1. Discrete vs. Continuous Compounding

For nominal rate $R$ compounded $m$ times per year over time $T$, terminal future value is $FV = PV \left( 1 + \frac{R}{m} \right)^{m T}$. Taking the limit as $m \to \infty$ yields continuous exponential growth:

$$FV = \lim_{m \to \infty} PV \left( 1 + \frac{R}{m} \right)^{m T} = PV \, e^{R T}$$

The Effective Annual Rate (EAR) relationship is:

$$\text{EAR} = e^R - 1$$

2. Continuous Cash Flow Streams

For a continuous cash flow rate $C(t)$ discounted at continuous rate $r$, the present value integral is:

$$PV = \int_0^T C(t) \, e^{-r t} dt$$

If $C(t) = C$ is constant, analytical integration yields:

$$PV = \frac{C}{r} \left( 1 - e^{-r T} \right)$$

3. Python Numerical Compounding Converter

import numpy as np

def convert_nominal_to_continuous(nominal_rate, m_frequencies=[1, 4, 12, 365]):
    """
    Compare discrete compounding EAR with continuous exponential limit.
    """
    results = {}
    for m in m_frequencies:
        ear = (1.0 + nominal_rate / m)**m - 1.0
        results[f"Compounded {m}x/yr"] = round(ear * 100, 4)
    results["Continuous Limit (e^R - 1)"] = round((np.exp(nominal_rate) - 1.0) * 100, 4)
    return results

if __name__ == "__main__":
    rates = convert_nominal_to_continuous(nominal_rate=0.08)
    for freq, ear_pct in rates.items():
        print(f"{freq}: {ear_pct}%")