Compounding and future value

Compounding and future value
Compound interest is where interest paid on an investment during the first period is added to the principal (the initial amount invested), and
the interest during the second period is earned on the original deposit plus the interest earned during the first period. Therefore, money
invested at compound interest accumulates at an increasing rate each period, exhibiting ‘exponential behaviour’. In contrast, in transactions
involving simple interest, the interest earned is based solely on the original deposit.
The following example illustrates the difference between calculating simple and compound interest.
An investment of $1000 that earns simple interest of 10% per annum, will pay 10% x $1000 = $100 interest per year, every year. After 20
years, the total future value of the investment will be $1000 (the original investment) + $2000 (20 years of $100 simple interest) = $3000 total.
A $1000 investment that earns compound interest of 10% per annum will pay $100 interest in the first year, which is then added to the
principal. That means in the second year the interest will be calculated on the new principal of $1100. So, 10% x $1100 = $110. Again, this is
added to the principal making it $1210, which in the third year will generate $121 of interest.
Now, you can see that the value of the interest changes every year, and it would be onerous to calculate the final value after 20 years by
calculating every single interest payment and updating the principal after each year (i.e. 1000 + 100 + 110 + 121 … ). To make the calculation
faster, we can use the following formula for calculating the future value (formula 5-1A on page 142 of the textbook):
In this formula, represents the nominal annual interest rate in decimal form (0.1 = 10%, 0.08% = 8% etc), represents the number of years
that the interest rate is compounded, and represents the value of the initial deposit or principal. We are solving for which is the
future value after years have passed.
To solve our above example, we would put our variables into the equation like so:
— Note: that 10% interest is equal to 0.1 when expressed as a decimal number.
— Solve the equation in the brackets first.
— Then solve the power (usually the key or ‘^’ symbol on a calculator).
When you compare simple vs compounding interest, you can see the power of compounding. The future value of the deposit that earned
compound interest is more than double that of the deposit only earning simple interest.
If you look at the step-by-step solution you will notice that at the last step, we multiplied (in this case $1000) by 6.7275. Now, if we
wanted to discover the future value of a different initial deposit invested for 20 years at 10%, we would still multiply it by 6.7275 as that side
of the equation will not change. In other words, any value invested for 20 years at 10% will grow in value around 6.7 times. This number is
called the future value interest factor and can be found for any value of interest for years using the second half of the future value
formula: .
Accounting for different compounding periods
What about financial instruments that have shorter compounding periods, such as monthly or quarterly? The more compounding periods
there are, the greater the exponential effect of compounding on the future value, which means we will need to adjust the formula. But first,
what are nominal rates?
Most of the time, especially in study problems, you will be provided with nominal interest rates for calculating the value of financial
instruments. A nominal interest rate is the sum of interest rates applied for each compounding period in a year. So, a nominal rate of 12%
compounded annually means the interest is calculated once and uses a value of 12%. A nominal rate of 12% compounded quarterly
means that the interest is calculated four times in a year at 3%.
Essentially, the nominal annual rate is being ‘parcelled out’ equally across each compounding period in a year.
Let’s look back at our example of $1000 invested at 10% for 20 years and imagine that the interest is calculated quarterly. We need new
values for the formula that account for more compounding periods at a fraction of the interest rate. To find the interest rate per quarter, we
divide the nominal rate by the number of periods it is parcelled out across per year .
We are also compounding 4 times per year, not once, so the total number of periods is now the number of years multiplied by the number
of compounding periods per year .
= × (1 + )

20 = 1000 × (1 + 0.1)
20
20 = 1000 × (1.1)
20
20 = 1000 × 6.72750

20 = $6,727.50

(1 + )

12% nominal annual rate = 3% 1st quarter interest + 3% 2nd quarter interest + 3% 3rd quarter interest + 3% 4th quarter interest

= = 2.5% per quarter

10% nominal rate
4 periods per year

× = 20 years × 4 periods per year = 80 compounding periods4/22/2021 Module 4: Time value of money
https://learning.aib.edu.au/mod/book/tool/print/index.php?id=96080 7/23
If we take these new values and plug them into the formula, we get:
Compared our previous answer of $6727.50 with the same nominal rate of 10% compounded annually, you can see the positive effect of
more frequent compounding periods on the future value, despite the more numerous smaller periods attracting a smaller interest rate.
Knowing this, is it any surprise that most credit cards compound interest daily?
Common values of include:
1 = compounding annually
4 = compounding quarterly
12 = compounding monthly
52 = compounded weekly etc.
We can also now update our original formula to include flexibility to deal with compounding periods other than annually:
Other applications
As the textbook indicates, the Future Value formula doesn’t only apply to compound interest problems but can be adapted for other
situations involving constant compounding growth. For example, if you know your current annual turnover is 23,000 units sold, and it has
been growing at a steady 5% per year, then you can use the formula to determine your projected sales in 10 years’ time.
Solving FV problems with Excel
If you are using Excel formulas to solve for FV, note that the variable names are different. The Excel formula looks like this: FV(rate, nper, pmt,
[pv], [type]), where:
rate = interest rate (as a decimal) per period = (or if compounding more frequently than annually)
nper = number of periods = (or if compounding more frequently than annually)
pmt = = regular payments, we will see this a bit later in the module when looking at annuities.
[pv] = = present value of original deposit. The square brackets mean it is an optional variable and will default to ‘0’ if left blank.
[type] is another variable we will see later in the module. It is a binary variable and only accepts 0 or 1 as its value. It is used in Excel to
differentiate between payments that are made at the beginning of the period (e.g. rent payments) and payments that are made at the end of
the period. If left blank, this variable defaults to ‘0’ and calculates for payments at the end of each period. You will need to set this variable to
‘1’ to solve problems where payments are made at the start of the period, also known as annuity due problems.
Tip: Spreadsheets on the go
If you choose to use the spreadsheet method of solving these problems, but sometimes find yourself with a computer, you can install the free
Google Sheets app on your smartphone (provided you have a Google account).
The technique and formula for solving financial problems in Sheets are the same as in Excel. The following link will create a blank Google
Sheets spreadsheet for you to test on a PC (note: you’ll need the app for mobile use): https://docs.google.com/spreadsheets/u/0/create?
usp=sheets_web
Watch
The LinkedIn Learning Compound interest and investing video explains compound inte