Grinold ch 14

[shanlane]

Hello,

I just watched the video for ch 14 for Grinold and there were a few things that stumped me. For instance, at one point you said that if the benchmark really over-performs, then the active return will be high. But the formula says that active return=portfolio return-benchmark return. Wouldn't this mean that the active return would be less if the benchmark return is high? I think you might have meant that the portfolio return would be high if the benchmark return was high, then again, you obviously know this material way better than I do.

Also, for the two formulas on slide 11 of the video, does theta equal alpha for the residual case? If so, what does it represent in the active case?

I just ask because if you re-arrange both of those equations you get different values for return on the portfolio:

Residual: return on portfolio=theta plus Beta*return of benchmark
Acive: return on portfolio=theta + return on benchmark+Beta*return on benchmark.

What is actually meant by the "risk aversion parameter"? I see some formulas that use it, but what does it actually represent and what does the amount represent? Does a risk aversion parameter of .05 really mean anything compared to a value of 0.1?

Sorry for the REALLY long question. The choice of GARP to include this chapter is really bad. You need a LOT of background to even understand what is going on and to understand what the variables represent.

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

I am still recording videos today and tomorrow (it is long and exhausting process), but your compound question will take me time to research etc (I agree Grinold 14 is really a pain). As always, if i could answer it briefly without references, I would do so immediately ... but i can't in this case ... I can definitely try to incorporate your questions into the upcoming video revision of Grinold 14. But, I will try to get to this question as soon as humanly possible, it just won't be immediate b/c it takes extra time. thanks for your understanding.

 

[shanlane]

I completely understand. If it could somehow be in the videos, that would be great.

Also (Moving on to Jorion), I noticed that you said the component VaRs, by construction, add up to the total VaR. Isn't this just an approximation because component VaR is really an approximation? Wouldn't the incremental VaR's actually add up to the total VaR? I know yo prob can't answer this right now either (although it may be slightly more straightforward than the previous one), but if this could also be briefly addressed in the videos it would be awesome.

Thanks!

Shannon

 

[David Harper CFA FRM]

I think i do address this in the video for Jorion Chapter 7, you are now referring to different chapter.

Incremental VaR does sort of approximate component VaR, but it's really rough when the trade is the entire position. Incremental VaR does NOT sum to diversified portfolio VaR, unlike component VaR. In a way, this betrays the imprecision of the approximation (the difference between the curve an a line isn't noticeable for short changes, but it's big for deducting the entire position). I'm not even sure why we (jorion) says they approximate each other, to think about ... but it's only component VaR that sums to diversified portfolio VaR, the incremental VaRs will sum to something different. If i had more time, i would just do a calc, but i am pretty sure about it, thanks

 

[David Harper CFA FRM]

Just real quick, look at Jorion Table 7-1. component vars sum to portfolio VaR : component (CAD) 105.630 + component (EUR) 152.108 = 257.738.
Incremental VaRs are 257.738 - 165 = 92.738 and 257.738 - 198 = 59.738; i.e., portfolio VaR - position = remaining INDIVIDUAL VaR.
Sum of those incremental VaRs = 92.738 + 59.738 = $152.476; i.e., not even close (in haste as i am recording, feel free to check my math)

 

[shanlane]

You do go over it in the other video, but you show how comp VaR is actually an approximation for incremental VaR (using marginal VaR, kind of like the delta normal appx or mod duration*delta y), not the other way around. And that it only really works, as you said, when there are many positions so that the linear approximations are close. I will look over it again, but it seems like if incremental VaR is the actual added VaR for every position and that Component Var is an approximation of that, then the incremental VaRs should add up to total VaR and the the sum of the component VaRs will give a very good approximation when there are lots of positions.

Just a thought.

Thanks again!

Shannon.

