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

Statistical estimation of expectation and sd of

This preview shows page 23 - 34 out of 39 pages.

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
Image of page 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
Image of page 24
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
Image of page 25
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
Image of page 26
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
Image of page 27
The Normal Distribution 28 Example: Normal Distribution with mean return 10% and standard deviation 20%
Image of page 28
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
Image of page 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
Image of page 30
Normal and Skewed Distribution 31 Risk averse investors care more about extreme negative values than extreme positive values When skewness > 0, SD overestimates risk
Image of page 31
Normal and Fat-Tailed Distribution 32 SD underestimates the likelihood of extreme values
Image of page 32
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.
Image of page 33
Image of page 34

You've reached the end of your free preview.

Want to read all 39 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes