# How did Dimson, Marsh and Staunton (2002) computed the equity index annual real return?

I was trying to read the triumph of the optimist, but it was almost impossible to see a well-written formula to show how the returns have been computed. In a simple sense, I do not know how the annual return index have been computed? I am also not sure whether natural logarithm has been applied to say (P(t)-P(t-1))/ln(p(t-1)). If anyone has encountered this argument please refer me to a page or help us understand whether the returns found in Ibbotson associate is natural logarithm or simply simple returns?

These data are commonly used for long-term investors such as in pension funds, They also seem to have been called low-frequency returns.

• The 2d half of the book (Chs.17-34) has detailed info on methodology IIRC (I don't have the book with me right now). Quote (from Dimson artice) The real return is defined as 1+Nominal rate of return divided by 1+Inflation Rate, minus one. /EndQuote. I believe the 'nominal rate of return' is what you call the simple return, no logarithms. For the US, the inflation rate that is used is the govt CPI rate. Mar 24 '17 at 18:19
• Hello noob2, thank you very much, this has been written in an article "Risk and Return in the 20th and 21st Centuries". Again Dimson et al. were not explicit regarding how the cpi were computed they did not mention say the base year right? remember that any cpi computation involves a base year. Having read your comment, I thought that the annual index total real rate of return = (1+total index return)/(1+inflation)-1. Where total index return accounts for dividends but this I have not seen explicitly in either their book or papers! Mar 24 '17 at 21:56
• (Obviously dividends are included, so it is indeed the Total Return that is used). Mar 24 '17 at 22:18
• All of the DMS returns are annually compounded. The base year for CPI is not relevant in this context, since it's a just matter of scaling (the YoY CPIs wouldn't change after the scaling). Mar 24 '17 at 22:46
• @HelinGai please may you confirm my answer below? Apr 10 '17 at 14:06

All of the DMS returns are adjusting for dividends. Hence dividends are accounted in the sample. Moreover, DMS have also accounted for inflation. Hence, the real total net equity index return, now and hereafter, $e_{t}$, may be mathematically defined as \begin{equation*} e_{t}=\frac{1+\frac{P_{t}+D_{t}}{P_{t-1}}}{1+\pi_{t}}-1 \end{equation*} where $P_{t}$, and $D_{t}$ refers to the price of the index and dividend. $\pi_{t}$ refers to the inflation between periods $t-1$ and $t$ \cite[see,][]{dimson2000risk}.

@article{dimson2000risk,
title={Risk and Return in the 20th and 21st Centuries},
author={Dimson, Elroy and Marsh, Paul and Staunton, Mike},