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# Holding Period Return Arithmetic and Geometric Mean

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Complete the integrative practical exercises found on IBM Data Excel File attached.

The ultimate objective of this exercise is to translate this data into standard measures of risk and return.
The accompanying Excel file provides you with monthly price and dividend data for IBM (Ticker symbol: IBM) from Yahoo Finance.

• Expand the spreadsheet and compute monthly HPRs for each of the 24 months using the HPR formula bellow and make sure to include the dividends.

• Calculate the arithmetic mean of the monthly HPRs for IBM.

• Calculate the geometric mean of the monthly HPRs for IBM.

• Expand the spreadsheet to calculate the standard deviation of the monthly HPRs using an approach similar to the one illustrated bellow:

Measuring Risk:

Total Risk = Variance (or standard deviation) of historical HPRs
Computing Variance: Recall 12 monthly HPRs for 2012. The arithmetic mean of monthly HPRs is 0.0272, or 2.72%.

Variance = Sum of squared deviations of actual returns from expected (arithmetic mean) HPR, divided by number

Example: Var = [(0.1271 - 0.0272)2 + (0.1883 - 0.0272)2 + ... + (-0.0907 - 0.0272)2] / (12 - 1)
Or in table form:

Month Monthly HPR Monthly HPR - Mean (Monthly HPR - Mean)2
Dec-11
Jan-12 0.127111 0.099921 0.009984
Feb-12 0.188311 0.16112 0.02596
Mar-12 0.105284 0.078093 0.006099
Apr-12 -0.02597 -0.05316 0.002826
May-12 -0.0107 -0.03789 0.001436
Jun-12 0.010853 -0.01634 0.000267
Jul-12 0.045822 0.018631 0.000347
Aug-12 0.093539 0.066349 0.004402
Sep-12 0.002796 -0.02439 0.000595
Oct-12 -0.1076 -0.13479 0.018169
Nov-12 -0.01241 -0.0396 0.001568
Dec-12 -0.09074 -0.11793 0.013908

The sum of squared deviations shown in the last column, divided by the number of observations minus 1 is the variance!

= 0.007778

Standard Deviation = Square Root of the Variance
= (0.007778)1/2 = 0.0882 or 8.82% per month.

• Calculate the monthly 5% VaR using the variance-covariance approach.

• Calculate the annualized version of your answers from parts 2, 3 and 4. (Your answers should be the annualized versions of only the final answers of parts 2, 3 and 4, not annualized versions of each monthly HPR.)

• Report your final answers for parts 2 through 6 in the yellow cells in column E of the spreadsheet.

Note:
You may notice that the examples in the online notes use a divisor of n-1 for calculation of variance and standard deviation instead of n. In statistics, this is appropriate if we have a sample of returns rather than the entire population of returns. While this difference is not a big deal if you have over 30 observations, it can make a difference if you have a modest number of monthly returns to work with. Use the divisor of n-1 for this exercise.