# Bond Valuation and liquidity

Assuming the market is perfect liquid, the bond price can be replicated and is related as follows:

$$\sum_{t=0}^{N}c_ne^{-Y(t_n-t)}=\sum_{t=0}^{N}c_nP_{t_n}=\sum_{t=0}^{N}c_ne^{-Y_{t,t_n}(t_n-t)}$$

The $P_{t_n}$ denotes the ask price of a zero coupon bond at maturity $t_n$ and $c_n$ corresponding payment of the bond at time $t_n$. $Y$ is the yield to maturity and $Y_{t,t_n}$ the yield curve. The idea is just to replicate the payment of bond by zero coupon bond at different maturity.

Question: Why does this equation holds only under assumption of perfect liquid market? What would change if we are in a illiquid market?

In practice, this equation won't even hold for the vast majority of bonds in the US Treasury market, which is the most liquid government bond market.

The chart below shows the spreads of US Treasuries relative to a fitted curve (more specifically, a model price is calculated for each bond by discounting its cash flows using a theoretical zero coupon curve. The difference between the model yield and the market quoted yield is shown in the chart): During times of stress, these spreads can become much larger. A similar chart for December 15, 2008 is shown below. Note the range on the y-axis: Some bonds, such as 10-year on-the-runs and the 15-year sector, traded very rich (at extremely negative spread), because of high demand from investors looking for safe and liquid instruments. By contrast, old 30-year that have rolled into the <10-year sector traded at very cheap levels (very positive spread), because people were dumping these papers and moving into more liquid instruments or cash.

• Sorry but I cannot really understand the graph, is it one graph or two separate? I cannot see the fitted curve somehow. I see only green data points. – quallenjäger May 7 '17 at 2:25
• It would be nice if you could provide some information of chart. I am still struggling to recognize it. What is on Y scale and X scale. Again thank you for the extensive answer. – quallenjäger May 7 '17 at 2:32
• I have updated both charts. Y is the spread (in basis points), X is time to maturity. – Helin May 7 '17 at 2:34
• These were original 30-year bonds that were issued 20+ years ago (at the time) and hence had <10-years to maturity. So they were very old and seasoned bonds. – Helin May 7 '17 at 2:37
• So overall, can I see it the market yield curve depends on liquid preference. If there is a perfect liquid money market account so that the investor doesn't care about the liquidity. Then the spread will be zero? – quallenjäger May 7 '17 at 2:40

Illiquidity is a quantifiable state.

The debt investor is not simply picking yield. She has preferences which generate illiquidity. Some can be modelled, and these may be: coupon (above or below par rate), convexity, demand for particular date (e.g. a liability matched investment), and repo income, inter alia.

Around this, other fast money investors may smooth these interplays to normalise the spread distortions. In this mechanism they provide liquidity to the market.

A limited investible volume, amount of financing and/or balance sheet available, and risk appetite of each party determines whether the illiquidity or liquidity factors dominate.