###### 4.6.5 Returns

In finance, a **return** is a measure of economic benefit from holding assets. If an asset’s value is EUR 50 one month and EUR 55 the next month, we might say the asset had a 1-month 10% return. Let’s extend this notion to quantities other than asset values. If an interest rate rises one day from .050 to .055, it is reasonable to say that the interest rate had a 1-day 10% return. In this context, return is no longer a measure of economic benefit, but merely a measure of change in a time series. Suppose the temperature at the summit of Mt. Washington rises over a day from 50° F to 55° F: isn’t it reasonable to say that the temperature had a 1-day 10% return? Accordingly, we treat returns as a purely mathematical notion.

Consider univariate time series ** x** = {

^{–α}

*x,*… ,

^{–2}

*x,*

^{ –1}

*x,*

^{ 0}

*x*}. We define two measures of return. The

**simple return**of

**over the period [**

*x**t*– 1,

*t*] is

[4.46]

The **log return** of ** x** over the period [

*t*– 1,

*t*] is

[4.47]

For small returns, these two metrics closely approximate each other. Both are widely used in finance. Each has advantages and disadvantages. Simple returns are appealing because they combine linearly across positions. If a portfolio is equally invested in two assets whose values experience respective simple returns of .08 and –.02 over a given period, the portfolio experiences a .03 simple return over the period, but this would not be true with log returns. On the other hand, log returns combine linearly over time. If a portfolio experiences a .05 log return followed by a –.04 log return, its log return over the entire period is .01. This would not be true with simple returns.

A shortcoming of returns is the fact that their calculation entails division. If a time series can take on the value 0, returns may be poorly behaved or undefined. An example might be a component of a time series representing the spread between spot prices of gold and platinum. At various times, this spread has been positive, negative, and even zero. For such components of time series, differencing may be used instead of returns.