Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

66 HANDBOOK OF PORTFOLIO MATHEMATICS

FIGURE 2.13 N′′(Z) giving the slope of the line tangent tangent to N′(Z) at

Z=+ 2

To determine what the slope of the N′(Z) curve is at+2 standard units:

N′′(Z)=− 2 / 2. 506628274 ∗EXP(−(2^2 )/2)

=− 2 / 2. 506628274 ∗EXP(−2)

=− 2 / 2. 506628274 ∗. 1353353

=−. 1079968336

Therefore, we can state that the instantaneous rate of change in the

N′(Z) function when Z=+2is−.1079968336. This represents rise/run, so

we can say that when Z=+2, the N′(Z) curve is rising−.1079968336 for

every 1 unit run in Z. This is depicted in Figure 2.13.

For the reader’s own reference, further derivatives are now given.

These will not be needed throughout the remainder of this text, but are

provided for the sake of completeness:

N′′′(Z)=(Z^2 −1)/ 2. 506628274 ∗EXP(−(Z^2 )/2)) (2.24)

N′′′′(Z)=((3∗Z)−Z^3 )/ 2. 506628274 ∗EXP(−(Z^2 )/2)) (2.25)

N′′′′′(Z)=(Z^4 −(6∗Z^2 )+3)/ 2. 506628274 ∗EXP(−(Z^2 )/2)) (2.26)

As a final note regarding the Normal Distribution, you should be aware

that the distribution is nowhere near as “peaked” as the graphic examples