I'm computing deltas for the 0-3% of the CDX.NA.IG4 index. I use market calibrated base correlations, roughly 9% at the 3% strike. However, I really use so-called normalized strikes, or strike / EL(Pool). Now, when I blip individual credit curves to get my deltas, the term EL(Pool) changes as well, thence my interpolated base correlation changes. I find that if I "freeze" my correlations to the initial values pre-blipping, the delta is about 12x, whereas if I adjust them as described above, I get closer to 15x deltas. Effectively, the latter incorporates a cross-derivative DP/DCorr * DCorr/DSpread. Which one is right? My instinct is the latter, as long as I define my correlation curve to normalized strikes.Views?

QuoteOriginally posted by: j20056I'm computing deltas for the 0-3% of the CDX.NA.IG4 index. I use market calibrated base correlations, roughly 9% at the 3% strike. However, I really use so-called normalized strikes, or strike / EL(Pool). Now, when I blip individual credit curves to get my deltas, the term EL(Pool) changes as well, thence my interpolated base correlation changes. I find that if I "freeze" my correlations to the initial values pre-blipping, the delta is about 12x, whereas if I adjust them as described above, I get closer to 15x deltas. Effectively, the latter incorporates a cross-derivative DP/DCorr * DCorr/DSpread. Which one is right? My instinct is the latter, as long as I define my correlation curve to normalized strikes.Views?This is simillar to the sticky strike/sticky delta in equities and you have to assume something about the dynamics of the correlation smile. and take the total derivative?

Having read various notes on computing tranche deltas using base correlations there doesn’t appear to be a single, standard approach. Maybe this shouldn’t be so surprising. That said, given the variations I would like to poll opinions on the application of the base correlation framework for calculating tranche deltas.Before jumping to computing tranche deltas I thought, for clarity and in the spirit of getting some discussion going, it would be useful to state definitions up-front and outline the steps I use to arrive at tranche deltas. Should any of this be incorrect please advise....and bear with me this is quite long winded.Base Correlation Framework:Several variations on a theme exist. The approach I have adopted is the tranche present value one.For a given equity tranche my present value nomenclature is as follows:PV(0-x%,p,s)denotes the present value of tranche 0-x%, evaluated assuming correlation value p for computation of the loss distribution and payments based on a spread of s.PV(0-x%,p_0-x%,s_0-x%) = 0 (fair spread)where p_0-x% is the implied correlation for the equity tranche 0-x% and s_0-x% is the market quoted (or equivalent, given there aren’t any quotes as such for equity tranches >3%).Assuming expected losses are additive, the PV of a non-equity tranche can be stated asPV(x-y%) = PV(0-y%,p_0-y%,s) – PV(0-x%,p_0-x%,s)andPV(x-y%) = 0 = PV(0-y%,p_0-y%,s_x-y%) – PV(0-x%,p_0-x%,s_x-y%)where the spread paid is the market fair spread for that non-equity tranche.Solving this equation for successive equity tranches enables one to extract the piecewise market base correlation skew.General Definition of Delta:Here the ‘tranche delta’ vernacular relates to the general notion of tranche leverage, or sensitivity to the underlying index. For a given bps shift in the underlying credits of the index, this amounts to the ratio of change in the present value of a $1 notional tranche to the change in present value of a $1 notional of the index.In the case of an at inception fair market priced trancheTranche_x-y%_MtM = PV*(s_fair_x-y%) Index_MtM = PV*(s_fair_0-100%)where PV* denotes loss distributions in the present value calculation have been computed under the new regime of underlying credit spreads, e.g. the 1bps bumped spreads.Tranche_x-y%_Delta = (Tranche_x-y%_MtM / Index_MtM) * (Index_Notional / Tranche_x-y%_Notional)Deltas Calculated Using Base Correlations:Taking 3-7% as an example. Under the base correlation framework, Tranche_MtM for a non-equity tranche is computed using the additive PV principle.The 3-7% tranche is assumed to receive the initial fair spread of s_3-7%. Hence the MtM under the underlying change in credit spreads3-7%_MtM = PV*(0-7%,p_0-7%,s_3-7%) – PV*(0-3%,p_0-3%,s_3-7%)index_MtM = PV*(0-100%,p_independent,s_0-100%)where * denotes loss distributions computed using the bumped underlyingsusing the general definition of tranche delta3-7%_Delta = (3-7%_MtM) / (Index_MtM) * (1/(0.07-0.03))Base Correlation Scaling When Calculating Deltas?:Dealers price off-the-run tranches by interpolation along the market skew curve to adjust for the risk associated with different attachment and detachment points. In the base correlation framework, scaling of first loss detachment points has been suggested as one approach to interpolate bespoke portfolio tranche pricing where the bespoke portfolio is similar in geographic composition and term to a market index. When calculating deltas, the underlying risk in the index changes. Is it customary to use this scaling logic when computing deltas? If so what form of scaling is recommended? Two come to mind: scaling by EL ratio; or scaling by index spread ratio?Example:Attached, sample tranche mid-price data taken from CDX.IG.NA 22 July 2005 and the associated underlying CDS mid-price spreads. In the same Excel sheet some corresponding tranche deltas from quote providers, along with theoretic values calculated using the base correlation approach outlined above. Detachment point scaling methods when determining skew correlations have been investigated.Referring to the computed values, regardless of scaling or not, calculated deltas do not fit the quoted deltas consistently across the capital structure. No apparent error exists in my underlying extracted base correlation skews, they match dealer quotes to the first d.p.I would be interested to know from others if they obtain similar deltas using the base correlation framework and the market data included in the Excel sheet. If so, why the noiseyness of delta values compared to the market quotes? Is this acceptable?

