In the previous lesson, we constructed the following formula to calculate the future value of an investment where compounding occurred every month:
$\text{FV} = \text{PV} \times \left( 1 + \text{i} \times \frac{\text{Months in period}}{\text{Months in year}} \right)^{\text{Number of periods}} $
This formula was derived from an application of the simple interest formula and we need to refine and generalise it so that we can employ it across multiple scenarios.
Refinement:
Our interest rate had to be apportioned and because each investment period was for one month, we multiplied it with 1/12. Instead of multiplying by 1/12 we will rewrite the formula as i/12:
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{12}} \right)^{\text{Number of periods}} $
Generalisation 1
Let’s generalise the formula by introducing a general term for the number of compounding periods per year. We’ll call it m:
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{Number of periods}} $
This first generalisation reflects the fact that the apportionment of the interest rate to the number of periods in a year means we have calculated the interest rate per period.
Generalisation 2
What happens if we invest funds for periods longer than one year?
The number of investment periods will be equal to m if the investment is for one year, by contrast if it was for 2 years, the number of investment periods would be 24 (12×2)
Let’s introduce the variable n to refer to the number of years and include it in the formula to allow for scaling across years:
General Formula
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}} $
A word of caution: The Number of periods variable has been replaced with $\text{m} \times \text{n}$. This isn’t exactly the same as Number of investment periods but provides a shortcut calculation for multiple year, multiple compounding period investments.
Market Quotes
It turns out that there are a number of acronyms used by market participants that indicate the number of compounding periods per year to investors:
| NACA | Annual compounding | m = 1 |
| NACS | Semi-annual compounding | m = 2 |
| NACQ | Quarterly compounding | m = 4 |
| NACM | Monthly | m = 12 |
All these acronyms start with NAC which means nominal annual compounding. By appending A for annual, S for semi-annual, Q for quarterly or M for monthly to this acronym, a new acronym is formed that informs investors how often compounding will occur.
Before proceeding any further let’s address the meaning of the NAC lettering:
Nominal Annual Compounding
Along with the compounding frequency a market participant will also quote an interest rate 1(i representing the annual interest rate in our formula)
This rate is typically referred to as a nominal rate because it is the rate used in our calculations but does not represent the effective rate that an investor will earn on their investment. The effective rate will depend on the number of compounding frequencies and is calculated by dividing the final future value inclusive of interest by the initial investment.
Base scenario:
Starting investment: $100 (PV)
Nominal interest rate: 12% (i)
Investment period: One year (n)
Next we cycle through NACA, NACS, NACQ and NACM to calculate the respective FV value with all variables (except m) kept the same.
In each instance we provide a visual reference to illustrate:
- The interest rate per period inside
- The number of periods
NACA (m=1)
| $ \left( 1 + \frac{\text{0.12}}{\text{1}} \right) = 1.12$ |
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}} $
$\text{FV} = \text{100} \times \left( 1 + \frac{\text{0.12}}{\text{1}} \right)^{\text{1} \times \text{1}} $
$ \text{FV} = 112 $
Effective Rate Calculation 2Refer to the derivation of the Effective Return formula below
$ \text{Effective Rate} = \frac{\text{FV}}{\text{PV}} – 1$
$ \text{Effective Rate} = \frac{112}{100} – 1$
$ \text{Effective Rate} =\text{0.12 or 12%} $
Now that we have calculated it, it may seem obvious. But this should reinforce the fact that an annual interest rate is a NACA rate 3Annual Interest Rate Assumption: We are assuming that compounding occurs at least annually
NACS (m=2)
| $ \left( 1 + \frac{\text{0.12}}{\text{2}} \right) = 1.06$ | $ \left( 1 + \frac{\text{0.12}}{\text{2}} \right) = 1.06$ |
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}} $
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{0.12}}{\text{2}} \right)^{\text{2} \times \text{1}} $
$\text{FV} = 112.36 $
Effective Rate Calculation
$ \text{Effective Rate} = \frac{\text{FV}}{\text{PV}} – 1$
$ \text{Effective Rate} = \frac{\text{112.36}}{\text{100}} – 1$
$ \text{Effective Rate} =12.36%$
NACQ (m=4)
Interest per period:
$ \left( 1 + \frac{\text{0.12}}{\text{4}} \right) = 1.03$
| 1.03 | 1.03 | 1.03 | 1.03 |
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}} $
$\text{FV} = \text{100} \times \left( 1 + \frac{\text{0.12}}{\text{4}} \right)^{\text{4} \times \text{1}} $
$\text{FV} = 112.55$
Effective Rate Calculation
$ \text{Effective Rate} = \frac{\text{FV}}{\text{PV}} – 1$
$ \text{Effective Rate} = \frac{\text{112.55}}{\text{100}} – 1$
$ \text{Effective Rate} = 12.55%$
NACM
Interest per period:
$ \left( 1 + \frac{\text{0.12}}{\text{12}} \right) = 1.01$
| 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 | 1.01 |
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}}$
$\text{FV} = \text{100} \times \left( 1 + \frac{\text{0.12}}{\text{12}} \right)^{\text{12} \times \text{1}}$
$\text{FV} =112.68$
Effective Rate Calculation
$ \text{Effective Rate} = \frac{\text{FV}}{\text{PV}} – 1$
$ \text{Effective Rate} = \frac{\text{112.68}}{\text{100}} – 1$
$ \text{Effective Rate} =12.68%$
Summary
| NACA | NACS | NACQ | NACM | |
| Nominal | 12% | 12% | 12% | 12% |
| Effective | 12.0% | 12.36% | 12.55% | 12.68% |
By definition, the nominal rate remains the same across the spectrum of compounding periods. From the summarised data we note that as the compounding frequency increase, the effective interest rate also increases.
Derivation of effective interest rate calculation
The effective interest rate refers to an annual interest rate. One that, if applied to one single investment period of one year and an annual interest rate, will result in a PV growing to the FV provided.
Starting with this earlier result:
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{m}} \right)^{\text{m} \times \text{n}} $
And knowing that m=1, our formula simplies to:
$\text{FV} = \text{PV} \times \left( 1 + \frac{\text{i}}{\text{1}} \right)^{\text{1} \times \text{1}} $
$\text{FV} = \text{PV} \times \left( 1 +\text{i} \right)$
Solving for i, results in the interest rate actually earned:
$ \text{i}_{\text{Effective Rate}} =\frac{\text{FV}}{\text{PV}} -1$