\%\times 1=\$6,\$100\times 6\%\times 2=\$12\text{, and so on})$

  • If the question asks for the total amount, add $P$ to $I$ at the end

    • Basically, use the formula $\color{orange}I=(P\times R\times n)+P$ instead

Compound Interest

  • Compound interest is where the interest in each period is calculated on the principle, PLUS any interest earned until that point
    Most loans, debts, and repayments are compound interest. If the type of interest isn’t specified, it’s almost definitely compound.
  • The formula for compound interest is $\color{orange}FV=PV(1+r^{n})$
    • $FV$ stands for “future value” or “final value” (same as $I$ in simple interest)
    • $PV$ stands for “present value” or “principal value”
    • $r$ is the rate per period (for example 6% per year)
    • $n$ is the number of periods passed

Increasing Future Value

  • There are 3 main ways to increase the future value of an investment:
    • Increase the principal value (basically, more money = more money)
    • Increase the frequency of the compounding periods (e.g. make $r$ per month rather than per year)
    • Increase the interest rate (7% is more than 6%)

Remember, if you’re the one paying for it (e.g. a loan or debt), you want to do the OPPOSITE of those.

Comparing Investment Strategies (Question Guide)

Usually, questions involving interest tend to involve comparing investment strategies. An example question would be:

Heidi goes to the bank to take out a loan of 1000 dollars over 3 years. The bank offers her two options: the first with a compounding interest rate of 5% per annum (compounding annually), and the second with a compounding rate of 4% per annum (compounding monthly). Which is the better deal for Heidi?

Toggle Solutions

  1. Figure out whether you need to find the smaller or larger value.
    • In the case of a “which is the better deal” question involving a loan, smaller is better.
  2. Calculate the future value of the first option
    • In this case, $FV=1000\times (1+0.05)^{3}=\$1157.63$
  3. Repeat step 2 for each option
    • Option 2: $FV=1000\times(1+\frac{0.04}{12})^{12\times 3}=\$1012.07$
  4. Answer the question.
    • Since Heidi will get a better deal from a lower final sum, option 2 is the better deal for her.

Investment Graphs

Simple Interest

  • Simple interest demonstrates a linear relationship, with the x-axis as $n$ (number of time periods), and the y-axis as $I$ (the interest earned).
  • To draw a simple interest graph:
    1. Construct a table of values for $I$ and $n$ using the simple interest formula.
    2. Draw a number plane with the $n$ horizontal axis and $I$ vertical axis, then plot the points.
    3. Join the points to make a straight line.
Example

Draw a graph of the simple interest earned over a period of 10 years, where the initial amount is $10, and the rate of interest is 8% p.a.

Compound Interest

  • Compound interest demonstrates an exponential curve, with the x-axis as $n$ (number of time periods), and the y-axis as $FV$ (the future value).
  • To draw a compound interest graph:
    1. Construct a table of values for $FV$ and $n$ using the compound interest formula.
    2. Draw a number plane with the $n$ horizontal axis and $FV$ vertical axis, then plot the points.
    3. Join the points to make an exponential curve.

Appreciation and Inflation

  • Appreciation is when an item increases in value.
  • The rate of financial appreciation can often be expressed using the compound interest formula.

$$\color{orange}{FV=PV(1+r)^{n}}$$

  • Inflation is when the value of money goes down. When inflation occurs, the price of goods and services increases.
    • Usually, inflation is between 2% and 3%.
  • This increase in the price of things can also be expressed using the compound interest formula.

Declining-Balance Depreciation

  • Declining-balance depreciation occurs when the value of the item decreases by a fixed percentage each time period.
  • Declining-balance depreciation has a slightly modified version of the compound interest formula:
    • $\color{orange}{S=V_{0}(1-r)^{n}}$
    • $S$ is the final or “salvage” value, $V_0$ is the initial value, $r$ is the rate of depreciation per time period, and $n$ is the number of time periods.

Reducing-Balance Loans

  • Reducing-balance loans are loans where the interest is calculated on the outstanding amount, rather than the total amount.
  • These use more complicated formulae, so you’ll typically be given a two-way table, which you can then use to determine the amount outstanding.
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