I get similar delta results to yours, to around 0.1-0.2. I didn't see base correlation curves in your data, but backed it out from the tranche levels you suppied then used them to calculate deltas.My model is a "market-standard" 1 factor semi-analytic Gaussian copula.Perhaps since May 05 some dealers either (i) adjust their deltas based on their judgment / forecasts of the market or (ii) are experimenting with different copulas.I assume that your dealers are not using the LHP model to quote with - this can skew the deltas.

Herbie,Thanks for your delta feedback. Did the numbers broadly agree across the range of different +/- bps changes for the different scaling approaches? I'd be interested in seeing your values if you get a spare minute to upload them.- Do you scale for the change in underlying risk level, either greater or lesser? If so, which scaling method do you favour? - Any feeling on what level of bps shift to use: 0.1bps; 1bps; 5bps; 10bps; other? - Should the same absolute bps shift be used across all credits in the collateral pool, e.g. imposing a 10bps shift on a 25bps credit and a 10bps shift on a 250bps credit just doesn't seem reflective of observed movements. - Would a % change (parallel across the term structure at the individual name level) be more applicable? - Do you average across the deltas computed for both a +ve and -ve bps shift, i.e. compute +1bps delta, compute -1bps delta and average? - Should a different level of bps shift be used when computing different tranche deltas, i.e. say a +1bps shift across all credits when computing a senior tranche deltas versus a 10bps shift across all credits when computing a junior tranche delta? Didn't provide base correlations because my experience is tranche pricing can be quite sensitive to the base correlation numbers generated by a particular model. Also some participants scale the underlying index credits to obtain the same spread for the complete index as the traded index spread quotes. Anyway, for completeness my skew values attached.I assume my dealers are using the same technology, i.e. market standard 1-factor Gauss semi-analytic models, but I will endeavour to check.Look forward to your comments.

Sorry, only ran the non scaled (basic) deltas. I not sure scaling is appropriate anyway, unless your portfolio is bespoke. The portfolio you look at is the index itself.Regarding your other questions, 'delta' is just a scenario - you can pick any you like. If you are concerned about hedging large moves in credit spreads and changes in base correlation, clearly a dv01 measure is not an appropriate hedge.In other words 'delta' here is not necessarily a risk management parameter - it is something that is actually traded. It is the hedge notional of the index based on the scenarios that concern your dealer (i.e. that they want to hedge), and their probabilities. I think that since May 05 dealers have become more sophisticated in identifying and understanding these scenarios.

