Gross Returns and Chain Relatives

Summary

The document clarifies terminology for the wealth multiplier associated with a period’s return. If wealth begins at one and a period earns a return, adding one to the rate gives the gross return, also called the gross rate of return. Multiplying successive gross returns compounds wealth across periods; the example illustrates this with consecutive gains.

It also gives a related statistical term: a link relative or chain relative. This describes adjacent observations connected by multiplication, as in constructing a price index or another time series. The note is terminological rather than a treatment of return estimation: it does not discuss fees, risk adjustment, losses beyond the basic convention, or how to compare investment performance. The useful distinction is that a simple return is a rate of change, while its gross form is the multiplicative factor applied to prior wealth.

Key ideas

• A period’s gross return is one plus its simple rate of return.
• Successive gross returns multiply to produce compounded wealth across periods.
• Link relative and chain relative are statistical terms for adjacent values connected multiplicatively.
• The document focuses on naming the quantities and does not address performance adjustments or risk.

Tags

# What is `1+ return` called?


# What is `1+ return` called?












Assuming day 1 my wealth is 1. At day 2 I earned 20%. So the rate of return is 0.2 and the wealth at day 2 is 1.2. At day 3 I earned 50 % again. So the wealth at day 3 is 1.8. I wonder what is the value of 1, 1.2, 1.5 for each is called in academic?

## Answer by rbm (score 4, accepted)

https://quant.stackexchange.com/a/38714

It's called 'Gross return' (a.k.a. 'Gross Rate of Return'). See Zivot Course Notes Eqn. 1.8

Also, as pointed by Alex C:

> ... in Statistics it is called a 'link relative' or 'chain relative' (because it links together (via multiplication) adjacent terms of the CPI index or other time series)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.