Dean wrote in his last posting that the exam question posed by Craig Turnbull’s thoughtful piece on Matching Adjustment was whether the MA, whose purpose is to provide a measure of long-term credit default risk, actually delivers a ‘good measure’ of this risk.
Turnbull sets out a deceptively simple looking problem. Suppose we hold a well-diversified portfolio of MA-eligible 10-year zero-coupon non-financial corporate bonds. All the bonds have a BBB public credit rating and a yield to maturity of 2.5%. The 10-year risk-free yield is 1.0% and so the bond credit spread is 1.5%. The problem is to work out this bond’s capital requirement.
He continues:
There is little doubt that, if we were starting with a non-S(olvency)II blank piece of paper, we would not assess long-term credit default risk by making the calculations prescribed in the SII MA. Instead, the long-term credit default risk and its capital requirements would surely be assessed by directly considering that risk.
As Dean put it to me, you don’t measure risk by following some formula designed by the EU, rather, you measure the risk.
That is, we would use some sort of model of the multi-year default experience of the bond portfolio and assess the probability distribution of the outcomes. The capital requirement would then be determined as [a] function of the losses in the tail (e.g. the 95th percentile of tail losses). …
Actuaries should recognise that this is a very difficult question to definitively answer: by its nature, it is extremely difficult to robustly estimate the tail probabilities associated with long-term financial market risks. …
In the specific example above, the estimation of the capital requirement to support the 10-year hold-to-maturity credit risk of the portfolio naturally requires assumptions for the 10-year default rate of the bonds and the recovery rate in the event of defaults. Most critically, however, it also requires assumptions about the nature and degree of co-dependency amongst the default experience of the bonds and the portfolio diversification that can therefore be projected to occur (in the tails). (My italics)
For example, suppose we define the capital requirement as the capital needed to cover the 95th percentile portfolio loss after holding the bonds until they mature after 10 years. Let’s further suppose there are 100 bonds in the portfolio and the 10-year default probability is assumed to be 5% (just to make the maths simple, rather than because that might be the ‘right’ number). If the bonds’ default experiences were assumed to be statistically independent, then the number of bond defaults in the portfolio has a straightforward binomial distribution with a mean of 5 and a 95th percentile of 9. If, to take the other extreme, the bonds are assumed to be perfectly correlated, then the mean is still 5, but the 95th percentile default outcome for the portfolio is not 9 but 100 defaults.
The ‘right’ answer for the 95th percentile estimate is no doubt somewhere between 9 and 100. But where? Many statisticians (and actuaries) will be pleased to attempt an answer to this question. A good answer will involve discussion of the relative merits of Gaussian and Gumbel copulas, calibration optimisation algorithms and suchlike. But before we turn our computers on, we should take a step back.
Then he and Dean explain that we don’t have long enough data samples to estimate this 95% VaR with any precision and they are of course right. From which Craig concludes:
It is simply not possible, I would argue, to infer a reliable estimate of the 95th percentile tail of the 10-year outcome from a statistical analysis of this data. To be clear, I am not suggesting there is a better way of estimating the statistic. Rather, I am suggesting it is simply not possible to obtain reliable statistical answers to this question … (My italics)
I couldn’t agree more. I would add however that it is not just that the datasets are not long enough. I have done simulations myself on the volatility of the correlation and I can tell you that existing datasets are nowhere near long enough to estimate the correlation with any precision even on the heroic assumption that the correlation is constant, which it is not. In fact, empirical correlations are famously volatile. But these points further reinforce Craig’s conclusion that it is not possible to obtain reliable statistical answers to this question.
Cue the Matching Adjustment. Craig then develops his example and shows that the Matching Adjustment reduces the Solvency II capital requirement on his bond portfolio by a factor of more than 4. Dean has worked through his example analysis carefully and confirmed that it is correct.
Craig then draws the inevitable conclusion:
to assert that the MA calculations result in an appropriately prudent amount of capital being held to support the long-term credit default risk within MA bond portfolios requires a large and unavoidable leap of extra-statistical faith.
Beautifully put!
Let me put it this way.
The statistics tell us that there is a lot of long-term risk and the implication is that we may as well acknowledge that we have to live with it. The risk is simply there.
So how does Matching Adjustment address this risk?
The answer is that it doesn’t. It just assumes most of it away as if by the wave of an actuary’s wand.
