Time Needed to Assure a Gain

We are assured that stocks will show a gain, eventually, even after adjusting for inflation.

How long do we have to wait? So far, the longest delay for the S&P500 has been 18 years. How about a confidence interval?

Data Collection

I brought up a table of S&P500 real returns for sequences lasting up to 60 years. I determined the final year with a loss for each sequence. About half of the time, there never was a loss.

I thinned the data to 1881-1990 in order to have sufficient data. I didn’t have P/E10 information for 1871-1880. Sequences are too short after 1990.

I plotted the Years to Last Loss versus 100E10/P for 1881-1990. There were many sequences on the horizontal axis (i.e., no loss ever). A straight line curve fit (regression equation) had an R-squared of 0.1529.

I thinned the data to exclude all sequences without a loss. There were 60 sequences remaining out of 110 from 1881-1990. The regression equation (Years to Last Loss versus 100E10/P) for what remained had an R-squared of 0.1585. It had the undesirable feature of becoming negative when valuations were high enough (P/E10 below 6.4).

I plotted the regression equation using P/E10 instead of the percentage earnings yield 100E10/P. It was a better plot. R-squared increased to 0.2089.

I narrowed the sequences to those beginning in 1921-1980. There were 33 sequences out of 60 that included a (real) loss. I plotted the Years to the Last Loss versus P/E10. R-squared jumped to 0.5513.

This plot was well behaved. The equation for the number of years until the final lost y = 0.8935x-7.4279 plus 4 and minus 7, where x is P/E10. This equation and these confidence limits tell us what happens ON THE CONDITION that there is a loss.

The probability that there is a loss is of the order of 33/60 or 55%. Using the standard Gaussian approximation for a binomial distribution, with a probability p=0.55 and q=1-p=0.45, the standard deviation as a percentage is the square root of pq/n = 0.55*0.45/60 = the square root of 0.0004125 = 2.0%. Placing the confidence limits at plus and minus two standard deviations, I calculate that the probability that there is a loss is 55% plus and minus 4%.

The Effect of Valuations

When P/E10=8, valuations are attractive. Assuming that there is a loss, the number of years until the last loss is -0.3 plus 4 and minus 7. The negative numbers correspond to not having a loss. The conditional losses are: Worst case, the maximum loss would occur in year 4. If merely unlucky, the loss would occur in year 2. Typically, there would be no loss in spite of the assumption (condition).

When P/E10=14, which is slightly higher than usual, and assuming that there is a loss, the number of years until the last loss would be 5.1 years plus 4 and minus 7. Worst case, the maximum loss would occur in year 9. If merely unlucky, the loss would occur in year 7. Most likely, it would occur in year 5. If lucky, it would occur in year 1.6. If very lucky, there would be no loss in spite of the assumption (condition).

When P/E10=20, which defines the hazardous region, and assuming that there is a loss, the number of years until the last loss would be 10.4 plus 4 and minus 7. Worst case, the maximum loss would occur in year 14.4. If merely unlucky, the loss would occur in year 12.4. Most likely, it would occur in year 10.4. If lucky, it would occur in year 6.9. If very lucky, it would occur in year 3.4.

When P/E10=28, as it has recently, and assuming that there is a loss, the number of years until the last loss would be 17.6 years plus 4 and minus 7. Worst case, the loss would occur in year 21.6. If merely unlucky, the loss would occur in year 19.6. Most likely, it would occur in year 17.6. If lucky, it would occur in year 14.1. If very lucky, it would occur in year 10.6.

At this point, we need to reconsider the probability that there is a loss. It is 55%. Yet, it is reasonable to expect the number to increase as prices become more expensive.

Refinement for High P/E10

The number of sequences that begin with P/E10=20 and higher is 15 out of 110 from 1881-1990. The ratio is 13.6%. The number of sequences that begin with P/E10=20 and higher is 10 out of 60 from 1921-1980. The ratio is 16.7%.

The number of sequences that begin with P/E10=20 and higher and also avoid a loss is 1 out of 15. It is the 1901 sequence. To calculate the probability that there is at least one year with a loss, I use p=0.067, q=0.933, n=15. By formula (without Yates’s correction), the standard deviation is 6.46%. The confidence interval is of the order of 80% to 100%, with a most likely value of 93%.

I determined the 1921-1980 regression equation when P/E10=20 and above. It is y = years until last loss = 0.0399*(P/E10)+12.906. R-squared is 0.0004.

Stated differently, when P/E10 is 20 or above, the number of years until the last loss is 12.9 (on average). The observed range is between 5 and 18. Using the LINEST function, the standard deviation is 3.99 years and the confidence limits (90%, by formula, multiply by 1.64) are plus and minus 6.55 years about the mean. According to formula (without any adjustments), the confidence limits are 6.35 years to 19.45 years.

Today’s Outlook

In the original calculations, the probability of experiencing a loss at some point in the future was about 55%.

The original calculation showed that, assuming that there is a loss, the most likely waiting time before showing a profit (in real dollars) is 17.6 years. The worst case would be 21.6 years. If only unlucky, the delay would be 19.6 years. An outcome of 14.1 years would be lucky. The best case outcome (delay), assuming that there is a loss, would be 10.6 years.

Using an alternative calculation that relies only on historical sequences with P/E10=20 and higher, the likelihood of a loss at some point in the future jumps to 93%. (The full range of possibilities is from 80% and 100%.)

Assuming that there is a loss, the most likely waiting time until showing a profit (after adjusting for inflation) is 12.9 years, with possible delays ranging from 6.35 years to 19.45 years.

My Assessment

The probability of being blindsided is around 10% or 20% regardless of the quality of a study. I assign unlikely outcomes to this category.

One set of estimates predicts delays from 10.6 years to 21.6 years before stocks have experience their last loss. Yet, its procedures assign a about a 50% probability that there will not be even one year of losses, regardless, even when starting from today’s valuations.

I recommend extending the lower side of this estimate below 10.6 years and assigning a very high probability of loss at some point, possibly with the last loss occurring within two or three years.

An alternative set of estimates predicts a delay between 6.45 years and 19.45 years. This alternative assigns near certainty to the chance that there will be a loss at some point in the future. The alternative possibility, that there will never be a loss, goes into the blindsided category.

Combining these approaches, I estimate that the last (real dollar) loss will occur at some point between 5 years and 20 years, most likely around 13 to 17 years.

Have fun.

John Walter Russell
December 14, 2006