 

[David Harper CFA FRM]

Hi Shannon - Yes, I agree with the logic of that in the case where the individual VaR is small relative to portfolio VaR (which may correlate with small position relative to portfolio) b/c the linear approximation (component VaR) should be pretty accurate to the non-linear incremental VaR. But that makes the summation statement very conditional on a scenario, IMO; using Jorion's example, we can find an example of where the difference is huge: individual VaRs sum to $152 MM versus $257. So, I don't see a general basis for the statement that individual VaRs sum to (diversified) portfolio VaR, since any large positions start to make it untrue. Although, I can agree with you that it should hold up in the special case of a relatively granular portfolio, I think. Thanks,

 

[shanlane]

Thank you. I do not wish to take up any more of your time as you have already been so generous in looking at my questions, but I believe you were adding individual VaRs, where my question concerned INCRIMENTAL VaRs, specifically, full revaluation incremental VaRs. I believe the math you did above (I know you are in a huge rush) was flawed. Total (diversified) VaR-individual VaR doesnt really mean anything. If you add the individuals, you will get undiversified VaR, not diversified VaR

Diversified VaR(port+a)-diversified VaR(port)=Incremental VaR(a)

So If you keep taking away assets and their incremental VaRs (which should be lower than their individual VaRs, unless there is perfect correlation), you will eventually end up with one asset left in the portfolio and its incremental VaR=diversified VaR=undiversified VaR.

Not looking for an immediate response, but I think my logic holds.

Looking forward to the new videos!

Thanks again!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Thanks, I really do appreciate your graciousness with respect to my time.

Can you see my primer here: http://forum.bionicturtle.com/threads/component-versus-incremental-value-at-risk-var-level-2.4961/
prompted by GARP's sample two-asset portfolio question here @ http://forum.bionicturtle.com/threads/l2-2-18-portfolio-construction-invest.4191/

In Jorion's case, when I subtract as follows:
257.738 Portfolio VaR - 165 Individual VaR = 92.738 Incremental VaR, I am relying on the fact there are only two assets, so the 165 Individual CAD VaR is the portfolio VaR after subtracting the EUR position!

Portfolio VaR [Portfolio] - VaR [Portfolio - EUR] = Incremental VaR due to elimination of EUR position; but in this case, equal to:
Portfolio VaR [CAD + EUR] - VaR [CAD only] = Incremental VaR due to elimination of EUR position
i.e., only b/c it's two assets can I use VaR [CAD only] to replace VaR[portfolio - EUR]

See if that holds up? Please check out GARP's sample above ... thanks!

 

[shanlane]

Very interesting. It is as if component VaR is static while incremental VaR is dynamic, if that makes sense. In other words, component VaR deals only with the portfolio we currently have, while incremental VaR compares two different portfolios. When there are not many assets, these two portfolios could be VERY different, thus the major difference between component and incremental VaR.

Not looking for a response here, just thinking out loud (or in print, anyway). If we considered a very large portfolio and then considered incremental VaR of every asset, one asset at a time and all of them based on the original portfolio, if the sum of those incremental VaRs would be the total VaR. It makes sense that the comp VaR and incremental VaR are not close for a small portfolio (and I believe it was actually stated a few times in your material that they are only close when the portfolio is large), but as the portfolio gets large it seems as if they would approach each other.

Just a thought.

Thanks again and happy daylight savings day!

Shannon

 

[Aleksander Hansen]

Very interesting. It is as if component VaR is static while incremental VaR is dynamic, if that makes sense. In other words, component VaR deals only with the portfolio we currently have, while incremental VaR compares two different portfolios. When there are not many assets, these two portfolios could be VERY different, thus the major difference between component and incremental VaR.

Not looking for a response here, just thinking out loud (or in print, anyway). If we considered a very large portfolio and then considered incremental VaR of every asset, one asset at a time and all of them based on the original portfolio, if the sum of those incremental VaRs would be the total VaR. It makes sense that the comp VaR and incremental VaR are not close for a small portfolio (and I believe it was actually stated a few times in your material that they are only close when the portfolio is large), but as the portfolio gets large it seems as if they would approach each other.

Just a thought.

Thanks again and happy daylight savings day!

Shannon

Component VaR just decomposes your existing portfolio into individual VaR contributions.
Incremental VaR just multiplies the contribution 'weight' of what you add with the size of that position when using local valuation and is a quick approximation.
Unless you have iid normal returns individual VaR components will not sum up to portfolio VaR (unless you account for codependence).

