# Markit recovery rates : assumed vs real

I often see two different recovery rates in Markit : real recovery rate and assumed recovery rate. What is the difference between them ?

This is indeed a markit vocabulary that spread worldwide. Both recoveries are indeed "often" equal, but there is nevertheless a huge difference between them : one is a pure quotation tool whereas the other is an average or selected market "prices" :

• the assumed recovery rate is only used for a quotation purpose : to do the (quoted spread,coupon) --> upfront and (upfront,coupon) --> quoted spread conversions
• the real recovery rate $R_{\textrm{real}}$ is used for pricing a cds outside of a conversion context : it is $\textrm{Notional}\times (1 - R_{\textrm{real}})$ that is payed is case of default.

For a non distressed name, assumed and real recovery are equal. If the name starts to be distressed (that is, if the market starts pricing default risk up) then the market starts "trading" recoveries on that name, and you have a bid/offer on the recovery such that $\textrm{Notional}\times (1 - \textrm{recovery})$ is going to be payed to the protection buyer in case of default. The real recovery quoted by Markit on a given day is then a average (on all contributors to markit) recovery of mid recoveries on bid/offer of recoveries provided by the contributors at the previous business date.

More details. (May the 9th, 2018.)

What Markit is quoting. For each 5-uple (Markit Ticker, Seniority, Currency, Doc Clause, Running Coupon) markit provides, among other things :

• (the methodology and the data required to bootstrap) a rate curve for the given currency, called the ISDA curve for the curreny (this curve is solely used for quoted spread <--> upfront conversion) ; googling "markit isda curve" is a good idea for getting Markit's official pdf about methodology, data fetching etc etc)
• upfronts and quoted spreads quotes at (best at) the 6m, 1y, 2y, 3y, 4y, 5y, 7y, 10y, 15y, 20y and 30y maturities (the maturity date corresponding to 3y is strictly next IMM date to settlement + 3y)
• a real recovery rate
• an assumed recovery rate

About conversions. I fix a settlement date $t_s$ (equal to trading date + one business day). I note

• $\textrm{UPF}^{\textrm{ISDA}}(\mathscr{C_{\lambda}}, \mathscr{C}_{r},c,R)$ the upfront (a.k.a. clean PV) at $t_s$ of a cds with coupon $c$ and payoff recovery $R$ (that is $1-R$ is paid at default) in the standard ISDA model for a default intensity curve $\mathscr{C_{\lambda}}$ and a rate curve $\mathscr{C}_{r}$.
• $\textrm{ParSpread}^{\textrm{ISDA}}(\mathscr{C_{\lambda}}, \mathscr{C}_{r}, R)$ the unique real number $S^*$ such that $\textrm{UPF}^{\textrm{ISDA}}(\mathscr{C_{\lambda}},\mathscr{C}_{r},S^*,R) = 0$.

1. From (upfront,coupon) to quoted spread.

How do we find the quoted spread QS associated to a given upfront $u$ and coupon $c$ ? We note $R_a$ (resp. $R_r$) the assumed (resp. real) recovery rate and $\mathscr{C}_{\textrm{ISDA}}$ the ISDA curve.

1. We find a $\lambda_0$ such that $$\textrm{UPF}^{\textrm{ISDA}}(\mathscr{C_{\lambda_0}},\mathscr{C}_{\textrm{ISDA}},c,R_r) = u$$ where $\mathscr{C_{\lambda_0}}$ is the flat default intensity curve with intensity constant equal to $\lambda_0$
2. The associated quoted spread QS is defined by $$\textrm{QS} := \textrm{ParSpread}^{\textrm{ISDA}}(\mathscr{C_{\lambda_0}},\mathscr{C}_{\textrm{ISDA}}, R_a)$$

Due to the nature of the equation to solve, sometimes one cannot solve the equation an find a $\lambda_0$. In this case, one can still talk about the upfront, but not about the QS. Markit could have quoted the upfront and not the QS but they choose to quote nothing.

2. From (quoted spread,coupon) to .

How do we find the upfront $u$ associated to a given quoted spread QS and coupon $c$ ?

1. We find a $\lambda_0$ such that $$\textrm{ParSpread}^{\textrm{ISDA}}(\mathscr{C_{\lambda_0}}, \mathscr{C}_{\textrm{ISDA}}, R_a) = \textrm{QS}$$ where $\mathscr{C_{\lambda_0}}$ is the flat default intensity curve with intensity constant equal to $\lambda_0$
2. The associated upfront $u$ is then defined by : $$u := \textrm{UPF}^{\textrm{ISDA}}(\mathscr{C_{\lambda_0}}, \mathscr{C}_{\textrm{ISDA}},c,R_r)$$

One sees indeed that the real recovery rate is always used in a pricing context, that is, when one calculates an upfront, whereas the assumed recovery is used in a quotation context.