# 1.6.3 Excess returns

Excess returns (ER) refers to total investment return that exceeds the rate of return on a security that is perceived to be risk free.

where:

denotes the excess return of the index at time *t*.

denotes the total return of the index at time *t*, described in total investment return

denotes the short-term (3-month) risk-free rate at time *t*.

Since the choice of risk-free security impacts the excess return, *Scientific Infra and Private Assets* uses two different methods depending on the reporting currency of the index.

## Local currency indices

When indices are computed to report only local returns (ignoring the effect of foreign exchange movements on returns for a given investor), the excess return of the index consists of a weighted average of the individual constituents' excess returns, given the 3-month risk-free rate in each country included in the index. Hence, the excess return is written:

where:

denotes the local excess return of the index at time t*n* denotes the number of constituents in the index

denotes the weight of constituent *i* at time *t-1*

denotes the excess return of constituent *i* at time *t*

## Single currency indices

If the index is reported in currency *j (e.g. USD, CAD, GBP, JPY, EUR or AUD) *then the reference 3-month risk free rate is that of the reporting currency.

where:

denotes the excess return of the index reported in currency *j* at time *t.*

denotes the total return of the index at time *t*, described in the chapter on total investment return.

denotes the risk-free rate in market *j* at time *t.*

## Mean excess return

Mean excess return is the average quarterly excess return of the index

where:

denotes the mean excess return of the index over the horizon.*t* denotes the number of periods in the time horizon.

denotes the excess return of the index at time *t.*