Python for Finance

Chapter 9
var=w1**2*var1 +w2**2*var2+2*w1*w2*rho*sigma1*sigma2
if(var<portVar):
portVar=var
finalW1=w1
#print(vol)
print("min vol=",sp.sqrt(portVar), "w1=",finalW1) ('min vol=',
('min vol=', ('min vol=', 9.3132257461547852e-10, 'w1=',
0.80000000000000004)
First, the result confirms our previous result with w1=0.8 and w2=0.2. In the
program, we have 1000 pairs of w1 and w2. A small value, called tiny, is
1/1000=0.001. The first pair of two weights is 0.1% and 99.9%. A very big number
is assigned to our solution variable, that is, as an initial value. In this program,
portVar=10. Other big numbers would work perfectly, such as 100. Here is the logic:
based on the first pair of w1 and w2, we estimate the portfolio variance. If this new
portfolio variance is less than portVar, we replace portVar with this new value
and record w1 as well. If the new portfolio variance is bigger than portVar, we do
nothing. Repeat the same procedure until we finish the loop. Here is an analogy.
Assume that we want to find the tallest person among 1,000 persons. Assume that
we have a variable call tallestPerson and its initial vale is 0.1 inch. Since every
person will be taller than this value, the first person's height will replace this value. If
the next person's height is higher than this variable, we replace it. Otherwise, we go
to the next one. The procedure is repeated until the last person. In terms of efficiency,
one small trick is to estimate var1 and var2 just once.
In finance, it is a convention to use both variance and standard deviation to represent
risk, since they describe uncertainty. Usually, we use standard deviation of returns
to represent the volatility. It is a good idea to look at the impact of correlation on
the efficient frontier. First, let's learn how to generate a set of correlated random
numbers. There are two steps involved:
1. Generate two random time series, x1 and x2, with a zero-correlation.
2. Apply the following formula:
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