Risk and Return-Historical Returns

There are several problems with using historical returns to predict expected returns and standard deviations that will actually be realized in the future. One of course is that the underlying data may not be normally distributed in the first place. This is the case with stock returns since one can only lose up to 100% but theoretically at least, have an infinite gain.

Another problem is of course that the future may have little in common with the past, and if that wasn’t enough, values of historical returns and standard deviations are highly susceptible to the time period examined. We will examine this last problem in detail below.

When making projections using returns and standard deviations, the expected total return for any time period n is simply (1+r)ⁿ –1. For example, if r = 10% and you want to know what total return you should expect over the next 10 years, the answer is simply 1.1¹⁰ – 1 = 1.5937 or 159.37%. However, you may want to know how confident you can be about that value. Unfortunately, the process of determining this is somewhat complicated even under the assumption of normality. This is because returns over time are not normally distributed, but lognormally distributed. Because of this, returns must be converted into their continuous counterparts to determine confidence intervals, and then converted back into discrete values.

Using our example above, suppose the expected return is 10% and the standard deviation is 15% and we want to find the 95% confidence interval 10 years from now around the 159.37% expected return. To do this, we must convert the return and standard deviation into their continuous counterparts. The equations to do this are the following:

1) u_c = ln(1 + u_d) - F²_c /2

2) F²_c = ln[F²_d/(1+u_d) + 1] and F_c = the square root of F²_c

where u_c = continuous expected return

u_d = discrete expected return

F²_c = continuous variance

F²_d = discrete variance.

Once these values are found, then apply the following for whichever time period you are looking for:

3) TAV = F_c/n^.5 where we will let TAV = Time adjusted standard deviation

To find the range of continuous values, one then uses the following:

4) CV= u_c +/- 1.96(TAV). The 1.96 comes from the standard normal distribution table for a 95% confidence interval.

Now convert these returns back into discrete values by using the inverse natural logarithm function where e = 2.71828

5) DV = e^CV –1 where I am letting DV stand for discrete values.

Finally, to find the range for any n, add 1 to DV and take to the nth power.

6) Range = (1+DV)ⁿ

For our example with a mean of 10% and a standard deviation of 15%, letting n =10,

U_c = ln(1+.1) - F²_c /2. To find F²_c , we must use equation (2) which is

F²_c = ln[.15²/(1+.1)² + 1] = .018424, (Note F_c = .1357) and u_c = ln(1+.1) – .18424/2 = .0861 or 8.61%

Plugging into equation (3), TAV = .1357/10^.5 = .0429 or 4.29%. To find our range of returns then, use equation (4) and we have

CV = .0861 +/- 1.96(.0429) which gives the return range of .00197 to .170226. Now convert back to discrete values using equation (5)

DV lower limit = e^.00197 – 1 = .001972 and the DV upper limit = e^.170226 – 1 = .185573. Almost done! Now attain the range via (6)

Range = 1.001972¹⁰ = 1.01982 for the lower limit and 1.185573¹⁰ = 5.48635 for the upper limit.

What does this mean? For every dollar you invest into an asset with a 10% expected return and 15% standard deviation, 10 years from now you can expect to have make 159% on your money or end up with $2.59 with a 95% confidence range of $1.019 to $5.48. In other words, you can expect to make 159%, but the 95% confidence range is from 1.982% to 448.635%.

SO, I’M PREDICTING THAT YOU MAY THINK THIS IS SOMEWHAT COMPLICATED AND DON’T WANT TO HAVE TO DO THIS EVERY TIME FOR DIFFERENT EXPECTED RETURNS AND STANDARD DEVIATIONS. I DON’T BLAME YOU. THUS, CLICK HERE TO LINK TO A SPREADSHEET THAT DOES IT ALL FOR YOU OUT TO 40 YEARS, or click the Risk and Return Spreadsheet link on my website.

Risk and Return Spreadsheet

YOUR ASSIGNMENT IS BELOW:

The table below gives the S&P 500 historical returns and standard deviations for a varying time period lengths that will be told to you later. For now, examine the table and answer the following questions:

	Avg. Return	Std. Dev.
Last 10 years	5.40%	14.70%
Last 20 years	10.00%	16.20%
Last 40 years	5.90%	21.20%

1.

a. Starting with $10,000, what is the expected terminal value 10 years from now using the average return value using the last 10 years of data? Hint: Simply compound $10,000 for 10 years at 5.4% per year.

b. What is the 95% range of values you can expect 10 years from now using the standard deviation number over the last 10 years? (Go to spreadsheet or work out from above using equations 1-6.

2. Now repeat parts 1 a-b using the last 20 years of data. Note you are still only projecting out to 10 years, your'e just using the 20 year numbers as your inputs.

3. Finally, repeat parts 1 a-b using the last 40 years of data.

4. Now that you have three terminal values and three ranges of values for your prediction over the next 10 years, which one do you think you should use and why?

5. Finally, compare your values to what actually occurred over the next decade. Keep in mind your values are based on arithmetic averages which tend to overstate actual realized returns. The decade in question happened to be 1970 to 1980 which was not one of the stock market’s better decades. The returns and standard deviations in the table are based on actual values over the previous 40 years.

The actual average arithmetic return for the 1970’s was 3.4% but the geometric annualized return for this decade was only 1.7%. The annual standard deviation for the 1970’s was 20.0% with a 32% return being the best year and –29% the worst year. $10,000 invested in 1970 would only have accumulated to $10,000(1.017)^10 = $11,836 by 1980. Did your range of values in each case encompass what actually happened? Can you think of anyway to improve your long-term projections?