YouTube T3-06: Minimum variance hedge

[Nicole Seaman]

The minimum variance hedge is based on the slope of the regression line. If we use the number of contracts implied by the minimum variance hedge ratio, then we are minimizing the volatility of the net position (i.e., the portfolio that consists of the exposure plus the hedge).

David's XLS is here: https://www.dropbox.com/s/qyd9j5n9i3z0ptx/031418-mvh.xlsx?st=dtgcafhz&dl=0

 

[jchun8523]

Hi there, would you mind clarifying the concept to me pls?

Can't tell the difference between these two formulas of MVP:

- correl(a,b)*stdev(a)/stdev(b)

- [variance(a) - correl(a,b)*stdev(a)*stdev(b)]/[variance(a) + variance(b) - 2*correl(a,b)*stdev(a)*stdev(b)]

When do I use each one of them?
thanks!

 

[gsarm1987]

Hi there, would you mind clarifying the concept to me pls?

Can't tell the difference between these two formulas of MVP:

- correl(a,b)*stdev(a)/stdev(b)

- [variance(a) - correl(a,b)*stdev(a)*stdev(b)]/[variance(a) + variance(b) - 2*correl(a,b)*stdev(a)*stdev(b)]

When do I use each one of them?
thanks!

@jchun8523 These are used for analytical methods

Its using of variance, covariance matrix concept, in that each asset (is an integrated part of portfolio) has its risk footprint and some correlations with other assets and even portfolio as a whole. These are analytical methods, where you establish the vol, corr etc from past and then apply them ahead for new decisions etc.

A risk management tool used to check the impact on overall portfolio risk when you add an asset class (say B in our example).

You must have heard the term Marginal contribution to risk, its done with help of the above: MCTR (beta X Vol PF). now if you want to find the average contribution to risk, just multiply MCTR with the weight of B you'd like to have in the portfolio.

Now back to your question above, observe:

A portfolio(PF) has A and b assets, each assets have their own returns and volatilies (Vol a and Vol b)
PF vol = vol a ^2 + volb ^2 - 2 cor(a,b)*vol a * vol b. (Notice the sign before '2' is negative to say if the asset is negatively correlated, example could be some hedge position)
Note: the last term is saying covariance (a,b), which is such that
Cov (a,b) = Corr (a,b) * Vol a * Vol b

If you want to find the Beta (which comes out of the regression, its the regression slope observed from the historical record of B and PF performances), of say Asset B, with respect to the whole portfolio, or say if the portfolio performs well, then how mauch assets B would have performed, or say Asset B has performed well and how much Portfolio overall would have performed, that relation is mapped with help of Betas. Think of CAPM, where you are assessing how would an asset respond when the market is performing in a certain way: simple example, if market returned 10%, your asset only returned 5%, this could imply a 50% beta or responsiveness to the market. Note, performance does not just return, we also have to take risk into consideration

Think about CAPM, Beta can be calculated as : Cov (asset B, Market PF)/variance of Market PF or Corr (b,market PF)*vol b/vol Market. As you know variance is vol square.
Now bringing context to portfolio above, assume portfolio is the market, Asset B is part of that market, then beta should be:

Beta B = Cov (b,PF)/vol of PF = Corr(b,PF)*Vol b * Vol PF/Variance of PF, simplify and you get Corr(b,PF)*Vol b/Vol PF.

 

[gsarm1987]

The minimum variance hedge is based on the slope of the regression line. If we use the number of contracts implied by the minimum variance hedge ratio, then we are minimizing the volatility of the net position (i.e., the portfolio that consists of the exposure plus the hedge).

David's XLS is here: http://trtl.bz/2FQxUnN

@Nicole Seaman Thank you for sharing, can you please also add a link to the spreadsheet used in the video?

 

[jchun8523]

@jchun8523 These are used for analytical methods

Its using of variance, covariance matrix concept, in that each asset (is an integrated part of portfolio) has its risk footprint and some correlations with other assets and even portfolio as a whole. These are analytical methods, where you establish the vol, corr etc from past and then apply them ahead for new decisions etc.

A risk management tool used to check the impact on overall portfolio risk when you add an asset class (say B in our example).

You must have heard the term Marginal contribution to risk, its done with help of the above: MCTR (beta X Vol PF). now if you want to find the average contribution to risk, just multiply MCTR with the weight of B you'd like to have in the portfolio.

