# Mismeasurement of Risk

Investors tend to consider risk as an outcome—how much could be lost at the end of an investment period? Risk is typically measured as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon. This perspective considers only the result at the end of the investment horizon, ignoring what may happen within the portfolio along the way. We argue that exposure to loss *throughout* an investment horizon is important to investors, and propose two new ways of measuring risk: within-horizon probability of loss and continuous value at risk (VaR). Using these risk measures, we reveal that exposure to loss is often substantially greater than investors assume.

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### Where is the danger in measuring risk at the end of an investment period?

Financial analysts worry that means and variances used in portfolio construction techniques are estimated with error. These errors bias the resultant portfolio towards asset for which the mean is over-estimated and variance is underestimated, which may lead analysts to invest in the wrong portfolio. Additionally, financial analysts worry that higher moments, such as skewness and kurtosis, are misestimated. In that case, extreme returns occur more frequently in reality than is implied by a lognormal distribution. These estimation errors often cause investors to underestimate the probability of loss, and to overestimate the probability of gain.

Rather than focusing simply on addressing these issues (though we do address them), we focus on what we believe to be a more fundamental cause of financial failure: Investors’ wealth is affected by risk throughout a period in which it is invested, but risk is generally measured only for the termination of the period.

**Figure 1 **demonstrates the distinction between risk based on ending outcomes and risk based on outcomes that may occur along the way. Each line represents the path of a hypothetical investment of 100 through four periods. The horizontal line at 90% represents a loss threshold( which equals 10% here). It is clear here than only one of the five paths beaches the loss threshold at the end of the horizon, which may cause some to believe that the likelihood of a 10% loss is 20%. However, four of the five paths breach that loss threshold throughout the investment horizon. So, if we care about the investment’s performance along the way, we will conclude that likelihood of a 10% loss is not 20%… but a whopping 80%.

### Why should we care about interim risk?

Investors care about exposure to loss throughout the investment horizon because there are often thresholds that cannot be breached if the investment is to survive to the end of the horizon. If survival is not the main concern, investors may be motivated to pay attention to within-horizon risk because they could be penalized for breaching a barrier. Consider the following:

**Asset management**. A client has a portfolio with a provision that it should not depreciate more than 10% over a 5-year investment horizon. Should the asset manager assume that the client will only review performance at the end of the investment period? Not likely. The client will review performance throughout the investment horizon… and terminate the manager if the portfolio dips below 90% of its value at inception. To limit the likelihood of termination, the manager should consider within-horizon risk.**Hedge-fund solvency**. A hedge-fund manager who believes that the likelihood of significant loss at the end of the investment horizon is slim, leverages the portfolio to increase expected return. However, a significant decline from the value of the underlying assets from inception to any point throughout the investment horizon is much more likely than the likelihood implied by the ending distribution of a hedge-fund’s assets. Additionally, significant interim loss could trigger withdrawals that might impair the hedge-fund’s solvency.**Loan agreement**. A borrower is required to maintain a particular level of reserves as a condition of a loan. If the reserves fall below the required balance, the loan is called.**Securities lending**. Many institutional investors lend their securities to others who engage in short selling. These investors are required to deposit collateral with the custodian of the securities. The required collateral is typically adjusted on a daily basis, to offset changes in the values of the securities. Suppose the investor wishes to estimate the amount of additional collateral that might be required at a given probability for the duration of the loan. This value depends on the distribution of the securities’ values throughout the term of the loan.**Regulatory requirements**. A bank is required to maintain a capital account equal to a certain fraction of its loan portfolio. A breach in this requirement will result in a fine. The probability that the bank will need to replenish the capital account to avoid breaching depends on the distribution of the ratio of the capital account to the loan portfolio throughout the planning horizon, not at the end of the horizon or a finite period within the horizon.

These examples are only a few of the many circumstances in which investors should pay attention to probability distributions that span the duration of their investment horizons.

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### How do you measure within-horizon exposure to loss?

To estimate probability of loss:

- Calculate the difference between the cumulative percentage loss and the cumulative expected return
- Divide this difference by the cumulative standard return
- Apply the normal distribution function to convert this standardized distance from the mean to a probability estimate

When VaR is to be estimated, we turn this calculation around by specifying the probability and solving for the loss amount:

To capture within-horizon variability, we use a statistic called “first-passage time probability,” which measures the probability (Pr_{w}) of a first occurrence of an event within a finite horizon.

This equation gives the probability that an investment will depreciate to a particular value over some horizon during which it is monitored continuously. You may notice that the first part of Equation 3 is identical to Equation 1 for the end-of-period probability of loss. It is augmented by another probability multiplied by a constant, and in no circumstances is this constant equal to zero, or negative. Therefore, the probability of loss throughout an investment horizon must always exceed the probability of loss at the end of the horizon. Additionally, within-horizon probability of loss rises as the investment horizon expands in contrast to end-of-horizon probability of loss, which diminishes over time.

