Interest rate percentages
When a lender deposit funds at a bank, the lender may expect to earn a return 1Return: a profit from an investment on the investment. For this reason we’ll use the terms lender and investor interchangeably.
The returns (or interest) paid by the bank will be directly proportional 2Proportional: having a constant ratio to another quantity to a pre-agreed rate of interest. Furthermore, the interest rate will be expressed as an annual percentage 3Annual percentage: the interest rate for a whole year.
Example
Suppose an investor transfers \$100 into a fixed deposit account on 1 January 2023 at 10% annual interest. In addition, the investor agrees to hold the investment until 31 December 2023.
These dates inform the period of the investment, in other words 12 months or exactly one year.
What will this investment yield in one year?
Without much calculation, we can quite simply infer that the interest from this investment will amount to $10 in one year’s time. In other words, 10% of the initial investment.
We may formalise the calculation with this formula:
$ \text{Interest amount} = \text{Investment} \times \text{interest rate} $
The length of time of an investment
Interest amounts earned from an investment (or borrowing) has to be adjusted to reflect the period 4period: a length or portion of time of an investment.
Example
\$100 is invested at a interest rate of 10% from 1 January to 29 February.
What interest amount will this investment yield?
The interest rate of 10% is quoted as an annual percentage or : 10% for 365 days. The number of days that the funds are invested are however not for a full year and needs to be apportioned accordingly.
This apportionment can be achieved by calculating the following fraction:
$ \frac{\text{Number of actual days in investment period}}{\text{Actual number of days in year}} $
In words, to calculate the fraction, we divide the number of days of the investment in relation to the days in the full one year period. The number of days in the investment period is 31 days for January plus 29 days in February, i.e. a total of 60 days (31+29) and the fraction becomes:
$\frac{60}{365} = 0.1644$
Continuing from our previous formula:
$ \text{Interest amount} = \text{Investment} \times \text{interest rate} $
\$100 invested at 10% gives \$10 per year (but the period must be 0.1644 of a year):
$ \text{Interest actually earned} = \text{Annual interest} \times \text{‘Fraction of year’} $
$ \text{Interest actually earned} = \text{10} \times 0.1644 = \text{1.644} $
This interest amount calculation required two steps:
- Calculate the annual interest
- Apportion the interest according to period of investment
In addition to the number investment days divided by number of days in the year apportionment, we may also do this according to months.
It is simple enough to perform this as a single calculation:
$ \text{Interest} = \text{Investment} \times \text{interest rate} \times \frac{\text{Number of actual days in investment period}}{\text{Actual number of days in year}} $
$ \text{Interest} = \text{100} \times \text{10%} \times \frac{60}{365} $
$ \text{Interest} = \text{100} \times \text{0.1} \times \text{0.1644} $
$ \text{Interest Amount} = \text{1.644} $
Calculating the investment period
In these examples we are counting the actual 5Actual: Existing in fact number of days in the investment and interest rate periods. This is one of several conventions 6Convention: a way in which something is usually done.
In the money markets a number of conventions exist to establish how to calculate the number of days in a year or investment period. These conventions apply to specific financial instruments encountered in the markets and are covered in the money market course.
All examples in this course adheres to the market convention: $ \frac{act}{act} $ which simply means we are concerned with the actual number of days in the investment period and the actual number of days in the investment period.
Simple interest
The interest calculations we have done thus far are known as simple interest calculations. This means that there won’t be interest on interest already earned i.e. interest that is capitalised 7Capitalise: Adding interest income to the capital or sum invested
This classification may seem a bit mysterious right now but will be clear after we consider the case where we do capitalise the interest and perform compound interest calculations.
Simple Interest Future Value Calculations
We conclude this lesson by calculating the total value of an investment at the end date. This total value at the end of the investment period is referred to as the maturity value and also as the future value.
For the purposes of our discussion ‘future value’ will be preferred to emphasise that this is the amount expected to be received in the future. This future value will consist of the principal 8Principal: Initial investment amount plus interest. In other words:
$ \text{Future Value} = \text{Present Value} \times \left(1 + \text{interest rate} \times \frac{\text{Number of actual days in investment period}}{\text{Actual number of days in year}} \right) $
This formula can be abbreviated as follows:
$ \text{FV} = \text{PV} \times \left(1 + \text{i} \times \frac{\text{Day count}}{\text{Annual basis}} \right) $
N.B. For practical purposes, the present value is an interchangeable term for the value of the initial investment on day 0.
Day count = The number of days from start to end of investment or borrowing period
Annual basis = The number of days in a year assumed to be 365 in this introductory course
Example 1
Company ABC invests $100 for 200 days at 5% simple interest per annum. What will the future value of this investment be?
$ \text{FV} = \text{PV} \times \left(1 + \text{i} \times \frac{\text{Day count}}{\text{Annual basis}} \right) $
This is what the respective input values will be:
$ \text{PV} = 100 $
$ \text{Day count} = 200 $
$ \text{i} = 5\% $
Plugging these values into the simple interest FV formula:
$ \text{FV} = \text{100} \times \left(1 + \text{5%} \times \frac{\text{200}}{\text{365}} \right) $
$ \text{FV} =102.74 $
Example 2
What if company ABC invested their $100,000 for three years at 5.5% simple interest per annum?
In this scenario, the period of investment is not a fraction of a year but rather, exactly 3 years and we can make a slight adjustment to our formulae to account for this:
$ \text{FV} = \text{100,000} \times \left(1 + \text{5.5%} \times \text{3} \right) $
$ \text{FV} = \text{116,500} $
Test:
$ \text{FV} = \text{PV} \times \left(1 + \text{i} \times \frac{\text{Day count}}{\text{Annual basis}} \right) $