Business risk concerns deviations of cash flow from expected levels. If we include unexpected cashflow in computations of capital, shouldn’t we also include expected cashflow? It seems sensible that a highly profitable firm would need less capital than a twin firm with equivalent risk but lower profits. After all, the probability of a given size end-of quarter loss will always be lower for the more profitable firm.

Unfortunately, it is not this simple. Risk grows at the square root of time while expected profits accrete linearly with time. The ratio of risk to expected return generally converges to infinity for time short intervals. As a result, the importance of earnings drift varies with the choice of horizon.

If all that mattered were the end-of-quarter results, then it would be this simple. But horizons in most risk computations are the least interesting part – horizons are really more of a standardization tool than anything else. Regardless of the horizon of a particular bottom-up VaR computation, what we really care about measuring is instantaneous risk. There are two reasons for this. First, we do not honestly think that the portfolio will remain constant and fixed over any reported horizon longer than a week or so (e.g., two-weeks, one month, etc.). Over time the actual distribution of the P&L will differ from the VaR because of intervening portfolio adjustments.

Second, and more important, is that the probability of breaching a given loss amount over a fixed period can be far higher than the probability of breaching that loss amount at the end of the period. Equity prices during 1987, the year of the crash, were unremarkable if one looks only at annual US returns. One could not guess how frightening and risky that Monday in October was given only the way things turned out. This is the first passage time problem.

First passage times are only rarely discussed in conventional VaR calculations based purely on riskless drift. It’s not that first passage times aren’t important for such calculations – they can be material. But most people know how to calibrate the correspondence between a standard VaR and a VaR that measures the probability of crossing the threshold at any time over the horizon. Once we introduce drift – non-zero expected profits – the relationship between these two VaR calculations varies dramatically based on expected profits.

To see all this, it’s best to consider an example. Let us take a standard VaR computation for a portfolio with a certain risk and expected return over a given horizon. The VaR merely tells us the losses on the portfolio that we expect to exceed a given percentage of the time at the end of the horizon. For instance, suppose a portfolio has a continuous expected return of zero and annualized standard deviation of 18% (roughly the historical average risk of the US stock market). A 95% VaR over a 1 day horizon would yield 1.54%, meaning that 95% of the time, we would expect the portfolio’s return to exceed a 1.54% loss over the next day.4

Next, let us calculate a VaR based on first passage time. This VaR will give us the loss on the portfolio that we expect to exceed a given percentage of the time from inception to any point during the specified horizon. One can think of this calculation as looking across portfolio paths and finding the value of the portfolio that we expected to exceed continuously. To continue with the same portfolio example, a 95% VaR based on first passage time over a 1-day horizon yields 1.83%. In other words, 95% of the time during that the day, the portfolio value was above a loss of 1.83%. This result shows greater dispersion than the end-of-day VaR because extreme end-of-day losses are quite likely to have been preceded by even more extreme losses during the day.

Percentage By Which First Passage Time VaR Exceeds Standard VaR

This table shows the sensitivity of VaR calculations to horizon. If we simply focus on risk, and leave expected returns out, the first line shows that there is a 15%-20% deviation between standard VaR measures and first passage time VaR. For example, the ratio for 1 day with a continuously expected return of zero is just 19% = 1.83% / 1.54% - 1. Increases in risk on the order of 19% are not insignificant. However, as the first line of the table shows, the increase is not very sensitive to horizon, so it is not particularly troublesome.

However, when expected returns are non-zero, the difference between these two risk measures becomes very sensitive to horizon. As we already know, over short periods of time, expected returns don’t matter for the risk computation – over a 1 day period, a 20% annual expected return is less than 10 basis points. So there is little effect on the VaR. However, at longer horizons, the two risk measures begin to diverge considerably. Indeed, over very long (e.g., 3 year) periods of time, the ratio computed in the table no longer makes sense. With a 20% expected return, the standard VaR shows a positive return, since the 95% level of the distribution is an absolute gain. However, this is not true for the first passage time portfolio, which continues to show large losses at 95%.

How can we make sense of these numbers? Of course, we don’t literally care about first passage time VaR. Such a computation is most useful if there is an inflexible and meaningful threshold that we don’t want to cross, trigger perhaps by a debt covenant, or a particular level of losses. No firm is literally going to be shut down because of a momentary breach of economic or regulatory capital requirements.

However, first passage time is in the spirit of what we are trying to achieve with capital measures. This is particularly the case because much of the concern with capital standards is driven by customer and rating agency expectations rather than solvency per se. One cannot know the point at which questions begin to be raised by customers and/or regulators about bank viability and liquidity. But surely that point matters for determining capital, and just as surely, it is not driven by end-of-period values.

With this in mind, we can return to our motivating question in this section: shouldn’t we have lighter capital requirements for a firm with higher profits but the same risks? If we take the first passage time concept seriously, we still see an impact at longer horizons of profitability on risk. The table below shows how first passage time VaR changes with expected return at different horizons. Naturally, there is no impact of expected returns at short horizons. But at quarterly intervals, there remains a substantial reduction in risk as expected profitability increases. This reduction is much smaller than under standard VaR measures, but it is noticeable.

It is interesting to note that at 1 quarter, the standard VaR computation used in Table 1 above yields 13.7% and 9.3% for expected returns of zero and 20%, respectively. Thus the standard VaR result with zero expected returns is essentially the same as the first passage time VaR result with a 20% return, which the table below reveals to be 12.8%. In this example, therefore, adding expected profits reduces losses by about the same amount as account for first passage time increases losses.

First Passage Time VaR

Before leaving the computation of risk issue, recall that by necessity we continue to assume that the portfolio of risks remains constant throughout the entire horizon and that the portfolio returns are instantaneously normal with constant mean and variance. As mentioned above, these approximations are likely to become greater sources of concern at horizons like a quarter or a year.

The important message from this section can be summarized as follows. Standard VaR measures basically estimate continuous risk over a short interval. Risk could be measured this way over longer intervals (e.g., 1 quarter or longer), but that would inappropriately emphasize the importance of an arbitrary an end-of-horizon date. As a result, expected returns – which matter only at longer horizons – don’t have a very important practical impact on VaR calculations. First passage times, however, are concerned with the entire horizon and therefore do not emphasize the end-of-horizon date so much. So first passage time probabilities are more appropriate for gauging risk over longer horizons when expected returns are material. Efficient capital markets may drive expected excess returns under risk neutral probabilities to zero. But efficient product markets in non-warehousing functions, such as origination, distribution, and servicing, are much less likely. Our measures of first passage times show that non-warehousing businesses can have a major impact on economic capital requirements of financial firms.

4 Because this is only an example, I assumed volatility is evenly spread across a 365-day year. As is well known, the volatility during days that the market is open is much higher than the days on which it is closed, so that this result understates the risk on a day when the market is open.