# Value the fair price of a 20y Bond

The on the run treasury bonds can only have the following maturities: $$2,3, 5, 7, 10, 30$$. With a $$4$$ year bond it's "easy" to evaluate it since we can compare it to the price of the $$3$$ and $$5$$ on the run bonds.

Though how people typically evaluate a bond with a maturity of $$20$$ years? If the bond has $$20$$ years it means it's an old $$30$$ years Treasury bond, so it has been issued $$10$$ years ago and is thus very illiquid. The two closest on the run bonds are the $$30$$ and the $$10$$ years bond. These bond are very far away in maturity from that $$20$$ year bond so the price of the $$10$$ and $$30$$ year bond probably don't reflect the price of the $$20$$ year.

So how people would typically evaluate the price of such a bond, and in the case of comparing to near on-the-run bond what strategy people typically use in the industry to take into account the illiquid aspect of the off-the-run?

There are on-the-run 20Yr UST bonds. The 20Yr New Issue is auctioned quarterly, with re-openings monthly. In fact there was one that was just auctioned on 2/21/24 and will settle tomorrow 2/29/24.

10Yr and 30Yr New Issue are also auctioned quarterly. When a new bond is auctioned, the previous on-the-run is what is referred to as the "olds", and after the next auction, it will be referred to as the double olds, or old-olds. The old-olds, olds, and on-the-runs are all considered to be liquid (although the on-the-runs are more liquid than the others). Those that like to stay liquid will do what is known as rolls; they will sell the olds for the on-the-runs. Less liquid bonds will trade at a higher bid-ask spreads, in smaller sizes, and less frequently. You may have to induce someone to buy the less liquid bonds by offering them cheaper, sometimes through the mid (ie. at a higher yield than might be prescribed by the yield curve.)

Any of these intermediate maturity points could be compared to a price derived from a well constructed yield curve, with adjustments for liquidity (for example, adding a yld premium to buy cheaper by looking at volumes, volatility etc.). Also, one could check the holders to see if the issue is held by real money or fast money investors.

For greater generality, I will ignore the US Treasury reference.

"Illiquid" may mean many things. Perhaps it means that the yield is not observable. Or perhaps is means that the price is observable and the bid-offer spreads is wider than for other instruments. Or perhaps it means that the instrument seldom trades and in small sizes, so trying to move a lot of it would move the yield by a lot.

Suppose we can observe mid yields for 10y and 30y, and nothing in between. We may want to interpolate the 20y mid yield - perhaps because there's a bond there whose yield is not observable, or perhaps to estimate what the yield might need to be to sell a new issue at par. To interpolate, you could linearly interpolate the yields, or assume constant forward, or use splines, or copy the shape of other yield xueves...

Once you have an interpolated mid yield, you wonder how low liquidity would affect it. Perhaps it would increase the yield. Or perhaps it would just widen the bid-ask spread.

You can and should test all such conjectures - apply your methodologies to yields that actually ate observable, and compare.