Time Value of Money and Interest Rates › Future Value and Compounding

With a positive interest rate and no withdrawals or offsetting fees, an interest-bearing deposit grows over time. Compounding adds earned interest to the balance so that it can earn interest in later periods. The same mechanism can increase an unpaid debt. We will build from simple interest to the continuous limit used in derivatives pricing.

Simple interest is the most basic form of interest calculation and is our conceptual starting point. Under simple interest, the interest accrues only on the original principal amount, never on accumulated interest. Think of it as a linear process: each period adds the same fixed amount of interest, regardless of how much has accumulated before. If you invest a principal $P$ at an annual interest rate $r$ for $t$ years, the future value is:

where:

• $FV$: future value of the investment
• $P$: principal amount (initial investment)
• $r$: annual interest rate (expressed as a decimal)
• $t$: time in years

The structure of this formula reveals its linear nature. The term $rt$ represents the total proportional increase: the rate $r$ multiplied by time $t$. When you add this to 1 (representing the original principal as 100%), you get the total growth factor. Multiplying by $P$ then scales this to your actual investment amount. Notice that time and rate enter symmetrically through their product, which is then added to 1. Doubling the time has exactly the same effect as doubling the rate.

For example, investing $1,000 at 5% simple interest for 3 years yields:

Notice that the interest earned ($150) equals the annual interest ($50) multiplied by the number of years (3). Simple interest grows linearly with time because this convention deliberately excludes interest on accumulated interest. Many products instead use periodic compounding, which we turn to next.

Under periodic compounding, each period's interest is added to the balance before the next period's interest is calculated. At a constant positive rate, with interest reinvested and no other balance changes, your original investment and its accumulated interest both earn returns. The balance then grows exponentially rather than linearly. At zero interest the balance stays constant; an admissible negative rate reduces it.

To understand this exponential behavior, consider what happens period by period. After the first compounding period, your balance is $P(1 + r/n)$. After the second period, interest is calculated on that updated balance, giving $P(1 + r/n)(1 + r/n) = P(1 + r/n)^2$. Each period multiplies the previous balance by the same factor. A factor greater than one produces growth; a positive factor below one produces decay.

For an investment compounding $n$ times per year at annual rate $r$, the future value after $t$ years is:

where:

• $FV$: future value after $t$ years
• $P$: principal amount invested
• $r$: nominal annual interest rate (stated rate)
• $n$: number of compounding periods per year
• $t$: time in years
• $(1 + r/n)$: growth factor per compounding period
• $nt$: total number of compounding periods

Here $n$ must be positive and the per-period growth factor $1+r/n$ must remain positive when the exponent $nt$ is not an integer. The product $nt$ measures the number of compounding periods; contracts that accrue only over completed periods need an integer period count or an explicit stub-period rule. These domain conditions are usually implicit in ordinary deposit and loan examples, but they matter when rates can be negative or time includes a fractional period.

The formula captures a key insight: each compounding period applies a growth factor $(1 + r/n)$, and these compound multiplicatively over $nt$ total periods. The rate per period is $r/n$ because we divide the nominal annual rate among $n$ periods, and the total number of periods is $nt$ because we have $n$ periods per year for $t$ years. With a positive nominal rate, positive principal, and positive horizon held fixed, more frequent compounding increases future value. The size of this increase depends on those parameters. Interest added to the balance earlier can itself earn interest over more of the remaining horizon.

Let's compare different compounding frequencies for the same nominal rate:

Taking the limit as the compounding frequency approaches infinity gives us continuous compounding. Rather than compounding at discrete intervals like monthly, daily, or even every second, continuous compounding assumes that interest accrues and is reinvested at every instant. It is a common mathematical convention in continuous-time derivatives models because exponential growth factors compose cleanly across time.

The continuous compounding formula emerges from taking the limit of the discrete formula as $n$ grows without bound:

where:

• $FV$: future value
• $P$: principal amount
• $r$: continuously compounded annual rate
• $t$: time in years
• $e$: Euler's number ($\approx 2.71828$)

This result relies on the fundamental limit $\lim_{n \to \infty}(1 + x/n)^n = e^x$, one of the most important limits in mathematics. To see why this limit makes sense intuitively, consider that as $n$ increases, each individual growth factor $(1 + r/n)$ gets closer to 1, but you apply more and more of them. These two effects, smaller factors applied more times, balance out to give a finite limit, and that limit happens to be the exponential function.

The exponential function $e^{rt}$ emerges naturally because continuous compounding represents the limiting case of infinitely many infinitesimally small growth events. This makes continuous compounding mathematically convenient: returns become additive in the logarithm, meaning $\ln(FV/P) = rt$. This logarithmic property greatly simplifies many calculations in derivatives pricing, because the return over multiple periods can be computed simply by adding up the returns over sub-periods, rather than multiplying growth factors together.