Jan 11

Capital-at-risk versus economic capital

by David Harper, CFA, FRM, CIPM

FRM |

The latest issue of the GARP Risk Review has two good articles with something in common, economic capital. One article explains an approach to economic capital and the other explains a way to calculate capital-at-risk. Economic capital is one of those key "at-risk" metrics; but most of the at-risk metrics share a single idea. I wanted to explain their commonality. If you keep in mind the commonality among the various at-risk metrics, before you tackle their important and complex differences, it should be easier to learn them.

The key idea is that potential loss is simply the likelihood that some asset will fall below some threshold level. In statistical terms, if the asset value's future outcomes can be represented by a distribution, we refer to the probability the asset will "fall into the tail." For example, if somehow our asset value could be represented by a standard random normal variable, the odds it will fall -1.5 standard deviations below mean are about 6.7%; i.e., =NORMSDIST(-1.5) = 6.7%:

image

The plot above is a density function. The area under the curve, in this example, is almost 7% of the total area under the curve. That's the cumulative distribution function (CDF). The essential idea that unites many of these "at-risk" concepts is a cumulative (loss) distribution. Specifically,

  • Value-at-risk (VaR): the traditionally short-term (e.g., daily) loss metric gauged with 95%, 99% or higher confidence level. For example, at 99% confidence, expected loss is 2.33 standard deviations to left of mean (expected value - 2.33 standard deviations). Therefore, 99% VaR is standard deviation (volatility) multiplied by -2.33. VaR is just a place on the distribution, near the tail. It's just a cumulative distribution function. Or I like to summarize VaR this way: scaled volatility. As in, scaled by the user-selected confidence and time horizon inputs.
  • Cash flow at risk (CFaR): same thing, really, except substitute a cash flow metric (cash flow or P&L, which is just the accrual analog) for the balance sheet metric (asset value). In traditional terms, a stock measure (point-in-time) versus a flow measure (over a period). Given that switch, we are still talking about a cumulative loss distribution.
  • Economic capital (EC): this is the same VaR concept in another context! It refers to the unexpected loss on a credit portfolio. Superficially, the key change is the distribution: we replace the normal distribution with a credit risk distribution. Unlike the symmetrical normal, credit portfolio outcomes are skewed because they have limited upside yet significant downside. The mean is an expected loss (EL), so what we get here is: economic capital (EC) = VAR - EL. See how essentially it is the same idea? Namely, how far to the left in the relevant distribution (how many standard deviations) before we exceed our confidence? Note: the time horizon is longer, too.
  • Capital-at-risk: same as economic capital where the difference is a much lower confidence. From Peter-Paul Hoogbruin's article in GARP Risk Review: "capital-at-risk is a statistical measure of the resources required to absorb unexpected losses over a given period (typically one year) with a 90% level of confidence. This measure is used in addition to economic capital, which measures risk at much higher levels of confidence (e.g., 99.95%)."

Even the Basel II regulatory framework, at its very foundation at least for advanced approaches in the U.S., has a single idea at its core: value-at-risk (i.e., cumulative loss distribution given time and confidence). Abstractly, this is the same as economic capital. The advantage, as in Basel, is capital can be aggregated (summed) across capital types. VaR or economic capital is a common currency for "apples-to-apples" comparison or aggregation.

Finally, this is meant to be a learning point. I don't mean to minimize the profound application differences. The details make for some big differences in practice, for example:

  • Just changing the confidence level can imply an audience shift. As Hoogbruin suggests, economic capital is a function of very high confidence; so it's really about "solvency or survival." Even if the distribution is unchanged, as we shift down to the 90% confidence of capital-at-risk, we start to measure something more interesting to shareholders.
  • Economic capital: as mentioned the application of VaR to a credit portfolio involves changing both the distribution and the time horizon. Going from a daily horizon to an annual horizon is non-trivial (probably not simply a matter of applying square root rule) because, as Jorion shows (Value at Risk, 3rd Edition, page 421) we care about today's economic capital as a function of next year's worst expected credit loss. So, there is nontrivial discounting
  • CFaR moves into operational realm an invites a host of operational considerations that are beyond financial engineering. Where VaR may inter-relate primarily to market liquidity risk (how does liquidity impact instrument price or saleable value?), CFaR concern funding liquidity risk (can our firm fund obligations)

Comments

  1. Kelly Stuart 12 Jan 2008

    Good timely topic.. The author points to a number of factors that very important.

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