Financial Math FM/Stochastic Interest


Stochastic interest

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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 investment

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Example. (some simple formulas) For the accumulation of a unit sum of money over the period   to  ,

  • the mean is   (for non-random interest, it is  )
  • the variance is  
  • the 2nd moment is  

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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Some information about 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:  

Motivation of using log-normal distribution

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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).

Examples

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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