The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 98 Friday, March 12, 2004 12:39 PM

98 The Mathematics of Financial Modeling and Investment Management

rS&P = –1.46% 1.93% 3.76% 6.06% 0.74% 7.09% , , , , , ,

7.80% 0.66% 10.87% 8.80% 5.89% , , , , , –5.88%

One can perform standard operations on n-tuples. For example,

consider the portfolio returns in the two 12-tuples. The 12-tuple that

expresses the deviation of the portfolio’s performance from the bench-

mark index is computed by subtracting from each component of the

return 12-tuple from the corresponding return on the S&P 500. That is,

rport –rS&P

= 1.10% 1.37% 2.95% 5.78% 0.51% 7.32% , , , , , ,^

7.13% 1.47% 9.54% 7.32% 6.19% , , , , , –4.92%

• – 1.46% , 1.93% 3.76% 6.06% 0.74% 7.09% , , , , ,
  7.80% 0.66% 10.87% 8.80% 5.89% , , , , , –5.88%

= 2.56% , –0.56%, –0.81% , –0.28%, –0.23% 0.23% , ,

• 0.67% 0.81% , , –1.33%, –1.48% , 0.30% 1.26% ,

It is the resulting 12-tuple that is used to compute the tracking error of a

portfolio—the standard deviation of the variation of the portfolio’s return

from its benchmark index’s return described in Chapter 19.

Coming back to the portfolio return, one can compute a logarithmic

return for each month by adding 1 to each component of the 12-tuple

and then taking the natural logarithm of each component. One can then

obtain a geometric average, called the geometric return, by multiplying

each component of the resulting vector and taking the 12th root.

Distance

Consider the real line R^1 (i.e., the set of real numbers). Real numbers

include rational numbers and irrational numbers. A rational number is

one that can be expressed as a fraction, c/d, where c and d are integers

and d ≠ 0. An irrational number is one that cannot be expressed as a

fraction. Three examples of irrational numbers are

2 ≅ 1.4142136

Ratio between diameter and circumference

= π ≅ 3.1415926535897932384626

Natural logarithm = e ≅ 2.7182818284590452353602874713526