Financial Math FM/Stochastic Interest

Stochastic interest

In this book, we have mainly discussed Template:Colored em (i.e. non-random) interest, and we will briefly introduce Template:Colored em (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 Template:Colored em for $t=1,\dots ,n$. Let $S_n$ be the accumulation of a single Template:Colored em sum of money invested for $n$ years, i.e. $$S_n=(1+I_1)\cdots (1+I_n).$$ Then, by independence, $$\begin{array}{cc}𝔼[S_n]& =(1+{\mu }_1)\cdots (1+{\mu }_n)\\ 𝔼[S_n^2]& =𝔼[(1+I_1{)}^2\cdots (1+I_n{)}^2]\\ & =𝔼[(1+I_1{)}^2]\cdots 𝔼[(1+I_n{)}^2]\\ & =({\sigma }_1^2+(1+{\mu }_1{)}^2)\cdots ({\sigma }_n^2+(1+{\mu }_n{)}^2)\\ \mathrm{Var}(S_n)& =𝔼[S_n^2]-(𝔼[S_n]{)}^2\\ & =({\sigma }_1^2+(1+{\mu }_1{)}^2)\cdots ({\sigma }_n^2+(1+{\mu }_n{)}^2)-(1+{\mu }_1{)}^2\cdots (1+{\mu }_n{)}^2\end{array}$$ For simplicity, further assume that $I_t$'s are i.i.d. (identically and independently distributed), with mean $\mu$ and variance ${\sigma }^2$. Then, $$\begin{array}{cc}𝔼[S_n]& =\underset{n\text{ copies}}{\underbrace{(1+\mu )\cdots (1+\mu )}}=(1+\mu {)}^n\\ 𝔼[S_n^2]& =({\sigma }^2+(1+\mu {)}^2{)}^n=(1+2\mu +{\mu }^2+{\sigma }^2{)}^n\\ \mathrm{Var}(S_n)& =𝔼[S_n^2]-(𝔼[S_n]{)}^2=(1+2\mu +{\mu }^2+{\sigma }^2{)}^n-(1+\mu {)}^{2n}\end{array}$$

Accumulation of investments with log-normal distribution

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 Template:Colored em distribution with Template:Colored em (Template:Colored em 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): $f(x)=\frac{1}{x\sigma \sqrt{2\pi }}\exp (-\frac{1}{2}{\left(\frac{\ln x-\mu }{\sigma }\right)}^2)$
• mean: $𝔼[X]=e^{\mu +\frac{{\sigma }^2}{2}}$
• variance: $\mathrm{Var}(X)=e^{2\mu +{\sigma }^2}(e^{{\sigma }^2}-1)=(𝔼[X]{)}^2(e^{{\sigma }^2}-1)$

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 Template:Colored em 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 Template:Colored em of independent normal random variables $\ln (1+I_1)+\cdots +\ln (1+I_n)$ is normally distributed Template:Colored em (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).