S P O N S O R ’ S S T A T E M E N T

There are several ways of looking at the credit spreads implied by bondprices and there is often some debate as to which represents the bestproxy for a credit default swap (CDS) spread. In this article, we look

at the most commonly used spreads, discuss the situations in which theydiffer and suggest which spread is the best proxy for a CDS spread.

Analysis of this kind is especially important in a steep yield curve envi-ronment and where bonds are trading at a significant premium or discountto par. The most commonly used bond spreads available on Bloomberg arethe asset swap (AS) spread, the I spread and the Z spread. For the sake ofsimplicity we refer throughout to annual 30/360 (unadjusted) yields andspreads and we assume that all bonds trades will settle on a coupon dateso there is no difference between clean and dirty prices.

Asset swap spreadAn asset swap is the term used to describe a trade where the bondholder buysa package consisting of a bond priced at par and a swap where the bond-holder pays the bond coupon in exchange for Libor plus the AS spread SAS.

Consider a risky bond with a notional of one and a market price M that islower than the riskless price R. The AS spread, when paid as an annuity,compensates the bond holder for the difference between the riskless priceand the market price:

(1)

where RD is the sensitivity of the riskless price to a change in the bond coupon.

I SpreadConsider a bond that pays a coupon C annually and denote its riskless yieldas Y. We can express the riskless price R as being the sum of the cashflowsdiscounted at Y, that is to say:

(2)

The I spread S I is a discounting spread and is the shift required in Y to solvefor the market price M, that is to say:

(3)

Z spreadThe Z spread is another discounting spread and its calculation is similar tothat of the I spread except that the Z spread is a shift of the whole risklesszero curve necessary to solve for the market price M. Let us denote zerorates corresponding to the bond payment dates as Z1, Z2,...Zn, then theexpression for the riskless price is:

(4)

and the Z spread is defined as the constant shift in each zero rate such thatdiscounting the bond cashflows at the zero rate plus the Z spread Sz givesthe market price M:

(5)

Note that when the bond has no embedded options, the Z spread is equalto the option-adjusted spread (OAS), which is quoted on Bloomberg.

Relationship between AS, I and Z spreads for a flat curveWhen the annual input rates on the yield curve are flat, these will be equal tothe annual zero rates and therefore equations 3 and 5 above are equivalentand thus the I and Z spreads are identical:

(6)S SI Z=

M CZ S Z Si Z

ii

n

n Zn= ⋅

+ ++

+ +=∑ 11

111 ( ) ( )

R CZ Zi

ii

n

nn= ⋅

++

+=∑ 11

111 ( ) ( )

M CY S Y SI

ii

n

In= ⋅

+ ++

+ +=∑ 11

111 ( ) ( )

R CY Yi

i

n

n= ⋅+

++=

∑ 11

111 ( ) ( )

S R MRDAS =−

Bond spreads as a proxy for credit default swap spreads

Mark Davies and Dmitry Pugachevsky of Bear, Stearns & Company analyse

the use of bond spreads and how they relate to credit default swap spreads

CREDIT DERIVATIVES

However, unless the bond is at par, the AS spread will not be the same asthe I or Z spreads. For a premium bond the AS spread is greater than the Iand Z spreads:

(7)

whereas for a discount bond the AS spread is less than the I and Z spreads:

(8)

To explain this, consider either the I or the Z spread since the analysis is thesame for both. So for the I spread, since the cashflows are discounted at yieldY plus I spread as in equation 3, and since curve is flat, Y is equal to Libor, anda bond that pays a coupon of Libor plus the I spread will be worth par. In theevent of default the coupon of Libor plus the I spread is no longer paid.

To replicate this for a bond with an arbitrary coupon we must introduce aswap of this coupon with Libor plus I spread. But for this strategy to be iden-tical to the floater above that pays Libor plus I spread, the swap must becredit-linked such that it terminates in the event of a default. This credit link-age of the swap is the crucial difference to an AS described above where thecashflows on the swap continue even after a default.

Analysis of this kind is important in a steep yield

curve environmentThus for a premium bond, where by definition the coupon must be greaterthan Libor plus the I spread, the swap where the bond holder is paying thecoupon and receiving Libor plus the I spread will have a negative value whentaken in isolation, such as after a default. Thus to compensate the bond-holder for the additional risk of having a swap that does not terminate ondefault, the AS spread must be higher than the I or Z spread. The oppositeis true for a discount bond.

This difference can be expressed as:

(9)

where MD is the sensitivity of the market price to a change in the bond

coupon and note that MD will always be lower than RD.

Relationship between AS, I and Z spreads for an upwardsloping curveWhen the riskless curve is upward sloping, zero rates increase with time andthe riskless yield Y will be bound by the first and the last zero rate:

(10)

To determine whether the I spread or the Z spread should be higher underthis scenario, let us consider the sensitivity of the bond price R in equa-tion 2 to a shift of the I curve, which is made of a flat yield Y, versus thesensitivity of R in equation 4 to a shift in the Z curve, which is composedof the increasing zero rates. This sensitivity is the dollar duration, andwhen this sensitivity is divided by the bond price R it is the modified dura-tion, which is effectively a weighted average payment time, where theweights are the discounted cashflows. The largest cashflow is the repay-ment of principal and interest at maturity and in the Z calculation, thiscashflow is discounted using the rate which, as we have seen above, isgreater than Y. Thus the weight of this largest term is smaller, whichresults in a lower sensitivity for the Z curve. To compensate for the samedifference between the riskless and the market prices, the Z spread mustbe higher than the I spread.

