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.)

 

Can anyone offer some help on part E? Thanks.

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

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?

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

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

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

please help to correct me if i'm wrong
we have Y-zero=200m, theta-zero=10m and var(et)=40^2
and the capital after one month is
Y-hat(1) = Y-zero + theta-zer0 which is equals to 210
so mean of capital after one month E(Y-hat(1)) = 210?
but what is the variance?

Ben, there is a hint for this in part B.  First of all, your variance at time zero is zero, as we are given the initial value.  The volatility of each month is independent, so in variance, this is additive.  For month one, there is only one month of volatility, so we square 40mil and take the square root.  For the second month, we would square 40mil for the first month, square 40mil for the second month, add the two months together and then take the square root of the sum.  Dot dot dot.  For the nth month, we square 40mil n times, add the n terms and take the root of the sum.  This leaves a cute, little formula

Variance(t) = 40mil² x t.  Thus: Standard Deviation(t) = 40mil sqrt(t)

RDH

Ray is right on in helping everyone and I mean no disrespect, but I don't think part E is a MLE problem, it's just simple optimization.  Just in case anyone was confused by that.

   -    Dave