Interpreting Recovery in Bond Total Return Swap Valuation
Summary
The document compares three formulas for valuing a bond total return swap (TRS) and asks why they treat recovery differently. The accepted explanation identifies the cash-flow components of the swap: changes in bond value, bond coupons and principal, recovery following default, and the financing cost of holding the bond. Under the described no-arbitrage setup, recovery is an inflow received in place of principal repayment, while financing is an outflow. This supports a formula that adds recovery to the return and bond cash flows, then subtracts funding.
The answer suggests that the other expressions may reflect a misreading of a source or a different definition of loss given default. In particular, loss given default can be expressed as notional minus recovery, so subtracting that loss is economically distinct from subtracting the recovery itself. The source does not fully reconcile each reference’s conventions or assumptions, so formula comparisons require checking how each defines principal, recovery, and default-related losses.
Key ideas
- A bond TRS can include bond price changes, bond cash flows, recovery, and funding cost.
- Recovery is an inflow in the described setup because it replaces principal repayment after default.
- Loss given default can be represented as notional minus recovery, so it is not the same as recovery.
- Differences between formulas may arise from definitions or assumptions that need checking in their sources.
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Full text
# Why these formulas for Bond Total Return Swap valuation are different on handling recovery
# Why these formulas for Bond Total Return Swap valuation are different on handling recovery
I studied some bond total return swap valuation. They are very similar but differ in the handling of loss given default and recovery. I feel confused and have no idea why and what’s the economic concept behind the formula.
(1) According to the discountingbondtrsengine.cpp of ORE( open source risk engine)
```
TRS=return leg + bond cash flows + recovery amount-funding amount
```
(2) According to the the paper, Migration plan of Risky Total Return Swap to Bond Return Swap
```
TRS=return leg + bond cash flows - recovery amount-funding amount
```
(3)According to the Chapter 7 of the book, Computational Finance using C and C#
```
TRS=return leg + bond cash flows - (bond notional - recovery amount) -funding amount
```
Note:
- a.return leg: the gain or loss due to bond price fluctuation.
- b.bond cash flows : all cash flows of the bond during TRS contract period.
- c.recovery amount :the residual value of the bond given default.
- d.funding amount : basically Libor or Sofr to financing.
I think there should be some linkage between three formula, or there may be some practical assumption behind them.
## Answer by D Stanley (score 2, accepted)
https://quant.stackexchange.com/a/81375
A bond Total Return Swap has three sources of inflows:
- Changes in value of the bond ("return leg")
- coupons from the bond and principal repayment at maturity ("bond cash flows")
- Recovery instead of the principal if the issuer defaults ("recovery amount")
And one source of outflow - the financing cost that is needed to borrow money to buy the bond (common in a no-arbitrage model).
So the `TRS=return leg + bond cash flows + recovery amount - funding amount` model is completely accurate.
I suspect that you're misinterpreting the second paper somehow to think that the recovery amount should be subtracted, and as Dimitri suggests, that Levy's book (which I do not have access to) quantifies a "Loss Given Default" that could be converted back to a recovery amount as (`Notional - Loss`)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.