R10.P1.T1.BODIE_CH10_DIVERSIFICATION_of_RESIDUAL_RISK

[gargi.adhikari]

Hi,
In Reference to R10.P1.T1.BODIE_CH10_DIVERSIFICATION_of_RESIDUAL_RISK :-
The Weighted-Variance of the Residual Risk = Avg-Variance of Residual Risk/ N =[ (Std-Dev of Residual Risk) ^ 2 / N ] / N
The Avg-Volatility = ( Std-Dev/ N ) = 40%
So, the Last term should be just (40% ) ^2 as the 40% is the AVG-Volatility..So the AVG-Variance is just the Square of the AVG-Volatility..?

 

[David Harper CFA FRM]

Hi @gargi.adhikari Sorry I don't follow This looks okay to me (except there is an interim typo, the '2' does not belong in '2σ^2e(i)' above, but it does not impact the final calculation thankfully). To illustrate, let's say that n = 3 equally-weighted securities in the portfolio, where for convenience we are assuming that each security has a non-systematic volatility, σ(e_i) = 40.0% per the slide above; then the nonsystematic variance of the portfolio, σ^2(e_p) = (1/n)*Σ[σ^2(e_i)/n] = (1/n)*average_σ^2(e_i). In this example of n = 3 and σ(e_i) = 40.0%, that's two equivalent ways to get the portfolio's nonsystematic variance:

• (1/3)*[40%^2/3 + 40%^2/3 + 40%^2/3] = 0.0533; but the point of the final step in the formula to simply to (please note the horizontal hat above the sigma that indicates average):
• (1/3)*average_σ^2(e_i) = (1/3)*40%^2 = 0.0533. The graph is plotting volatility not variance of course.

So this looks okay to me. Here is the XLS in case you think i'm missing something: https://www.dropbox.com/s/oj21bgsa2rdjzcj/0929-bodie-residual.xlsx?dl=0 Thanks!

 

[gargi.adhikari]

Got it Thanks so much @David Harper CFA FRM

 

[gargi.adhikari]

Hi @David Harper CFA FRM - my apologies for nudging you over this again...I was revisiting this topic..and I seem to have some hiccups over the calculation below.
Issue # 1 : 40% is the Avg Volatility for the Non-Sytemic-Firm-Specific-Risk. So as per the Screenshot 2 , we should divide the 40%^2 by N= the No of assets in the Portfolio.But N is not given here....
Issue # 2 : Also, the 40%^2 /N does not seem to be added to { ((.50)^2) * ((25%)^2) } = which itself evaluates to 1.5625..... So the Avg Volatility for the Non-Sytemic-Firm-Specific-Risk is not factored into the calculation at all...am I missing a point here....?
(I found it a bit hard to decipher the spreadsheet - as it was not labelled and I had a hard time figuring what is what...wish I could- so sorry to be taking up some bandwidth over this...)
Screenshot 1:

Screenshot 2:

 

[David Harper CFA FRM]

Hi @gargi.adhikari No worries!

1. Re issue #1, yes, that is correct per the slides that the non-systematic (aka, idiosyncratic) risk enters the portfolio variance as σ^2/N. Just so we have perspective, this is just because per single-factor (~ capm), E(r) = α + β*F + e; and we can take the variance (but make a key assumption of independence between β and e which zeros out the covariance term) so that σ^2[E(r)] = σ^2[α + β*F + e] = σ^2(β*F) + σ^2(e) which is just the variance(a+b) = variance(a) + variance(b) if (a) and (b) are independent. But β is a constant, so that σ^2[E(r)] = β^2*σ^2(F) + σ^2(e). Just to de-mystify the portfolio variance, here. Re: "But N is not given here...." N is the x-axis (it is in the time, apologies, it should label the axis). The plot/XLS is showing the volatility of the portfolio as it converges toward sqrt[β^2*σ^2(F)] = β*σ(F) = 0.50*25% = 12.50% as the number of stocks, n, increases (to the right).
2. The plot/XLS does add the idiosyncratic component per (eg) =$D$5^2*$D$6^2+$D$7^2/B9. This is why the first dot is located at 41.9% = sqrt[0.50^2*25%^2 + 40%^2/1] and the second dot (n = 2 stocks ) is located at 30.9% = sqrt[0.50^2*25%^2 + 40%^2/2]. The first slide below, by virtue of the right-area, means to suggest convergence toward the 12.5% because, as (n) gets larger, the idiosyncratic component approaches zero,; i.e., σ^2/n → 0 as n → ∞. I hope that clarifies, thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so so much for clearing up the above- Totally Get it now !!THANK YOU !!!!!!!!!!!!!!!!