Future Value and Compounding | CFA Level I
In today’s lesson, we’ll cover the time value associated with a single cash flow or lump sum investment. We’ll learn how to calculate the future value of a single cash flow, understand the concept of effective annual rate, and explore compounding, including continuous compounding.
Understanding Interest, Compounding, and Future Value
When you deposit $100 in a bank, the bank rewards you with interest. For example, let’s assume the bank offers a fixed 5% interest rate at the end of each year. After 1 year, you’ll have your initial $100, plus the 5% interest ($5). This is called Simple Interest.
Now, if you leave the money in the bank for another year, you’ll receive not only interest on your initial $100 but also on the $5 interest you earned the previous year. This is the magic of compounding! For instance, if your $100 investment compounded annually at 5% for 100 years, it would grow to a whopping $13,150!
Key Terms and the Future Value Formula
- Present Value (PV): The initial investment, e.g., $100 in our example.
- Interest Rate (r): The rate at which the investment grows per period, e.g., 5% per year.
- Number of Periods (N): The total number of periods the investment will grow for.
- Future Value (FV): The expected ending value of the investment after N periods.
The general formula for future value is:
FV = PV * (1 + r)^N
Now, let’s apply this formula in a fun exercise:
Did you get the answer? Drawing a timeline can help clarify the situation and make solving time value of money problems easier. In this case, the investment period (N) is 3 years, and the annual interest rate (r) is 8%. Plug the values into the formula to find the answer.
Considering Different Compounding Frequencies
What if interest is paid more frequently, like semi-annually or monthly? In such cases, make sure to adjust the interest rate (r) and the number of periods (N) to match the payment frequency.
The more frequent the compounding, the higher the effective interest amount. This brings us to the concept of Effective Annual Rate (EAR).
Calculating the Effective Annual Rate (EAR)
The EAR helps us compare interest rates compounded at different frequencies on an even platform. To calculate it, use the following formula:
EAR = (1 + r/N)^N – 1,
where N is the number of compounding periods per year.
EXAMPLE
Let’s say Piggy Bank offers a 4.95% annual interest rate, paid monthly, while Porky Bank offers a 5% annual interest rate, paid semi-annually. If you want to save $100,000 for 2 years, which bank should you choose?
Using the EAR formula, we can compare the two offers:
- Piggy Bank: EAR = (1 + 0.0495/12)^12 – 1 ≈ 5.064%
- Porky Bank: EAR = (1 + 0.05/2)^2 – 1 ≈ 5.063%
Surprisingly, Piggy Bank’s offer is slightly better, even though Porky Bank advertises a higher annual rate. The EAR helps you make informed decisions by comparing apples to apples.
Continuous Compounding: The Ultimate Frequency
What if interest compounds continuously? In this case, the formula for the future value changes slightly:
FV = PV * e^(r * N),
where “e” is the mathematical constant (approximately 2.718).
Let’s practice with another example:
EXAMPLE
Lee invests $50,000 in an account with a guaranteed 4% annual interest rate and continuous compounding. How much total interest can Lee expect to earn by the end of 6 years?
Using the continuous compounding formula, we find that Lee’s investment will grow to approximately $63,562, with $13,562 being the total interest earned.
Recap and Next Steps
In this lesson, we learned about present and future values, the importance of adjusting the interest rate (r) and the number of periods (N) to match the payment frequency, and how to calculate the effective annual rate (EAR) for comparing different compounding frequencies.
As you progress, always remember to use the correct r and N values. It may seem simple now, but many candidates trip up under time constraints in exams.
Join us in the next lesson, where we’ll reverse the order and examine the present value of a single cash flow. Happy studying!
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