Craig Turnbull argues whether the matching adjustment results in EU life insurers holding a prudent amount of capital to support the long-term credit default risk in their fixed income portfolios
Solvency II’s matching adjustment (MA), and the British actuarial profession’s defence of it, have been in the financial press recently. To the extent that the profession wishes to defend the MA as a matter of actuarial principle, we must provide a clear explanation of the apparent logical contradiction at the core of its treatment of credit risk capital: that the capital required to support the risk of adverse asset outcomes can be (partly) created by assuming those same assets perform well.
Such an explanation doubtless must rely on a distinction between the short-term risk horizon of Solvency II and the long-term nature of the risks involved in the cash-flow matching of (illiquid) life annuities. This is the essential rationale for the MA and its application to annuity business.
The potential difficulty here, however, is that the net credit risk capital requirement, i.e. the solvency capital requirement (SCR, measure of short-term market value-based tail risk capital) less the MA (capitalisation of long-term expected credit return from holding matching bonds to maturity), is not (to me, anyway) an intuitive or necessarily meaningful measure of the capital required to support long-term credit default risk.
A simple example of MA and credit risk capital
Let us take a very simple, stylised example to illustrate how the MA impacts on credit risk capital requirements. This will necessarily involve a rather convoluted series of sums, but the calculation is not particularly complex.
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 MA fundamental spread at end-March 2019 is given by Eiopa as 0.57% for such bonds. The MA rules allow the bond spread net of the fundamental spread to be added to the risk-free discount rate for the purposes of the liability valuation.
So the MA allows a switch from the risk-free matching bond to the BBB bonds to increase the liability discount rate from 1.00% to 1.93% (= 2.50% - 0.57%). This reduces the value of a liability cash-flow paying 1 in 10 years from 0.905 to 0.826, i.e. it essentially creates 0.079 of capital.
Under the standard formula, the credit SCR for these corporate bonds is 20%. So, the insurer has a credit capital requirement of 0.2 x 0.826 = 0.165. If we express this market value stress as an increase in the bond spread, then the bond spread increases by 2.31% from 1.50% to 3.81%.
"MA rules can have a highly material impact on the capital required to support the credit risk of bond portfolios"
The standard formula rules allow the insurer to assume that this credit spread increase is not wholly incorporated as an increase in the MA fundamental spread that is used in the stressed liability valuation.
For the BBB bonds, the standard formula allows firms to increase the fundamental spread by 75% (rather than 100%) of the stressed increase in the bond spread. So, in our example, the fundamental spread increases from 0.57% to 2.30% (= 0.57% + 75% x 2.31%).
As a result, the liability discount rate increases from 1.93% to 2.51% (= 1.00% + 3.81% - 2.30%) in the credit spread stress test. The liability value is therefore reduced in the credit spread stress test from 0.826 to 0.781 (whilst the asset value is reduced by 20% from 0.826 to 0.661). The resulting SCR is 0.781 – 0.661 = 0.120.
The net capital requirement for bearing credit risk in the MA asset portfolio in this stylised example is therefore 0.120 (SCR) – 0.079 (MA) = 0.040. In the absence of the MA (and, indeed, the volatility adjustment and the interest rate swap spread which will have to be topics for another day!), assets with a market value of 0.905 would be stressed by 20%, generating a capital requirement for the credit risk in the bond portfolio of 0.181.
So, in our example, the MA has effectively reduced the net capital requirement arising from bearing credit risk in the bond portfolio by a factor of more than 4. This factor will vary as a function of credit rating and duration. It will also vary as market spreads and fundamental spreads move around over time. Nonetheless, the example suggests the MA rules can have a highly material impact on the capital required to support the credit risk of bond portfolios.
Measuring long-term credit risk
Having established that the MA can reduce the capital strains associated with taking bond credit risk very substantially, the next question that arises is: if the MA is intended to provide a measure of long-term (hold-to-maturity) credit default risk rather than the Solvency II SCR’s short-term market price volatility basis of risk, does the above set of sums deliver a good measure of this long-term risk?
There is little doubt that, if we were starting with a non-Solvency II blank piece of paper, we would not assess long-term credit default risk by making the calculations prescribed in the Solvency II MA. Instead, the long-term credit default risk and its capital requirements would surely be assessed by directly considering that risk.
"It is extremely difficult to robustly estimate the tail probabilities associated with long-term financial market risks"
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 function of the losses in the tail (e.g. the 95th percentile of tail losses). What answers would such a model deliver, and how would they compare with the credit risk capital requirements generated by the Solvency II MA?
Actuaries should recognise 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. Indeed, one of the drivers of the Solvency II SCR’s one-year VaR approach was a belief that it would be possible to estimate one-year tail risk more objectively / reliably / scientifically than 10 or 20-year tail risk.
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).
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).
"Before we turn our computers on, we should take a step back"
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.
The statistician / actuary will probably advocate using something like 30 years of historical data, because older data is either difficult to access or is not judged relevant to the estimation of future risk because the economic world is inherently non-stationary – and so what happened in the 1960s and 1970s arguably isn’t an informative guide to our question of what may happen in the 2020s. So, our calibration data set essentially consists of 3 non-overlapping 10-year paths taken from a non-stationary population.
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 (this need not imply long-term ‘real-world’ probabilistic models are useless; to me, it means they are useful as analytical tools that can produce deep insights in the hands of those who understand their limitations; but, for the avoidance of doubt, this also suggests they are not useful when merely used as answer-generating machines).
So what?
Where does this leave us? As illustrated by the simple example above, the MA can materially reduce the capital strains associated with taking credit risk in bond portfolios.
A fundamental argument in favour of this result is that, for insurers with illiquid liabilities, the capital required to support long-term financial market risk should be lower than the capital required for short-term risk - because short-term market prices are too volatile, and hold-to-maturity investors do not need to concern themselves with this short-term excess volatility.
"Of course, credit risk is only one dimension of the assessment of the capital requirements of MA business such as annuities.
Over the past 40 years or so, some financial economists have produced plenty of academic studies of empirical financial market price behaviour that can lend some support to this argument. The MA calculation is one route to practically effecting this reduction in capital requirement while remaining within the overarching framework of Solvency II.
And, of course, credit risk is only one dimension of the assessment of the capital requirements of MA business such as annuities. Regulators have various other mechanisms and margins at their disposal that may be used to ensure the overall capital position of MA business is held at appropriate levels.
However, to assert 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.
Ultimately, this is a difficulty that can only be mitigated by reducing the amount of credit risk that is taken in the asset portfolios that back fixed policyholder liabilities.
If long-term credit risk is as good a bet as the MA suggests, perhaps the best solution for the policyholder would be for life assurers to deliver an alternative to life annuities in the form of a long-term longevity protection product in which the policyholder rather than the shareholder participates in the investment return of the invested premium.
Craig Turnbull is a Fellow of the Institute and Faculty of Actuaries and author of "A History of British Actuarial Thought". The views expressed herein are not intended to represent those of his employer.
