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Interests

What Is Interest?


We've all heard of interest rates—on a mortgage, a credit card, or a loan. But what does interest really mean?
Interest is essentially the “rent” you pay for borrowing money. It is the extra amount you pay to use someone else's money for a certain period of time. The percentage used to calculate this extra amount each year is called the interest rate.
Example of interest:
Imagine you borrow \(\dollar 100\) today and promise to pay it back in one year. If you return exactly \(\dollar 100\) after one year, there's no interest. However, the lender might want to be compensated for letting you use their money.
They may ask for a percentage as interest. For example, at a \(10\pourcent\) interest rate per year, the interest paid is:$$\begin{aligned}\text{Interest Paid} &= \text{Percentage of the Original Amount} \\ &= \text{Interest Rate} \times \text{Original Amount} \\ &= 10\pourcent \times 100 \\ &= \frac{10}{100} \times 100 \\ &= 10~\text{dollars}\end{aligned}$$Therefore, after one year, you owe:$$\begin{aligned}\text{Amount at Year 1} &= \text{Original Amount} + \text{Interest Paid} \\ &= 100 + 10 \\ &= 110~\text{dollars}\end{aligned}$$So you would pay back \(\dollar 110\) instead of \(\dollar 100\). The extra \(\dollar 10\) is the interest—the cost of borrowing for a year.

Definition Principal
The principal is the original amount of money that is either invested or loaned.
Definition Interest
Interest is the cost paid for borrowing money or the amount earned from lending or investing money.

Simple Interest


Suppose you borrow \(\dollar 100\) with an interest rate of \(10\pourcent\) per year. With simple interest, the interest is calculated only on the initial amount each year.
  • \(\begin{aligned}[t]\text{Total interest after 1 year} &= 10\pourcent \times 100 \\&= \frac{10}{100} \times 100 \\&= 10~\text{dollars}\end{aligned}\)
  • \(\begin{aligned}[t]\text{Total interest after 2 years} &= 2 \times 10\pourcent \times 100 \\&= 2 \times \frac{10}{100} \times 100 \\&= 20~\text{dollars}\end{aligned}\)
  • \(\begin{aligned}[t]\text{Total interest after 3 years} &= 3 \times 10\pourcent \times 100 \\&= 3 \times \frac{10}{100} \times 100 \\&= 30~\text{dollars}\end{aligned}\)
These observations lead to the simple interest formula:$$\text{Simple Interest} = \text{Number of years} \times \text{Interest rate} \times \text{Principal}$$

Definition Simple Interest
The simple interest is calculated each year as a fixed percentage of the principal (original amount) borrowed or invested. With simple interest, the same amount of interest is added every year.
Proposition Simple Interest Formula
The simple interest, denoted by \(I\), is calculated as:$$I = t \times r \times P$$where:
  • \(P\) is the principal (original amount)
  • \(r\) is the interest rate per year (expressed as a decimal, e.g. \(3\pourcent = 0{.}03\))
  • \(t\) is the time (in years)
The final amount, denoted by \(A\), is:$$A = P + I$$So we can also write:$$A = P(1 + rt)$$
Example
Find the simple interest on a principal of \(\dollar 500\) at a rate of \(3\pourcent\) per year over \(5\) years. Then find the final amount.

Write the rate as a decimal: \(3\pourcent = 0.03\).$$\begin{aligned}[t]I&= t \times r \times P\\ &= 5 \times 0.03 \times 500 \\ &= 5 \times 15 \\ &= 75~\text{dollars}\end{aligned}$$The final amount is:$$\begin{aligned}[t]A &= P + I\\ &= 500 + 75\\ &= 575~\text{dollars}\end{aligned}$$

