Ralph Vince - Portfolio Mathematics

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

58 HANDBOOK OF PORTFOLIO MATHEMATICS

Thus, we can say that the height of the curve at X=−1 is .2419705705.

The function N′(Z) is also often expressed as:

N′(Z)=EXP(−(Z^2 /2))/

√

8 ∗ATN(1)

=EXP(−(Z^2 /2))/

√

8 ∗. 7853983 (2.15b)

=EXP(−(Z^2 /2))/ 2. 506629

where: Z=(X−U)/S (2.16)

and ATN( )=The arctangent function.

U =The mean of the data.

S=The standard deviation of the data.

X=The observed data point.

EXP ( )=The exponential function.

Nonstatisticians often find the concept of the standard deviation (or

its square,variance) hard to envision. A remedy for this is to use what is

known as the mean absolute deviation and convert it to and from the stan-

dard deviation in these equations. The mean absolute deviation is exactly

what its name implies. The mean of the data is subtracted from each data

point. The absolute values of each of these differences are then summed,

and this sum is divided by the number of data points. What you end up

with is the average distance each data point is away from the mean. The

conversion for mean absolute deviation and standard deviation are given

now:

Mean Absolute Deviation=S∗

√

2 / 3. 1415926536

=S∗. 7978845609 (2.17)

where: M=The mean absolute deviation.

S=The standard deviation.

Thus, we can say that in the Normal Distribution, the mean absolute

deviation equals the standard deviation times .7979. Likewise:

S=M∗ 1 /. 7978845609

=M∗ 1. 253314137 (2.18)

where: S=The standard deviation.

M=The mean absolute deviation.

So we can also say that in the Normal Distribution the standard devi-

ation equals the mean absolute deviation times 1.2533. Since the variance