RiskMetrics™ —Technical Document

46 Chapter 4. Statistical and probability foundations
The relative price change, or percent return 2 ,
R t
, for the same period is
[4.2]
P
R t
– P t – 1
t
= ----------------------
P t – 1
If the gross return on a security is just 1 + R t
, then the log price change (or continuously-compounded
return), , of a security is defined to be the natural logarithm of its gross return. That is,
r t
[4.3]
r t
= ln ( 1 + R t
)
=
⎛ P t
⎞
ln⎜-----------
⎟
⎝P t – 1 ⎠
= ( p t
– p t – 1
)
where p t
= ln ( P t
) is the natural logarithm of P t
.
In practice, the main reason for working with returns rather than prices is that returns have more
attractive statistical properties than prices, as will be shown below. Further, returns (relative and
log price changes) are often preferred to absolute price changes because the latter do not measure
change in terms of the given price level.
To illustrate the different results that different price changes can yield, Table 4.1 presents daily
USD/DEM exchange rates for the period 28-Mar-96 through 12-Apr-96 and the corresponding
daily absolute, relative, and log price changes.
Table 4.1
Absolute, relative and log price changes*
Date
Price
(USD/DEM), P t
Absolute price
change (%), D t
Relative price
change (%), R t
Log price
change (%,) r t
28-Mar-96 0.67654 0.427 0.635 0.633
29-Mar-96 0.67732 0.078 0.115 0.115
1-Apr-96 0.67422 −0.310 −0.458 −0.459
2-Apr-96 0.67485 0.063 0.093 0.093
3-Apr-96 0.67604 0.119 0.176 0.176
4-Apr-96 0.67545 −0.059 −0.087 −0.087
5-Apr-96 0.67449 −0.096 −0.142 -0.142
8-Apr-96 0.67668 0.219 0.325 0.324
9-Apr-96 0.67033 −0.635 −0.938 −0.943
10-Apr-96 0.66680 −0.353 −0.527 −0.528
11-Apr-96 0.66609 −0.071 −0.106 −0.107
12-Apr-96 0.66503 −0.106 −0.159 −0.159
* RiskMetrics foreign exchange series are quoted as USD per unit foreign currency given that
the datasets are standardized for users whose base currency is the USD. This is the inverse of
market quotation standards for most currency pairs.
As expected, all three series of price changes have the same sign for any given day. Also, notice
the similarity between the log and relative price changes. In fact, we should expect these two
return series to be similar to one another for small changes in the underlying prices. In contrast, the
absolute change series is quite different from the other two series.
2 Although it is called “percent return,” the relative price change is expressed as a decimal number.
RiskMetrics —Technical Document
Fourth Edition