Why these formulas for Bond Total Return Swap valuation are different on handling recovery

Question

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 

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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 1

A bond Total Return Swap has three sources of inflows:

1. Changes in value of the bond ("return leg")
2. coupons from the bond and principal repayment at maturity ("bond cash flows")
3. 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)