A.     What is the distribution of capital after one month? (What is the type of distribution, such as normal, lognormal, uniform, or something else? Use the characteristics of a white noise process. What is the mean of the distribution after one month? Use the starting capital and the drift. What is the standard deviation after one month? The volatility is the standard deviation per unit of time, not the variance per unit of time. It is ó, not ó².)

B.     What is the distribution of capital after six months? (The serial correlation is zero, so the capital changes are independent and the variances are additive. Derive ó² for one month from ó, add the ó²’s for six months, and derive the ó after six months.)

C.    What is the distribution of capital after one year?

D.    What are the probabilities of insolvency at the end of six months and one year? (You have a distribution with a mean ì and a standard deviation ó. Find the probability that a random draw from this distribution is less than zero. Use the cumulative distribution function of a standard normal distribution. Excel has a built-in function for this value.)

E.     At what time in the future is the probability of insolvency greatest? (Write an equation for the probability as a function of (i) the mean of the distribution at time t and (ii) the standard deviation of the distribution at time t. To maximize this probability, set its first derivative to zero, and solve for t.)

peat, I would suggest a cup of coffee and some reading glasses.

A random walk is a white-noise process (and thus a normal distribution) with a drift (a mean which increases linearly with time) and volatility (a much cooler discription for the standard deviation).  We are given the intial mean, its "slope" and the standard deviation with respect to time: 200m, 10m, and 40m.  It is straightforward to derive their equations from there to answer the first few questions.  Remember we are given the standard deviation, not the variance, so be careful when attempting to find the new standard devation for one and six months out.  The last two questions are fun, use the standard normal tables or excel to help find your answers.  I think the final question relates to maximum likelihood estimation, but we talk about that in the previous posts.

Other than that, I'd also suggest the beach and perhaps a picnic for the weekend, enjoy!

RDH