# Callable Bond – Part 1: YTW vs OAS

Callable bond: a credit perspective – Part 1: YTW vs OAS

Bonds with callable feature are very common in the HY space with close to 65% and 35% of all new US and European HY bonds are callable. These bonds tend to have a call schedule (rather than a single call date and price) with credit component more of a concern than the fluctuations in interest rate. This is a topic falls in an area somewhere between quants and fundamental analysts and tends to ignore by many. I intend to look closer to it in this series of articles. Yield-to-worst (YTW) and option-adjusted spread (OAS) are the commonest analytics being used. In part one, I will explain how to calculate YTW and OAS and how should we interpret them.

Before jumping into the calculation, let’s first look into a typical callable HY bond.
The example shown here is a USD denominated fixed rate bond issued by Numericable (NUMFP 4 7/8 2019 ISIN:US67054LAA52). It is a 5-year bond with coupon fixed at 4.875% with the first call date set at 2-year after issuance (15 May 2016) at 103.656 (“5NC2”). If the bond has not been called, the issuer can call it later on at 101.828 in 15 May 2017 or at 100 in 15 May 2018 (first par call date). The proceeds of this bond are used to fund the takeover of SFR.

From YTM to YTW
The bond yield is the simplest bond analytic. For a bullet bond, all we need is the bond current price, the bond maturity, coupon rate and frequency to calculate the yield. For a callable bond, we can repeat the yield calculation assuming the bond is outstanding until the maturity (yield-to-maturity, YTM) and redeemed early at each of the call dates (yield-to-x-call, YTCx). We can determine whether and when a bond issuer would call once we calculate all the yields (YTM and YTCs). We assume that the issuer would refinance early and call the bond if one of the YTC is lower than that of the YTM, or else, it will repay the bond at maturity. From the investor’s view point, the bond will be refinanced at the worst (ie lowest) yield to him/her. Hence it is called the yield-to-worst (YTW).

A worked example for the NUMFP 4 7/8 2019 bond:
Valuation date = 15 May 2014
Bond price = 104.
Yield-to-1st-Call: set bond maturity to 15 May 16 for 103.656 => YTC1= 4.53%
Yield-to-2nd-Call: set bond maturity to 15 May 17 at 101.828 => YTC2 = 4.03%
Yield-to-last-Call: set bond maturity to 15 May 18 at 100.000 => YTC3 = 3.79%
Yield-to-maturity: set bond maturity to 15 May 19 at 100.000 => YTM = 3.99%
Yield-to-worst: min(YTC1, YTC2, YTC3, YTM) = 3.79%. In other words, it is expected to be called around the third call date in 2018. Graphically the relationship between YTC and YTW is shown in Figure 1.

YTW is easy to calculate and intuitive. It is intuitive in the sense that, if the YTW is at the Yield-to-2nd-Call, the second call date would be the most likely time when the issuer redeems the bond. Following the methodology, a series of other analytics can also be derived e.g. from z-spread to z-spread-to-first-call, z-spread-to-maturity, z-spread-to-worst.

YTW however has a number of major drawbacks:
1) Fail to account for the shape of the yield curve: YTW calculation implies the issuer should call the bond and refinance early when YTC < YTM. This is true if the yield curve is flat. When the yield curve is steep, a fixed rate borrower that pays a coupon above spot interest rate early on will get compensated in the later years. Consider the interest rate is at 0% in the first 5 years and 10% thereafter, a 10-year bond carrying 4.85% coupon issued by an entity with negligible credit risk would be valued at par. The issuer would be indifferent to a par call at 5-year if the bond price is at par. However when the price of the bond rises slightly (say to 100.1), the above calculation  would suggest that the issuer will call at 5-year and refinance. This is unrealistic as we can see the issuer would need to borrow at 10% after call the bond early at 5-year at par. The issuer is clearly better off to pay 4.85% for the last 5 years and to wait till maturity. It suggests the forward interest rate between the call and maturity rate is especially important as the front end of the yield curve is artificially depressed in many countries which adopts quantitative easing and various expansionary monetary policy.

2) The price-yield relationship derived from YTW calculation (Figure 1) has a separate curve segment for each additional call date. The durations can be obtained by differentiating the price-yield curve. While the price-yield relationship may seem plausible, the first derivative is discontinuous and these curve segment intersection points with the calculated duration jumps from one value to another. Since the call date is still some time away in the future, it is unintuitive to have these abrupt jumps in sensitivity.

