Bodie's Notes question 7...Help needed!

[shivanin]

I don't understand how this works?

"For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $100,000 position (either long or short) in each stock. Net market exposure is zero, but firm-specific risk has not been fully diversified. The variance of dollar returns from the positions in the 20 stocks is: 20 × [(100,000 × 0.30)] = 18,000,000,000 The standard deviation of dollar returns is $134,164. "

How was $100,000 position calculated for 20 stocks? And from where $134,164 value came from?

 

[David Harper CFA FRM]

Hi @shivanin Who's boris? The interim formula appears to be missing something as 20 × [(100,000 × 0.30)] doesn't equal $18 million. But i do notice that if the pairwise correlation is zero, and the volatility of each stock is the same 20%, then portfolio dollar standard deviation = 30%/sqrt(20)*(20*$100,000) = $134,164; ie., if ρ=0 with identical or average per-security standard deviation of σ_%(i)^2, then σ_%(p)^2 = σ_%(i)^2/n. So the answer would seem to verify that the assumptions include ρ=0. Oh, and $134,164 is the dollar volatility, and if you square it, you get the dollar variance whish is $18,000,000.

... and actually, i can see what boris (?) meant to do with the interim calculation:
$100,000*30 = $30,000 is the per-position dollar volatility. If we assume ρ=0, then portfolio dollar variance = [($100,000*30)^2]*20 = $18,000,000,000 portfolio dollar variance; then sqrt($18,000,000,000) = $134,164.08 portfolio dollar volatility. That's just using var(a+b) = var(a)+var(b) when the cov(.) term is zero.

... and actually again, the point that "but firm-specific risk has not been fully diversified" is an advanced way to introduce the ρ=0 assumption: if all of the risk is firm-specific, then ρ=0.

 

[shivanin]

Sorry for the typo David....I ment Bodie's note. Question 7.

I still don't get where $100,000 come from?

Note from Nicole: I updated the title from "Boris's notes" to Bodie's notes for search purposes.

 

[David Harper CFA FRM]

@shivanin oh Bodie, thanks, that explains the high-level introduction of the ρ=0 assumption ! The $100,000 is given as an assumption: For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $100,000 position (either long or short) in each stock.

 

[shivanin]

Ohh... Got the calculation... But since 30% is volatility per stock should'nt we first do (30%)^2 and then multiply by $100,000 for variance of each stock?
Y are we doing ( 100,000*30%)^2?

 

[David Harper CFA FRM]

variance(aX + bY) = a^2*Variance(X) + b^2*variance(Y); assuming ρ=0. It's the same as when we square the weights in the familiar portfolio = σ(p)^2 = w(a)^2*σ(a)^2 + w(b)^2*σ(b)^2 + 2*w(a)*w(b)*cov(a,b)

 

[shivanin]

That clarifies it perfectly.... Thanks a lot David... For timely response to the doubts. Very helpful.

This makes it clear... Choosing BT for FRM study was a great decision for me.