Credit spread curves are essential for analyzing lower-grade bond markets and for the construction of trading strategies that are based on carry and relative value. However, simple spread proxies can be misleading because they assume that default may occur more than once in the given time interval and that losses are in proportion to market value just before default, rather than par value. A more accurate method is to estimate the present value of survival-contingent payments – coupons and principals – as the product of a risk-free discount factor and survival probability. To this, one must add a discounted expected recovery of the par value in case of default. This model allows parametrically defining a grid of curves that depends on rating and maturity. The estimated ‘fair’ spread for a particular rating and tenor would be a sort of weighted average of bonds of nearby rating and tenor.

The below post consists of quotes from the paper and a few additional sources (linked next to the quote). Headings, cursive text, and text in brackets have been added. Also, mathematical symbols used in sentences in the original paper have been replaced by text for easy readability.

This post ties in with this site’s summary on price distortions.

### The importance of credit spread curves

“The credit-riskiness of a bond is…encapsulated by a quantity known as the spread which, loosely, indicates how much yield it has by comparison with a risk-free bond of the same maturity. The notion of a __credit spread curve is fundamental in fixed income investing, but in practice it is not ‘given’ and needs to be constructed__ from bond prices either for a particular issuer, or for a sector rating-by-rating.”

“Credit investors need to answer a variety of questions. How has a particular subset of the universe, e.g. BBB miners in the 5–10y maturity bucket, traded over the last few years? Where is it now relative to history? Is the BBB mining curve flat or steep today by comparison with history? How have A/BBB vs BB/B miners traded in the last few years? [These questions] are essentially matters of data aggregation, fitting or parametrisation, taking into account rating or maturity. They might therefore seem rather trivial: why cannot we simply take a set of bonds with the desired characteristics—rating, maturity range…and then compute the average spread (weighting by issue size and duration would be common practice)? The problem with this is that over time bonds enter and exit the bucket, for various reasons: new bonds being issued; existing ones being redeemed or called; ratings changing; maturity steadily declining so that the bond moves into or out of the desired maturity range. This will create jumps in the average spread. Also if the desired bucket is too narrowly defined…we might find no bonds at all.”

“There is no law that states that bonds should trade monotonically with credit rating, and in practice we can easily find examples where a BBB name trades tighter than a BBB+ one (at the same maturity point), as the market spread is in a sense current, and the rating may be out of date, or perceived as such. All this indicates that __we need to produce a parametrically defined grid of curves that are of a sensible shape and vary monotonically with rating__ (i.e. do not cross). Then __the estimated spread for a particular rating and tenor is a sort of weighted average of bonds of nearby rating and tenor__.”

[__Carry], rolldown and RV require the construction of an issuer’s curve__. For carry it is less obvious, but there is a subtlety…But this [issuer’s curve] is in practice a mythical beast, and needs to be built from available bond data.

### The trouble with the academic approach

“Academic treatises [price] a contingent claim in terms of a ‘default-adjusted discount rate’ (dr).”

“…with r the riskfree short rate, h the hazard rate (rate of default) and L the loss given default…The loss mechanism is wrong…The financial interpretation…is that __default may occur more than once in the given time interval and that losses are in proportion to the market value of the asset just before default__. But this bears no resemblance to reality: a bond can only default once, and the loss is a proportion of the par value, as that is the bankruptcy claim…__This error is fundamental because the coupon stream and principal payment are different claims and require different discounting__, and…there is a tendency for premium bonds to trade at a higher yield than discount bonds of the same maturity…the par/non-par problem.”

### Principles for estimating credit spread curves

“Here in basic terms is what we regard as the correct approach:

- We should
__discount cash flows using risk-free discount factors and survival probabilities.__Methods such as Z-spread are unsuited to bonds trading away from par…Also, no bond spread definition is compatible with CDS spread… - We do not need hazard rate models…we simply declare that the present value of a risky dollar…is the product of a risk-free discount factor and the survival function…
- It is necessary to
__parametrise the survival function__using an appropriate monotone-decreasing function… - When there are too few bonds to fit a curve for an issuer, or where we want to think about relative value,
__we index credit quality using ratings__. We can then think about how a name trades relative to its rating curve…When fitting multiple rating curves we need to do them all at once… - One gains a much simpler implementation by assuming the coupons to be continuously paid…
- We fit to prices (or CDS present values). However, we want to plot the spread versus maturity and this necessitates what
__we are calling the par-adjusted spread__, which removes the par/non-par effect.”

