Portfolio Mgmt: standard deviation

CFA Level I Portfolio Management
#1

For the question below, did I get lucky with the answer or is what I did an alternate method? Thanks!

Q)

Assets A (with a variance of 0.25) and B (with a variance of 0.40) are perfectly positively correlated. If an investor creates a portfolio using only these two assets with 40% invested in A, the portfolio standard deviation is closest to:

What I did -

.4 * (square root of .25) + .6 (square root of .4) = .5795

Answer per book:

The portfolio standard deviation = [(0.4)2(0.25) + (0.6)2(0.4) + 2(0.4)(0.6)1(0.25)0.5(0.4)0.5]0.5 = 0.5795

#2

^Yes, that is purely coincidence.

You need to know the formula for the standard devation of a 2-asset portfolio. You will see it no fewer than 5000 times over the next few years. This is one of those formulas that you need to know as well as you know your own name.

#3

i think you did it right. Since the correlation is perfectly positive, i.e it is equal to 1, the formula would go like this:

{weight squareA*varianceA+ weight squareB*varianceB+2weightA*weightB*S.dA*S.dB(1)}^0.5 [1 because correlation is perfectly positive]

Now when you solve this you’ll find it equal to [(a+b)^2]^0.5 and hence

[(weightA*S.dA+weightB*S.dB)^2]^0.5

The sqaure root gets cancelled by the square and you are left with the same thing you solved above.

Hope it makes sense.

#4

It’s not purely coincidence that his method worked. In the case of two perfectly positively correlated assets, the standard deviation of the portfolio will be a weighted average of the standard deviation of the individual assets (with weights based on relative market values). But this holds ONLY when the assets are perfectly positively correlated. If the two assets are less than perfectly positively correlated, the porfolio standard deviation will be less than this weighted average.

If you were to graph this out, portfolios consisting of two perfectly correlated assets will plot (with returns on the Y axis and standard deviations on the X-axis) on a straight line between the return/standard deviations of the individual assets. If the assets are less correlated, the line will plot on a curved line to the left and above the straight line.

To see this, set up an excel spreadsheet showing a graph of the risk-return possibilities for a two asset portfolio with weights from 100% in A to 100%in B, and A abd B being perfectly correlated, Then redo it with different correlations between A and B. Once I have my undergrads do this exercise, the see it pretty quickly.