Now back to your question above, observe:

A portfolio(PF) has A and b assets, each assets have their own returns and volatilies (Vol a and Vol b)
PF vol = vol a ^2 + volb ^2 - 2 cor(a,b)*vol a * vol b. (Notice the sign before '2' is negative to say if the asset is negatively correlated, example could be some hedge position)
Note: the last term is saying covariance (a,b), which is such that
Cov (a,b) = Corr (a,b) * Vol a * Vol b

If you want to find the Beta (which comes out of the regression, its the regression slope observed from the historical record of B and PF performances), of say Asset B, with respect to the whole portfolio, or say if the portfolio performs well, then how mauch assets B would have performed, or say Asset B has performed well and how much Portfolio overall would have performed, that relation is mapped with help of Betas. Think of CAPM, where you are assessing how would an asset respond when the market is performing in a certain way: simple example, if market returned 10%, your asset only returned 5%, this could imply a 50% beta or responsiveness to the market. Note, performance does not just return, we also have to take risk into consideration

Think about CAPM, Beta can be calculated as : Cov (asset B, Market PF)/variance of Market PF or Corr (b,market PF)*vol b/vol Market. As you know variance is vol square.
Now bringing context to portfolio above, assume portfolio is the market, Asset B is part of that market, then beta should be:

Beta B = Cov (b,PF)/vol of PF = Corr(b,PF)*Vol b * Vol PF/Variance of PF, simplify and you get Corr(b,PF)*Vol b/Vol PF.

Good evening,

thanks for the reply and sorry for my ignorance, but I couldn't fully understand whether they represent the same formula or I need to use them in different occasions.

I did understand that Beta represents:
- correl(a,b)*stdev(a)/stdev(b) or COV (a,b)/ variance of b (such as market)


- [variance(a) - correl(a,b)*stdev(a)*stdev(b)]/[variance(a) + variance(b) - 2*correl(a,b)*stdev(a)*stdev(b)] and this second part represents de VAR(a-b).

But in order to find the MVP, maybe use either formula?

thanks again!

 

[gsarm1987]

Good evening,

thanks for the reply and sorry for my ignorance, but I couldn't fully understand whether they represent the same formula or I need to use them in different occasions.

I did understand that Beta represents:
- correl(a,b)*stdev(a)/stdev(b) or COV (a,b)/ variance of b (such as market)


- [variance(a) - correl(a,b)*stdev(a)*stdev(b)]/[variance(a) + variance(b) - 2*correl(a,b)*stdev(a)*stdev(b)] and this second part represents de VAR(a-b).

But in order to find the MVP, maybe use either formula?

thanks again!

No worries, your first expression looks like beta between a and b, the second expression appears to be variance of B over variance of entire portfolio. the negative sign before 2, implies opposite positions or negatively correlated positions. Looks like a hedging scenario. It resonates with delta hedging. Say if you have X number of spots, how many number of Futures would you need to hedge. If you dont mind, can you share where you've read these two expressions, please give page reference or screen shot. ill have a look, may be i need to revise something

 

[Nicole Seaman]

@Nicole Seaman Thank you for sharing, can you please also add a link to the spreadsheet used in the video?

@gsarm1987 The spreadsheet link is in the original post above.

 

[jchun8523]

No worries, your first expression looks like beta between a and b, the second expression appears to be variance of B over variance of entire portfolio. the negative sign before 2, implies opposite positions or negatively correlated positions. Looks like a hedging scenario. It resonates with delta hedging. Say if you have X number of spots, how many number of Futures would you need to hedge. If you dont mind, can you share where you've read these two expressions, please give page reference or screen shot. ill have a look, may be i need to revise something

sure!

Might be wondering if I need to use the second formula just to find the minimum variance on the CAPM Model and the first one on a hedge position.

would you mind share some light over it?


thanks!!


T3.CH8 (page 12)


FormulaSheet (16/154)

 

[gsarm1987]

we cannot interchangeably use these formulas, because:

When completely hedged, any change in market would bring about zero change in the Asset position. This is the case with the minimum variance porfolio. However, the question is not related to that Efficient frontier thing.

Instead its about measuring the hedging ratio (between asset and its derivative bet on its opposite exposure). If you recall, the efficient frontier portfolio, it doesnt allow shorting. Shorting kicks in when you introduce that tangential CAPM line, which is obviously not at the point where MVP lies, but is on the same curve, rather at different point.

In our case we are shorting a future/forward, to make a hedge, therefore its not in the context of MVP. in the same formula sheet that you have mentioned, see pages 39 and 40. thats what we would use.


Plz let me know if that makes it clear

 

[jchun8523]

we cannot interchangeably use these formulas, because:

When completely hedged, any change in market would bring about zero change in the Asset position. This is the case with the minimum variance porfolio. However, the question is not related to that Efficient frontier thing.

Instead its about measuring the hedging ratio (between asset and its derivative bet on its opposite exposure). If you recall, the efficient frontier portfolio, it doesnt allow shorting. Shorting kicks in when you introduce that tangential CAPM line, which is obviously not at the point where MVP lies, but is on the same curve, rather at different point.

In our case we are shorting a future/forward, to make a hedge, therefore its not in the context of MVP. in the same formula sheet that you have mentioned, see pages 39 and 40. thats what we would use.


Plz let me know if that makes it clear

thanks for helping me out!