“A good part of the __[risk-free rate’s] complexity…is rooted in whether the investor is ‘real-money’ or ‘levered’__…The vast majority of bond investing is done by real-money accounts such as pension funds, insurance companies and sovereign wealth funds. For them, the relevance of a spread measure is simply to quantify the excess yield over a Treasury bond of the same maturity. To hedge interest rate risk, they short Treasuries or the futures. In that case the appropriate risk-free rate is simply the Treasury yield…[For] levered investing the correct discounting rate is OIS and depends on the collateralization of the borrower. However…a hedge fund will typically have a chunk of cash to invest and also a leverage facility, which it can use on-demand. When operating without leverage the risk-free rate is the same as for real-money firms, but when the leverage facility is used, the funding rate and collateralisation come into play.”

### A basic formula

“We write B(T) for the risk-free discounting curve and Q(T) for the survival curve. The present value of a survival-contingent payment occurring at some future date the product of discount factor and survival probability. This allows the coupon stream cj to be valued as a [simple] sum-product:”

“If…we approximate this as a continuous payment stream [with coupon c] then it [can be written by using an integral]:”

“The principal repayment is simply the discount factor times the survival probability at maturity if there is no possibility of recovery. “

“In reality if default occurs we can present a claim for the principal amount and expect some proportion, the recovery rate R) to be honoured. To value this extra amount, we divide the time frame [0, T] into slices and note that the probability of default in [each] interval (t, t+ dt) is [the decrease in the survival rate (−dQ(t)). Multiplying by B(t) and integrating gives a term for recovery value (RV):”

“Adding the parts gives the model price of the bond:”

“This formula, and the reasoning behind it, __emphasises an important distinction between the principal and the coupon stream: the former is partly recoverable, the latter not.__ Hence the coupon stream is a riskier claim on the firm than the principal repayment, and this is the root of the par/non-par problem.”

### The “par/non-par problem” and the case against conventional spreads

“The difference [between academic and practioners’ models] is that __[the academic approach] treats the coupons and principal as essentially the same, but the market clearly does not.__ If two bonds of the same maturity but different coupon trade at the same yield, then [according to the academic approach] the discounting mechanism is the same for the principal as for the coupon, i.e. just using a risky discount factor, and so the market is pricing zero recovery… Bonds with a high dollar price trade at a higher yield or spread than those with low dollar price (at the same maturity point), even though they do not necessarily offer better value…But in general the market does not price bonds this way, and so the high-coupon bond trades at a higher yield.”

“__When building curves we should not, therefore, simply use yield or yield-related spread measures __such as **Z- or I-spread**. Although it is fiddly to attempt to infer the market-implied recovery when we have many bonds of different maturity and dollar price, we should make a better attempt than simply assuming the recovery to be zero. Otherwise, high dollar price bonds always look cheap, when in fact much of the cheapness is illusory.”

“The benchmark spread or **spread to Treasury** is __the most rudimentary of the yield-based measures, and is almost universal in USD markets__. It is simply the yield difference between the bond in question and the benchmark Treasury bond which is not maturity-matched. Thus bonds of maturity 7–15y are all quoted off the 10y Tsy. This is helpful for pricing on the day of the trade but it is a useless construct for analytical work. As a bond’s maturity moves down to around 7-year, it jumps to being quoted off the 5-year Treasury, which typically has lower yield than the 10-uear, and so the spread to Treasury suddenly jumps up even if the bond price has not moved.”

“The **asset-swap spread** of a fixed-rate bond can be thought of in a couple of equivalent ways. One is that I can hand over the coupons and an upfront amount 1 − P/100 (if this is negative, I receive money) to a swap counterparty in return for LIBOR plus a spread, which is the asset- swap spread…[The swap spread] is linear in the bond price….and is zero if the bond trades flat to LIBOR but [depends inversely on the] swap PV01, which has nothing to do with the bond’s credit quality.”

“Finally the par **CDS spread** is the value of the running spread that makes a CDS contract value to par. It is similar to the Z-spread in the sense that if the CDS trades off the same survival curve as the bond (i.e. there is no basis between the two markets)…As the survival probability declines, and with it the bond price, the par spread increases and is a convex function of the bond price…The CDS spread hits infinity when the bond price hits recovery; so typically it exceeds the Z-spread.”

### Par-adjusted spreads

“One might suppose, given that valuation of bonds can be done without reference to any kind of spread measure (one needs only the risk free discount factor, the survival probability and a recovery assumption), that we can scrap the whole idea of spread altogether. That causes a problem, however. It is more natural to think in spread terms, and it is fundamental that we graph spread versus maturity or duration…[And appropriate approach is] the__ par-adjusted spread measure [which] takes recovery and deviation from par into account __[and is explained in detail in the paper].”

