PDF | This paper derives a tractable, arbitrage-free valuation model for corporate coupon bonds that includes a more realistic recovery rate.

**Table of contents**

- The Valuation of Corporate Coupon Bonds
- Your Answer
- credit - Bond recovery rate with coupon - Quantitative Finance Stack Exchange

W e assume that traded in the economy are default-free zero-coupon bonds of all. The market is assumed to be frictionless and competitive. The default-free money market account earns in terest continuously at the. W e initialize. W e let the time t value of a default-free zero-coupon bond paying a dollar at time.

The b ond pays the C dollar coupons. If default happens prior to the. If default does not happen, the face value of L dollars. As an approximation, we assume that if default happens within the time in-. In practice, a portion of the next coup on payment after default repre-. W e ignore this residual. We assume that these state v ariables. We include the default-free spot.

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Examples of additional state v ariables that could b e included in this. Intuitively ,. In particular,. W e want to v alue the risky coup on bond in an arbitrage-free market. Hence, w e. Existence of an Equivalent Martingale Me asure. There exists an e quivalent prob ability measure Q such that. Equivalen t means that both the probability measures P and Q agree on zero.

It is well known that this assumption implies that the market. See Jarrow and Protter [21]. P see Bremaud [5], p. T o value the risky coupon bond, we need one additional assumption. We assume. By the second fundamental theorem of asset pricing see. Jarrow and Protter [26] , completeness implies that the risk-neutral probability. Q is unique and risk-neutral valuation applies.

Hence, we can apply risk-neutral. The discount rate is the default-free spot rate r t. The adjustment. The risk-neutral probabilities include the required risk premium. Note as discussed above that no recov ery payments are included for any coupon. T o empirically implement and estimate expression 2 , we need to add more struc-. This section adds this structure. This section presents the additional structure imposed on the default and recovery. F or this structure we need the following.

Let F r. This assumption is imposed for analytic tractability. It states that under the. This implies that these. How ever, observations of these stochastic processes. And under the statistical probabili-. Hence, nonzero pairwise. Next, we add the following assumption on the reco very rate process. This assumption states that, after adjusting for risk, the best estimate at time t.

W e note this assumption. The risk adjustment under Q will typically lo wer. This observation will prov e useful in a subsequent section. These more basic securities will enable us to provide an intuitiv e interpretation. The value of this security at time. Hence, these are risky zero-coupon bonds that have a zero recovery rate in.

The value of this. The time t value of the left and righ t sides of expression 5 is. This represents the present v alue of the t k maturity zero-coupon bond at time s ,. The right side. This expression is used below. This section derives the empirical v aluation formula used in the estimation under. From expression 2 , using the independence assumption,.

Using expression 6 , we see that the last term in this expression is the sum of. This expression has an intuitive. It shows that a risky c oupon bond c an always be de compose d into a portfolio of. This expression faciliates the computation utilized below. Indeed, the default-free. In addition, in this form it is easy to see that the value of this coupon bond is. Let D t, t k denote the time t value of such a zero-coupon bond promising.

Then, using the same mathematics as. The additional terms in the full-coupon recovery. Lastly , to facilitate the estimation of the intensity process, we assume that.

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## The Valuation of Corporate Coupon Bonds

This assumption enables the estimation of the default intensities using historical. It is important to note, as discussed in Jarrow, Lando, and. Y u [24], that this assumption do es not imply that risky coupon bonds earn no risk. Quite the contrary. But, the timing. In the full recovery coupon model, one gets a. T o estimate the default probabilities, we use a proportional hazard rate mo del. F or an application of such a hazard. Corporate bond markets are illiquid relative to T reasury bonds or exchange traded. Jacquier, and Jarrow [8]. To incorporate such a liquidity discoun t, we use the dis-.

W e do not apply a liquidity dicount. Discounting the zero-recov ery zero-. The F t measurability of the liquidity parameter. This is the valuation. Similarly , taxes paid on coup ons and capital gains. Consequently , to determine a market price, an equilibrium model is needed.

Equilibrium mo dels. F urthermore, an argument can be made that the marginal trader, who determines the mar-. Here, we note that many institutions pa y small. T o clarify the anticipated pricing errors between the zero coupon recovery model,. The pricing error. F or illustrative purposes we make the following simplifying assumptions: This implies that the pricing error is.

The par value of. For short maturity. The largest pricing error is equal to. There is a relation between the pricing errors, the default probability , the. From expression. The full-coupon recovery model has a fourth component, the present v alue of the. This fourth component is also. How ever, this payment still needs. For longer periods, a similar logic applies. It then follows that the pricing error is exactly proportional to the product of. In contrast, the total pricing error.

The reason is that the probability of survival depends on the default. Ignoring discounting and the adjustment for the probability. For a bond receiving N coupon payments the factor. This means that the approximate total error is equal to. T able 1. It serves as a reasonable predictor of the actual pricing error expected. It also provides a goo d approximation for the actual. Later, we will use these insights. T o summarize, the pricing error is zero if the recovery rate, the default prob-.

The error grows approximately with the. Pricing with T wo Spread Curves. Pricing bonds using the same credit spread term structure is based on the incorrect. Howev er, this does. Instead, one needs to use two. If there is a pricing error using the coupon recovery model to price bonds, then. T able 2 provide some illustrative examples of spread curves. We use the same. As long as there is positive recov ery, coupon.