Hi vespGL150,I came across some of the same questions you mentioned below. Regarding the way of shifting the underlyings, I talked to a few dealers and was told they all simply use the additive shift (i.e., they shock all underlying reference entities by 1 bp). Shifting the underlying differently (such as by 1% change) will result a different delta.You had a very good summary of Base Correlation framework. I have a question regarding computing delta and hope you guys can shed some lights.Say I am interested in computing the delta of the 3~7% tranche. Under the base correlation framework, the delta of the 3~7% tranche is simply the difference between the delta of the ficticious 0~7% tranche and the delta of the 0~3% tranche.I know how to compute the 0~3% delta and I know I have to taken into account the upfront payment factor. My question is when we compute the 0~7% tranche delta, do we need to assume that there is also an upfront payment associated with the ficticious 0~7% tranche? It buzzles me because unlike 0~3% tranche, the 0~7% doesn't really exists and I don't know whether I should incorporate the upfront payment factor when compuing its delta.Any insights would be greatly appreciated.Thanks,Ted

guoted,as the delta for the upfront amount is zero (cash has no sensitivity to credit spreads) you don't need to worry about it. (There's no upfront for the 3-7 tranche anyway.)You need to price the 0-3 and 0-7 tranches with about 110bps running (or whatever your quote for the 3-7 is) within your delta calculation.Just to clarify, the 0-3 tranche delta differes from the delta you get for the 0-3 tranche within the 3-7 tranche delta calculation. (As the premium legs differ.)

Hi Complexity,Thank you for your reply. I probably didn't ask my question very clearly in the previous post... I'll try to rephrase it again here.As I understand, under the base correlation framework, the delta of 3~7% tranche is simply the detla of the 0~7% tranche minus the delta of the 0~3% tranche. The delta of the 0~3% tranche is calculated by dividing the tranche DV01 by the index DV01. Since the tranche DV01 of the equity tranche is affected by the upfront payment, I made sure my calculation reflects that. The next step would be to compute the delta of the 0~7% tranche, which is a ficticious tranche. Just like before, the delta of the 0~7% tranche is equal to the DV01 of the 0~7% tranche divided by the index DV01. However I am not sure about how to compute the DV01 of the 0~7% tranche mainly because i don't know what upfront payment assumption is made for this ficticious tranche that doesn't really exist. For example, if I also assume that there's a 500 bps running and an upfront payment for this tranche, I'll get one answer. If I assume all the premium is paid on a running basis, i'll get another answer for the tranche DV01.... Am I taking the wrong approach? How would you do the calculation? It would be great if you could provide some details.Thanks again,Ted

Ted,the 0-3 equity tranche is different from the 0-3 tranche used within the 3-7 portfolio. They have the same default leg but different premium legs. As I mentioned before, the upfront payment has zero spread sensitivity. Thus, it does not matter whether you include it in your delta calculation or not. There are 3 legs in the 0-3 tranche: cash + premium leg + default leg. You calculate DV01's as follows: DV01_tranche = DV01(cash) + DV01(prem leg) + DV01(default leg) = DV01(prem leg) + DV01(default leg). The delta exchange would then be DV01_tranche / DV01_index.Assmue the 3-7 tranche trades at 110 bps. Then, you carry out all tranche calculations at 110bps. DV01_tranche37 = DV01_tranche07 - DV01_tranche03. You could also rewrite it as follows: DV01_tranche = DV01_defaultleg + DV01_premiumleg = DV01_defaultleg + 1/shift * (BPV_up - BPV) * 110bps. Or, DV01_tranche37 = (DV01_premiumleg07 - DV01_premiumleg03) + 1/shift * ( (BPV_up07 - BPV_up03) - (BPV_07 - BPV_03) ) * 110bps.For the equity 0-3 piece you'd compute: DV01_equity = DV01_defaultleg + 1/shift * (BPV_up - BPV) * 500bps

