Chapter 12 The Normal Probability Model Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Black Monday (October, 1987) prompted investors to consider insurance against another “accident” in the stock market How much should an investor expect to pay for this insurance? Insurance costs call for a random variable that can represent a continuum of values (not counts) of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Percentage Change in Stock Market Data of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Prices for One-Carat Diamonds of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable With the exception of Black Monday, the histogram of market changes is bellshaped The histogram of diamond prices is also bell-shaped Both involve a continuous range of values of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Definition A continuous random variable whose probability distribution defines a standard bell-shaped curve of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Central Limit Theorem The probability distribution of a sum of independent random variables of comparable variance tends to a normal distribution as the number of summed random variables increases of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Central Limit Theorem Illustrated of 45 Copyright © 2011 Pearson Education, Inc 12.1 Normal Random Variable Central Limit Theorem Explains why bell-shaped distributions are so common Observed data are often the accumulation of many small factors (e.g., the value of the stock market depends on many investors) 10 of 45 Copyright © 2011 Pearson Education, Inc 12.3 Percentiles Quantile of the Standard Normal Find new mean weight (µ) for process 16 − µ = −2.5758 ⇒ µ = 16 + 0.2( 2.5758) ≈ 16.52 0.2 31 of 45 Copyright © 2011 Pearson Education, Inc 4M Example 12.2: VALUE AT RISK Motivation Suppose the $1 million portfolio of an investor is expected to average 10% growth over the next year with a standard deviation of 30% What is the VaR (value at risk) using the worst 5%? 32 of 45 Copyright © 2011 Pearson Education, Inc 4M Example 12.2: VALUE AT RISK Method The random variable is percentage change next year in the portfolio Model it using the normal, specifically N(10, 302) 33 of 45 Copyright © 2011 Pearson Education, Inc 4M Example 12.2: VALUE AT RISK Mechanics From the normal table, we find z = -1.645 for P(Z ≤ z) = 0.05 34 of 45 Copyright © 2011 Pearson Education, Inc 4M Example 12.2: VALUE AT RISK Mechanics This works out to a change of -39.3% µ - 1.645σ = 10 – 1.645(30) = -39.3% 35 of 45 Copyright © 2011 Pearson Education, Inc 4M Example 12.2: VALUE AT RISK Message The annual value at risk for this portfolio is $393,000 at 5% 36 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Multimodality More than one mode suggesting data come from distinct groups Skewness Lack of symmetry Outliers Unusual extreme values 37 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Normal Quantile Plot Diagnostic scatterplot used to determine the appropriateness of a normal model If data track the diagonal line, the data are normally distributed 38 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Normal Quantile Plot (Diamond Prices) All points are within dashed curves, normality indicated 39 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Normal Quantile Plot Points outside the dashed curves, normality not indicated 40 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Skewness Measures lack of symmetry K3 = for normal data z + z + z K3 = n 3 n 41 of 45 Copyright © 2011 Pearson Education, Inc 12.4 Departures from Normality Kurtosis Measures the prevalence of outliers K = for normal data z + z + + z K4 = −3 n 4 n 42 of 45 Copyright © 2011 Pearson Education, Inc Best Practices Recognize that models approximate what will happen Inspect the histogram and normal quantile plot before using a normal model Use z–scores when working with normal distributions 43 of 45 Copyright © 2011 Pearson Education, Inc Best Practices (Continued) Estimate normal probabilities using a sketch and the Empirical Rule Be careful not to confuse the notation for the standard deviation and variance 44 of 45 Copyright © 2011 Pearson Education, Inc Pitfalls Do not use the normal model without checking the distribution of data Do not think that a normal quantile plot can prove that the data are normally distributed Do not confuse standardizing with normality 45 of 45 Copyright © 2011 Pearson Education, Inc ... to pay for this insurance? Insurance costs call for a random variable that can represent a continuum of values (not counts) of 45 Copyright © 2011 Pearson Education, Inc 12. 1 Normal Random... Copyright © 2011 Pearson Education, Inc 12. 1 Normal Random Variable Prices for One-Carat Diamonds of 45 Copyright © 2011 Pearson Education, Inc 12. 1 Normal Random Variable With the exception of... Inc 12. 1 Normal Random Variable Definition A continuous random variable whose probability distribution defines a standard bell-shaped curve of 45 Copyright © 2011 Pearson Education, Inc 12. 1