Financial Math FM/Stochastic Interest

Stochastic interest
In this book, we have mainly discussed (i.e. non-random) interest, and we will briefly introduce (i.e. random) interest, by regarding the interest rate as a random variable. We use the following notations:


 * $$I_t$$: interest rate random variable for the period $$t-1$$ to $$t$$
 * $$\mu_t$$: mean of $$I_t$$
 * $$\sigma_t^2$$: variance of $$I_t$$

Accumulation of single payment over several time periods
Assume that $$I_t$$ are for $$t=1,\ldots,n$$. Let $$S_n$$ be the accumulation of a single sum of money invested for $$n$$ years, i.e. $$ S_n=(1+I_1)\cdots(1+I_n). $$ Then, by independence, $$ \begin{align} \mathbb E[S_n]&=(1+\mu_1)\cdots(1+\mu_n)\\ \mathbb E[S_n^2]&=\mathbb E[(1+I_1)^2\cdots(1+I_n)^2]\\ &=\mathbb E[(1+I_1)^2]\cdots\mathbb E[(1+I_n)^2]\\ &=\left(\sigma_1^2+(1+\mu_1)^2\right)\cdots\left(\sigma_n^2+(1+\mu_n)^2\right)\\ \operatorname{Var}(S_n)&=\mathbb E[S_n^2]-(\mathbb E[S_n])^2\\ &=\left(\sigma_1^2+(1+\mu_1)^2\right)\cdots\left(\sigma_n^2+(1+\mu_n)^2\right)- (1+\mu_1)^2\cdots(1+\mu_n)^2 \end{align} $$ For simplicity, further assume that $$I_t$$'s are i.i.d. (identically and independently distributed), with mean $$\mu$$ and variance $$\sigma^2$$. Then, $$ \begin{align} \mathbb E[S_n]&=\underbrace{(1+\mu)\cdots(1+\mu)}_{n\text{ copies}}=(1+\mu)^n\\ \mathbb E[S_n^2]&=(\sigma^2+(1+\mu)^2)^n=(1+2\mu+\mu^2+\sigma^2)^n\\ \operatorname{Var}(S_n)&=\mathbb E[S_n^2]-(\mathbb E[S_n])^2=(1+2\mu+\mu^2+\sigma^2)^n-(1+\mu)^{2n} \end{align} $$

Some information about log-normal distribution
If $$Y=\ln X$$ has a normal distribution with mean $$\mu$$ and variance $$\sigma^2$$, then $$X$$ has a distribution with ( mean/variance generally) $$\mu$$ and $$\sigma^2$$. The following are some properties of random variables following log-normal distribution with parameters $$\mu$$ and $$\sigma^2$$:
 * probability density function (pdf): $$f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\ln x-\mu}{\sigma}\right)^2\right)$$
 * mean: $$\mathbb E[X]=e^{\mu+\frac{\sigma^2}{2}}$$
 * variance: $$\operatorname{Var}(X)=e^{2\mu+\sigma^2}\left(e^{\sigma^2}-1\right)=(\mathbb E[X])^2\left(e^{\sigma^2}-1\right)$$

Motivation of using log-normal distribution
Let's apply log-normal distribution to stochastic interest. If $$1+I_t$$ follows a log-normal distribution with parameters $$\mu$$ and $$\sigma^2$$, then $$\ln(1+I_t)$$ will be distributed with mean $$\mu$$ and variance $$\sigma^2$$.

Then, considering the natural logarithm of accumulation of a single investment of one unit for a period of $$n$$ time units, we have $$ \ln S_n=\ln((1+I_1)\cdots(1+I_n)) =\ln(1+I_1)+\cdots+\ln(1+I_n). $$ Assuming $$I_t$$'s are independent, $$\ln(1+I_t)$$ will also be independent. If we further assume that $$1+I_t$$'s are also log-normally distributed with parameters $$\mu_t$$ and $$\sigma_t^2$$, then $$\ln(1+I_t)$$'s are normally distributed with mean $$\mu$$ and variance $$\sigma^2$$x, and the of independent normal random variables $$\ln(1+I_1)+\cdots+\ln(1+I_n)$$ is normally distributed (which is a well-known result about normal distribution). That is, $$ \ln S_n=\ln(1+I_1)+\cdots+\ln(1+I_n)\sim N(\mu_1+\cdots+\mu_n,\sigma_1^2+\cdots+\sigma_n^2). $$ Thus, if we apply log-normal distribution to stochastic interest, we can obtain this nice result ($$\ln S_n$$ follows a simple normal distribution).