RiskMetrics™ —Technical Document

70 Chapter 4. Statistical and probability foundations
4.5.2.3 One-tailed and two-tailed confidence intervals
Equation [4.44] is very important as the basis of VaR calculations in RiskMetrics. It should be recognized,
however, that there are different ways of stating the confidence interval associated with
the same risk tolerance. For example, since the normal distribution is symmetric, then
[4.45]
Probability ( r t
< – 1.65σ t
+ µ t
) = Probability ( r t
> 1.65σ t
+ µ t
)
= 5%
Therefore, since the entire area under the probability curve in Chart 4.18 is 100%, it follows that
[4.46a]
Probability (–
1.65σ t
+ µ t
< r t
< 1.65σ t
+ µ t
) = 90%
[4.46b]
Probability (–
1.65σ t
+ µ t
< r t
) = 95%
Charts 4.19 and 4.20 show the relationship between a one-tailed 95% confidence interval and a
two-tailed 90% confidence interval. Notice that the statements in Eqs. [4.46a] and [4.46b] are
consistent with Eq. [4.45], a 5% probability that the return being less than −1.65 standard
deviations. 25
Chart 4.19
One-tailed confidence interval
Standard normal PDF
0.40
0.30
0.20
95%
0.10
0.00
5%
-5 -4 -3 -2 -1 0 1 2 3 4
-1.65
Standard deviation
25 The two statements are not equivalent in the context of formal hypothesis testing. See DeGroot (1989, chapter 8).
RiskMetrics —Technical Document
Fourth Edition