The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 97 Friday, March 12, 2004 12:39 PM

Principles of Calculus 97

a quantitative framework, real numbers represent the result of observa-

tions (or measurements) in a simple and natural way.

n-tuples

An n-tuple, also called an n-dimensional vector, includes n components:

(a 1 , a 2 , ..., an). The set of all n-tuples of real numbers is denoted by Rn.

The R stands for real numbers.^3

For example, suppose the monthly rates of return on a portfolio in

2002 are as shown below along with the actual return for the S&P 500

(the benchmark index for the portfolio manager):^4

Month Portfolio S&P 500

January 1.10% –1.46%

February 1.37% 1.93%

March 2.95% 3.76%

April 5.78% 6.06%

May 0.51% 0.74%

June 7.32% 7.09%

July 7.13% 7.80%

August 1.47% 0.66%

September 9.54% 10.87%

October 7.32% 8.80%

November 6.19% 5.89%

December –4.92% –5.88%

Then the monthly returns rport for the portfolio can be written as a 12-

tuple and has the following 12 components:

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

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

Similarly, the return rS&P on the S&P 500 can be expressed as a 12-

tuple as follows:

(^3) Where the components of an n-tuple are only integers, the set of n-tuples is denoted
by Zn , Z representing zahlen, which is German for integer.
(^4) The monthly rate of return on the S&P 500 is computed as follows
Dividends paid on all + Change in the index
-----------------------------------------------------------------------------------------------------------------------------the stock in the index value for the month -– 1
Value of the index at the beginning of the period