Flash quiz 2008 Market (Round 1) - Question 7

[sathyat]

Hi David,
Q) If a trader wants to increase the beta of a portfolio by using S&P500;index futures, he/she should:
The Answer says- Take long Positions.

Let us say the beta of the portfolio is already more than 1. say 1.3. And he takes long position in S&P;Index. Let us say that the initial portfolio is worth 10 million, and he hedges for 1 million.

Now the new beta is: (10*1.3 + 1*1)/(10 + 1) = 1.27.

Now the beta has decreased.

Based on this, how can we say that a long position in S&P;will always increase the beta ?

Regards,
Sathya

 

[David Harper CFA FRM]

Hi Sathya,

Aren't you recomputing beta for a $1 investment in the S&P;, rather than a futures *notional* of $1 MM?
I think the new beta = [1.3(10MM) + 1MM]/$10 MM = 1.4 beta; i.e., i don't think the $1 MM goes in the denominator.

You point give me pause, I admit (b/c it means portfolio beta has no limit, as my question implies), but on second thought, Hull's 3.4 etc indeed implies that via the *leverage* of futures, beta can be increased ad infinitum. If we rearrange 3.4, then long futures contracts always add beta equal to: Value of Futures contracts/Portfolio (as above we are adding $1/$10 = 0.1)

David

 

[phwdisse]

Dear David,

I have question about all the formulas in the different reading.

Market risk
The beta to hedge a equity stock. In your answer of 6 august you said it was :“Value of Futures contracts/Portfolio (as above we are adding $1/$10 = 0.1)” . Which I can understand.

There is also the Beta β i = sum(C i ,C )/VAR(C) which is the Beta mentioned in “Contribution of Division CAR” .

Investment Risk

In investment risk there is a beta mentioned in marginal Var ∑w/(w’∑w).

How are these beta related?

I hope you can help me with this.


On page 22 of the Investment risk notes there are two formulas about portfolio variance. Isn’t there missing a 2 in both of the formulas.

Regards Paul

 

[David Harper CFA FRM]

Paul,

(sorry for delay). I thought your point about the commonality of beta deserved a screencast.
Please view here (9 minutes). I hope this is helpful.

In short, these beta are all: covariance/variance. Even Jorion's, which is in matrix format. And Stulz is using different measures (cash flows) but his also covariance/variance.

Re page 22, yes that is an error. Yikes, ugly. Thank you for pointing it out.

David