 

[shanlane]

Interestimg.

Thank you.

Shannon

 

[David Harper CFA FRM]

Incremental VaR just multiplies the contribution 'weight' of what you add with the size of that position when using local valuation and is a quick approximation.

@ahansen: I think you reversed incremental with component? i.e., the incremental is the only one of Jorion's VaR concepts that doesn't rely on the local valuation (beta and marginal VaR).

I think Shannon's reference to Component VaR as "static" is really good: as Jorion's Figure 7-4 shows, the Component VaR is a function of the (straight line) linear approximation, so I think it's better to say that "Component VaR approximates Incremental VaR, for small changes to the position relative to the portfolio" rather than "Incremental VaR approximates Component VaR" because Component VaR is only accurate instantaneously in the CURRENT portfolio. Incremental VaR, by definition a full repricing, is always accurate.

I was thinking about a really simple example; e.g., a two-asset portfolio, both assets = $100, both volatilities = 10%, correlation = 0. And to keep it simple, VaR confidence 84%, so the deviate = 1.0, so basically it's just a two-asset volatility:

• Individual VaRs ($volatility) = $10.0 = $100 * 10% * 1
• Portfolio VaR ($volatility) = $14.1 = $200 * 7.07% portfolio volatility
• Component VaRs are both = $7.07 = 50% of 14.1 = 1.0 beta * 7.07%/10%
• Incremental VaRs are both = $4.14 = $14.1 - $10; i.e., the drop in portfolio VaR when you subtract one position

 

[Aleksander Hansen]

@ahansen: I think you reversed incremental with component? i.e., the incremental is the only one of Jorion's VaR concepts that doesn't rely on the local valuation (beta and marginal VaR).

I think Shannon's reference to Component VaR as "static" is really good: as Jorion's Figure 7-4 shows, the Component VaR is a function of the (straight line) linear approximation, so I think it's better to say that "Component VaR approximates Incremental VaR, for small changes to the position relative to the portfolio" rather than "Incremental VaR approximates Component VaR" because Component VaR is only accurate instantaneously in the CURRENT portfolio. Incremental VaR, by definition a full repricing, is always accurate.

I was thinking about a really simple example; e.g., a two-asset portfolio, both assets = $100, both volatilities = 10%, correlation = 0. And to keep it simple, VaR confidence 84%, so the deviate = 1.0, so basically it's just a two-asset volatility:

• Individual VaRs ($volatility) = $10.0 = $100 * 10% * 1
• Portfolio VaR ($volatility) = $14.1 = $200 * 7.07% portfolio volatility
• Component VaRs are both = $7.07 = 50% of 14.1 = 1.0 beta * 7.07%/10%
• Incremental VaRs are both = $4.14 = $14.1 - $10; i.e., the drop in portfolio VaR when you subtract one position

I don't have Jorion's book in front of me but I seem to recall there being a slight difference in how Jorion and Carol Alexander define the terms.
That being said, I think of it as this:

1. Component VaR just decomposes your existing portfolio into individual VaR contributions so that is compatible with your static and current portfolio. Note that component VaR is not necessarily a linear function, or a first-order approximation.
2. You can derive incremental VaR using either local or full valuation:

• For local, you the increment to portfolio VaR is obtained by multiplying the gradient vector with the respective additional holding for a quick approximation (and subtract the old VaR). Hence, you assume that the gradient vector and the VaR decomposition is invariant to the change [which is incorrect but can be close for small quantities] and that the marginal component is linear.
• For full valuation, by definition, you run the entire VaR calc again, this time with the incremental holding. The increment is then VaRnew - VaRold.

Not sure if we are talking across one another, or if we have two different definitions in mind?

Bascially, incremental VaR would be used by a trader to perform a local valuation to assess the impact of a trade based on the existing component VaR and gradient vector.
The actual increment[al] to VaR would later be calculated using full valuation.

