K9 asks whether I believe some or all of the following.
1) the Merton model is inappropriate here;
2) that the BoE has parameterised the Merton model incorrectly; or
3) that this “residual” bit of the spread is accounted for something other than credit and liquidity.
All three as it happens.
The insight of Merton is to model the equity of a company as a call option on the company’s assets, struck at the value of debt – or (equivalently) as ownership of the assets combined with a put option struck at the same value. Default is the event of the market value of assets reaching the present value of debt, i.e. technical insolvency.
The main (but not the only) inputs of the model are the value of assets, the imputed volatility of assets, and the leverage, here defined as the value of debt divided by value of assets.
Regarding question (1) above, i.e. whether the Merton model is inappropriate, well clearly not wholly appropriate, although I like the insight. The option has to be a form of barrier option, exercised when the asset value touches a level which would trigger default, which is not the value of debt, but rather a value somewhat below, on the assumption that a firm with long-dated debt – such as a life assurance company – can limp along for many years while technically insolvent. The version of the model used by the Bank (the Leland Toft model) employs a barrier option, but calibrating the level of the barrier is a black art, and the function used to determine it is much far more complex than the barrier model itself. A model which involves such a degree of guesswork is hardly ‘appropriate’.
Regarding question (2), whether the BoE has parameterised the model correctly, all the evidence suggests that it has not. See the chart above, showing the decomposition not of sterling corporate bonds, but Euro denominated. The orangey bit is supposed to represent the illiquidity spread, but look what happens during the 2002 crisis, when the model spread rises to a level higher than the market observed spread, even through a period of extreme turbulence. That is, there is a considerable ‘illiquidity spread’ of about 75bp in the period before the 2002 crisis, when the market was smooth and presumably liquidity high, then as soon as the disruption happens, i.e. at a time when you would expect liquidity to dry up, the model spread explodes, and illiquidity turns negative!
A negative crisis-driven illiquidity spread fits uneasily with the claim in the Bank’s 2007 bulletin (p.533) that ‘compensation for bearing non-credit related illiquidity risk appears to have been a particularly important driver of high-yield spreads’ (my emphasis). They are referring to the 2007 crisis, but why should the completely opposite explanation apply to the 2002 crisis? Why should illiquidity go negative in the one case, but strongly positive in the other? Some explanatory device is clearly wanted, but we are not given one.
Alternatively, the model is simply wrong.
There is more. The main determinants of the Bank’s model are equity market volatility and leverage. There is public domain data available on the former (using equity option implied volatility), but leverage is more guesswork, moreover the Bank seems to have ignored an important source of error that I touched upon here in the context of Peter Feldhuetter’s work. Peter found (i) that many Merton models are calibrated against default data from periods when default experience was benign. When we extend the data to include periods before the 1940s, the situation is different. And (ii) convexity bias is a significant source of error. Market spreads are from bonds issued by firms with widely different leverage, but the Bank’s model uses a single value. Since an option model by definition has gamma or curvature, the spread is not exactly proportional to leverage, and will rapidly rise as the option gets closer to the money. Taking the function of the average rather than the average of the function leads to what is called Jensen’s inequality.
the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation
Peter discusses the inequality at length in his paper on the Credit Spread Puzzle (link). I managed to reproduce his results (roughly) by running the Bank’s model with leverage set at a higher level of 70%. See the chart below, where you see that there is no illiquidity effect, and indeed the modelled spread over the crisis period is higher than the observed spread.
Which takes me to the question (3) whether the residual bit of the spread ‘is accounted for [by] something other than credit and liquidity’. Well quite. If the Bank’s model has been wrongly parameterised, then the residual that you see when leverage is set to 41.6%, and which disappears when leverage is set somewhat higher, is simply the result of a mistake, not of anything real.
I did raise this issue while at the Bank, and there was no real dissent. It was explained that the model was simply used as an illustration. I had the sense (and this is very Bank of England) that it was not meant to be taken that seriously. Well fine, but the problem is that it was taken very seriously by the architects of the Matching Adjustment. I can remember any number of discussions when charts produced by that wretched model were waved around as though proving some decisive fact, and of course it was decisive.
The Matching Adjustment is with us now, and perhaps we shall have to live through some very bad things as the result of a possible calibration error.