# Compute the P&L from Risky Annuity in CDS market (buying protection in 10y and selling protection in 5y )

I have a question an example from the Credit Derivatives Handbook of JPMorgan in 2006.

This table is located in the first order effects part in the sensitivity to spread changes. And the goal is to see what happens if the curve (of CDS spread) moves in a parallel fashion.

Exhibit 10.5 : iTraxx Main Europe Long Risk (Sell protection) sensitivities to parallel curve shift

iTraxx Main 5 y :

Risky Annuity : 4.38

Notional (S) : 10 000 000

Approx P+L for 1 bp widening (S) : -4.380

iTraxx Main 10 y :

Risky Annuity : 7.9

Notional (S) : 10 000 000

Approx P+L for 1 bp widening (S) : -7.910

I don't understand this following sentence : "If we have a parallel move wider in spreads ((S(t+1) - S(t) is same for both legs) the Mark-to-Market of curve trade buying protection in 10y and selling protection in 5y in equal notionals of $10mm will be negative as the Risky Annuity is larger in the 10y leg than the 5y leg." But I don't understand why we have a negative Mark to Market since we are buying a protection in 10y (so if the spread moves up => we have a CDS with low spread in comparison of the CDS market so the MTM is = (S(t+1)-S(t))x7.91x10mm) and selling a protection in 5y (bad situtation for us because the spread goes up and it's not the case of my CDS contract, so for me MTM = -(S(t+1)-St)x4.38x10mm). We can conclude that the MTM to my position is : MTM = (S(t+1)-S(t))x3.53x10mm. In my opinion, the MTM is positive, what is wrong with my reflexion ? ## 2 Answers We recently had another question Receiver Swap Long vs Short the rate? about the meaning of being "long" or "short" interest rates, and I proffered the same advice - avoid using jargon because jargon is confusing and unhelpful. Reference obligations - bonds and loans - can be quoted as prices, or spreads over benchmarks, or yields. Traditionally, most investment grade debt is quoted as yields or spread over benchmarks, and higher-yield debt is quoted as price, but conversion from one to another is trivial. Likewise, credit indices can be quoted as prices or spreads. Traditionally, investment grade credit indices are quoted as spreads, and high-yield credit indices are quoted as prices, but conversion from one to another is again trivial. In practice, like single-name credit default swaps, the indices are traded with standard running spread and varying upfront fee. If the overall credit quality worsens, then the credit-risky debt becomes less valuable, and the credit protection becomes more valuable, meaning that the prices of both reference obligations and of credit indices will go down, while all spreads and yields will go up. Conversely, if the credit quality improves, then the credit protection becomes less valuable. To express the view that the credit quality will improve, you can: "buy" a bond, "buy" credit index, "sell/write" credit protection (index or single-name), etc. Further, using the jargon that I dislike, you're "long" bond, credit quality, index price, but "short" credit risk, credit protection, and credit spread. (CFA poseurs might say:) Conversely, to express the view that the credit quality will worsen, you can: sell a bond, sell credit index, buy credit protection, etc. In practice, if you buy credit protection, and pay some upfront fee, and the credit quality immediately worsens, and you sell the same credit protection, then you'll receive a larger upfront fee than you had paid. The table in Exhibit 10.7 of JPM' excellent training materials says: "Long Risk (Sell Protection) Sensitivities to Parallel Curve Shift". If you sell/write credit protection - in other words, go "long" credit quality and index price, and "short" the credit spread - then "Approx P+L for 1bp widening" is negative: for \$10 million notional amount, the P+L would be -\$4,380 for 5 year tenor and -\$7,910 for 10 year tenor. It is normal to have larger sensitivities at longer tenors.

The exercise further discusses a hypothetical trade that combines selling 5 year credit protection - same direction as the table - and buying 10 year credit protection - the opposite direction from the table, so you must flip the sign - for the same notional amount. When the credit spread widens 1 basis point on both tenors, you have -\$4,380 loss on the 5 year contract, in which you sold protection, and \$7,910 profit on the 10 year contract, in which you bought protection. The important point to note here is not the sign of the net P+L, but the fact that you're not flat if both 5Y and 10Y credit spreads change by the same number of basis points. As you noticed, using "long" and "short" in this context can cause confusion.

The paper further makes the point that you can minimize the P+L under the risk scenarios in which both 5Y and 10Y credit spreads change in parallel by the same number of basis points. Trade different notional amounts for 5Y and 10Y protection, so that the ratio of the notional amounts is simply the ratio of same-notional factor sensitivities. But you would not be flat under other, non-parallel risk scenarios, e.g. if every spread widens by 10%.

As far as I know, the e-mail address of the lead author on the first page still works. He's been there for close to 40 years.

• Long isn't used by me (but here long risk=sell protection), it's from JPMorgan: Credit Derivatives Handbook, Exhibit 10.5: "If we have a parallel move wider in spreads (S(t+1)-S(t) is same for both) the MTM of curve trade buy protection in 10y and sell protection in 5y in notionals of \$10mm will be negative as the Risky Annuity is larger in the 10y than the 5y". We buy protection in 10y and sell in 5y. I don't understand why MTM is negative if we have a parallel move wider in spreads. IMO buying a protection in 10y and selling a protection in 5y yields (delta spreads)x(7.9-4.38)x10mm Commented Aug 27 at 19:49
• The key point of the paragraph is not the sign of the P+L, but that the net P+L is non-zero. The authors use language like "inoculate", "credit duration", and got confused by their usage of "long credit" - long credit index price versus long credit index spread. A more important point is that you're still not "inoculated" against large non-parallel movs. In their example, if 5Y jumps from 35 bps up 6 bps bps in a day, then expect the 10Y jump from 60 bps to be more like 10 bps than like 6 bps. Historical data will tell you how spreads moved in the past, but the next stress may differ. Commented Aug 28 at 15:39
• This is a comment to an answer or rather a new question. Pls either place it as a comment to the relevant answer or start a new question. Commented Sep 2 at 18:44
• Had to edit down the original text quite a bit but it sort of fits now Commented Sep 3 at 19:45

In finance, when you are "long", you benefit if prices increase. The fixed income world is no exception. When you buy a bond, you benefit if interest rates decrease. When you sell credit protection, you earn coupon and benefit if the issuer gets into better economic health, namely if their credit spread decreases.

When an equity trader says "the market rallies", it means stock prices have gone up. When a govies trader says "the market rallies", it means treasury yields have gone down. When a credit trader says "the market rallies", it means credit spreads have gone down (either CDS or other corporate bond metrics).

Key investors from The Big Short were short by being "long protection", i.e. they benefited if the (mortgage securities) market evolved negatively, either in prices terms (lower bond prices imply higher CDS fair premia, i.e. they bought credit protection when it was cheaper than the market conditions) or in default terms (default trigger the payment of the protection by the protection seller).