Effective Rate of Interest: Usage and Examples

Actuarial Concepts
Written By Eryk Faracik

In the previous post on Actuarial Concepts, I introduced the effective rate of interest, its definitions, formulas, and why it’s not the same as the nominal rate of interest. However, the most valuable aspect is knowing how to use the effective interest rate to calculate the value of money over time, which I will demonstrate in this post.

After reading the first post about the effective interest rate (if you haven’t, I encourage you to check it out—here’s the link!), you should now have a better understanding of what it is and how to calculate it.

The effective interest rate represents the actual amount by which money will grow over a specific period. The most commonly used version is the annual effective rate of interest, where interest is compounded once a year. In this case, the nominal rate of interest would be the same as the effective rate.

However, if there are more or fewer compounding periods, these values will differ. When the annual nominal interest is compounded m times a year, we can calculate the effective interest rates for those periods as well as for the entire year. Once we determine the correct effective interest rate for a given period, we can then use it—but how? I’ll show you—let’s dive in!

Usage

In actuarial science, the effective rate of interest can be used in many ways, most commonly for calculating the present value and accumulated value (also known as the future value) of a series of cash flows.

Let’s start by clarifying what a cash flow is. Any money spent or earned can be considered a cash flow. For example, your monthly salary or dividends are cash flows. Of course, some cash flows will have a negative sign, as they represent money spent rather than earned.

Moving on, the accumulated value is the amount of money after it has earned the effective interest over a certain period. Conversely, the present value is the value of future cash flows at a specific point in time before they occur.

The process of calculating the accumulated value is called accumulating (surprising, right?), while the process of calculating the present value is called discounting.

I will start by describing the formula and process for calculating the accumulated value and then move on to the present value.

Accumulated Value

Let’s start with the general formula:

Formula 1. Accumulated value at year t+k for a deposit made in year t.

To calculate the accumulated (or future) value of money deposited at a given time t, we multiply it by (1+effective interest rate) raised to the power of the number of compounding periods. Between years t and t+k, there are k years. If there are m compounding periods per year, the total number of compounding periods is mk.

Most of the time, tt represents the present moment, so it is equal to 0. However, the general formula can be used to calculate the accumulated value of a cash flow at any given point in time.

Present Value

The formula for present value is the inverse of the accumulated value formula; instead of multiplying the deposit by (1+effective interest rate), we divide it (discount it). It can be written as:

Formula 2. Present value at year t of a cash flow occurring in year t+k.

Similarly to the accumulated value, year tt is often equal to the current moment, or year 0. However, we can calculate the present value of future cash flows at any given point in time.

Alright, the formulas have been shown. Let’s see them in action with the two examples below.

Examples

Example 1.

Mary deposited $100 into her savings account today. She plans to wait 7 years before withdrawing her money. The annual effective interest rate for this savings account is 7.5%, compounded at the end of each year. Calculate the amount of money Mary will have after 7 years.

The first step, which is helpful in financial mathematics problems, is to organize all the information provided in the problem into a graph—more specifically, a timeline.

Image 1. $100 is deposited at time 0 (today) with an annual effective interest rate of 7.5%, and accumulated values at time 1 and time 7 (the end of the first and seventh years, respectively).

Mary deposited $100 at time 0 (today), and her money earns an annual effective interest rate of 7.5%. The compounding period here is one year. We are tasked with calculating the amount of money Mary will have after 7 years, which is the accumulated value of her money at t=7.

We know the initial deposit amount, the number of compounding periods (7), and the annual effective interest rate. Let’s substitute these values into Formula 1:

So, the answer is that the amount Mary will have in her savings account after 7 years is $165.90.

At the end of this post, I’ve attached two links for present value and future value calculators. Using the future value (accumulated value) calculator, we can solve this problem by entering the correct information: the number of periods (N), the starting amount (PV), and the effective interest rate (I/Y), and then clicking the Calculate button.

In our case, only one deposit is made at the beginning, so in the periodic deposits (PMT) field, enter 0. If no periodic deposits are made, there is no difference in the accumulated value whether PMT is set at the beginning or end of the compounding period. I chose the beginning because Mary deposited $100 today, which is the start of the first year.

Image 2. Using the future value calculator for the accumulated value calculation, we can determine the future value as well as the interest earned during the compounding period.

Example 2.

John deposited money into his savings account three years ago. His current balance is $3,000. The annual nominal interest rate, compounded monthly, was 12% throughout the 3-year period. What was the amount John initially deposited?

Similarly to the previous example, let’s place the values on a timeline.

Image 3. $3,000 is accumulated at the end of 3 years (36 months) from an unknown amount X deposited at time 0. Calculate the accumulated value at the end of one year (12 months) given an annual nominal interest rate compounded monthly, equal to 12%.

$3,000 is the accumulated value of John’s initial deposit after three years. However, the compounding period is not one year. This can be determined by the definition of the annual nominal interest rate. It is stated that the interest is compounded monthly, so the compounding period is one month. The total number of compounding periods is equal to the number of months in a year (12) times the number of years (3), which gives 36 periods.

To calculate the j effective interest rate for one month, let’s use the formula from the last post and divide the annual nominal interest rate by the number of compounding periods in one year, which is the number of months:

Having calculated the monthly effective interest rate and gathered all the other necessary information, we can plug it directly into the present value formula:

So, the answer is that John initially deposited $2,096.77 into his savings account.

Similar to the accumulated value case, to solve this problem we can also use a specific calculator by entering the future value (FV), number of compounding periods (N), the effective interest rate (I/Y), and then clicking the Calculate button.

Image 4. Calculation of the present value using the present value calculator allows us to see, in addition to the total amount of the present value, the interest earned during the 36 months.

Summary

The effective rate of interest is used to calculate the present and accumulated values of cash flows. The present value represents the current value of future cash flows, while the accumulated value represents the value money will have after earning interest over time.

The formulas for present and accumulated values are inverses of each other, allowing us to calculate one based on the other. The process of earning interest is called accumulating, while the inverse process is called discounting.

Let me know if any of you found this post helpful. If you want to learn more about the effective rate of interest, I encourage you to check out the links attached below and explore some Financial Mathematics textbooks on your own!

Eryk Faracik
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