I'm having some problems getting started.  Any suggestions?

Luke, double check your work.  I think you canceled out a factor of ten somewhere that you shouldn't have.

And, yes, the probability does increase with time, but only to a point (which we're finding here) until it begins to decrease.  Let me know if you have further questions.

RDH

For the mean I have 200 + 10*t and for stddev 40*sqrt(t)

I took the derivative of mean/stddev as functions of t, set equal to zero and got t = 2. but when i checked in excel, Pr(X < 0) is increasing with t. What am i doing wrong?

ehezel, we want P(Y_t < 0).  Since Y_t follows a normal distribution, the easiest way to find this is to convert it to a standard normal distribution. So, P(Y_t < 0) = P(Y_t - mu_t < 0 - mu_t) = P[(Y_t-mu_t)/sigma_t < -mu_t/sigma_t] = P(Z < -mu_t/sigma_t).

In order to maximize (MLE wasn't my strongpoint in Stat Theory) this probabily, we need to take the derivative with respect to time t of (-mu_t/sigma_t) and set it equal to zero.  Those two functions should be very easy to derive from the given information.  You get a nice round number, which can be verified through brute force in Excel using =NORMDIST(0,mu_t,sigma_t,TRUE).  Let me know if you have any further questions.

RDH

Can anyone offer some help on part E? Thanks.

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.)