Why does the carry of a bond not adjust for defaults?
Why does the carry of a bond not adjust for defaults?
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carryquestionman · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: carryquestionman
Original post: https://quant.stackexchange.com/questions/85310
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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Ordinarily, the carry of a bond is computed by keeping 'everything fixed', but moving the bond in time, and then repricing it.
However, say there is a certain probability of the bond defaulting.
During the carry period, the bond may default with that probability. Therefore, we ought to remove this probability of default (adjusted for loss given default). After all, if you do not do this, you are essentially assuming that something
has
changed, namely that the default probability has collapsed to zero.
So, in my opinion, the carry ought to be adjusted downwards, by -
$PD \cdot LGD$
. Yet generally this is not part of the "carry" discourse? Why not?
Furthermore, I am not sure if
$PD$
ought to be the risk-neutral default probability or the real-world default probability.
As carry is supposed to represent what will "actually" happen, I think real-world probabilites ought to be used?
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Dimitri Vulis · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: Dimitri Vulis
Original post: https://quant.stackexchange.com/a/85313
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Suppose that you have some bond or loan from time
$T_0$
to time
$T_1$
, and you want to explain the P&L from having this bond. You decompose the P&L into components to help you accomplish whatever the goal of your P&L explanation. For example, if you have material interest rate risk in a fixed-coupon bond, then you want to see how much IR changes contributed in various ways. Conversely, if you hedge and have no material IR risk contributing to the P&L, then you don't care about slicing and dicing the immaterial P&L from that. Or, if you hedge the credit risk with CDS, then you may like to see how much P&L came from the changes in bond credit curves / CDS curves / basis between them, otherwise you don't care.
The mark to market of a performing bond or loan is based on a dirty price, which consists of a clean price and the accrued. The clean price already includes the market participants' views on interest rates, probabilities of default, losses given default, liquidity premiums - including the possibility of the coupons not being paid.
You can decompose the P&L due to the passage of time, for example, into the cost of financing the position, which is sometimes ignored but can be substantial, the changes in the accrued, the coupons actually paid and scheduled to be paid between times
$T_0$
and
$T_1$
, and any other breakdowns that helps achieve whatever your goals are.
You should read this 1999 paper by Duffie and Singleton "Modeling Term Structures of Defaultable Bonds"
https://doi.org/10.1093/rfs/12.4.687
which deals a lot with discounting future cash flows of credit-risky bonds using PD and LGD. There are also great papers by Tomasz Bielecki et al about it. If you then find a way to adjust a carry for PD/LGD that would be more useful to you than then plan old carry based on promised cash flows, then go ahead and use it. You way want to reflect on what problem you're trying to solve with this.
Edit: n Bielecki/Duffie-Singleton world, to obtain a fair dirty price, you pv the future cash flows like you would pv a CDS - from interest rates, PD, and LGD. That's what the price at which the instrument would actually trade, but you may need to put clean price = dirty price - accrued on a trade ticker.
Similarly, a bond's yield is the internal rate of return of the promised cash flows. Instead of the promised cash flows, you can adjust all the cash flows by PD and LGD, and calculate their internal rate of return, and term that the risky yield. But what good would that do you?
In case of a floater whose coupons is not set yet, which @Jaood mentioned, there are multiple methodologies for projecting the unset coupons - you can assume that the forwards are realized, which makes the most sense; but you can also assume that the rates remain unchanged, e.g. the 1 year rate today will be exactly the same in the future; and there are other variations which may work better depending on what you want to do with the result.
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Quoted from Forex.com.bd-Editorial External answer — Quantitative Finance Stack Exchange Author: Dimitri Vulis Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/a/85313 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Suppose that you have some bond or loan from time $T_0$ to time $T_1$ , and you want to explain the P&L from having this bond. You decompose the P&L into components to help you accomplish whatever the goal of your P&L explanation. For example, if you have material interest rate risk in a fixed-coupon bond, then you want to see how much IR changes contributed in various ways. Conversely, if you hedge and have no material IR risk contributing to the P&L, then you don't care about slicing and dicing the immaterial P&L from that. Or, if you hedge the credit risk with CDS, then you may like to see how much P&L came from the changes in bond credit curves / CDS curves / basis between them, otherwise you don't care. The mark to market of a performing bond or loan is based on a dirty price, which consists of a clean price and the accrued. The clean price already includes the market participants' views on interest rates, probabilities of default, losses given default, liquidity premiums - including the possibility of the coupons not being paid. You can decompose the P&L due to the passage of time, for example, into the cost of financing the position, which is sometimes ignored but can be substantial, the changes in the accrued, the coupons actually paid and scheduled to be paid between times $T_0$ and $T_1$ , and any other breakdowns that helps achieve whatever your goals are. You should read this 1999 paper by Duffie and Singleton "Modeling Term Structures of Defaultable Bonds" https://doi.org/10.1093/rfs/12.4.687 which deals a lot with discounting future cash flows of credit-risky bonds using PD and LGD. There are also great papers by Tomasz Bielecki et al about it. If you then find a way to adjust a carry for PD/LGD that would be more useful to you than then plan old carry based on promised cash flows, then go ahead and use it. You way want to reflect on what problem you're trying to solve with this. Edit: n Bielecki/Duffie-Singleton world, to obtain a fair dirty price, you pv the future cash flows like you would pv a CDS - from interest rates, PD, and LGD. That's what the price at which the instrument would actually trade, but you may need to put clean price = dirty price - accrued on a trade ticker. Similarly, a bond's yield is the internal rate of return of the promised cash flows. Instead of the promised cash flows, you can adjust all the cash flows by PD and LGD, and calculate their internal rate of return, and term that the risky yield. But what good would that do you? In case of a floater whose coupons is not set yet, which @Jaood mentioned, there are multiple methodologies for projecting the unset coupons - you can assume that the forwards are realized, which makes the most sense; but you can also assume that the rates remain unchanged, e.g. the 1 year rate today will be exactly the same in the future; and there are other variations which may work better depending on what you want to do with the result.
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