(11)

It also can be shown for a par bond that under this scenario the ASspread is lower then the I spread. To demonstrate this, consider the swapthat forms part of the asset swap package where the bondholderexchanges the coupon for Libor plus the AS spread. The upward slopingyield curve implies that the forward Libor rates will increase over time andtherefore if there is a default at some future point, this swap will have apositive value because the average of the remaining libor payments will behigher than the average of all the libor payments at the inception of theswap. Since the credit-linked swap (described in the flat curve analysisabove) of Libor plus I spread does not continue to get paid post default,the asset swap spread will be lower to take into account this potentialfuture positive value of the swap.

Thus for an upward sloping curve and a par bond, the relationshipbetween spreads is:

S SI Z<

Z Z Z Z Y Zn n1 2 1< < < < <...

S S MMD RDAS Z− = − ⋅ −( ) ( )1 1 1

S S SAS I Z< =

S S SAS I Z> =

115 110 105 100 95 90 85 80

Approximation 0.50% 1.10% 1.77% 2.51% 3.33% 4.27% 5.34% 6.60%

CDS Spread 0.51% 1.13% 1.80% 2.56% 3.40% 4.35% 5.44% 6.73%

Z spread 0.52% 1.13% 1.78% 2.47% 3.21% 4.00% 4.85% 5.77%

I spread 0.51% 1.12% 1.76% 2.44% 3.17% 3.94% 4.78% 5.68%

AS spread 0.54% 1.15% 1.76% 2.36% 2.97% 3.58% 4.19% 4.80%

Values of AS, I, Z and CDS spreads at different market prices

S P O N S O R ’ S S T A T E M E N T

(12)

Similarly, for a downward sloping yield curve – that is to say, zero ratesdecrease with time – and a par bond:

(13)

CDS spreadFor the analysis that follows, we assume for the sake of simplicity that thereis no basis between cash and derivatives and that the riskless curve is flat.

Consider a bondholder who owns a floater valued at par. As per the flatcurve analysis above, for a par bond the AS, I and Z spreads are all equal. If thisbondholder wishes to hedge this floater by buying credit protection in the formof a CDS, for this strategy to be arbitrage free, the CDS spread must equal allthe other spreads:

(14)

For a non-par bond these spreads will not be equal so we must now con-sider which of the bond spreads is the best proxy for a CDS spread.

Consider a par bond and a discount bond that have the same maturityand are issued by the same entity. Under this scenario the risk of default andtherefore the CDS spread is identical for both. If we assume that the bond-holder's strategy is to buy just enough credit protection through a CDS tohedge the credit exposure of the bond, therefore the amount of protectionneeded for a bond trading at a discount will be less than par. Since the dif-ference in the cost of protection between the par and discount bonds shouldbe reflected in the Z spread otherwise there would be an arbitrage, the Zspread for a discount bond will be less than the Z spread for a par bond,which equals the CDS spread. We can therefore expand expression (8) to get:

(15)

Conversely for a premium bond:

(16)

It follows from both these expressions that in a flat curve either the I or the Zspread is a better proxy to the CDS spread than the AS spread.

In a steepening curve, however, the Z spread is a better approximation tothe CDS spread, although it is possible to find examples of premium bondswhere the AS spread or I spreads are actually closer to the CDS spread, buteven in these cases the Z spread is very close to the CDS spread. Thereforethe best overall proxy, without making any adjustments, for the CDS spreadis the Z spread.

There is a relatively simple adjustment to the Z spread to obtain analmost exact value for the implied CDS spread. It is possible to derive a for-mula1, which only uses data quoted by Bloomberg such as modified dura-tion and convexity and a single assumption for recovery. This formula canbe easily applied:

(17)

The difference between the spreads increases with a bigger recovery and abigger discount or premium of the bond. Recovery does matter since therecovery on a bond will be a percentage of the face value of the bond irre-spective of the price at which the bond was purchased, whereas for a CDScontract the recovery is a percentage of the face value of the contract, whichwill match the market price of the bond at the time the strategy is executed.

Test resultsLet us consider a 10-year bond that pays a 7% coupon semiannually with a30/360 daycount basis, and assume that recovery is 40%. The table andgraph give the values of the AS, I, Z and CDS spreads, and an approxima-tion from equation (17) for different market prices for the US swap curve fromSeptember 18, 2003 (10-year swap rate = 4.64%).

It can be seen that throughout the whole range of prices, the Z and theI spreads stay close to each other, and that the Z spread is closer to theCDS spread than the AS spread. But for big discounts and premiums eventhe Z spread is not a good enough proxy, and thus the adjustment in theform of the formula (17) is necessary. Note that all yields and spreads arealso calculated based on semi-annual frequency and a 30/360 basis. ■

1 See Bear, Stearns & Company's working paper Comparison of Risky Bond Spreads

S S M ConvModDurCDS Z≈ ⋅ +

⋅ − ⋅− ⋅ ⋅

[ Re cov ( / )( Recov)

]1 1 11 2 2

S S S SAS I Z CDS> = >

S S S SAS I Z CDS< = <

S S S SAS I Z CDS= = =

S S SAS I Z> >

S S SAS I Z< <

CONTACTS

Mark Davies

Tel: +1 212 272 0400

e-mail: mdavies@bear.com

Dmitry Pugachevsky

Tel: +1 212 272 2883

e-mail: dpugachevsky@bear.com

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

80 90 100 110 120

Bond price

Sp

read

Approximation

CDS spread

Z spread

I spread

AS spread

Spreads vs. Market Price