### Curve fitting

“In general a constant forward hazard rate will not be sufficient to capture the term structure accurately, so we need to fit a [hazard] curve. We first deal with the case where we have one name, and enough bonds to make it sensible to fit a curve. If this is not so then __we need to fit multiple names and maturities in the sector and, in effect, interpolate based on internal or external credit rating__. After [we have fitted a hazard curve] we can always adjust one parameter so as to exactly fit the model curve to a given bond.”

“A model that seems to offer the right amount of flexibility, while giving a sensible shape for Maturity is [formulated below]:”

“It is necessary and sufficient for all three parameters to be positive, and they have a convenient interpretation: a, b are respectively the forward hazard rates for maturity converging to zero and infinity respectively, and s influences the shape of the curve between these limits. Typically a is smaller than b, resulting in an upward-sloping spread curve, but for dubious credits we will have the opposite. We suggest restricting the range of the time-scaling coefficient s to [0.05, 0.2]

[The above model] __cannot fit a humped forward hazard curve, as it has too few parameters. But parsimony confers two advantages: robustness and explainability__. The need for robustness is amply demonstrated later on. The curve’s shape is determined by two influences: the perceived credit quality of the issuer over different time horizons, and supply and demand in the market. It is unlikely that either can give rise to highly nuanced curves with subtle shapes (except as we have said for distressed credits): for one thing, there is simply too much uncertainty in the future profitability and leverage of the issuer. Our view is that in general anything more complex than a simple upward- or downward- sloping curve is likely to be overfitting.”

“The more general form of the model…fits many ratings at once, and allow the [maturity] parameters a, b to be rating-dependent…It is a useful feature of the way that credit markets seem to work that, roughly, spreads vary in geometric progression across the linear rating scale..In between…ratings, we use logarithmic interpolation; outside, we use logarithmic extrapolation.”

### Calculating carry, rolldown and relative value

“It may be helpful to define [key] terms:

- Carry is the profit and loss (PnL) contingent on the yield of the bond remaining fixed. For a par bond this is synonymous with the coupon, but for a bond trading > 100 the bond price will decrease and for one trading < 100 it will increase as a result of the pull-to-par-effect. The carry arises from two sources: a pure interest-rate component and a credit spread component.
__Rolldown is the PnL…that arises a result of maturity reduction, with the curve assumed to remain fixed__. Usually curves are upward-sloping and so the yield reduces, and the rolldown is positive, but when the curve is inverted the effect will be negative. of the bond while the curve remains fixed.__Credit relative value (RV) expresses, in spread or price terms, the degree to which a bond offers good value relative to its peers__.”

*Carry, rolldown and relative value are all calculated based on the model hazard curve.*

**Credit carry**is the change in value, plus accrued coupon, in the event that the spread remains unchanged…This__includes the pull-to-par. Hence the carry is not simply spread__multiplied with time…This calculation, despite being superficially trivial, requires a model curve.**Rolldown**is the effect of the spread changing from rolling down the model curve. This is the__spread change multiplied by the risky PV01 on the future date__, assuming that all curves are unchanged.**Relative value**is the__effect of moving towards the model curve__at the end of the time period.- Total return is the sum of carry, rolldown and relative value.

“[There is] a subtle matter worthy of attention. Suppose a bond trades at a significant spread to its corresponding curve. We can say that this gives rise to extra carry by comparison with bonds that lie on the curve, but also that the bond offers relative value, by being ‘cheap to the curve’. However, in quantifying the total return we must be careful not to double-count. An obvious way to avoid this is to say that over a period the bond earns extra carry than a bond on the curve, but the relative value component is obtained at the end of the period. Loosely this means that the relative value is the spread difference multiplied by the forward duration, not as seen today: otherwise we will double-count.”

“The issuer relative value ‘mean-reverts until it doesn’t’. A standard pastime of bond investors is buying bonds they consider ‘cheap to the curve’ and selling those that are ‘rich’. The __reason that mean reversion does not occur immediately is that there is no consensus as to where the curve is at any moment__; the work here shows how to do it in a way that we consider to be superior. __This method of investing works well until an accident befalls a particular credit__. Then it will trade wider and wider and always appear cheap, as the rating typically moves later than the price. __When this happens, the relative value is likely to exhibit short-term momentum__, as the move away from its original curve tends not to occur in one big jump. Accordingly, the role of a fundamental analyst is to determine, when an issuer starts to trade cheap.”