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# Basket Tranches value and risk

#### gautamFRM

##### New Member
Hi David,

When we find correlation in CDS, I understand following :
high correlation makes senior tranches (i.e., high n to default for basket CDS) more expensive (all other things equal) and junior tranches (e.g., 1st to default) less expensive;
low correlation makes senior tranches less expensive and junior tranches more expensive. You can see in my post that the binomial distribution can be used to illustrate this. Also, In terms of payoffs of basket CDs,
More Risky will have more payoff and more price/Value

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So my confusion is:

in case of higher correlation, senior tranche will have more value or less value ?
A: Senior is more risk ... so more payoff/more price,
But in the following question: you said "senior tranche decreases in value due to higher correlation"

I think i am missing something major here.

Should I think in terms of risk or value ?

Thanks,
Abhi

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Abhi,

I think it can get confusing. Can we assume a two-bond portfolio, where each bond has PD = 10%. Consider a 2nd-to-default basket CDS.
• If default correlation = 0, 2TD triggers with probability = 10%*10% = 1.0%.
• If default correlation increases to 100%, probability of 2TD trigger increases to 10.0%; i.e., as correlation increases, the probability that both bonds default increases (here, the senior nth-to-default tranche triggers)
• So the common (robust) dynamic is: higher default correlation increases (decreases) the risk [i.e., probability of defaults that reference] of the senior (junior) tranche
GARP sometimes prefers to refer to value of the tranche rather than the spread (cost of protection) as in the errata question here that you linked to, specifically:
The fund’s chief economist predicts that the default probability will decrease significantly and that the default correlation will increase. Based on this prediction, which of the following is a good strategy to pursue? Correct answer: d. Sell the senior tranche and buy the equity tranche.

