Problem 23			Due 4/7/00

	In this problem we examine constrained 
optimization.  Suppose you are saving money for a 
particular purpose, say college education for your 
children.  You have been given a windfall (you won 
the $50,000 dollar prize in the lottery) and you want to 
invest it wisely.  You have a choice of three 
investment vehicles.  The first choice is a T-bill 
account which pays out at 7%, and is very secure (e.g.,
it pays out at a fixed rate with no variability).  The 
second is a stock market account that averages a yield 
of 10% but has a standard deviation of 10% as well 
(e.g., 69% of the time it will have a yield between 0% 
and 20%, and about 15% of the time it will actually 
lose money in a given year).  The third is an 
aggressive growth fund investing in Asian stocks and
securities which has an average annual yield of 15%,
but a volatile standard deviation of 25%.  You may assume
that the covariance of the variability in the funds
from year to year and between funds is zero (not a
great approximation, but it makes things alot easier
to calculate).

	You are to determine an investment portfolio 
mix for your $50,000 initial investment which does 
two things:  First, you want to maximize your 
expected yield.  Second, you require that after 20 years 
(when your children are in college) you have at least a 
97.5% chance that you will have the $150,000 you 
expect that college will cost.  How much of your initial 
investment do you put into each of the three 
vehicles?

	How does your answer change if you have 
$75,000 to invest and you require a 97.5% chance that 
you will have $100,000 in five years?


Hint:  Because the variability of the investments is
large, you will need to calculate the variance and
standard deviation of the log of the return after 20
years rather than the return itself.  This quantity
is much better behaved, and you can then use standard
formulas for error propagation.  Otherwise, it would
predict that you could wind up with a negative account
balance!!

      You need to be careful in picking the correct
range for an initial value of P (assuming that you 
use penalty functions).  Also watch out for negative
values of investment - you will need to restrict the
values so that they are all positive, otherwise fmins
will make some positive and others negative.  Physically
this corresponds to taking out a loan at one rate and
investing it at another, but that is not what we are
trying to do here!