To carry out the MA analysis, we start with the assumption that we have a better handle on short-term risks than long-term ones. So we (or I should say, they) then project from the more certain short-term to the less certain long-term. And how do they do that? They invoke a set of Solvency II rules, as if (and this is the actuary’s wand bit) those rules somehow cut through all the longer-term uncertainty fog and reduce it to something much clearer, i.e., to something much less uncertain. As if that achievement were not remarkable enough, those rules go further and precisely split the credit spread into that portion, the Matching Adjustment proper, which they know will be obtained with absolute cross my heart certainty, and the remaining portion, the Fundamental Spread, which is the risky part of the spread.
The only trouble is that this solution is all make believe, because this problem of the riskiness of long-term risky investments cannot be solved, in the sense that the MA would solve it, by reducing a lot of uncertainty to a little bit of uncertainty.
The point is that you can’t break down the spread over risk-free into a risky part and a non-risky part, because all the spread over risk-free is risky. That’s why it is the spread over risk-free.
So if you ‘believe’ in MA, you believe that a bunch of dubious rules concocted out of thin air by a bunch of committees operating under all sorts of political, commercial and other pressures have somehow managed to ‘solve’ a problem that we know cannot be solved: they have created certainty where none existed before.
If you believe that, you really do believe in magical thinking.
All MA does is give the appearance of a solution.
Still, you can believe in MA if you wish. You can even escalate it to the status of a Fundamental Actuarial Principle, but all that does is make matters worse. Are they really going to teach this stuff? Presumably so. Can’t have a Fundamental Actuarial Principle and not teach it. If they are going to teach it, then they need to examine it too and I can see the exam questions already:
Q1. Explain why the Matching Adjustment is a fundamental principle of actuarial science. (Hint: a very good answer will also explain clearly and in plain English why the MA is not one of the weirdest emanations of the human mind.)
Q2. It is often said that one should not count one’s chickens before they hatch. Explain why the Matching Adjustment is an exception to this adage.
Q3. (Advanced candidates only) Explain the apparent paradox that the Matching Adjustment is a Fundamental Actuarial Principle but is totally absent in financial economics.
Good luck with all that, but please don’t write in asking for the model answers. What you have here is an actuarial version of the Doctrine of Papal Infallibility, minus not just the all-important guidance of the Holy Spirit but common sense too. Whilst they are at it, they may as well chuck in an Anti-Modernist Oath as well.
There is also an irony here. One of the reasons commonly given by actuaries for invoking the MA is that short-term prices are more volatile than long-term ones. As David Wilkie once put it, “The actuary is … saying that the market has temporarily got it wrong, but that, in due course, it will get it right.”1. Hold on, you say. Didn’t you say earlier that the MA is justified by the short-term being more certain than the long-term? Yes. But now you are saying that MA is justified by the long-term being more certain than the short-term. Yes, that is true too. It’s all quite simple really.
BBC journalist Howard Mustoe has a nice take on MA in his BBC Radio 4 programme, The Equity Release Trap, first aired on August 7th last year. To quote from the transcript:
29:00 Howard: [racecourse sounds in background] I tried this idea out at Leicester racecourse with Neil our bookie. He didn’t like it. [To Neil] ‘What would happen if the rules changed so that the people who make the rules decided that you could pay out for me and a few of my friends if we have this level of certainty’?
Neil: Well you can’t possibly know it’s really going to win. So there’s absolutely no point in me giving your money before the race, because it might not win. There’s no such thing as a racing certainty. Despite any rules [laughs].
Neil is entitled to scoff. If the asset return is not guaranteed, then he would be unwise to pay out the winnings before the race has finished. That’s why he waits for the race to end. Because he has never heard of Matching Adjustment, he doesn’t even pay out some of the winnings in advance. What would he know about betting?
But with MA it is not so much that the return is not guaranteed, but rather that according to statistical analysis, the return is highly uncertain. Yet MA claims to have somehow done away with most of this uncertainty, even though that uncertainty is statistically irreducible. So either the statistics is wrong or the “fundamental actuarial principle” of MA is wrong. Take your pick, but they can’t both be right.
MA also violates the fundamental accounting principle that risky profits should not be booked until they have been realised. After all, what happens if you book those profits because you thought they were certain and then they are never realised?
Or look at it through the perspective of (basic) economics. Either returns are risk-free or they are risky. If they are risk-free, you know you will get them and you discount at the risk-free rate. If they are risky, you don’t know if you will get them but you are promised a higher return for bearing the risk, and you discount at that higher return. That’s it.