Quantitative Methods - Statistics and Market Returns (cont.)

Start Time - 11:15

Continuing on through Reading 7.

Coefficient of Variation

• Interpreting standard deviation depends on units of measurement and location of means
• E.g. standard deviation does not change if you add a fixed $750 million to each observation in a set of data
• One standard deviation will mean a lot more % difference in the set with smaller numbers
• Coefficient of Variation (CV) is one measure of relative dispersion i.e. relative to a reference value or benchmark
• Formula: CV = standard deviation/sample mean
• This permits direct comparison and is a scale free measure
• Mean is in the same units as the standard deviation (e.g. % for returns)
• In context of risk/return, risk is standard deviation and sample mean is the return

Sharpe Ratio

• Note that the inverse of the CV is a measure of return to risk - each unit of standard deviation represents an x % return
• To be more precise with this you must recognize there is a risk free component of return, hence the Sharpe Ratio
• Sharpe Ratio = (Mean Portfolio Return - Mean Risk Free Return) / standard dev of portfolio return
• Numerator is 'mean excess return'
• Generally you want to maximize your Sharpe Ratio for a given standard deviation
• Caveats to Sharpe Ratio
• Negative Sharpe Ratios - may occur in bear markets
• When comparing negative, 'larger' (meaning less negative) does not mean better performance (doubling risk might move Sharpe Ratio from -1.0 to -0.5)
• Should generally try to increase the evaluation period to get positive ratios, and failing that use a different measure
• Sharpe considers only one measure of risk, standard deviation of return
• This makes sense for strategies with symmetric return distributions
• But Options for example have assymetric return distributions
• Also might have frequent small gains and infrequent large losses
• When a strategy is working, high sharpe ratio, but nonrepresentative

Symmetry and Skewness in Return Distributions

• In variance, for ex, since deviations are squared, you don't distinguish between pos and neg
• Normal Distribution - three properties:
• Median = mean
• Completely described by mean and variance
• 68% of observations within +/-1 standard deviation of mean, 95% w/in 2 std devs, and 99% within 3 std devs
• Skewness
• Positive skew - long tail on the right ('points to the positive') (skew right)
• Mode<median<mean because you have many high positive values skewing the numbers
• Negative skew - long tail on the left (skew left)
• Mean<median<mode
• Measure of Skewness
• Similar to measure of variance - use each observation's deviation from mean
• Skew = sum of cubed deviations from mean / cube of std dev
• Measure is again free of scale
• For sample skewness
• Must multiply the above by n / [(n - 1) (n - 2)]
• As n becomes large this term just approaches 1/n
• For reference, for any n>100, a skewness of +/- 0.5 would be unusually large
• Skewness is a real pain to calculate - see spreadsheet

Kurtosis

• Skewness is one way you can be different from a normal distribution
• Kurtosis is another - this measures how 'peaked' the distribution is - more 'peaked' 
• More peaked than normal - leptokurtic (excess kurtosis > 0)
• Less peaked - platykurtic (plateaus are wide) (excess kurtosis < 0)
• Same as normal - mesokurtic (excess kurtosis = 0) 
• If you are more peaked you will have fatter tails
• Kurtosis is one level up from skewness - so you raise to the FOURTH power in the calcs
• Kurtosis = sum of deviations from mean^4 / fourth power of std dev
• For a normal distribution, kurtosis = 3
• 'Excess kurtosis' = kurtosis - 3
• For sample kurtosis you must multiply by (n(n+1)) / [(n-1)(n-2)(n-3)]
• For sample excess kurtosis then you subtract 3(n-1)^2 / [(n-2)(n-3)]
• Another pain in the but calculation - probably no way this appears on the exam as a full calculation

Using Geometric and Arithmetic Means

• Geometric is appropriate for past performance, Arithmetic for future (A for Ahead)
• Remember, arithmetic is always greater or equal to geometric
• Geometric is appropriate for past because it captures how total returns link over time
• Better to use a semilogarithmic scale (normal scale on x axis, logarithmic scale for y axis) when graphing past performance than an arithmetic scale
• On this scale, equal vertical movements capture equal percentage changes
• Differences in lns of 10-fold increases is about 2.30.  E.g. ln10-ln1 = 2.30, and ln1000-ln100=2.30.
• Forward looking
• Essential difference is uncertainty
• Suppose you invest 100,000 and face equal chances of gaining 100% or losing 50%
• Geometric would give the median/mode ending value of 100,000
• But arithmetic mean gives a better prediction of mean ending wealth

• You can see based on the above that the average after 2 periods is $156,250 (all four outcomes are equally likely)
• The more uncertainty there is, the larger the divergence of arithmetic above geometric 

End of Reading 7.

1:15 pm

About 2 hours