Integral Calculus and Differential Equations › Continuous Compounding and the Exponential Function

Continuous compounding is an important application of integral calculus in finance. Interest accrual, compounding, and scheduled crediting depend on the contract. Continuous compounding supplies a mathematical convention for the models below.

Discrete compounding at a constant nominal annual rate $r$, $n$ times per year for $t$ years, gives the following value at compounding dates, where $nt$ is an integer:

$$FV = PV \cdot \left(1 + \frac{r}{n}\right)^{n \cdot t}$$

Each completed period credits interest at rate $r/n$. The comparison below fixes this nominal quote as frequency changes; between credit dates, its curves use fractional-power interpolation, not continuous crediting. As the compounding frequency $n$ increases, moving from annual to quarterly to daily to hourly, this expression approaches a limit. Taking $n \to \infty$ yields continuous compounding:

$$FV = PV \cdot e^{rt}$$

where:

• $FV$: future value of the investment
• $PV$: present value (initial investment)
• $e \approx 2.71828$: Euler's number, the base of natural logarithms
• $r$: continuously compounded annual rate in $e^{rt}$; the preceding discrete formula uses the same number as a nominal annual quote, not an effective annual return
• $t$: time in years

The exponential appears because the balance changes in proportion to its current value at a constant rate. A positive rate produces growth, zero rate leaves the balance constant, and a negative rate produces decay. The continuous formula is a limit of multiplicative discrete factors, not literal infinitely frequent contractual payments.

Every solution of $y'=ky$ with constant $k$ has the form $y(t)=A e^{kt}$. An initial value fixes $A$; in particular, $e^t$ solves $y'=y$ with $y(0)=1$.

Let's derive the formula for a positive initial balance $V_0$ and constant continuously compounded rate $r$. The balance $V(t)$ changes at rate $r$ times its current value, growing for $r>0$ and shrinking for $r<0$:

$$\frac{dV}{dt} = rV(t)$$

where:

• $V(t)$: account value at time $t$
• $\frac{dV}{dt}$: instantaneous rate of change of account value
• $r$: continuous interest rate
• $r \cdot V(t)$: interest earned per unit time, proportional to current balance

Solving this differential equation gives an exponential family; the initial balance selects one path. On the positive-balance branch, separation of variables moves the $V$ and $t$ terms to opposite sides:

$$\frac{dV}{V} = r \, dt$$

Integrating both sides:

$$\int \frac{dV}{V} = \int r \, dt$$ $$\ln|V| = rt + C$$

Here $C$ is the integration constant. Because we assumed a positive balance, exponentiating $\ln|V|$ gives $V$, rather than $-V$:

$$V(t) = e^{rt + C} = e^C \cdot e^{rt}$$

Setting $t = 0$ and letting $V(0) = V_0$ be the initial value:

$$V_0 = e^C \cdot e^0 = e^C$$

Therefore, $e^C = V_0$, and the solution is:

$$V(t) = V_0 \cdot e^{rt}$$

The final formula also solves the ODE directly for zero or negative initial values; $V_0=0$ gives the separate equilibrium $V(t)=0$. With a time-varying deterministic coefficient, $y'=k(t)y$ instead gives $y(t)=y(0)\exp(\int_0^t k(s)\,ds)$. Random shocks require a stochastic model rather than this ODE alone.