### No, the stock market doesn't really earn 10% per year

**Matthew Martin**1/19/2014 05:54:00 PM

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First things first: even if the only analysis you are interested in is the average returns, you should know that the average is actually only 7.3% (nominal) per year, not 10%. To get the 10% figure, you have to surreptitiously exclude data from both before 1975 and after 2007. We have a word for an oft repeated claim that excludes contrary data: derp.

Ok, so the actual average return is 7.3%, not 10%. We can just use the new number right? Actually, no, it's highly misleading to say that the average return is 7.3%--even though that is literally the number you get when you calculate the mean, that is not a valid calculation at all:

There are two things wrong with saying that the average growth rate of the Dow is 7.3% per year. First off, it is wrong. As the trend line shows, the actual figure should be 6.76%, reflecting the fact that the computed average is actually biased towards rare big-percentage-gain years that are really little more than trend-reversion after depressions. That is, recoveries tend to be more diffused than downturns, meaning that a simple average gives too much weight to recovery years. Yet, even if we correct for this bias, saying that the market grows at a certain percent per year is still misleading because such a statement depends entirely on the reference year. We can see this in the graph: the blue dashed line shows the extrapolated stock index value assuming 7.3% per year from present (which is itself both too high and growing too fast, as the solid trend line shows), while the dashed yellow line shows the forecast we would have gotten if we had extrapolated the same way back in 2007. Even if you buy the premise that the stock market reverts to a trend growth rate, for financial planning purposes knowing the trend line is at least as important as knowing the trend growth rate--if we believe that the current level of the stock index is above trend, then regardless of what the trend growth rate is we ought to be expecting less than those returns.

The graph above shows three different trend forecasts. The solid line is the "actual" time trend as estimated by OLS. The blue dashed line is a trend forecast by assuming the average annual growth rate will continue from today on, while the yellow dashed line is the trend we would have forecast back in 2007 by assuming a 7.3% average growth rate from then on. On the graph, the difference between these forecasts may not seem so significant, but they are. As a thought experiment, consider a person investing his retirement portfolio in the stock market today, for a predicted retirement in 2050. The blue line would overstate his returns by more than 215 percentage points.

The reality might even be worse than that. The estimated trend in the graph above may not be stationary. In a stationary model, R tells me that the "actual" trend is only 5.92% returns per year, barely half the advertised rate. (R code after the jump). That means that using the average would overstate the returns on his portfolio by a whopping 463 percentage points!

Bottom line is that if you take all the financial advice at face value--that you'll earn 10% per year--then chances are you'll end up scraping by on woefully inadequate savings in your golden years.

The data is here https://drive.google.com/file/d/0B8-nvwPMR6AyVzBGaHg1VXRXVjQ/edit?usp=sharing you may have to download the data and change the filepath to get this to work though. Data is the log levels of the DJIA from Fred

R code (requires forecast library):

dowJones<-scan("dowJones<-scan("https://drive.google.com/file/d/0B8-nvwPMR6AyVzBGaHg1VXRXVjQ/edit?usp=sharing",skip=0)

dow<-ts(dowJones,frequency=1,start=1959)",skip=0)

dow<-ts(dowJones,frequency=1,start=1959)

library(forecast)

dowArima=auto.arima(dow,ic="bic")

dowForecast<- forecast.Arima(dowArima,h=100)

dowForecast

dowArima

Max1/20/2014 02:53:00 AMMatthew Martin1/21/2014 03:08:00 PMJ Marshall1/24/2014 12:38:00 PM