Overview
Because money can generally not be made without some money to begin with, it is customary to measure the performance of an asset by the following formula for return
{% return = amount \, gained (or \, lost) / invested \, amount %}
The formula divides the amount that was gained (at the end of the period) by the amount that was invested (at the beginning).
Relative Return
Given a number of units {% U %} of an asset at time {% t %}, {% U(t) %}, then the return, measured in units of the asset is given by
{% r_1 = \frac{U_1(T) - U_1(t)}{U_1(t)} %}
If the units are units of money, then this is the start price return given above.
Given two assets, {% U_1 %} and {% U_2 %}, one can define the numeraire process to be
{% S(t) = \frac{U_1(t)}{U_2(t)} %}
{% r_2 = \frac{S(t)}{S(T)} (1 + r_1) - 1 %}
This means that
{% 1+r_1 = (1+r_2)\frac{S(T)}{S(t)} = (1+r_2)(1+i) %}
{% i = \frac{S(T)}{S(t)} - 1 %}
(This argument follows the logic in Bouchouev)
Fixed Income Returns
Fixed Income instruments make a periodic payment of interest to the holder of the security. When these securities are traded, it is the market convention to quote the price of the bond without including any accrued interest. This price is known as the clean price. This is as opposed to the dirty price, which includes the accrued interest.When a bond is traded at the clean price, it is the agreed upon convention that the seller owes the buyer the accrued value of the next coupon. As such, when calculating return using the clean price of a bond, it is important to include any accrued interest as follows:
{% \frac{Ending Market Price + Accrued Income}{Starting Market Price + Accrued Income at Start} - 1 %}