GARP 2020 Books - Question 5.10

[DShim27]

I'm having trouble understanding this question from the 2020 text.

If volatility A = 20% and volatility B = 40% and a portfolio is formed with half the money invested in each stock, then portfolio volatility must be

B. between 10% and 40%

Can anyone please help?

 

[amit.m.sharma]

Portfolio volatility = SQRT (w1^2*sigma1^2 + w2^2*sigma2^2+ 2*rho*w1*w2*sigma1*sigma2)
w1 =w2 = 0.5
sigma1 = 0.2
sigma2 = 0.4
rho can be anywhere between -1 and +1. So, calculate the portfolio volatility when rho = -1 and again when rho = +1. This will give you the two limits for the portfolio volatility.

I don't have the text but when I did the calculations, the answer I got was between 10% and 30%.

 

[DShim27]

This is very helpful. Thank you, Amit!

 

[David Harper CFA FRM]

I agree with @amit.m.sharma portfolio volatility here ranges from

• sqrt[0.50^2*0.20^2 + 0.50^2*0.40^2 + 2*0.50*0.50*0.20*0.40*(-1.0)] = 10.0%
• sqrt[0.50^2*0.20^2 + 0.50^2*0.40^2 + 2*0.50*0.50*0.20*0.40*(+1.0)] = 30.0%

 

[DShim27]

Thank you, David.

If we remove the "sqrt" in the equation, would that give us the variance?

So volatility^2 = variance?

 

[David Harper CFA FRM]

Yes @DShim27 that's always true!

 

[amit.m.sharma]

Not sure if this is already covered in the notes but a portfolio with 50% investment in 2 stocks simplifies the calculation for the range of volatility of the portfolio.
Portfolio volatility = SQRT (w1^2*sigma1^2 + w2^2*sigma2^2+ 2*rho*w1*w2*sigma1*sigma2)
if w1 = w2 = 0.5,
Portfolio volatility = SQRT (0.5^2*sigma1^2 + 0.5^2*sigma2^2+ 2*rho*0.5*0.5*sigma1*sigma2)
= 1/2SQRT (sigma1^2 + sigma2^2+ 2*rho*sigma1*sigma2)
If rho = +1, Portfolio volatility = 1/2*SQRT*(sigma1+sigma2)^2 = 1/2*(sigma1+sigma2)
If rho = -1, Portfolio volatility = 1/2*SQRT*(sigma1-sigma2)^2 = 1/2*|(sigma1-sigma2)|

So the range for sigma1 = 20%, sigma 2 = 40% is
upper limit = 1/2*(20%+40%) = 30%
lower limit = 1/2*|(20%-40%)| = 10%

 

[anandcp]

I'm having trouble understanding this question from the 2020 text.

If volatility A = 20% and volatility B = 40% and a portfolio is formed with half the money invested in each stock, then portfolio volatility must be

B. between 10% and 40%

Can anyone please help?

I thought it was 30%.
But the book says it is between 10% and 40%.
Both are choices.

 

[amit.m.sharma]

If you remove the restriction of the weighting then the portfolio volatility ranges between 10% and 40%. The volatility is lowest when rho =-1 and w1*(1-w1) is maximized. Using derivatives you can show the maximium occurs when w1 = 0.5.
So the lowest volatility is 10% and it occurs when w1=w2=0.5 and rho = -1.
The highest volatility is 40% and it occurs when w1=0 and w2 = 1
I doubt this is what GARP had in mind as they have fixed the weighting at 0.5 each.

 

[David Harper CFA FRM]

But @amit.m.sharma I agree with you that's not what the question had in mind, but if the weights are flexible then I find the MVP to be at weight(A) = 2/3, weight(B) = 1/3, ρ(A,B) = -1.0 with variance and standard deviation equal to zero (not 10%).

 

[amit.m.sharma]

But @amit.m.sharma I agree with you that's not what the question had in mind, but if the weights are flexible then I find the MVP to be at weight(A) = 2/3, weight(B) = 1/3, ρ(A,B) = -1.0 with variance and standard deviation equal to zero (not 10%).

Agreed. In my excitement to solve the problem quickly I didn’t realize that I should have looked at minimizing the function w1^2*0.04+(1-w1)^2*0.16-2*w1*(1-w1)*0.08
Differentiating by w1 and setting to 0 gives
0.08*w1+0.32*(1-w1)*(-1)-0.16*(1-2*w1) = 0
i.e., (0.08+0.32+0.32)*w1 = 0.48
So, w1 = 0.48/0.72 = 2/3