 # Three fixed income quickies

1. Why is this true: “Securities with longer effective durations have larger OAS and option costs because of their higher interest rate exposure.” Longer duration means more interest rate risk but how does that translate to larger OAS and OC? 2. Why is this true: “For a given duration, the greater the convexity, the lower the interest rate risk.” I see this mathematically but not graphically. Doesn’t convexity make the curve “more curvy” meaning that for a given change in yield, price would change more? 3. Schweser graph page 298 Book 5 shows that plain vanilla corporate should be valued via a z-spread, but earlier (page 201) they value a plain vanilla corporate using a binomial model. Will z-spread and binomial model give the same answer because option cost is zero, and which one’s more technically correct?
1. Larger effective duration means the bond expected life is longer. In effect there has to be a larger OAS to compensate the buyer for longer exposure to interest rate risk. The OC is also larger since now the option is outstanding for a longer period. 2. higher Convexity - if int rate rise the value of the bond will decrease less the value of other bonds. If int rate fall the value of the bond will increase more than the value of other comparable bonds. 3. Z spread for option free bonds and binomial to calc the value of the option.

I thought plain vanilla corporate (with no option) should always be valued w a Z spread Also, can you please summarize “negative convexity” I still get tripped up on that many thanks

just came across this: the security exhibits negative convexity, meaning that the opportunity for gains when rates fall is taken away.

Negative convexity- Callable bonds exhibit negative convexity because when yields decrease the value of the bond will increase but the increase is capped @ call price. you own a 6% callable @ 105 bond. If rates decrease to 4% the bond will increase in value > than par, let’s say 111 and the call will be exercised. so the value to the bondholder is capped at 105. value of the call option to the holder (corporation/issuer) 6.

OAS compensates for credit risk and liquidity risk, I do not see a reason why OAS should be always higher when effective duration is longer

If you own a bond for a longer period of time, you must get paid more to do so. Think of it in pure time value terms, if I have two bonds from the same Issuer with the same exact coupon and one has a duration longer than the other, the longer duration bond will have the higher OAS; i.e. the bond holder needs to be compensated more in return for the longer exposure. Think of convexity as how fast the speed of price-changes are to interest rates. So a callable (par call) bond that is priced at 70 when rates are 5%, is now worth 90 when rates are 4%. Now let’s say that same bond is worth 100 when rates are at 1%, here you can see that the rate of change in the price (first it was +20pts for -1% change in rates now it’s +10pts for -4% change in rates) has slowed down considerably, hence its “negative convexity” since the rate of change is decreasing.

Thanks for the help. Just want to comment on the three original points. 1. Ok I can see that higher duration means higher OAS because OAS is the compensation for more interest rate risk. But intuitively, it doesn’t make sense why cost would go up too (this negates your higher compensation via the higher OAS). And mathematically it doesn’t make sense either for both OAS and OC to go up: OAS = Z - OC 6 = 10 - 4 If OAS goes up 1: 7 = 10 - 3 So higher OAS means lower Option Cost holding Z constant. 2. Very clear now–thank you 3. You say “Z spread for option free bonds and binomial to calc the value of the option”. However, like I said, on page 201 of Schweser Book 4 they value the option free BOND, not the option.

1. with a callable bond, you’ve sold a call option to the issuer of the bond. If that call option has a longer duration than another one, all else being equal, the call option will be more valuable. Just like any two options with two different maturities would have different time values. Keep the reasoning as simple as that. With your equation above you are oversimplifiying the relationship between OAS and Z and OC, if a security has a longer duration the Z spread will also be higher (you need to discount cfs at a higher rate since they are longer now). 3. The figure on p. 201 is simply showing how to value an option-free bond using the binomial model. P. 298 is telling you that Z-Spread is the appropriate spread measure to use for option-free bonds. Two totally separate concepts.
1. thanks, got it 3. if page 298 is telling me that the z-spread it appropriate to use for option-free bonds, then why is page 201 showing me how to value an option free bond with a binomial model? how is this 2 different concepts?

If a question asked, “what is the appropriate spread measure to use to value a plain vanilla corporate bond?”, the answer would be “Z-Spread” - this is the goal of p. 298 If a question asked, “how do you value an option-free bond with the binomial model?” you would follow the steps outlined on p. 201. This page is coming up with a value based on a given interest rate tree, that’s all. It doesn’t say that this is a z-spread, simply asking for the value based on the various nodes in the tree. It shows the pure mechanics of how to value a bond using the binomial model - it’s meant as a warm up to show the steps in valuing a bond using the binomial model. It’s using an option free bond for simplicity.

ok youre not wrong. my only contention is that on page 298 they should put option free bond valuation under the category of both z-spread and binomial model since clearly both can be done, as they show on page 201

p. 298 is telling you which spread measure to use, not which model to use. You can’t produce a Z-Spread from a binomial model, by definition a z-spread assumes zero interest rate volatility.

ok fine but you can still value that option free bond with a binomial model (independent of the z-spread valuation)?

yes, you can value any bond (or any security for that matter) using the binomial model if you want to value something with assumptions around interest rate paths.