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Context: I have bond A from say Apple, Apple also issued different types of bonds , namely B , C, D, E bonds. Bonds A B C D E are all same, except, they were issued at different times, have difference size and different maturity. I am trying to find: What is the yield difference between yield of A and yield curve of Apple comprised of bonds B C D E. Because Bonds B C D E and A were issued in different time, with different maturity and different sizes, i acknowledge that there is difference in liquidity between these bonds, and just comparing yield of A at time t to yield curve of firm at time t will be fallacious. I have bid ask spread of all bonds but no trade volume.

Question: How can I come up with a metric that gives me a liquidity of yield curve? that takes into account difference in issue size and maturity profile.

Prop1: I was thinking of finding distance of bonds B,C,D,E with bond A.

As in:

d(B)= bond A maturity - bond b maturity

d(C)= bond A maturity - bond C maturity etc...

Then Liquidity of synthetic bond ( yield curve) will be

= d(B) / (d(A)+d(B)+d(C)+d(D)) * Liq of bond B + d(C)/(d(A)+d(B)+d(C)+d(D)) * Liq of bond C +...etc for bond D and E.

So its weighted on how close this particular bond is to bond A, the closer this bond is in terms of maturity , the more weight does it have.

But this only accounts for maturity profile ( distance from bond A), How can I mathematically do this so that it also accounts for size of bonds ( more weight on liquidity of bond that is closest to size of bond A) and perhaps issuance date ( closest to issuance of bond A)

Thanks so much for reading and possibly answering :) Cheers!

PS: if you have a better way of finding weighted liquidity of portfolio(?), then by all means, i wanna hear it

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  • $\begingroup$ To help would you please explain what is your definition of liquidity (a score or time to liquidate or executed price to mid or ....) and how you calculate it for bonds you are able to do. $\endgroup$ – Jiem Aug 17 '18 at 22:21
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Bonds A B C D E are all same, except, they were issued at different times, have difference size and different maturity.

I found that it's rare for the bonds to be all the same except for maturity and size. Even the most similar bonds could have some obscure difference. And it's entirely possible that some bonds are more liquid that others, for various reason in addition to the unique feature. It could be that some dealer runs the market for the bond, or burst of interest because of some news, or simply because someone just want to unwind.

I have bid ask spread of all bonds but no trade volume.

if the bid/ask are both tradable, I feel that you could utilize the mid with some level of confidence (if the bid/ask diff is not too crazy), and the bid/ask diff could contain some information about the liquidity (as premium).

Once you have the mids, you can start generating some capital structure and go from there

But this only accounts for maturity profile ( distance from bond A), How can I mathematically do this so that it also accounts for size of bonds ( more weight on liquidity of bond that is closest to size of bond A) and perhaps issuance date ( closest to issuance of bond A)

I don't think there's a standard way to define bond-level liquidity. The problem with the term liquidity in general is that it depends on your perspective and scope. Also relative liquidity is probably easier to define than absolute (e.g. BBG's bond liquidity score is a relative metric cross-sectionally)

However, what you've described are totally legit: new issuance are pretty liquid in general, the demand is high, people flip trades, blah, blah. But over time, people tend to hold the bonds to earn carry. So time since issuance definitely matters. On the other hand, issue size (or amount outstanding really) also indicates how likely there will be some interest in buying/selling the bond.

Of course, more tangible liquidity factor would be market trades and quotes. It's no different from any other financial instruments. If what you are trying to figure out is "how long I can unwind my position for what price impact" kind of liquidity metric, then bond probably A has very little to do with B, C, and D.

How can I come up with a metric that gives me a liquidity of yield curve? that takes into account difference in issue size and maturity profile.

To your main question, I don't think it's very meaningful to say the liquidity of a yield curve. The liquidity profile is more relevant at bond level instead of a curve. So I'd focus on market data instead of the other bonds from the same issuer (they could be indicative, but imho their impacts are indirect)

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One way to do it is to compute a credit adjusted Zspread, roughly:

  1. Discount use UST
  2. Assume a constant default intensity and back it out from say bond A
  3. Use the UST curve and the default intensity from A to compute zspread on bond B, C, D and E (i.e., find out the zspread on UST curve that will match market prices with the default intensity).

The credit adjusted zspread (=0 for A) will more or less reflect the liquidity premium/risk.

There is an issue with this approach though: we don't know what's the right recovery ratio. I'd use something inline with what the issue size says. But maybe someone with a better credit background than I have would be able to help elaborate.

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