The rate duration is defined as 1/P dP/dr and rate risk as dP/dr whereas the (credit) spread duration D is defined as D=1/P dP/ds and spread risk as dP/ds, with P = price, r=interest rate and s=credit spread. There are a few ways to define bond convexity. I find effective convexity the most intuitive (effective convexity first derivative of duration = (D_ – D+)/dY. Analytical solution is available for bullets. For bonds with more complicated payoff profile, these analytics are often calculated by finite-difference.

Interest rate tree to OAS
The alternative approach is to price the bond using a one-factor interest rate tree. It is single factor in the sense that only the short rate is modelled as a stochastic process. The detail construction of one-factor interest rate tree is covered in many quant textbooks (e.g. John Hull’s Options, Futures and Other Derivatives 6th Ed, Ch28) and is not repeated here. In summary, typical implementation would result in a trinomial (i.e. with each interest rate node leading to three paths: up/flat/down) recombinant (i.e. a path goes up one step and down another will be met with the one goes sideway for two steps in order to keep the total number of node tractable) interest rate tree. The tree is calibrated such that the interest rate and volatility are consistent with what implied from the market price of traded instruments (deposit rate and swaps for the interest rate and interest rate cap, floor or swaption for the volatility). As a feature of the mean reverting interest rate behaviour, some edge nodes lead to a sideway path and two pointing downwards (or upwards).

Once we calibrate the interest rate tree, we can record the callable conditions, the scheduled principal and coupon payment to each node as shown in the stylized 3-period trinomial recombinant interest rate tree (Figure 2). A worked example is provided for node F at period-2. The bond pays a coupon of c in each of the earlier periods, is repaid at par with coupon c at period-3. It is callable at period-2. With this knowledge, we can work backwards and generate the probability weighted discounted cash flow for each of the period-2 nodes including node F. The discounted future cash flow is then compared with the call price. Since the call price at node F is higher than the future cash flow, the issuer will not call the bond. The present value for the rest of the period-2 nodes can be computed in the same fashion. We then step back to period-1 nodes and so on become backing up with the risk-free bond price for the single node at period-0. For bond with finite credit risk, the calculation is repeated but with each cash flow is discounted by the sum of the interest rate and the OAS.

As an example, the price-OAS and spread duration-OAS plots are shown in Figure 3. For fixed rate bond, the rate duration and spread duration are the same. Black-Karasinki one-factor interest rate model is being used. The interest rate tree is calibrated with USD interest rate instruments.  Figure 3a:  Price-OAS plot.  Figure 3b: Spread Duration – OAS plot for a fixed rate bond (NUMFP 4.875 2019 as of 2014-05-15)

Pro:
1. The yield curve shape and interest rate volatility are properly accounted for.
2. The analysis is easily extendible to more complex bonds (floating rate, fixed-to-float, presence of cap/floor, callable / puttable bond….)
Con:
1. More complicated to calculate.
2. OAS is model and data dependent: the choice of interest rate model (Hull-White, Black-Karasinki, Cox-Ingersoll-Ross, multi-factor model etc) and volatility instrument (backing out the rate volatility from cap, floor, or swaption, from the at-the-money or out-of-money one) will give different OAS. It is no longer unique and makes comparison more difficult.
3. Not as intuitive. In the case of YTW, we can make an educated guess whether (and when) the bond might be called by comparing YTW to the YTCs. In the case of OAS, it is not clear. While we may calculate the probability weighted average for each period by exampling the nodes in the interest rate tree one by one, it would make the analysis more and more complicated.

Floating rate note

Some HY bonds are issued in form of floating rate note (FRN). While the interest rate sensitivity is greatly reduced, the credit spread sensitivity is relatively unaffected. Yield-to-worst is not applicable to FRN. We may extend the discount margin calculation but OAS is more straight forward measure here.

The price-OAS, spread duration-OAS plots are shown in Figure 4 for a typical callable HY FRN. The rate duration is low, about half of the interest rate reset period. The bond we modelled is ARGID Float 12/15/19 (ISIN: US03969AAF75), a senior secured 5.5-year FRN with Libor+3% coupon denominated in USD. It can be called by Jun16, Jun17 or Jun18 at 102, 101 and 100 price respectively.  Figure 4a: Price-OAS plot. Figure 4b: Spread Duration – OAS plot for a floating rate note (ARGID float 2019 as of 2014-05-15)

1. Yield-to-par-call and yield-to-maturity use the same parameters for the bond model with the time to call date taken as the bond maturity for the former. As all else being equal, long maturity bond has higher duration. Since the yield-to-par-call and yield-to-maturity calculations take the same input except the maturity with the former using the time to call date and the later using the original bond maturity, we would expect the same price increase would result in larger yield fall in the case of YTC calculation => hence YTC < YTM.