Question

Measuring standalone risk using realized data

Returns earned over a given time period are called realized returns. Historical data on realized returns is often used to estimate future results. Analysts across companies use realized stock returns to estimate the risk of a stock.

Five years of realized returns for Blue Llama Mining Inc. (Blue Llama) are given in the following table:

	2012	2013	2014	2015	2016
Stock return	11.25%	7.65%	13.50%	18.90%	5.85%

Also note that:

1.	While Blue Llama was started 40 years ago, its common stock has been publicly traded for the past 25 years.
2.	The returns on Blue Llama's equity are calculated as arithmetic returns.

Given this return data, the average realized return on Blue Llama Mining Inc.’s stock is   .

The preceding data series represents   of Blue Llama’s historical returns. Based on this conclusion, the standard deviation of Blue Llama’s historical returns is   .

If investors expect the average realized return on Blue Llama Mining Inc.’s stock from 2012 to 2016 to continue into the future, its expected coefficient of variation (CV) is expected to equal   .

Answer 1

Sample mean: Is the average of the data set used in the experiment.

Formula: Sum of all the returns / no. of data sets in the sample

Average realized return = [11.25% + 7.65^ + 13.50% + 18.90% + 5.85% ] / 5 = 57.15% / 5 = 11.43%

Standard deviation is the volatility in the returns from the average return, calculated as:

Square root of sum of squared differences divided by (no. of data sets in the sample minus one)

• Standard deviation, s = SQRT [ { (11.25% - 11.43%)^2 + (7.65% - 11.43%)^2 + (7.65% - 11.43%)^2 + (7.65% - 11.43%)^2 +(7.65% - 11.43%)^2 } / (5 -1) ]
• s = SQRT [ { (-0.18%)^2 + (-3.78%)^2 + (2.07)^2 + (7.47%)^2 +(-5.58%)^2 } / (5 -1) ]
• s= SQRT [ { 0.00000324 + 0.00142884 + 0.00042849 + 0.00558009 + 0.00311364} / (5 -1) ]
• s = SQRT[ {0.01055430} / 4 ]
• s = SQRT[ 0.00263858 ]
• s = 0.051367

standard deviation of the historical returns = 5.14% (rounded to 2 decimals)

Coefficient of variation

Measures the variability of the data set expressed as the % of mean. It is used to compae two data sets.

CV = standard deviation / mean = 0.051367 / 0.1143 = 0.4494 = 44.94%