A comparison of alternative risk capital definitions
Posted on 07-03-2011 | 0 comments
There is no universal definition for risk capital. European early adopters of risk-based capital methods (including Solvency II) have adopted a measure which requires sufficient capital to remain solvent (on a market-consistent basis) with 99.5% confidence at a 1-year horizon. In the US, firms and regulators have leaned towards a quite different measure which depends on the behaviour of the balance sheet over its lifetime ‘run-off’. These alternative definitions are difficult to compare directly. Indeed, the theory behind these measures is not well developed. However, we can say some things about the different measures – their sensitivities and how they will behave relative to each other.
Consider a very basic example. Suppose an insurer holds a single unhedged position in a written 10-year put option with a strike at 90% of the current (total return) index level. We assume that the only source of uncertainty is equity risk and calculate some alternate measures of capital as follows:
- VAR measures at 1, 2 and 3 year horizons
- A run-off capital requirement (the PV of the shortfall at the specified confidence level)
- A conditional tail expectation (CTE)
All measures are calculated at various confidence levels between 90% and 99.9%. The results plotted below are for the PV of capital in excess of the basic value of the option. I have highlighted 3 points which represent measures commonly used in different parts of the world:
- 1-year 99.5% VAR measure (14.9%)
- 95% run-off measure (15.9%)
- 98% CTE measure (17.6%)
You can see that, for this example, the capital requirements are broadly similar. Now, the quantity of capital required to support a risk exposure of this type will be critically dependent on the assumptions made about equity risk. For the 1-year VAR measure, the tail of the distribution of equity returns at the end of the first year of projection will be important. For the two alternative measures, the tail of the terminal 10-year distribution will be of most importance. For the analysis below we have assumed that equity returns follow a process where volatility is stochastic (i.e. it varies through time) but it is initialised at a neutral level. However, there are other assumptions we might make which, as you will see, have a material impact on absolute and relative capital requirements.
Consider the following alternatives:
- Volatility is stochastic but its initial value is set to be LOW
- Volatility is stochastic but its initial value is set to be HIGH
- Equity returns are assumed to incorporate some ‘mean reversion’ which will limit the tails of long-horizon equity distributions.
The chart below plots the results from the first chart in the first column. You can see that the capital requirements are pretty consistent assuming constant equity volatility. Now consider the additional columns. When initial volatility is reduced (year 1 volatility falls from 20.5% to 17.6%) the 1-year VAR capital is reduced by nearly 20% to 12.3%. The run-off measure is very slightly affected by the lower year-1 volatility. In much the same way, an elevation in initial volatility lifts the 1-year VAR measure to around 25% and modestly raises the run-off measures. Finally, the ‘bubble’ plot assumes mean reversion so that the annualised volatility of the 10-year return falls to 16.4% and the 95th percentile total return is moved from -42% (under the base case) to -26%. Unsurprisingly, this has a material impact on the run-off capital measures reducing them both to 3.9% (95% run-off) and 6.6% (98% CTE).
The story is a familiar one. 1-year VAR capital is very sensitive to initial market conditions. ‘Point-In-Time’ (i.e. conditional) estimates for the returns distribution will deliver far more variable (and pro-cyclical) capital requirements than ‘Through-The-Cycle’ (i.e. unconditional) measures. VAR capital will be sensitive to short-term movements in asset prices driven by changing risk and liquidity premia. By contrast, run-off capital measures will tend to be less sensitive to initial market conditions but very sensitive to the sources to which the modeller attributes volatility. If short-term price changes are attributed to transitory changes in risk and liquidity premia these assumptions will have a material impact on the run-off measures of capital requirements.
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