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Yield calculation question?

kamikade

Joined
4/21/10
Messages
47
Points
18
Hello everyone,

I came up with a yield calculation question today. It does not seem to be hard but somehow I was stuck with it. Hope someone can shed a light on this.

If you are given 2 bonds at different maturity dates and different coupon yields. Say, first bond matures on June 1, 2019, 3 3/4 coupon and has a yield to maturity of 1.5478. A second bond matures on June 1, 2020, 3 1/2 coupon and has a yield to maturity of 1.6790. If you want to determine yield of a bond that matures on Dec 1, 2019. How would you do that? ( Does the answer change if that bond has zero coupon or 3 3/4 or 3 1/2 coupon?) I guess this is a little bit unique since the third bond matures half-year before and after the first and the second bond. So is the approach different if the third bond is on Oct 1,2019?

Thank you,

Kami
 
Assuming they are the same issuer, the rough and dirty way that is often done on desks is to linearly interpolate the yields between the two bonds you are given. The approach is often considered good enough given the coupons are relatively close. You would have to make an adjustment is the one had a much bigger coupon, like a 8%, but then it would be off the run and not really appropriate to use for pricing. The better way would be to strip them and price each coupon separately.
 
Thanks Julian, I did the same thing as well. I was just wondering if it is a good way to solve this problem. So they use the same technique on trading desk if 2 bonds have the same (or close to be the same) structure?

I also thought about this: Calculate the forward rate R of the entire time between the first and the second bond (total return of the second bond if we buy today, divide by the total return of the first bond if we buy today). Then discount the forward rate by the time between the second bond and the third bond over the time between the first and the second bond (1+R to the power of (T3-T2)/(T2-T1)) to get a factor Q. Then divide the total return of the second bond by that factor Q. That would be approximately equal to the total return of bond 3. Hence, we can calculate the yield to maturity of bond 3.

I tested the 2 solution and I find that if the gap between bond 1 and bond 2 is small, then the difference of yield between the 2 bonds is less than 6-7 significant digit numbers. However, as the gap increases and the yield curve is extremely steep, then the calculated yield using the linear interpolation is always less than the yield calculated using the second approach (ranging from 10 basis points to 40 basis points). Given that I have to quote the yield of the third bond, do you think it is accurate to use the second approach?
 
kamikade said:
Thanks Julian, I did the same thing as well. I was just wondering if it is a good way to solve this problem. So they use the same technique on trading desk if 2 bonds have the same (or close to be the same) structure?
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Yes, at least they do on mine. I think its pretty standard for FI trading. Though I must say that usually the linear interpolation is to price a bond with some credit over the curve. This would be the case if the client wanted the price in spread over the curve. I.e. your unknown bond is issued by a different entity than your two know bonds, and you have an idea of the spread of a comparable bond that you will use to back out a credit rating.

kamikade said:
However, as the gap increases and the yield curve is extremely steep, then the calculated yield using the linear interpolation is always less than the yield calculated using the second approach (ranging from 10 basis points to 40 basis points). Given that I have to quote the yield of the third bond, do you think it is accurate to use the second approach?
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I would gather that this is more of an academic question? In reality if you wanted to be precise, you would be able to bootstrap a curve of on the runs and would value each coupon of the unknown using these zero rates. Or if you were pricing a new issue it would likely be based of the spread of comps and your indications of interest. New issue pricing is much more art than science in my short experience (and in my market). Also, I am starting my career on the market and the curves I have been exposed to have been pretty flat, so I am not sure if the methodology would change if the curve was steeper, though I would hazard a guess that it wouldn't.
 
There are a lot of variations on even linear interpolation. What exactly do you interpolate linearly? It can be spot rates, forwards, bond prices themselves.

And then there are higher order interpolation methods.
 
JulianT: yeah thats what I do too, I quote a comparable yield of government bond then add the spread to get the interest rate for clients. There are usually bonds with the same (or close to be the same) structure with small gap between the two maturity date for us to calculate a comparable yield. However, the other day, there was an odd request and we could not find good sample bonds to calculate comparable yield. I noticed the yield was quite odd so I was trying to figuring why. I probably try to ask a prof at school about that some day.

Yike Lu: I interpolate on the yield to maturity (Yield at higher maturity - yield at lower maturity)/(numbers of years between the 2 maturity dates). Then I just multiple that number by # of years between the lower maturity and the third bond, and add that to the yield to maturity of the lower maturity date bond.
 
Oh sorry, my question was rhetorical - there can be many things you can interpolate linearly. I believe the problem with doing it on YTM is it can potentially introduce forward arb (I haven't done FI stuff in a while).
 
Yike Lu said:
I believe the problem with doing it on YTM is it can potentially introduce forward arb (I haven't done FI stuff in a while).
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This is true for building a swap curve as you get a jagged (discontinuous) forward curve, not a nice property. Though when trading cash bonds, its usually not much of a concern.
 
JulianT said:
This is true for building a swap curve as you get a jagged (discontinuous) forward curve, not a nice property. Though when trading cash bonds, its usually not much of a concern.
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Why not, the arbitrage should still hold? Buy one, short the next. If it's a corporate, you might only have an approximate match and there'd be default risk, so I'd assume you'd hedge that with CDS. Again, I'm only spitballing, FI is not my field.
 
I haven't done too much on this side, I'll look into it more, but my understanding is the discontinuities are more significant in the long end. As such you tend to have a wider gap between the bond, thus introducing more curve risk. The buying one and shorting the next is commonly referred to as a roll if both instruments are cash bonds, and is used to make bets on the curve flattening or steepening.
 
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