Solution

p1 = sympy.Symbol("p1")
p2 = sympy.Symbol("p2")
def twoChoice(ret1,ret2,var1,var2,cov,goalReturn):
    retEquation = sympy.Eq(goalReturn,p1*ret1+p2*ret2)
    totalPercent = sympy.Eq(1,p1+p2)
    dictionary = sympy.solve((retEquation,totalPercent))
    percent1 = dictionary[p1]
    percent2 = dictionary[p2]
    variance = percent1**2*var1+percent2**2*var2+2*percent1*percent2*cov
    print("Put "+str(percent1*100)+"% in asset 1.")
    print("Put "+str(percent2*100)+"% in asset 2.")
    print("Your variance will be "+str(variance))

twoChoice(.03,.09,.05,.07,.03,.06)

We could also solve instead for a variance we want by feeding the portfolio variance equation.

p1 = sympy.Symbol("p1")
p2 = sympy.Symbol("p2")
def twoChoice2(ret1,ret2,var1,var2,cov,goalVar):
    varEquation = sympy.Eq(goalVar,p1**2*var1+p2**2*var2+2*p1*p2*cov)
    totalPercent = sympy.Eq(1,p1+p2)
    dictionary = sympy.solve((varEquation,totalPercent))
    return dictionary

twoChoice2(.03,.09,.05,.07,.03,.045)

You’re probably confused now, why is there two different weights that cause the same variance? This is because of the covariance term, it introduces non-linearity so in this case we can have more than one combination that causes the variance.

Let’s create a function that prints out the combinations.

def twoChoice2(ret1,ret2,var1,var2,cov,goalVar):
    varEquation = sympy.Eq(goalVar,p1**2*var1+p2**2*var2+2*p1*p2*cov)
    totalPercent = sympy.Eq(1,p1+p2)
    dictionaries = sympy.solve((varEquation,totalPercent))
    for dictionary in dictionaries:
        percent1 = dictionary[p1]
        percent2 = dictionary[p2]
        ret = percent1*ret1+percent2*ret2
        print("Put "+str(percent1*100)+"% in asset 1.")
        print("Put "+str(percent2*100)+"% in asset 2.")
        print("Your return will be "+str(ret))
twoChoice2(.03,.09,.05,.07,.03,.045)