Overview
In the previous lesson we investigated simple interest and in particular focused on how to calculate the future value of an investment in a simple interest scenario.
$\text{FV} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)^{\text{Number of periods}} $
In the international money markets there are a few types of financial instruments that pay simple interest. All of those that do, are of very short-term maturity and are covered in the money markets mini-course.
The norm in the money markets however is for investments to pay interest on interest. In other words to capitalise 1Capitalise: To convert into capital interest. At the end of certain pre-agreed periods (monthly, quarterly, annually etc.) interest amounts will be added to the principal amount and subsequent interest calculations will be performed on this aggregate 2Aggregate: Calculated by the combination of several separate elements; total principal
Example
In order to illustrate the concept of compounding, let’s start with an often encountered real scenario:
Suppose that a corporate investor places \$ 1,000,000 on a fixed deposit with an investment bank on 1 January 2024. Furthermore, the bank agrees to an interest rate of 10% and that the interest will be added every month.
A month-by-month calculation of the future value will look something like this:
January
The 10% interest rate is quoted on an annual basis and is exactly the same assumption as in the previous lesson.
The agreement also stipulate that the interest is to be added monthly. We can use the same formula from the previous lesson to calculate what the FV of the investment will be after each month.
Let’s start with the the first interest payment:
$\text{FV}_{\text{Jan}} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)$
$\text{FV}_{\text{Jan}} = \text{1,000,000} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Jan}} =\$ 1,008,333.33 $
This value now becomes the PV or initial investment at the start of the second month (February).
February
For this second month, we now calculate what the future value will be at the end of February. The interest from the previous month has been capitalised, meaning it has been added to the initial investment. Instead of \$ 1,000,000, the PV for this calculation is \$ 1,008,333.33
$\text{FV}_{\text{Feb}} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)$
$\text{FV}_{\text{Feb}} = \text{1,008,333.33} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Feb}} =\$ 1,016,736 $
This is now the starting or PV value for the roll-on into the third month:
March
$\text{FV}_{\text{Mar}} = \text{\$ 1,016,736} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Mar}} = \text{\$ 1,016,736} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Mar}} =\$ 1,025,209 $
A pattern emerges, and we apply it to the rest of the calculations:
April
$\text{FV}_{\text{Apr}} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)$
$\text{FV}_{\text{Apr}} = \text{\$ 1,025,209} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Apr}} = \$ 1,033,752$
May
$\text{FV}_{\text{May}} = \text{\$ 1,033,752} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{May}} = \$ 1,042,367$
June
$\text{FV}_{\text{Jun}} = \text{\$ 1,042,367} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Jun}} = \$ 1,051,053$
July
$\text{FV}_{\text{Jul}} = \text{\$ 1,051,053} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Jul}} =\$ 1,059,812$
August
$\text{FV}_{\text{Aug}} = \text{\$ 1,059,812} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Aug}} =\$ 1,068,644$
September
$\text{FV}_{\text{Sep}} = \text{\$ 1,068,644} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Sep}} =\$ 1,077,549$
October
$\text{FV}_{\text{Oct}} = \text{\$ 1,077,549} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Oct}} =\$ 1,086,529$
November
$\text{FV}_{\text{Nov}} = \text{\$ 1,086,529} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Nov}} =\$ 1,095,583$
December
$\text{FV}_{\text{Dec}} = \text{\$ 1,095,583} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
$\text{FV}_{\text{Dec}} =\$ 1,104,713$
That was exhausting! In order to calculate the compound interest value after 12 months, we needed to perform twelve calculations. One for each of the months!
This method is utterly impractical, nevertheless illustrated an important fact:
If we are to calculate the future value of an investment where the interest is compounded, each period’s FV is the next period’s PV. This is because we are capitalising interest every step of the way. This is also the secret to how we will bring this ridiculously calculation down to manageable size:
Ideally we want to start with the initial investment amount and calculate the future value at the end of all investment periods in a single formula.
Let’s revisit our earlier calculation but this time, start with the last term:
$\text{FV}_{\text{Dec}} = \text{PV}_{\text{Nov}} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
The PV November value in turn was calculated using the formula:
$\text{FV}_{\text{Nov}} = \text{FV}_{\text{Oct}} \times \left( 1 + \text{0.1} \times \frac{\text{1}}{\text{12}} \right)$
Each of the starting value can be traced back to the previous month’s FV value. This fact enables us to rewrite the 12 formulas into a single formula:
$\text{FV}_{\text{Dec}} = \text{PV} \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
$ \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
$ \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
$ \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
$ \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
$ \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) \times \left( 1 + \text{0.1} \times \frac{1}{12} \right) $
The $ \left( 1 + \text{0.1} \times \frac{1}{12} \right) $ term appears continuously and can be restated using an exponent:
$\text{FV}_{\text{Dec}} = \text{PV} \times \left( 1 + \text{0.1} \times \frac{1}{12} \right)^{12} $
That’s neat! Just like that we have a manageable, easy calculation that compounds the interest.
Let’s test it quick with our example scenario:
$\text{FV}_{\text{Dec}} = \text{1,000,000} \times \left( 1 + \text{0.1} \times \frac{1}{12} \right)^{12} $
$\text{FV}_{\text{Dec}} = 1,104,713 $
Ok, let’s now restate the formula in a general form so we never have to do a gazillions calculations when we can just do one:
General Formula
$\text{FV} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)^{\text{Number of periods}} $
Depending on the way the information pertaining to the length of each compounding period is received, the way we calculate the fraction will have to adjusted accordingly. We can think of these compounding periods as steps and it is important that the steps be of exactly the same length. In our example we used months, which means we worked with 12 steps.
It is also crucial at this time to point out that the interest amount is divided by the number of steps (compounding periods) to arrive at interest rate per period.
Another assumption we make is that the interest is compounded at the end of each period, whether the interest has been paid or not.
See you in the next lesson, but first, let’s ensure your calculation of a similar scenario is correct.