A higher default probability makes all spreads higher; in the. T able 1: T able 2: The details of the estimation procedures are as follows. The full amount paid is. We compare the full amoun t paid the present value. Survivor option bonds distort bond prices both b ecause they. The survivor option feature.

We also exclude some. W e used the U. Treasury yields reported daily by the U. Departmen t of the. T reasury 10 and derived the maximum smoothness T reasury forward rate curves. Using these historical forward. A typ-. The state variables used in the Kamakura hazard rate estimation. Given the zero coupon bond prices and the default process ab ove , w e estimate. W e compared the mo del values expression 11 to the mark et prices using a.

T o compute expression 11 , we discretized time. Then, we used the parameter. This section illustrates the valuation model by computing the model v alues for six. States, Canada, and Japan. The issuers were John Deere Capital Corporation,. Mitsui Financial Group Inc.

F or each issuer, we estimate the. We use August 30 instead of the. The parameter estimates and statistics for each of these companies are con-. We discuss eac h issuer in turn. T able 3: Ticker Issuer N Rating 1 Y ear. R 2 Mean Abs. C Citigroup Inc. WMT W almart Inc. W e did. Figure 1 shows the decomposition of each bond into coupon securities, recovery. The valuation model had. The market implied recovery rate was 0. The 1 year default probability was 0. Figure 1: There were 24 senior non-callable bonds traded for F ord Motor Credit Company.

Again, we did not use bonds issued by other legal entities in the F ord family. Figure 2 shows the decomposition of each bond. The R 2 for the reduced form. The recovery rate w as The 1 year default probability was 3. Figure 2: Next we turn to Citigroup Inc. There were 13 senior non-callable bonds traded for.

Figure 3 again shows the decomposition of each bond. For Citigroup. The recovery rate w as 0. Figure 3: Bank of Nov a Scotia. There were 9 senior non-callable bonds traded for Bank of Nov a Scotia. W e now turn to Figure 4, which shows the decomposition of each.

The R 2 for the reduced form bond valuation w as Figure 4: There were 9 senior non-callable bonds traded for Sumitomo Mitsui Financial. Group Inc. Figure 5 shows the decomposition of these b onds. The recovery. The 1 year default. Figure 5: Sumitomo Mitsui Financial Group Inc. There were 2 senior non-callable bonds traded for W almart Inc. Figure 6 shows. Figure 6: W almart Inc. The present v alue of the coup on payments, if no default, is.

The present v alue of the recovery paymen t, in the event of. F our of the six issuers have nonzero recov ery rates. Finally , the unexplained. This implies, of course, that.

### Your Answer

This section provides a comparative analysis of the reduced form v aluation model. One, the full-coupon recovery model, which is the tra-. In section 3. T wo, a ratings-based valuation model, which is a special case of the full-coupon. The ratings-based v aluation mo del is consistent with numerous. Relative to the. To obtain the ratings-based v aluation,. T reasury zero coupon bond prices, pro duced the minimum. It is important to emphasize that. W e begin with a comparison to the ratings-based valuation model.

Using the. This is our sample. Note that the ratings-based credit spread is the same. A comparison of the statistics in Table 4 with the zero coupon. As seen b elow, the mean pricing errors are. T able 4: Ticker Issuer Rating Credit. The distribution of the pricing errors for all of the bonds including the bonds. Figure 7: The pricing errors from the reduced form valuation methodology are shown. The pricing errors from the ratings-based valuation. The pricing error statistics. The R 2 for the reduced form valuation model. Next we turn to a comparison of pricing errors between the reduced form.

The results for August 30,. The two models are identical, of course,. Consequently , the two models will give similar v alues for issuers. T able 5: Statistics for the Full-coupon Recov ery Model. Ticker Issuer Mean. Figure 8: After the restrictions discussed above dropping callable bonds etc. For all of these observ ations we. In addition, we hav e calculated the full-coup on recovery and the rat-. W e therefore restrict attention.

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After these restrictions we are left with about W e perform a comprehensive analysis of the reduced form pricing. Pricing errors in many cases are substantial. Table 6 reports summary statis-. The median coupon rate is 2. Using these estimates. This calculation provides. T able 6: Pricing Error Summary Statistics. These n um-. If bonds. As done in the previous subsection 6.

T able 7: Table 7 reports these results. By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. How should we receive the recovery? The time of receiving the recovery value could vary. Generally, the developed markets convention is that all the coupon that has been accrued and not paid yet just disappears, but the remaining notional is accelerated - becomes due immediately.

To illustrate, suppose for concreteness that the issuer has two bonds.

## credit - Bond recovery rate with coupon - Quantitative Finance Stack Exchange

If the issuer defaults, then the coupons don't matter, the maturities don't matter. The current interest rate levels don't matter. Now, if instead of a bond you issue a bespoke credit-linked note, then you can specify anything you like on the term sheet. In particular, you can say that in case of a credit event, the coupon accrues until the day of default like running spread of a credit default swap. You can also link the recovery to interest rates in some way. Although this is very seldom done, a good analytics library should have the flexibility to support this.

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