Jumbo jets and fighter planes: exploring the Point-In-Time approach
Posted on 13-01-2012 | 0 comments
In previous articles, we have discussed the relative merits of the Point-in-Time (PIT) and the Through-the-Cycle (TTC) views of risk and capital; in a new research report, we analyse these approaches from the perspective of policyholder protection. This entry summarises the main findings, namely that PIT capital regimes can in principle simultaneously provide a higher degree of policyholder protection and a lower long-run average capital requirements than the TTC approach, although there are practical difficulties which must be considered.
Figure 1: Comparison of TTC and PIT Value-at-Risk at 99.5% over one day for the S&P 500
PIT capital takes account of all available current information to assess capital requirements for the current period. By contrast, TTC capital aims to set capital requirements for a typical year. These two approaches turn out to deliver both different levels of average capital and potentially different levels of protection for policyholders. In our report we develop and illustrate three key results as follows.
Result 1: The long-term average capital requirement using the PIT approach is typically lower than the capital requirement using a TTC approach.
We know that the average of a function of a random variable is not the same as the function of the average of that random variable. In a similar manner, the capital required for average risk is not the same as the average capital for time-varying risk. In fact, we have observed that long-term average PIT capital requirements are typically lower than TTC capital requirements. This is not always the case but in all the realistic examples we have investigated we have found this result to hold.
To illustrate this we can use real data on equity returns. Figure 1 shows daily S&P 500 losses (negative returns) from 1990 to 2011, along with TTC and PIT “capital requirements” of 99.5% Value-at-Risk over a one-day horizon. The frequency of losses exceeding available capital is 0.5% per day in both cases. The TTC case holds a constant level of capital, while the PIT case dynamically adjusts capital requirements according to forecasts of volatility. The average PIT capital requirement is 2.94%, significantly lower than the TTC capital requirement of 4.17%.
Intuitively this can be thought about using the following analogy: if you were flying a jumbo jet through a mountain range, how low would you dare fly? The chances are you’d want to fly at, or about, the height of the highest mountain in the range. How about if you were flying in a nimble fighter jet? You could fly lower through most of the range, and pull up over the high peaks whennecessary (Figure 2). Your average height would be lower in the second case than the first (although the ride might be more nail-biting!).
Figure 2: TTC is like a jumbo jet, PIT is like a fighter jet. You can fly lower on average with PIT, although at times you may fly higher.
Result 2: The PIT approach using Value-at-Risk provides the optimum protection to policyholders, in terms of the Expected Policyholder Deficit, for a given average level of capital support over time.
The Expected Policyholder Deficit (EPD) is the expected value of policyholders’ losses through insolvency. The EPD is often used as a measure of policyholder protection, as it considers both the severity and the frequency of insolvency. It turns out that, for any given long-term average level of capital support, the policyholders’ total expected deficit is minimised if this capital is held on a PIT basis, using Value-at-Risk as a risk measure.
This says that if an insurance company wishes to maintain a long-term average level of capital support of $100 million, say, then it is much more beneficial to policyholders for the insurer to hold more capital when riskiness is high, and less when riskiness is low, such that the average level of capital is $100 million, than to hold $100 million at all times. This is a very general result which does not make strong assumptions about the particular distributional form of the model.
In the example of equity returns above, if we scaled up the PIT capital requirement so that the average PIT capital is the same as the TTC capital requirement of 4.17%, the Expected Policyholder Deficit in the PIT case is 0.0011% per day while in the TTC case it is over 7 times higher at 0.0078% per day.
Result 3: The PIT approach can simultaneously give a higher degree of policyholder protection and lower long-run average capital requirements compared to the TTC approach.
What is the outcome of combining Result 1 and Result 2? Do the costs and benefits associated with the different approaches to capital just cancel out? If TTC capital requirements are higher than average PIT capital, does this offset the lower Expected Policyholder Deficit for PIT for the same average level of capital? The answer is no, not in general. Even when TTC capital requirements are higher than average PIT capital requirements, the Expected Policyholder Deficit tends to be lower in the PIT case than the TTC case.
To achieve the same degree of policyholder protection, in terms of Expected Policyholder Deficit, TTC capital requirements may have to be substantially higher than long-run average PIT capital requirements.
Table 1 summarises the average capital requirement, probability of insolvency and EPD for the following four cases in our S&P 500 example:
- TTC(I): Through-the-Cycle VaR at the 99.5% level
- PIT(I): Point-in-Time, scaled to match the same average capital requirement as TTC(I)
- PIT(II): Point-in-Time VaR at the 99.5% level
- TTC(II): Through-the-Cycle, scaled to match the same EPD as PIT(II)
|
|
TTC(I) |
PIT (I) |
PIT (II) |
TTC (II) |
|
Average Capital Requirement |
4.17% |
4.17% |
2.94% |
5.05% |
|
Probability of Insolvency |
0.49% |
0.12% |
0.49% |
0.26% |
|
Expected Policyholder Deficit |
0.0078% |
0.0011% |
0.0044% |
0.0044% |
Table 1: Comparison of results of different capital approachesapplied to daily S&P 500 returns
Summary
Our analysis and investigations suggests that, in principle, the PIT approach to capital requirements provides a higher level of policyholder protection than the TTC approach. To put it simply, with TTC, when policyholders lose, they tend to lose big. Further, PIT capital requirements tend to be lower in the long-run average sense than TTC capital requirements – capital is committed when it is needed most. Capital has an opportunity cost and so shareholders will prefer a company to operate at lower capital levels if they can achieve the same objectives. In practice, however, we do acknowledge that there are a number of operational difficulties in carrying out the PIT approach. In particular, PIT capital requirements can be very volatile, and a company could find itself having to raise significant levels of capital at certain points in the “cycle”. Thought needs to be given to how a company could manage these variations in capital requirements i.e. how the costs of raising and distributing capital influence the optimal strategy and how the volatility in capital requirements may enforce cyclicality.
Finally, these results do not incorporate the impact of model risk (i.e. the possibility of mis-forecasting risk) which would both tend to dampen the movements in PIT capital. In terms of our airplane analogy, if it is foggy, or if the instruments in the cockpit of the plane are faulty, the PIT pilot may be wise to fly higher to give themselves more of a buffer should a mountain suddenly appear.
Contrasting the differing requirements of the various stakeholders in an insurance company, there turns out to be a direct conflict between policyholder protection and the commercial difficulties of implementing a PIT capital management strategy in practice. We believe that the framework summarised here and presented in detail in our research note provides an important reference point for firms and policymakers in respect of a fundamental question in the capital management debate.
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