Risk, Return and Time Trade-off

A better understanding of the statistics of dice rolling will help you understand the risk, return and time trade-offs of investing.

The length of the holding period (time horizon) can reduce risk by allowing the outcomes to regress to the expected average or expected distribution. Portfolios of various risk and time horizons can be compared to rolling dice. Each combination of dice has an expected average and a standard deviation, similar to, but not exactly like, portfolios of indexes. Each combination of risk, return and time has a bell shaped curve that allow you to visualize the characteristics of that combination.

Three dice have an expected return or average of 10.5 and a standard deviation of approximately 3. See the population characteristics below right. A portfolio 10, held for 3 years has about the same risk, with a slightly lower average or expected return of 9.6%. At the other end of the spectrum, the only way to get a portfolio 70 risk level down to a risk of 3.0 is to hold it for 20 years. Each twenty-year period, as a whole, will be approximately equal in risk to a single roll of the 3 dice (please note the market has no absolute constraints on risk like the dice. In fact, the market has infinite risk. So the cumulative risk reducing effect of 20 rolls (years) of portfolio 70, equals one roll of 3 dice. However, the expected annualized return of portfolio 70 is 13.5, and the dice is 10.5. So each time you roll the dice, you must add 3 to the outcome for the portfolio 70 expected annualized return. In other words, the bell shaped curve for portfolio 70 would be shifted 3 units to the right of the three dice roll.

You can review this dice roll analysis here. You can compare a portfolio's risk over time at the Index Portfolio Time Series.

   
 
   
As Charles Ellis stated in Winning the Loser's Game, "The average long-term experience in investing is never surprising, but the short-term experience is always surprising." The confusion of most investors is derived from their inability to look at large sets of data about stocks, times, managers, or styles. Sampling error is the reason for the confusion. With a small sample of data, such as 60 months of S&P 500 returns or 60 rolls of 5 dice, a poor representation of the long term results is obtained. This lack of representation of the long-term characteristics of the 5 dice or the market is known as random drift and in the casino they refer to it as good or bad luck. This is similar to saying that an investor feels confident about the future outcomes of a certain time period based on a recent short-term experience, such as 60 months of S&P 500 data. The many histograms shown below illustrate this point. The 954 months S&P 500 returns on the right is all the data we have at this time. It bears a striking resemblance to 1,000 rolls of 5 dice on the left. We also obtained 5 samples of 60 rolls of 5 dice and 5 sample periods of 60 months of S&P 500 returns. The samples of both the dice and the market have a similar random configuration and neither look like the near 1,000 sample size. If you place your mouse over the bottom buttons, you can see overlays on the opposite chart. If you have any questions, please call an index funds advisor at 888-643-3133 for an explanation of this chart.  
 
   
 
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