Compound Interest


If you leave money in the bank for a period of time, the interest earned is automatically added to your account. After the interest is added, it also begins to earn interest in the next time period. This process is called compound interest.
Example of compound interest:
\(\dollar 1\,000\) is placed in an account that earns \(10\pourcent\) interest per annum (p.a.), and the interest is allowed to compound over three years. This means the account is earning \(10\pourcent\) p.a. in compound interest.
We can illustrate this in a table:
Year Amount Interest Earned
0 \(\dollar 1\,000\) \(10\pourcent\) of \(\dollar 1\,000 = \dollar 100\)
1 \(\dollar 1\,000 + \dollar 100 = \dollar 1\,100\) \(10\pourcent\) of \(\dollar 1\,100 = \dollar 110\)
2 \(\dollar 1\,100 + \dollar 110 = \dollar 1\,210\) \(10\pourcent\) of \(\dollar 1\,210 = \dollar 121\)
3 \(\dollar 1\,210 + \dollar 121 = \dollar 1\,331\) ---
After 3 years, there will be a total of \(\dollar 1\,331\) in the account, meaning you have earned \(\dollar 331\) in compound interest.
You can also calculate the final amount using a different method:
  • \(\begin{aligned}[t]\text{Amount after 1 year}&= \text{Initial amount} + \text{Interest on the initial amount} \\&= \text{Initial amount} + \text{Interest Rate} \times \text{Initial amount} \\&= 1\,000 + 0.1 \times 1\,000 \\&= 1\,000 \times (1 + 0.1) \\&= 1\,000 \times 1.1\end{aligned}\)
  • \(\begin{aligned}[t]\text{Amount after 2 years}&= \text{Amount after 1 year} + \text{Interest on the amount after 1 year} \\&= \text{Amount after 1 year} + \text{Interest Rate} \times \text{Amount after 1 year} \\&= 1\,000 \times 1.1 + 0.1 \times 1\,000 \times 1.1 \\&= 1\,000 \times 1.1 \times (1 + 0.1) \\&= 1\,000 \times 1.1^2\end{aligned}\)
  • \(\begin{aligned}[t]\text{Amount after 3 years}&= \text{Amount after 2 years} + \text{Interest on the amount after 2 years} \\&= \text{Amount after 2 years} + \text{Interest Rate} \times \text{Amount after 2 years} \\&= 1\,000 \times 1.1^2 + 0.1 \times 1\,000 \times 1.1^2 \\&= 1\,000 \times 1.1^2 \times (1 + 0.1) \\&= 1\,000 \times 1.1^3\end{aligned}\)
These observations lead to the compound interest formula (for interest compounded once per year):$$\text{Final amount} = \text{Initial amount}\times (1 + \text{interest rate})^{\text{number of years}} ,$$where the interest rate is written as a decimal (for example, \(10\pourcent = 0.10\)).

Definition Compound Interest
Compound interest is interest that accumulates on both the principal (initial amount) and the previously accumulated interest.
Proposition Annual Compound Interest Formula
The final amount \(A\) of an investment with interest compounded annually is:$$A = P (1 + r)^t$$where:
  • \(P\) is the principal (initial amount),
  • \(r\) is the annual interest rate (as a decimal),
  • \(t\) is the time (in years).
Example
Find the final amount for compound interest on a principal of \(\dollar 500\) at a rate of \(3\pourcent\) per year over \(5\) years.

$$\begin{aligned}[t]A &= P (1 + r)^t \\ &= 500 \times (1 + 0.03)^5 \\ &= 500 \times 1.03^5 \\ &\approx 579.64\end{aligned}$$The final amount is approximately \(\dollar579.64\).

Compound Interest per Period


Compound interest can be calculated more frequently than once a year, such as monthly or quarterly. In such cases:
  • the annual interest rate is divided by the number of compounding periods per year;
  • the exponent (time) is multiplied by the number of compounding periods.

Proposition Compound Interest Formula per Period
The final amount \(A\) after \(t\) years with interest compounded \(c_y\) times per year is:$$A = P \left(1 + \frac{r}{c_y}\right)^{c_y t}$$where:
  • \(P\) is the principal,
  • \(r\) is the annual interest rate (as a decimal),
  • \(c_y\) is the number of compounding periods per year,
  • \(t\) is the time in years.
The total number of compounding periods is \(n = c_y t\).
Example
Calculate the final amount on a principal of \(\dollar 1\,000\) at an annual rate of \(5\pourcent\) compounded quarterly over \(2\) years.

$$\begin{aligned}[t]A &= 1\,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 2} \\ &= 1\,000 \left(1 + 0.0125\right)^{8} \\ &= 1\,000 \times (1.0125)^8 \\ &\approx \dollar 1\,104.49\end{aligned}$$The final amount is approximately \(\dollar 1\,104.49\).

Using the Time Value of Money Solver

Method Finding Any Value Using a TVM Solver
The TVM Solver (Time Value of Money) can be used to find any unknown variable if all the other variables are given.
  • \(PV\) is the principal (present value), considered as an outgoing cash flow, and is entered as a negative value (\(PV = -P\)).
  • \(FV\) represents the final amount (\(FV = A\)).
  • \(I\pourcent\) is the annual interest rate \(r\) expressed as a percentage (e.g. \(5\pourcent\)).
  • \(C/Y\) is the number of compounding periods per year (\(C/Y = c_y\)).
  • \(n\) is the total number of compounding periods, not the number of years (\(n = c_y \times t\)).
  • \(PMT\) and \(P/Y\) are not used in this case. We set \(PMT = 0\) and \(P/Y = C/Y\).
Example
Find the final amount on a principal of \(\dollar 23\,000\) at a rate of \(3.45\pourcent\) over \(6\) years compounded quarterly.

