In Section 5.2.2 we argued, when trying to separate the interest rate from the credit component in the discounting of a bond, that other factors next to credit contribute to the correction to the usual discount factors: one of these was liquidity. Liquidity has a great impact on the pricing of bonds (see Schwarz  or Acharya and Pedersen ), impact that presents itself, in a more or less explicit way, in the form of a spread over a risk-free rate.
An investor choosing between two bonds would apply the same reasoning valid for any asset. Between two assets the preference would go toward the one offering an easy secondary market, that is, the possibility to exchange the asset for cash in an easy and fast manner. The more difficult one must compensate with a more attractive price and/or a higher coupon.
There are many concatenated issues around this. There is a link (see Brunnermeier and Pedersen ) between the funding liquidity of an institution and market liquidity in general (traders are unable to fund their positions) that translates into liquidity premia on assets in general. Another interesting point is made by Adrian and Shin  on the connection between leverage and liquidity. This in particular will be important shortly, appearing in the form of market presence, when discussing the pricing of bonds in emerging market currencies. A common feature throughout the literature is to stress an inverse relationship between liquidity and volatility.
In this section we will try to give a few examples of situations where we might need to price bonds with very low liquidity.
Pricing at Recovery
In Section 3.1.2, when introducing the concept of credit default swap, we mentioned the recovery rate of a bond. In the past the protection buyer would exchange the bond itself for the par value and the protection seller would be left with whatever was left of the bond, the recovery rate.
The recovery rate is not a clear concept and, as we have said before, it is not a value that, at least in the context of CDSs, is traded in itself: it is always assumed. When an entity is rather safe from default, a CDS is quoted, sometimes using the rule of thumb that a sovereign offers a recovery rate of 25% and a corporate a recovery rate of 40%. This is because, in case of corporate defaults, there is usually a well-structured process in place where investors can hope to seize a decent amount of assets. This process, because of their rarity and because by definition it would be a transnational one, is not in place for sovereign defaults.
This simple rule of thumb is valid when we are estimating the recovery rate of an entity that is still in good credit standing. As the entity's credit deteriorates however, the concept of recovery, becoming more real in the mind of investors, starts becoming more precise.
Let us place ourselves in the position of a protection seller. The more the default of the entity we are asked to provide protection on seems certain, the more we are going to charge an amount very similar to the one we are eventually going to offer, that is, the difference between par and the recovery rate. The more we are edging closer to default, the more the credit default swap price and the bond price itself will make the recovery rate value almost explicit. This of course is because the element we are usually solving for, default probability, becomes almost an input and frees the role of unknown variable. In this situation a bond is said to trade at recovery, meaning that its price is close to its (assumed) recovery rate.
In Figure 5.5 we are showing quotes of corporate bonds trading at recovery (respectively AIFUL Corporation, Clear Channel Communications Inc., Energy Future Holdings Corporation, and The PMI Group Inc.) and in Figure 5.6 we are showing quotes for Greece government bonds, also trading at recovery.
The first thing we notice is, of course, how low the prices of these bonds are. Values of this order of magnitude are usually easier to find in the corporate bond world than in the sovereign one: Greece is, hopefully, an exception.
FIGURE 5.5 A collection of corporate bonds pricing near recovery. Source: Thomson Reuters Eikon.
At this point we need to state that the concept of trading or pricing at recovery does not necessarily fall within the discussion of illiquidity: the bid-offer spreads shown for the quotes in Figures 5.5 and 5.6 are quite tight despite the dire financial situation of the respective issuers. However, it is not unusual, particularly in the situation of emerging markets, for the two phenomena to appear simultaneously, that is, a drying up of the market combined with a worsening credit situation.
FIGURE 5.6 A collection of four Greek government bonds pricing near recovery. Source: Thomson Reuters Eikon.
FIGURE 5.7 CDS quotes of entities trading at recovery. Source: Thomson Reuters Eikon.
It is useful to combine the information given by Figures 5.5 and 5.6 to the one shown by Figure 5.7 where we show the three-year CDS quote for the Hellenic Republic of Greece and the 10-year CDS quote for The PMI Group Inc. In a deteriorating credit situation we appreciate the importance of the new CDS quotation we discussed in Section 3.2.3 where the protection buyer pays a standard coupon plus an up-front fee. As the default of the entity on which protection is traded becomes more and more certain, the protection seller needs to make sure that the funds needed to offer protection are there to compensate the buyer.
From Figure 5.5 we see that the mid price for the PMI Group bond expiring on April 15, 2020, and paying a coupon of 4.5% is 31.2733. From Figure 5.7 we see that the 10-year CDS on PMI Group trades with a quarterly coupon with an annual rate of 500 bps and an up-front payment of 64.230%. This means that as a protection buyer we are asked to pay upfront 65.48% (the up-front payment plus a quarter of the coupon). In the case of a very likely default within a short period of time (let us imagine a probability of 95%) we are in the situation where a CDS simplifies to
from which it follows that the recovery is 31.07%, which is close to the bond price.
The same could be said for the Greek bond. Let us consider one of the bonds shown in Figure 5.6, for example, the bond paying a 5.5% coupon and expiring on August 20, 2014: the mid price is 41.256. From Figure 5.7 we see that the Hellenic Republic three-year CDS quoted up-front consists of an annual coupon paid quarterly of 500 bps and an up-front payment of 55.042%. This means that in order to buy protection against the default of Greece we need to pay up-front 56.292%. Again, assuming a very likely default in the near future (let as assume a 95% probability of default) the CDS reduces to
56.292% = (1 – R) 95%
from which it follows that the recovery is 40.75%, close to the bond value.
We need to stress again that the phenomenon of trading at recovery is linked first and foremost to distressed debt rather than illiquid debt per se. However the concept of pricing at recovery is an important tool that can be used to price illiquid debt of which, almost by definition, there is little market information.
To calibrate a model, as we have shown in Section 3.2.2, means to imply model variables from market variables and often is not unlike trying to cover something with a very small blanket. Usually there is a one-to-one relationship between the number of model variables and the number of market variables: when this one-to-one relationship is missing we make do with some sort of bootstrapping. In the case of CDSs, not only do we need to resort to bootstrapping, but we also need to assume the value of the recovery rate.
To summarize: we use the market variable, the CDS rate, we assume the recovery rate and we imply the model variable, the hazard rate, or survival probability. This process exists in what we could call the investment grade world, that is, the realm of good debt. When we move toward highly illiquid and/or distressed debt, things simplify considerably. Instead of assuming the recovery rate, we assume the survival probability. There could be two situations we might be dealing with: the situation of an illiquid bond issued by an entity whose CDS rates is more available than the bond price and the situation where neither the bond price nor the CDS rates are available.
In the first situation we apply the principle we have applied in the examples above (with the difference, of course, that in the examples above we had all the needed market information and we were simply proving the validity of the argument): we assume an almost certainty of default, we use the CDS (up-front) quote and we obtain the recovery rate that, because of the near certain default, must be close to the bond price.
In the case where even the CDS rates are not easily found, a trader would take an even cruder approach: he would estimate the recovery rate (assessing the realistic chances of an investor to recover some of the issuer's assets) and would use that value as bond price. This practice was often used to price emerging markets bonds during the peak of the 2007 to 2009 financial crisis.