Investments, tenth edition

  Mean and Standard Deviation Estimates 

from Higher-Frequency Observations 

 Do more frequent observations lead to more accurate estimates? The answer to this ques-

tion is surprising: Observation frequency has no impact on the accuracy of mean estimates. 

It is the  duration  of a sample time series (as opposed to the  number  of observations) that 

improves accuracy. 

 The total 10-year return divided by 10 is as accurate an estimate of the expected annual 

return as 12 times the average of 120 monthly returns. The average monthly return must 

be consistent with the average 10-year return, so the extra intra-year observations yield 

no additional information about average return. However, a longer sample, for example, a 

100-year return, will provide a more accurate estimate of the mean return than a 10-year 

return,  provided  the probability distribution of returns remains unchanged over the 100 years. 

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6/18/13   8:03 PM

6/18/13   8:03 PM

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P A R T   I I

  Portfolio Theory and Practice

This suggests a rule: Use the longest sample that you still believe comes from the same 

return distribution. Unfortunately, in practice, old data may be less informative. Are return 

data from the 19th century relevant to estimating expected returns in the 21st century? 

Quite possibly not, implying that we face severe limits to the accuracy of our estimates of 

mean returns. 

 In contrast to the mean, the accuracy of estimates of the standard deviation and 

higher moments (all computed using  deviations from the average ) can be made more 

precise by increasing the number of observations. Thus, we can improve accuracy 

of estimates of SD and higher moments of the distribution by using more frequent 

observations. 

 Estimates of standard deviation begin with the variance. When monthly returns are 

uncorrelated from one month to another, monthly variances simply add up. Thus, when the 

variance is the same every month, we annualize by:  

10

      s

A

2

5 12s

M

2

.  In general, the  T -month 

variance is  T  times the 1-month variance. Consequently, standard deviation grows at the 

rate  of     

"T,  that is:    s

A

5

"12s

M

.  While the mean and variance grow in direct proportion 

to time, SD grows at the rate of square root of time.