Using the TVM Solver with:$$n = 6 \times 4 = 24,\quad I\pourcent = 3.45,\quad PV = -23\,000,\quad C/Y = 4,\quad (PMT = 0,\ P/Y = 4)$$we find the final amount:$$FV \approx \dollar 28\,264.50$$

Inflation


Inflation is the rate at which the general level of prices for goods and services is rising, and, as a result, purchasing power is falling. Central banks attempt to limit inflation — and avoid deflation — in order to keep the economy running smoothly.

Definition Inflation Rate
The inflation rate is the percentage increase in the general price level from one period to another, typically measured annually.
The future price of a good or service, accounting for inflation, can be calculated using a formula similar to that of compound interest. This reflects how prices grow over time when subjected to a constant inflation rate.
Proposition Effect of Inflation on Prices
The price of goods increases over time due to inflation:$$FV = P \left(1 + r\right)^t$$where:
  • \(P\) is the present price,
  • \(r\) is the inflation rate per period (often per year),
  • \(t\) is the time in periods (typically years).
Example
If the inflation rate is \(2\pourcent\) per year, what will be the price of a \(\dollar 100\) item after \(5\) years?

$$\begin{aligned}[t]FV &= 100 \left(1 + 0.02\right)^5 \\ &= 100 \times (1.02)^5 \\ &\approx \dollar 110.41\end{aligned}$$The price will be approximately \(\dollar 110.41\) after \(5\) years.

Definition Real Value of an Investment
The real value of an investment is its final (nominal) amount adjusted for inflation to express it in terms of today's purchasing power. In other words, it tells us what the future sum is truly worth in present-day terms.
To find the real value, we reverse the inflation formula: instead of projecting a present value into the future, we discount a future value back to present-day purchasing power.
Proposition Formula for Real Value of an Investment
The real value \(RV\) of a final amount \(A\) after \(t\) years at an inflation rate \(r\) per year is:$$RV = \frac{A}{(1 + r)^t}$$
Example
An investment grows to \(\dollar 10\,000\) after \(5\) years. If the inflation rate is \(2\pourcent\) per year, find the real value of the investment in today's dollars.

$$\begin{aligned}[t]RV &= \frac{10\,000}{(1 + 0.02)^5} \\ &= \frac{10\,000}{(1.02)^5} \\ &\approx \dollar 9\,057.31\end{aligned}$$The real value is approximately \(\dollar 9\,057\).

Loans


Imagine you take out a loan of \(\dollar 10\,000\) with an annual interest rate of \(5\pourcent\). You agree to make a regular repayment of \(\dollar 1\,000\) at the end of each year.
Interest is calculated on the amount you still owe (the balance) before your payment reduces it.
Let's track the balance of the loan:
  • After One Year:
    • Interest charged: \(5\pourcent \text{ of } 10\,000 = 0.05 \times 10\,000 = \dollar 500\).
    • Balance before payment: \(10\,000 + 500 = \dollar 10\,500\).
    • Balance after payment: \(10\,500 - 1\,000 = \dollar 9\,500\).
  • After Two Years:
    • Interest charged (on the new balance): \(5\pourcent \text{ of } 9\,500 = 0.05 \times 9\,500 = \dollar 475\).
    • Balance before payment: \(9\,500 + 475 = \dollar 9\,975\).
    • Balance after payment: \(9\,975 - 1\,000 = \dollar 8\,975\).
We can summarize this in a table:
Year Start Balance Interest (\(5\pourcent\)) Repayment End Balance
1 \(\dollar 10\,000\) \(\dollar 500\) \(\dollar 1\,000\) \(\dollar 9\,500\)
2 \(\dollar 9\,500\) \(\dollar 475\) \(\dollar 1\,000\) \(\dollar 8\,975\)
Notice that the interest amount decreases each year because the balance is getting smaller. This means more of your \(\dollar 1\,000\) payment goes towards paying off the actual loan principal in the second year compared to the first.

