 Estimating the Real Rate of Return on Stock Index

Judicious use of the rate of return framework on equity, or "cost of capital" facilitates decision-making within the financial industry by providing a metric for comparing different stock markets between them but also a comparison to other asset classes.

Article also available in : English | français Dividends of a company or an index are known and available data, that make the difference between price and total return performance. the total return performance is divided into two : payments of dividends and changes in stock prices. the stock’s performance is available to an investor who buys a stock and then reinvests the dividend in shares. Similarly the total return performance of a stock index like the S&P is available, it is reproduced by trackers. When you look at the stock or index history, the total return performance is the most stable and unchanged data, while the split between dividends and price movements is disturbed by buy-backs, and periods of over or under-investment.

This paper presents a method for calculating a series of real return, using only historical prices and total return performance as data. there are methods of calculation based on the present value of dividends or earning forecasts, but they are problematic both in theory and practice. This paper wants to avoid the debate on the validity of the Price Earning as a measure of profitability, and accounting biases affecting financial data. Staying in the actuarial rate framework, prevent us from looking beyond the link between past performances and rate of return (estimate of future return, similar to the bond yield to maturity).

### ILLUSTRATION USING THE S&P 500

We want to approximate the S&P 500 since 1946 (data from Shiler) with a growing annuity. Knowing the price, the next dividend and growth rate of dividends, the Gordon formula provides the corresponding yield to maturity, which is the return an investor would expect by investing in the index.

Step one : Obtaining historical rate of return series from a performance series

The idea is to look for rates that explain the performance, by exploiting the link between performance and rate of return, given the large actuarial formula:

Performance during ?t = RR before x ?t – Sensitivity rate x Variation RR during ?t

RR = Rate of Return
For example, if the rate of return is the same at the beginning and end of year, the annual performance equals that same rate; with this, and the assumption on the growth in dividends over the given period, we solve a small puzzle: let’s give us any rate in 1946, we will get step by step the rate of subsequent years, with the annual performance already known, and by using the above equation.

We will notice that this algorithm diverges , except for a range of rates starting in 1946. Because starting form a low are, we must continuously imagine rate cuts or the S&P performance. Conversely, starting form a high return, from year to year we need stronger rates increases in explaining performance.

Step two : Getting historical rate of return series form price and performance series

Then, with this family of solutions, again for a given growth, we obtain a simple way to choose the best series, leaning on prices. That is

• a total return performance equal to that of the S&P every month
• a price always close to that of the S&P
• Implicitly, as dividends make the difference between price and performance, dividends on the S&P are close to the growing annuity income

We then state its real yield to maturity that we know by calculation is that of the S&P

Last Step : Choosing the growth rate

The final challenge is to determine the growth rate, that is an important parameter. several criteria can be taken into account to obtain a value:

• The dividend growth (data unfortunately altered)
• Earnings growth (more reliable than dividends)
• Per capita consumption as an indicator of Balanced Growth

The result is that one obtains a series of historical rate of returns to date (page 10), explaining exactly the S&P year to year performance, although explaining its price history, starting only with price and past performance, leaving aside earnings, theoretical debates and problems with accounting numbers. Calcul du taux réel du S&P 500
Etude complète d’Arnaud Motte sur le calcul de taux réel du S&P 500
October 2011

Article also available in : English | français Footnotes

 The buy-backs move the dividend / price distribution at the expense of dividends, but they do not change the total return performance..

 The dividend / price distribution is disrupted when the company is reinvesting rather than distribute dividends. In his work with Miller in 1953-54, Modigliani showed that the value of the shares is unchanged in case of profit distribution or reinvestment (within a simplified framework without tax friction and saving scale)

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