Explanation: The decrease in probability of default would increase the value of the equity tranche. Also, a default of the equity tranche would increase the probability of default of the senior tranche, due to increased correlation, reducing its value. Thus, it is better to go long the equity tranche and short the senior tranche.
One challenge (for me, at least) is due to the different between the tranche (which is analogous to a bond) and protection on the tranche (which is a CDS, just a basket CDS). Consider the analogous bond (~ tranche) and CDS (~ senior basket CDS on tranche):
• Start with bond (tranche) at value of $X due to 3% spread (over riskfree) such that CDS spread =3%. Now assume risk of bond increases: • If bond gets riskier, 3% spread increases to (eg) 5% --> price (value) of bond decreases. But, hedged by increase in CDS spread to 5% and increase in price of CDS. I hope that helps, thanks, #### maheshs ##### Member Hi David The screen grab is from the credit risk notes pg 57 (Malz chapter 9) I don't fully understand the statement "But when default rates are relatively high, an increase in correlation can materially increase the IRR of the equity tranche but also increase the losses to the senior bond tranche" I understand the bit about losses to the senior tranche but the not the part that refers to the increase in IRR of equity tranche. I find it contradictory to the question raised earlier in the thread that talks about increase in value of equity tranche when the default probability reduces - perhaps, i am mixing up things. If you get a chance, can you help clarify this please? Thanks Mahesh #### ShaktiRathore ##### Well-Known Member Subscriber You are confused between default rates and correlation. And high value does not imply high IRR. I would think and understand the above results as follows: 1)a At high correlation b/w the tranches and high default probability: High default rate of probability implies high risky equity tranche, because as default rate of senior tranches increases implies risk of losses to equity tranche increase due to higher correlation so that there are losses to senior tranche but at the same time equity tranche IRR increases, as risk increases return increases same logic implies that as risk of equity tranche increase IRR increases. b) at high correlation and low default probability b/w tranches: this implies that senior tranches becomes less risky due to low default probability which implies due to high correlation the riskiness of the equity tranche also decreases so that the equity tranche value increases you can think of it as that future same cash flows of the equity tranche are discounted now at low discount rate due to reduction in risk so that the value of equity tranche increases. This does not imply high IRR but infact if i am not wrong a low IRR. thanks #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi maheshs, I agree with ShaktiRathore: I don't see the conflict unless we confuse default rate (PD) with default correlation. Ceteris paribus, a higher default probability will correspond to a lower value (or higher spread) for all tranches (equity, junior, and senior). This is illustrated in Malz page 326, see below, by continuously lower values as we move to the right (the X-asix is default probability). Keep in mind that he graphed equity value, but junior/senior as losses, so their higher plots correspond with lower values. Unlike default probability (which is directionally the same for all tranches), default correlation equivocates for the junior (mezzanine; some authors will say mezzanine is "complicated") and is DIFFERENT for the Equity and Senior tranches. This is also illustrated by Malz (below), who is also (I think) consistent with the above thread. I think we can still comprehend it, by analogy, with the same 2-bond portfolio above. That is, a portfolio contains two bonds, each with PD = 10%. Consider two extreme cases: • Zero correlation: A first-to-default (the imperfect analog to equity tranche) has trigger probability = 1-(1-10%)^2 = 19%. A second-to-default (analog to senior tranche) has trigger probability = 10%^2 = 1.0%. • Now spike up to perfect correlation of 1.0 (an "increase in default correlation"). Notice the opposite direction of the two tranches: the "equity" tranche has trigger probability = 10%, which is a DECREASE from 19%, but the "senior" tranche trigger probability of 10% is an INCREASE from 1%! In this way, with increasing correlation, the value of equity increases (PD decreases) while the value of senior decreases (PD increases) The simple dynamics of the 2-bond example appear to be consistent with Malz below: • With respect to equity value, for any given default probability (x value), an increase in correlation (rho) implies an increase in value (compare three lines) • With respect to senior value, for any given default probability (x value), an increase in correlation (rho) implies an increase in LOSSES, which corresponds to a decrease in value Malz page 326: #### k.simpson ##### New Member I have a variant on the OP's question. Suppose that you are of the view that correlation is higher than the market thinks it is. There is a CDS basket available on 100 names (equal value), divided into senior and subordinated baskets (sub detaches at 20). What would be the trade most consistent with your belief? Should you: A) Short the senior and go long the subordinated basket? B) Long the senior and go short the subordinated basket? My reasoning (assuming I'm confident my view will play out) was to go long the senior (if the market assigns less chance of the senior tranche being impaired, then this protection is undervalued by my view. It is cheap so buy it), and short the sub (again, the market views the subordinated tranche as more risky than I see it, due to my higher correlation belief, so I will sell them protection). The answer was A, the opposite. Am I missing something fundamental? Is it somehow that buying the basket is actually selling protection, and that's why the answers are opposite? Any clarity would be greatly appreciated! TIA #### k.simpson ##### New Member The question also mentions that this is a basket of 100 single name CDS swaps. As far as I understand it, then, that's essentially a standard basket that is then tranched into a senior/sub combo. Any suggestions on this and where I might have gone wrong? ShaktiRathore, your help was valuable last time Thanks if you can offer it again. I'm thinking that my confusion may not be from not understanding the implications of rising/falling correlations, but rather the trading aspect. #### Johnson ##### New Member Hi David, I understood the effect on the values of the tranches with increasing correlations and default probability. Malz has in the same chapter page 331 provided the impact on the credit var with changing correlations and default prob. I am not able to understand it. How is it impacted? Thanks #### Unusualskill ##### Member Hi Abhi, I think it can get confusing. Can we assume a two-bond portfolio, where each bond has PD = 10%. Consider a 2nd-to-default basket CDS. • If default correlation = 0, 2TD triggers with probability = 10%*10% = 1.0%. • If default correlation increases to 100%, probability of 2TD trigger increases to 10.0%; i.e., as correlation increases, the probability that both bonds default increases (here, the senior nth-to-default tranche triggers) • So the common (robust) dynamic is: higher default correlation increases (decreases) the risk [i.e., probability of defaults that reference] of the senior (junior) tranche GARP sometimes prefers to refer to value of the tranche rather than the spread (cost of protection) as in the errata question here that you linked to, specifically: One challenge (for me, at least) is due to the different between the tranche (which is analogous to a bond) and protection on the tranche (which is a CDS, just a basket CDS). Consider the analogous bond (~ tranche) and CDS (~ senior basket CDS on tranche): • Start with bond (tranche) at value of$X due to 3% spread (over riskfree) such that CDS spread =3%. Now assume risk of bond increases:
• If bond gets riskier, 3% spread increases to (eg) 5% --> price (value) of bond decreases. But, hedged by increase in CDS spread to 5% and increase in price of CDS.
I hope that helps, thanks,

@David Harper CFA FRM

Came across this post while studying. May I ask how would you explain this dynamic: Higher default correlation decreases the risk of the junior tranche.

For my understanding, "higher default correlation increases the risk of the senior tranche", I explain using this idea: When there is a higher possibility of defaulting together for junior and senior tranche. Thus, when junior tranche defaults, senior tranche is also more likely default, resulting in investors in the senior tranche suffer loss, thus is risker (higher spread) for senior tranche.

However, I cannot think of a way to explain the idea in similar manner for this "Higher default correlation decreases the risk of the junior tranche"? Thank you!