Agreed on the delta simply being a 'scenario'. - I guess I was curious if anyone had any thoughts on the consensus / best practice regarding what level of bps shift to use respectively for junior, mezz and senior tranches.- Also, while a parallel 1bp shift to all credits is simple to apply is it realistic / sensible? Any thoughts on parallel shifts of different magnitudes to credits of different quality in the portfolio? I understand that this may give a different delta, but is it a more robust delta that is more reflective of actual behaviour?Turning to the issue of scaling when calculating deltas. - My understanding of scaling is that it is a means to account for the fact that a 3-7% tranche in a 100bps index is a very different beast, in terms of its characteristics (i.e. equity, mezz or senior), to a 3-7% tranche in a 25bps index. - Scaling (EL, spread, other?) of equity tranche detachment points to determine correlations to be subsequently used in the base correlation pricing framework is a crude means of accounting for this. - While I have reservations of its extensibility from say the CDX to significantly different portfolios (e.g. try pricing iTraxx tranches using a scaled CDX) or one of different tenor, I would have thought that a portfolio of identical tenor comprised of identical names would be about as good as you could hope for in terms of suitability for scaling. - When computing tranche deltas if you are shocking the underlying credits by say 10bps this creates a material change in risk of the index. I would have thought that if one then used the raw (unadjusted) skew to compute the equity tranche MtMs this wouldn't implicitly capture the behavioural difference of the tranche in the new shocked index regime? Any thoughts?- Clearly the smaller the underlying credit shock, the smaller the impact of the change in tranche characteristics. However, if the shock is too small, I would imagine numerical noise will start to creep in. This then comes back to the choice of delta 'scenario' and my question on 'best practice' above.This relates back to j20056's original point on this thread. Thoughts on this would be greatly appreciated.

Hi Complexity,Thank you so much for taking time to answer my question. I think I understand the 'big picture' now.However I do have a few questions regarding your post and hope maybe you could clarify a little further:1) In calculating the DV01 of the 3~7% tranche, according to your approach, shouldn't the formula be: DV01_tranche37 = DV01_DefaultLeg37 + DV01_PremiumLeg37 = (DV01_DefaultLeg07 - DV01_DefaultLeg03) + (DV01_PremiumLeg07(110bps) - DV01_PremiumLeg03(110bps)) = (DV01_DefaultLeg07 - DV01_DefaultLeg03) + [PV_PremiumLeg07_afterShift(110bps) - PV_PremiumLeg07_beforeShift(110bps)] - [PV_PremiumLeg03_afterShift(110bps) - PV_PremiumLeg03_beforeShift(110bps)]Your formula was "DV01_tranche37 = (DV01_premiumleg07 - DV01_premiumleg03) + 1/shift * ( (BPV_up07 - BPV_up03) - (BPV_07 - BPV_03) ) * 110bps" and I am not sure whether they are the same or not (I wasn't sure what 'BPV' stands for ... )? Could you kindly confirm it please?2) Another question: in the above formulas (whether yours or mine), are ALL those DV01s computed based on $1 notional? I personally found the scalings a bit confusing when we do delta calculations...Regards,Ted

Ted, BPV stands for basis point value. I.e. the value of a 1bp premium leg.

Hi vespaGL150;Thank you for detail information about cdo delta calculation. I don't know how to calculateDV01(0-100%) as there is no market quote are available for CDX(30%-100%). Do we justsupport the spread for 30%-100% zero.Thanks.

the 30-100 spread can be calculated synthetically. basically the duration and notional weighted sum of tranche spreads should equal the index spread. this is true since you should have the same amount of risk in the index as you do if you bought all the tranches on the index. you should get a spread of around 5bps or so for 30-100%. as far as calculating deltas i think most people do it by shifting all the underlying curves by 1bp. this isn't very satisfying in my experience since when the cdx sells off it isn't because all names sell of by 1bp, it's usually because a couple of names blow out (i.e. gmac). doing a proportional shift is better. in other words there are two ways to move the index by 1bp - shift all the names by 1bp or shift the higher names by >1bp and lower names by <1bp. this works a lot better.