Definition Loan
A common way to finance large purchases like houses or cars is through a loan. The financial institution lends money to the borrower, who repays the amount plus interest through regular payments over a set period. This process is called amortisation.
Interest is calculated on the reducing balance of the loan, meaning you pay interest only on the amount you still owe.$$\text{Interest Paid} = \text{Total Repayments} - \text{Principal}$$
Method Calculating Loan Repayments with a TVM Solver
We can use the TVM Solver to calculate the regular payment amount (\(PMT\)) required to pay off a loan.
  • \(N\): Total number of payments (\(N = \text{number of years} \times \text{payments per year}\)).
  • \(I\pourcent\): Annual interest rate as a percentage.
  • \(PV\): Present Value. This is the loan amount. Since you receive this money, it is positive.
  • \(PMT\): Payment amount. Since you pay this to the bank, it is negative.
  • \(FV\): Future Value. To pay off the loan completely, this should be \(\mathbf{0}\).
  • \(P/Y\): Payments per Year (e.g., 12 for monthly).
  • \(C/Y\): Compounding Periods per Year (usually the same as \(P/Y\)).
Example

Erica takes a personal loan of \(\dollar 16\,500\) to buy a car. The loan is for 4 years at \(5.5\pourcent\) p.a. compounded monthly.
  1. Calculate her monthly repayment.
  2. Calculate the total amount she repays over the 4 years.
  3. Calculate the total interest she pays.

  1. Using the TVM Solver:
    • \(N = 4 \times 12 = 48\)
    • \(I\pourcent = 5.5\)
    • \(PV = 16500\) (positive, money received)
    • \(FV = 0\) (loan paid off)
    • \(P/Y = 12\), \(C/Y = 12\)
    Solving for \(PMT\), we get \(PMT \approx -383.74\). So, the monthly repayment is \(\dollar 383.74\).
  2. Total Repayment = Monthly Payment \(\times\) Total Months $$ 383.74 \times 48 = \dollar 18\,419.52 $$
  3. Total Interest = Total Repayment - Principal $$ 18\,419.52 - 16\,500 = \dollar 1\,919.52 $$

Annuities


Imagine you have saved \(\dollar 300\,000\) for retirement. You put this money into an account that earns \(5\pourcent\) interest compounded annually. You plan to withdraw \(\dollar 25\,000\) at the end of each year to live on.
Let's see what happens to your savings:
  • After One Year:
    • Interest earned: \(5\pourcent \text{ of } 300\,000 = 0.05 \times 300\,000 = \dollar 15\,000\).
    • Balance before withdrawal: \(300\,000 + 15\,000 = \dollar 315\,000\).
    • Balance after withdrawal: \(315\,000 - 25\,000 = \dollar 290\,000\).
  • After Two Years:
    • Interest earned (on the new balance): \(5\pourcent \text{ of } 290\,000 = 0.05 \times 290\,000 = \dollar 14\,500\).
    • Balance before withdrawal: \(290\,000 + 14\,500 = \dollar 304\,500\).
    • Balance after withdrawal: \(304\,500 - 25\,000 = \dollar 279\,500\).
We can summarize this in a table:
Year Start Balance Interest (\(5\pourcent\)) Withdrawal End Balance
1 \(\dollar 300\,000\) \(\dollar 15\,000\) \(\dollar 25\,000\) \(\dollar 290\,000\)
2 \(\dollar 290\,000\) \(\dollar 14\,500\) \(\dollar 25\,000\) \(\dollar 279\,500\)
Because you are withdrawing more (\(\dollar 25\,000\)) than you are earning in interest (\(\dollar 15\,000\)), the balance decreases. Eventually, the money will run out. This is how an annuity works.

Definition Annuity
An annuity is essentially the reverse of a loan. Instead of borrowing a lump sum and paying it back, an individual deposits a lump sum into an account and then makes regular withdrawals over a fixed period. The remaining balance in the account continues to earn interest.
This is commonly used for retirement funds (superannuation), where a person saves during their working life and then "rolls over" that lump sum into an annuity to provide a regular income.
Method Calculating Annuities with a TVM Solver
When performing annuity calculations:
  • \(PV\): The lump sum initially deposited. Since you are putting this money into the account (away from you), it is entered as negative.
  • \(PMT\): The regular withdrawal amount. Since you receive this money, it is positive.
  • \(FV\): The amount remaining in the account at the end. Usually, we calculate how long the money lasts until it runs out, so \(FV = 0\).
Example

Heather retires at age 65 with \(\dollar 900\,000\) in her savings fund. She rolls this money into an annuity fund earning \(4\pourcent\) p.a. compounded monthly. She wants to withdraw \(\dollar 5\,400\) per month to live on.
How long will her money last? Give your answer in years and months.

Using the TVM Solver:
  • \(I\pourcent = 4\)
  • \(PV = -900000\) (money invested)
  • \(PMT = 5400\) (monthly income)
  • \(FV = 0\) (money runs out)
  • \(P/Y = 12\), \(C/Y = 12\)
Solving for \(N\), we get \(N \approx 243.7\).This represents approximately 243 full withdrawals.To convert to years and months:$$ 243 \div 12 = 20 \text{ years with a remainder of } 3 \text{ months} $$So, the money will last for 20 years and 3 months.