We can also use Equation 3 to estimate continuous value at risk. Whereas conventional VaR gives the worst outcome at a chosen probability *at the end *of an investment horizon, continuous VaR gives the worst outcome at a chosen probability from inception *to any time during* an investment horizon. We cannot, however, solve for continuous VaR analytically- we must use numerical methods. We set Equation 3 equal to the chosen confidence level and solve iteratively for *L*. Continuous VaR equals *–L *times initial wealth.

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### How can I apply within-horizon probability of loss and continuous VaR?

**1. Currency Hedging. **Suppose we allocate a portfolio equally to Japanese stocks and bonds, represented by the MSCI Japan Index and the Solomon Brothers Japanese Government Bond Index. Table 1 shows, based on monthly returns from January 1995 to December 1999, the standard deviations and correlations of these indexes together with the risk parameters of the Japanese yen from a U.S. dollar perspective.

Let us assume further that the underlying portfolio has an expected return of 7.50 percent, hedging costs equal 0.10 percent, and our risk aversion equals 1.00. Based on these assumptions, the optimal exposure to a Japanese yen forward contract is -87.72 percent. The expected return and risk of the unhedged and hedged portfolios are shown in table 2.

Now, let us estimate the probability of loss for the unhedged and hedged portfolios. Table 3 shows the likelihood of a 10 percent or greater loss over a 10-year horizon at the end of the horizon and at any point from inception throughout the horizon for an unhedged and optimally hedged portfolio of Japanese stocks and bonds.

If we were concerned only with the portfolio’s performance at the end of the investment horizon, we might not be impressed by the advantage offered by hedging. But, if we instead focus on what might happen along the way to the end of the horizon, the advantage of hedging is much more apparent.

Even with the foreknowledge that we are more likely than not at some point to experience a 10 percent cumulative loss, we may consider such a loss tolerable. But what about a loss of 25 percent or greater? Again, calculating the probabilities indicates that, although the impact of hedging on end-of-period outcomes is unremarkable, it vastly reduces the probability of a 25 percent or greater loss during the investment horizon. Although many investment programs might be resilient to a 10 percent depreciation, they are less likely to experience a decline of 25 percent or more without consequences.

Now, let us compare VaR measured conventionally with continuous VaR for the hedged and unhedged portfolios. Table 4 reveals that the improvement from hedging is substantial whether VaR is measured conventionally or continuously. For example, measured conventionally, hedging improves VaR from a 5 percent chance of no worse than a 14.68 percent loss, to a 5 percent chance of no worse than a 26.52 percent *gain*. More important, however, is the substantial difference between VaR measured conventionally and VaR measured continuously. Continuous VaR is more than twice as high as conventional VaR for the unhedged portfolio, and when the portfolio is hedged, continuous VaR shows a substantial loss compared with a substantial gain when it is measured conventionally.

**2. Leveraged Hedge Fund**. Now consider the implications of these risk measures on a hedge fund’s exposure to loss. Supposed we are interested in a hedge fund that uses an overlay strategy, which has an expected incremental standard deviation of 5 percent. This hedge fund also leverages the overlay strategy. Table 5 shows the expected returns and risks of the hedge fund and its components for varying degrees of leverage.

The data in Table 5 assume that the underlying asset is a government note with a maturity equal to the specified three-year investment horizon and that its returns are uncorrelated with the overlay returns. Managers sometimes have a false sense of security because they view risk as an annualized volatility, which diminishes with the duration of the investment horizon, but as we have noted, the fund’s assets may depreciate significantly during the investment horizon. Figure 2 compares the likelihood of a 10 percent loss at the end of the three-year horizon with its likelihood at some point within the three-year horizon for various leverage factors (e.g., 2 to 1). Figure 2 reveals that the chance of a 10 percent loss at the end of the horizon is low, but there is a much higher probability that the fund will experience such a loss at some point along the way, which could trigger withdrawals and threaten the fund’s solvency.

The same issue applies if exposure to loss is perceived as VaR. Figure 3 shows the hedge fund’s VaR for various leverage factors measured conventionally and continuously. Whereas conventional VaR for leverage factors less than 6 to 1 is negative (a gain) and still very low for leverage factors up to 10 to 1, continuous VaR rangers from approximately 10 percent of the portfolio’s value to approximately 40 percent of its value.

## Conclusions

Investors measure risk incorrectly if they focus exclusively on the distribution of outcomes at the end of their investment horizons. This approach to risk measurement ignored intolerable losses that might occur throughout an investment period, either as the result of the accumulation of many small losses or from a significant loss that later (possibly, too late) recovers.

To address this shortcoming, we have introduced two new approaches to measuring risk—within-horizon probability of loss ad continuous VaR. Our applications of these measures in reasonable scenarios illustrates vividly that investors are exposed to far greater risk throughout their expected investment periods than end-of-horizon risk measures indicate.

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