In this book, we have mainly discussed deterministic (i.e. non-random) interest,
and we will briefly introduce stochastic (i.e. random) interest,
by regarding the interest rate as a random variable. We use the following notations:
- : interest rate random variable for the period to
- : mean of
- : variance of
Accumulation of single payment over several time periods
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Assume that are independent for .
Let be the accumulation of a single unit sum of money invested for years, i.e.
Then, by independence,
For simplicity, further assume that 's are i.i.d. (identically and independently distributed),
with mean and variance . Then,
Accumulation of investments with log-normal distribution
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If has a normal distribution with mean and variance ,
then has a log-normal distribution with
parameters (not mean/variance generally) and . The following are some properties of random variables following log-normal distribution with parameters and :
- probability density function (pdf):
- mean:
- variance:
Let's apply log-normal distribution to stochastic interest.
If follows a log-normal distribution with parameters
and , then
will be normally distributed with mean and variance .
Then, considering the natural logarithm of accumulation of a single investment of one unit for a period of time units,
we have
Assuming 's are independent, will also be independent.
If we further assume that 's are also log-normally distributed with parameters and ,
then 's are normally distributed with mean and variance x,
and the sum of independent normal random variables
is normally distributed with mean
and variance (which is a well-known result about normal distribution).
That is,
Thus, if we apply log-normal distribution to stochastic interest, we can obtain this nice result ( follows a simple normal distribution).
Example.
(a) It is given that follows log-normal distribution with parameters and ,
and .
Compute and .
Solution: (a) Based on the given mean and variance,
we have and So,
(b) It is further given that is the annual yield rate for the th year, 's are i.i.d., and follow the distribution mentioned above.
Compute the mean and variance of the accumulation of for years,
and the probability that its accumulated value will be less than .
Solution: (b)
Since ,
So, follows log-normal distribution with parameters and . Thus,
its mean and variance are as follows:
- mean:
- variance:
Then, we can compute the probability by