Friday, December 14, 2012

Mirror, mirror on the wall. How shall we define "fair value", if at all?

Today I managed to sneak away from the office for some quiet contemplation on the matter of pricing accuracy. I don't think anyone noticed I was gone.

Does anyone care about the accuracy of their vendor prices?

Pricing accuracy in the bond markets ought to be a hot topic. After all, how can one cross trades effectively, or monitor customer markups (something we know isn't really done) or optimize one's portfolio, or perform any number of front, middle or back office activities with confidence? Many industry participants bemoan the poor quality of third party bond pricing and perhaps unremarkably, no vendors of bond or CDS prices dare to quantify their accuracy. The question of accuracy is never broached except through vague references to confidence.

On the vendor side it is sometimes argued that customers are insensitive to pricing quality and the service is therefore sticky. But that argument presumes there will be no material change in market structure or competitive forces - an argument that was admittedly correct in the past. Major buy side firms are gearing up to better quantify their relative transaction costs (it provides an excellent marketing opportunity if nothing else). Others wish to use their inventory to generate alpha. And many are looking for lower cost means of making markets or supervising the same.

So the relevant question in a couple of years might not be "is accuracy of end of day pricing important?" but "why would I buy an additional service for end of day pricing only that is less accurate than the real-time services I have recently taken on? Who would want to generate day one P/L issues for themselves, for example?

Thus in the interest of fighting accuracy apathy, either real or perceived, we consider here two simple targets that vendors might be asked to aim at when pricing the "fair market" value of a bond. They are imaginatively called the "fair value target" and the "fairer value target". The first is easier to explain. The second slightly more logical. Both are flawed, but let's not make the perfect the enemy of the good.

A Fair Value Target

The Fair Value Target at time is a "size", "money" and "time" weighted average of the subsequent interdealer trades, where loosely speaking, "money" = \(\int\) "size". Specifically, if we fix some moment \(t\) at which a price is supplied by a vendor and consider the \(J\) subsequent interdealer trades (say \(J=25\)) one might compute $$ FVT(t;J) = \frac{ \sum_{j=1}^{J} p_j s_j e^{-(t_j-t)} e^{-M^-_j} } { \sum_{j=1}^{J} s_j e^{-(t_j-t)} e^{-M^-_j} } $$ where \(p_j\), \(s_j\) and \(t_j\) are the price, size and time of subsequent interdealer trades with time measured in business days. This is just an exponentially decaying weighted average where imminent trades are weighted more heavily than distant ones. But there is also an additional "money" decay term I have thrown in - at the very least to provoke discussion amongst the \(2\frac{1}{2}\) readers of this blog. Here $$ M^{-}_j = \frac{1}{c}\sum_{k=1}^{j-1} s_k$$ is the cumulative trading volume up to but not including the trade in question, and we set \(c=$1,000,000\), say, so that "money" is measured in millions. Some motivation for this additional term comes from the notion of a risk limit or rather, the notion that a sufficient volume of trading is sufficient to establish a market price (and therefore render subsequent trades irrelevant). For example it seems unreasonable to argue that $20M of thursday's trades should be included in the assessment of accuracy of Tuesday night's end of day price if there was, say, $10M of trading on Wednesday.

A Fairer Value Target?

At time of writing the Fair Value Target has been road tested with valuation specialists at a couple of bulge bracket banks, and with traders looking to auto-quoting bonds. It has proven reasonably popular, though that may reflect more on the gaping void in this space than anything else. The fair value target is open to several critiques and I decided to mention one here before anyone else noticed: the slightly unnatural use of \(M^{-}_j\). Indeed the fair value target, as written above, is not invariant to splitting of future trades. We can easily fix this, however, by integrating in money instead of time. Thus we might write $$ FVT'(t;J) = \frac { \int_{m=0}^{M^{+}_J} p(m) e^{-m}e^{-(t(m)-t)} dm } { \int_{m=0}^{M^{+}_J} e^{-m}e^{-(t(m)-t)} dm } $$ where \(M^{+}_J := M^{-}_{J+1}\) is the total amount of money under the bridge up to and including the \(J\)'th trade, \(p(m)\) is the price when \(m\) dollars of trading has occurred, and $t(m)-t$ is the time we have progressed when \(m\) dollars of trading has occurred. The reader may verify that this amounts to the following changes in the fair value formula when expressed as a sum over future trades: $$ FVT'(t;J) = \frac{ \sum_{j=1}^{J} p_j e^{-(t_j-t)} \left( e^{-M^-_j} - e^{-M^{+}_j} \right)} { \sum_{j=1}^{J} e^{-(t_j-t)} \left( e^{-M^-_j} - e^{-M^{+}_j} \right) } $$ where again, \(M^{+}_j\) is shorthand for \(M^{-}_{j+1}\), the money that has flowed under the bridge by the time the \(j\)'th trade is "over".

Which target is best?

Now I personally prefer the aesthetics of the "fairer" over the "fair", but admittedly the former is slightly harder to explain. As an aside I certainly make no claim that either is optimal for assessing the de-noising of multivariate data with serial correlation, for example, such as appears to characterize the corporate bond market. But if we restrict ourselves to univariate statistics and specifically the "fair" and the "fairer" as defined above, then as a pragmatic matter I have not yet pushed for the "fairer" over the "fair". I'm just not sure it makes any difference to the results so why limit the audience? Simplicity is a virtue.

For those who may be interested here is a histogram of the ratio of the two fair value proxies when computed for a reasonably large collection of trade time series. The ratio is reasonably close to unity, though I am not suggesting we dismiss the difference on this basis. My temporary opinion is based more on the fact that overall results tend to be robust to much bigger changes in the target than the ones we contemplate.

So who wins?

This is, as yet, no competition in the real-time bond pricing space. We can say, however, that Benchmark's realtime "Magenta Line" sub-sampled at the end of the day is roughly \(25\) percent more accurate than any other end of day data your author has been able to get his hands on, including that supplied by vendors and also internal marks from banks. Perhaps the next step would be for audit firms to perform their own, independent analysis.

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