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I am learning the finance topic value at risk and trying to understand the number of standard deviations.

I dont quite get why was 1.65 used to multiply against the std dev instead of 1.644853 as shown in the calculator below (used to retrieve the z-score from the probability)

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enter image description here

BruceET
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ilovetolearn
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1 Answers1

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In the first panel you posted, $Q$ seems to be the probability above $z$ under a standard normal curve. So the program gives $Q = 0.049471$ for $z = 1.65.$ That is, if $Z \sim \mathsf{NORM}(0,1),$ then $P(Z > 1.65) = 0.049471.$ Then $P(Z \le 1.64) = 1 - 0.049471 = 0.950529,$ by the complement rule of probability.

I'm not sure I understand how your program is working, so I'll give some results from R statistical software. I have used it a lot and trust its output. In R, pnorm gives the probability below the cutoff value inside (), and qnorm gives the cutoff that has the probability inside () below it. This matches the usual definition of quantile in statistics books.

pnorm(1.65)
## 0.9505285      # P(Z < 1.65)
pnorm(-1.65)
## 0.04947147     # P(Z < -1.65)
qnorm(.05)
## -1.644854      # q such that P(Z < q) = .05
qnorm(0.950529)
## 1.650005       # q such that P(Z < q) = .050529
qnorm(0.04947147)
## -1.65          # q such that P(Z < q) = .04947149

Because the density function of $Z$ is symmetrical, it is also true that $P(Z < -1.65) = 0.049471.$ The figure might be misleading you in two ways. First, it does not stress that $z = -1.65,$ that $z = \mu - 1.65\sigma$ (note the $-$ sign in the figure). Second, the tail probability is shown as $5\% = 0.05$ while the printout (in very tiny type) shows $0.049471 \approx 0.05$ instead of exactly $0.05.$

Is it possible that your figure has a missing minus sign in the box for "Given z"?

In the second panel, you enter $Q = .95$ and get the answer $z = 1.644853.$ Now $Q$ seems to be defined as the probability below the requested cutoff. That is, $P(Z \le 1.644853) = 0.95.$

R statistical software works the same here as it did above:

qnorm(.95)
## 1.644854     # c = 1.644854 is the cutoff with P(Z < c) = 0.95
pnorm(1.644854)
## 0.95         # P(Z < 1.644854) = 0.95
pnorm(1.645)
## 0.9500151    # P(Z < 1.645) = (nearly) 0.95: '1.645' is the value printed in many books

Anyhow, the two results you show from your program seem inconsistent. Maybe your program changes the meaning of $Q$ depending on whether $z$ is positive or negative. Maybe your first (graph) and second (Calculate Q) images are not from the same session. Maybe there is some 'setting' for the program you don't know about or did not mention. Maybe the picture you captured for the top panel has a missing minus sign.

BruceET
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