@
I think it's better to say that "Component VaR approximates Incremental VaR, for small changes to the position relative to the portfolio" rather than "Incremental VaR approximates Component VaR"

Just to add to this:
I would say that Component VaR is not an approximation for incremental VaR, but is rather just marginals due to infinitesimally small perturbations.

I agreee that incremental VaR does not approximate Component VaR - it approximates the resulting portfolio VaR [when using local valuation], based on the Component VaR. The Component VaR and the Portfolio VaR will both change, althought using local valuation we 'pretend' that the Component VaR stays the same and is accurate for perturbations that are not infinitesimally small.

 

[David Harper CFA FRM]

I don't have Jorion's book in front of me but I seem to recall there being a slight difference in how Jorion and Carol Alexander define the terms.
That being said, I think of it as this:

1. Component VaR just decomposes your existing portfolio into individual VaR contributions so that is compatible with your static and current portfolio. Note that component VaR is not necessarily a linear function, or a first-order approximation.
2. You can derive incremental VaR using either local or full valuation:

• For local, you the increment to portfolio VaR is obtained by multiplying the gradient vector with the respective additional holding for a quick approximation (and subtract the old VaR). Hence, you assume that the gradient vector and the VaR decomposition is invariant to the change [which is incorrect but can be close for small quantities] and that the marginal component is linear.
• For full valuation, by definition, you run the entire VaR calc again, this time with the incremental holding. The increment is then VaRnew - VaRold.

Not sure if we are talking across one another, or if we have two different definitions in mind?

Total agreement. Not different definitions, I think this is perfectly consistent with Jorion.

IMO, your "2.Local" scenario (approximating incremental VaR with the marginal VaR/gradient) highlights the current debate. It seems clear to me, per your 2.Local that incremental VaR can be approximated. But to generalize this notion to the assertion that incremental VaR ~ Component VaR violates your "for small quantities" criteria.

That's what I meant by linear approximation, (and I'm really just looking at Jorion's Figure 7-4): as marginal VaR is the derivative (change in portfolio VaR w.r.t. change in position), the use of component VaR to approximate incremental VaR uses:
Component VaR = Marginal VaR * Component position

• This is, of course, correct w.r.t component VaR as it "decomposes your existing portfolio into individual VaR contributions so that is compatible with your static and current portfolio."
• But, to use this as an approximation of incremental VaR is to make an incorrect linear approximation: to infer the marginal VaR (constant slope) holds up for the entire subtraction of the position.

Given that, I further agree with your final conclusion (as better than mine) that ultimately neither component approximates incremental nor vice versa.

 

[shanlane]

Great points. The only reason that I brought up the approximation, is that in the case where the change is small (say 1 asset from a portfolio of 100) then the incremental would be very close to the component. Again using Jorion's 7-4, the component VaR (being Marginal VaR*amount of position) gets asymptotically closer to incremental VaR as the change in position gets smaller. I know I am making some gross oversimplifications, but I actually just noticed that Jorion says (p 175) "The incremental VaR is the change in VaR... This is approximated by the component VaR."

I learned a lot from this little discussion.

Thanks!

Shannon

 

[David Harper CFA FRM]

Not to belabor, I just think that it is worth understanding why ahansen's statement ("Component VaR is not an approximation for incremental VaR") is better than the apparently contradicting Jorion ("The incremental VaR is the change in VaR... This is approximated by the component VaR") on page 175. Note that Jorion is saying, btw, component $152,108 approximates incremental $92,738 [sic]. On a portfolio VaR of 257.7, that's an approximation with a ~ 23% error!

The weakness (IMO) with Jorion's statement is that component-as-a-proxy-for-incremental implies a subtraction of the entire position, which may violate the "small quantities" criteria (that's where i tend to say linear). Jorion knows what he's doing: the numbers don't approximate, he knows, as his Figure 7-4 conveys blatantly, it's approximate only for small quantities. I think it's better to say something like "component will not approximate incremental EXCEPT when the component is small [i.e., the error in assuming constant marginal VaR won't be too large]" ...

thanks for great discussion, I learned a lot!