Chapter 1

Value-at-Risk

1.1 Measures

Measures are widely used in science and in every-day activities. While it is common to speak of measuring things, we actually measure attributes of things. For example, we don’t measure a meeting, but we may measure the duration of a meeting or the size of a meeting. Duration and size are attributes.

A measure is an operationally defined procedure for assigning values. An attribute is that which is being measured—the object of the measurement.

A highway patrolman points a Doppler radar at an approaching automobile. The radar transmits microwaves, which are reflected off the auto and return to the radar. By comparing the wavelength of the transmitted microwaves to that of the reflected microwaves, the radar generates a number, which it displays. This entire process is a measure. An interpretation of that number—speed in miles/hour—is an attribute.

There are measures of length, temperature, mass, time, speed, strength, aptitude, etc. Assigned values are usually numbers, but can be elements of any ordered set. Shoe widths are sometimes assigned values from the ordered set {A, B, C, D, E}.

Let’s consider our first exercise.