Interpreting Forward Rates from Risky Discount and Credit Curves
Summary
The document examines whether forward rates derived from a risky discount curve can be interpreted like forwards from a risk-free curve. It distinguishes using a curve to value instruments from treating its implied forwards as an arbitrage-free forecast of future yields. One answer argues that a risky curve mixes rate and credit assumptions, so its forwards depend on how those components are modeled and interpolated. It suggests building a risk-free discount curve first, then deriving a credit curve from risky bonds using that discounting basis.
A second answer describes curve-based valuation as useful when enough bonds are available, while stressing that implied forwards represent the market’s current view rather than a guaranteed forecast. The discussion also considers CDS curves and implied default risk. It offers no quantitative derivation or empirical test, and the CDS interpretation is qualified by comparable liquidity premia. The answers reflect modeling opinions and assumptions, rather than a universal rule for constructing or interpreting risky curves.
Key ideas
- A risky discount curve can combine interest-rate and credit effects, making its forwards dependent on modeling assumptions.
- A risk-free curve can serve as the discounting basis for separately constructing a credit curve.
- Implied forwards are market-implied quantities, not assured forecasts of future yield-curve shapes.
- A risky curve can support bond valuation when enough issues are available to construct it.
- Interpreting CDS-implied forwards as default-risk information depends on assumptions such as comparable liquidity premia.
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# Implied term structure from risky discount curve: does it make sense? # Implied term structure from risky discount curve: does it make sense? We know that, taken every discount curve, it's possible to calculate its forward rates according to our tenor preferences. We know also that it's actually possible to extract an implied term structure from every discount curve simply using the appropriate forward rates. This is indeed true for risk free discount curves, because the arbitrage-free assumption which returns the forward rates is fair. Now let a risky discount curve, that is, every discount curve obtained from liquid issues' yields possibly interpolated and bootstrapped... and so on. Questions: - does it make sense to extract an implied term structure from it like we're used to do with a risk free zero rate curve? - Is the implied term structure obtained in such a way an arbitrage-free forecast of the future yield curve shape? - If not so, is the assumption that allows to calculate the forward rates valid for risk-free curves only? - Does it make sense to obtain an implied term structure from, say, a CDS spread curve? Why? - If the answer to 5. was affirmative, would that implied term structure be an arbitrage-free forecast of the issuer's default risk? ## Answer by Lipton (score 2, accepted) https://quant.stackexchange.com/a/37886 I think your questions above are all manifesto of a question on modeling assumption: should we assume a credit term structure in valuing risky/corporate bonds? My personal opinion is yes. The framework I prefer to have is: - Risk free rates/discounting curve from treasury bonds (or other assumptions you find valid) - Credit curve from the risky bonds (using the discount curve from the 1st step) Why does this matter to your question? You can certainly build a discount curve and compute forwards etc based on risky bonds. The question is what that means? When you compute forwards on the discount curve for example, you're basically doing interpolation using rates assumptions you derived from the discount curve. But that could be "incorrect" if you buy in the credit assumption (as described above). In other words, with the credit assumption, the interpolations (say getting forward yield, etc) are all different. So to me, the answers to your questions are: - Does it make sense to extract an implied term structure from it like we're used to do with a risk free zero rate curve? -- No - Is the implied term structure obtained in such a way an arbitrage-free forecast of the future yield curve shape? -- No - If not so, is the assumption that allows to calculate the forward rates valid for risk-free curves only? -- Well...technically no but that's my explanation above. - Does it make sense to obtain an implied term structure from, say, a CDS spread curve? Why? -- Yes - If the answer to 5. was affirmative, would that implied term structure be an arbitrage-free forecast of the issuer's default risk? -- Yes, at least implied by bond prices (assuming they have comparable liquidity premium, etc) ## Answer by Fred (score 2) https://quant.stackexchange.com/a/35283 1.Yes, for some bonds where you have enough issues to build a curve you are able to value them using the curve instead of just relying on the price per issue. 2.It's not a forecast of the future of the yield curve shape, it's a snapshot of the current markets view of the future. 3.No, see answer for Q1 4.Yes, to be able to value contracts on a curve instead of just on price (so you could see where a contract 'should' be) 5.I think the following post will point you in the right direction ( How to compute the implied probability of default from a CDS spread? ).
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