# What is the difference between Option Adjusted Spread (OAS) and Z-spread?

I am preparing for the CFA level 2 exam, I got confused by the concept Z-spread and OAS.

When a call option is added to a bond, since it is not favorable to the bond buyer, they would require more spread (which is the OAS) for this instrument in order to get more discount on the bond price.

So to me, Z spread should be less than the OAS.

But this is not what has been discussed in the book. Can someone help with this?

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To sum up what is discussed in the CFA curriculum, it discusses 3 types of spreads. They are used to compare a risky bond to a Treasury bond (assumed to be risk-free).

Simply computes the difference between the YTM of the risk-free bond and the YTM of the risky bond.

The major problem of this measure is that it doesn't take into account the shape of the spot yield curve.

This spread is actually a single value that needs to be added to every spot yield of the curve in order to make the present value of the risky bond equal to the present value of the risk-free bond.

That is, if the present value of the risky bond is $v_b$, then the Z-spread $z$ is the value such that:

$$v_b = \sum_{i=1}^N \frac{C}{(1+R_f(0,i)+z)^i}+\frac{FV}{(1+R_f(0,N)+z)^N}$$

where $R_f(0,t)$ is the spot rate for maturity $t$ for a risk-free bond (not annualized).

This still doesn't take into account any embedded option to the risky bond.

This last spread is used to measure the impact of the optionality in the bond. It is defined as follows:

$$\text{OAS}=z-o$$

where $o$ is the the price of the embedded option.

For callable bonds, the option benefits the issuer (it allows him to buy back the bonds if rates go down, i.e. bond prices go up), and $o>0$ hence $\text{OAS}<z$.

For putable bonds, the option benefits the bond owner (it allows him to sell back the bonds if rates go up, i.e. bond prices go down), and $o<0$ hence $\text{OAS}>z$.

It there is no embedded option, then $o=0$, $\text{OAS}=z$.

I believe spreads are to be understood as being used to measure the risk inherent to the core bond, and not to the option that are embedded to it. Hence, if the bond was callable, you required more yield for the bond, but the "core" bond only required a spread equal to the OAS and not really the spread computed by the Z-Spread approach.

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Mathematically, there is an interesting paper related to the non-linearity of this relationship (practically credit spread/o), in "Next Generation Models for Convertible Bonds with Credit Risk", p70, at wilmott.com/pdfs/030813_ayache.pdf –  user7056 Aug 14 at 12:37

The z-spread and OAS both are measures of the difference in price between an ABS and a zero-risk bond. The OAS and z-spread are not spreads that a bond with and without options should require, they are two ways of looking at the same bond.

The CFA material states that the z-spread is equal to the OAS when

1. There is no prepayment option
2. There is a prepayment option, but the borrowers tend not to exercise that option.

The z-spread essentially ignores the option, so a larger spread is needed to explain the price of a bond if there is an option that has a good chance of being exercised.

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The Z Spread is the constant spread above the treasury curve that compensates the bond holder for credit, liquidity and option risk.

The Option Adjusted spread is simply the Z- Spread excluding the premium to compensate for the option risk.

Thus, the OAS is the spread above the treasury curve that compensates for credit and liquidity risk only.

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The question remains unanswered...

What you basically state by OAS = z - o is that OAS is less than the z-spread for callable bonds. Now if you compute the PV of the bond by discounting it by respective yields (Treasury yield + spread) you will find that the the bond discounted using the z-spread will be cheaper than the one discounted by OAS. It does not make sense because as a rational investor I demand a larger discount for the call option that I provide the issuer with.

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Very insightful discussion. Let me try if I can address this issue. What victor has challenged makes perfect sense, but I think we when calculate z-spread, we just ignore the option. So when we actually calculate call embedded bonds, investors are looking for larger discount to compensate, so the spread is larger which reflect on the market price. And this price is what we use to calculate z-spread. So you will see larger z-spread version put embedded one with smaller z-spread. And Z = OSA + Option. OSA is a ideal spread which is measuring the spread between the fair value and market value. Like if you don't have the call option, the bond price will be higher and the discount rate will be lower, and the spread is smaller. The difference is the price(spread) of option. For call is positive and for put is negative. I hope this can clear some misconceptions.

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I think you are just inverting the stuff. (Like I did) You consider Z-spread what, in fact, is OAS.

OAS: discount to add to treasury excluding the option price. (Z-spread of an equivalent bond with no option)

Z-spread: discount to add to treasury for the option bond (thus accounting for the option).

It is misleading because "Option Adjusted" calls for a spread adjusted to reflect the value of the option in the Bond and we instinctively go for it as the actual spread to use for the bond having the option. And, off course, it should reflect a higher discount for a call option, reducing the price of the issue.

They should call it: "Option Excluded Spead" or something like this...

Take care,

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