Statistical estimation of expectation and SD of returns Past returns data Good

# Statistical estimation of expectation and sd of

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Statistical estimation of expectation and SD of returns: Past returns data Good estimate if 1. States of the market and associated returns are constant over time 2. Large enough sample Challenges: time varying market states, choice of sample period, etc. 23
Estimate of Expected Returns Arithmetic Average of rate of return: It is an unbiased estimate of expected returns. Geometric (time-weighted) average return: TV = Terminal Value of the Investment g = geometric average rate of return 24 r ˆ E ( r ) 1 n r ( t ) t 1 n
Estimate of Variance and SD Estimate of variance: Estimate of standard deviation: They are unbiased estimates of variance and standard deviation. 25 ˆ 2 1 n 1 [ r ( t ) t 1 n r ] 2 ˆ 1 n 1 [ r ( t ) t 1 n r ] 2
Probability Distribution of States 26 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Probability distribution (Density plot) Probabilities (bar heights) sum up to 1 Possible HPR realizations
Probability Distribution of States 27 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0 0.05 0.1 0.15 0.2 0.25 A finer distribution Recession Recession Mild recession Normal Boom
The Normal Distribution 28 Example: Normal Distribution with mean return 10% and standard deviation 20%
Why the Normal Distribution? A good approximation: The mean of a large number of independent movements is well approximated by a Normal Distribution Central Limit Theorem Convenience: Normal distribution is symmetric A Normal Distribution is completely characterized by its mean and standard deviation The weighted average of two normally distributed variables is still normally distributed 29
Deviations from Normal Distribution What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skewness and kurtosis Skewness: a measure of asymmetry. Normal has zero skewness. Kurtosis: a measure of tails (fat or not). Normal has zero kurtosis. 30 Skew average ( R R _ ) 3 ˆ 3 Kurtosis average ( R R _ ) 4 ˆ 4 3
Normal and Skewed Distribution 31 Risk averse investors care more about extreme negative values than extreme positive values When skewness > 0, SD overestimates risk
Normal and Fat-Tailed Distribution 32 SD underestimates the likelihood of extreme values
Value at Risk (VaR) Value at risk (VaR): the quantile of a distribution below which lies q % of the possible values of that distribution, usually 5% Interpretations: 5% VaR means 95% of returns will exceed this VaR, only 5% will worse than VaR Measure of downside risk, worst loss that will be suffered with given probability VaR of Normal Distribution: VaR (0.05, normal) = Mean + (-1.65) SD since for standard normal distribution, P ( X <-1.65) = 0.05.

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• Fall '09
• Xia