Idealization Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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The formula for compound interest is

A = P(1 + r/n)^{nt}

. Explain what idealisations are involved and how the continuous compounding limit

A = Pe^{rt}

arises as

n \to \infty

1
Idealisations: the interest rate $r$ is assumed constant; the bank credits interest exactly $n$ times per year with perfect precision; no fees or taxes are modelled.
2
As $n \to \infty$ (compounding every instant): use the limit $\lim_{n\to\infty}\left(1+\frac{r}{n}\right)^n = e^r$ .
3
Therefore $A = P\left(1+\frac{r}{n}\right)^{nt} = P\left[\left(1+\frac{r}{n}\right)^n\right]^t \to Pe^{rt}$ .

Answer

A = Pe^{rt} \text{ (continuous compounding idealisation)}

Continuous compounding is an idealisation of the real compounding process. The mathematical convenience of

e^{rt}

justifies the approximation in many financial models.

About Idealization

Replacing a messy real-world object or process with a perfect, simplified version that captures its essence while ignoring complications.

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