The concept of confidence intervals is new in accounting, but it is rapidly increasing in importance. Whether in physics, engineering or financial analysis,
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a considerable number of problems involve the estimation of variables of a given population distribution, which is often assumed on the basis of sample(s). In esti- mating the behaviour of a given variable, the analyst will be typically concerned with obtaining from the sample under study:
● One single value, usually the mean, x–, which is the best estimate of the variable’s central tendency, and
● A measure of dispersion characterizing the range within which would fall other values of the variable under study (more on this later).
The mean, also known as the expected value, is just a point estimate; it is a stat- istic inadequately fulfilling the analyst’s objective to learn about the variable’s behaviour, because nothing travels in a straight line. At best, the area under the curve from one leg up to the mean represents 50% of values. And the other 50%?
By itself, the expected value gives us no indication about the range within which we have a certain confidence that the different values of the variable will be included, nor does it provide any evidence about the shape of the distribution. In fact, this point estimate may be the arithmetic mean of the values in the distribu- tion, or the mode (point of highest frequency), or the mid-range (halfway between the higher and lower value).
Mathematically speaking, one way to proceed with a study of the distribution of events (or values) is through the method of maximum likelihood. Under fairly general conditions, this works on the assumption that for relatively large samples the obtained estimates will be approximately normally distributed, and that the higher frequency events (and their estimates) will be closer to the mean value.
Another way to proceed is the method of moments. With this, we obtain esti- mates of the population parameter(s) by equating a sufficient number of sample moments to the population moments – the mean being the first moment of the distribution. Notice, however, that these approaches still leave open the need to estimate the distribution’s dispersion – which has been classically done through the variance, the second moment of the distribution (the standard deviation s is the square root of the variance).
The mean and standard deviation are shown in Figure 4.5, which represents a normal distribution. One of the problems with this approach with respect to risk management is that while we assume that values and events are normally dis- tributed (which is itself an approximation), we also concentrate on values or events of higher frequency – though we know that risk events at the long leg of the distribution have low frequency but high impact.
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A realistic distribution of risk events is shown in Figure 4.6. While the main body is assumed to be normally distributed (it is leptokyrtotic rather than nor- mal), it would be a mistake to end our analysis at 3s, within which lies a little over 99% of the area under the curve. This will leave out the spikes at the right leg, representing risks of major impact.
Moreover, because we want to do a solid job, we would like to have an estimate of how many values from the risk distribution are included in our study. This is provided by confidence intervals, which have become a key element in research and analysis. From physics and engineering to finance, this concept is important in all statistical studies:
● Values falling outside the confidence interval are not relevant in our study, though their after-effect can be significant
● Those values falling within the confidence interval are characterized by a margin of uncertainty commensurate to their frequency (or distance from the mean).
In simple terms, the confidence interval presents an estimate of how many values around the mean are likely to be included in the study, and must be taken into account in the evaluation of results. A confidence level of 95% means that 5% of all values are likely to fall outside its borders.
This statistic is called a confidence interval because we can be reasonably con- fident that the population mean and other values fall within that interval, even if not all values are included. Confidence limits are the end lines of the interval in question, as can be seen in Figure 4.7 through an example on the spillover of mar- ket volatility in the 1990–96 timeframe (notice that at any particular time the dis- tribution of yield is not normal).
This approach provides commendable results if the estimates of a popula- tion’s parameters through the proxy of sample statistics are accurate enough.
Approximations derived from samples should be close to the mean value, and if repeated approximations are made these should cluster together about their own mean.
Here, in a nutshell, is how the level of confidence is chosen, keeping in mind that in every statistical reference there exists variation. We can say that in lieu of determining from the sample one single value as an estimate of the unknown parameter k, we determine two values less than and more than k:
K⬘ ⭐k⭐K⬙ (4.2)
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This is done in such a way that there exists a given probability 1⫺α, that K⬘ ⭐k⭐K⬙:
● The interval between K⬘and K⬙is known as the confidence interval, and
● These two values, K⬘and K⬙, are the confidence limits of the given parameter.
Statistically speaking, we do not know whether this particular interval covers k, but we do know that the probability of having drawn a sample interval which
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1s* 1s
3s 3s
Standard deviation from mean
* One standard deviation
The large majority of values
But here lie the higher impact events Frequency
2s 2s
Figure 4.5 We assume risk events are normally distributed, but this is not always true High
Frequency
Low
Low Impact
( just note difference)
High Figure 4.6 A practical example from real-life risk analysis: distribution of risk events with long leg and spikes
does not cover k is α(also known as the level of confidence, producer’s risk or Type I error). In the example with yield volatility confidence intervals α⫽0.05, and this corresponds to a confidence level of 95%.
In the operating characteristics (OC) curve in Figure 4.8, we also observe a Type II error, consumer’s risk, or β.8 Let’s take credit rating as an example:
βwould stand for the probability the bank would give a loan when it shouldn’t, given the borrower’s low creditworthiness. By contrast, αwould be the likeli- hood of rejecting the loan requested by a creditworthy borrower.
Given a certain sample and its statistics, in a wide number of applications we can determine the limits that may be expected to contain the values of the credit- worthiness parameters of the population. Here the word ‘expected’ includes in itself a certain risk – namely, the risk that some values of the parameter considered at a certain test would not be included within the computed range.
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Correlation coefficient
Monthly averages in percent
1990 1992 1993 1994 1995 1996
⫹0.4
⫺0.2
⫹0.1
⫹0.3
0
⫺0.1
⫹0.2
⫹0.8
⫹0.5
⫹0.7
⫹0.6
95% Confidence interval
1991
Figure 4.7 Spillover of yield volatility from the American debt securities market to the German market (Source: Deutsche Bundesbank, by permission)
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● Ifα⫽0.05, as in the preceding example,
● Then chances are that 5% of the parameter values would fall outside the confidence limits and 95% would fall within.
For α⫽0.01, the corresponding chances are 1% and 99% respectively. Other things being equal, a confidence interval calculated with α⫽0.01 is much broader than the one calculated with α⫽0.05; the latter, too, is broader than one computed with α⫽0.10. A higher level of confidence means a smaller α. Entities that choose a higher confidence level are essentially more conservative because they want to be sure their confidence limits will not be exceeded by real-life events.
In conclusion, operating characteristics curves, like the one in Figure 4.8, are at the root of the process of inference and, at the same time, map the fact that stat- istical inference involves risks. Risks taken with statistical techniques is the
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90%
0%
High PA Probability of acceptance
Credit rating: High, Medium, Low (just note difference) 80%
60%
40%
30%
20%
10%
50%
70%
r0
Medium Low
b Type II error
r1 r2 r3 r4
a⫽ 0.05, Type I error level of confidence Add-on rate
Interest rates for a loan r0⬍ r1⬍ r2⬍ r3⬍ r4
Accept
prime rate Reject
Figure 4.8 Accept or reject: using an operating characteristics curve in deciding on a loan
price we pay for the use of analytics. This price is much less than the one we will be eventually paying by using rule of thumb and speculation.