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Stochastic Models Decisions Walt Pohl Universität Zürich Department of Business Administration February 28, 2013 Making Decisions The ultimate purpose of models in business is to make decisions. Project Good Outcome Bad Outcome A 50 million 0 million B 15 million 15 million Which is the right decision? Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 2 / 11 Mean The simplest criterion, and the one most often used outside of financial applications, is the use the mean. Examples: Expected waiting time for customers. Expected cost. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 3 / 11 Return on Stocks versus Bonds Here are the stats for the monthly return on US stocks and US government bonds since 1920. Mean Std. Dev. 1% 5% Stocks 0.009 0.054 -0.150 -0.078 Bonds 0.003 0.003 0 0 If you buy bonds, are you making a mistake? Maybe. Or maybe you are just risk averse. Everything else being equal, you prefer a certain outcome over an uncertain one. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 4 / 11 Mean-Variance Analysis In the 50s, Markowitz suggested that we also consider the variance. Stocks have much more variance than bonds: a monthly standard deviation of 5% versus less than 1%. Stocks-versus-bonds face a risk-versus-return tradeoff. Stocks have higher return, but more risk of dropping. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 5 / 11 Limitations of the Variance Variance does not account for “fat tails”: the chances of rare but disastrous outcomes. Consider these two distributions: Normal with mean zero and variance 2. t with 3 degrees of freedom. They both have mean zero and variance 2. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 6 / 11 Fat Tails Here are the lower quantiles for t versus Normal. 1% 5% Normal -2.33 -1.65 t -4.54 -2.35 Under the t distribution, 1% of the time you will lose twice as much than you would under the normal distribution. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 7 / 11 Value at Risk Value-at-risk (VAR) is a rule widely used by banks. You consider how sensitive to extreme bad outcomes you are. Bad outcome is how much you would lose an extreme quantile such as the 1% or 5% quantile occurred, relative to the mean. (This quantile is known as the confidence level.) Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 8 / 11 Value at Risk, cont’d For stocks and bonds, the historical VARs are, assuming that you invest 1 US dollar at the beginning of the month: 1% 5% Stocks -0.159 -0.087 Bonds -0.003 -0.003 Note that this depends on the window: here monthly. A bigger window would see bigger numbers. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 9 / 11 Expected Shortfall VAR suffers from one obvious shortcoming. Suppose that you have an investment that 99.9% of the time earns you 1 dollar, and 0.1% of the time you lose 1 trillion dollars. Unless you pick a very small confidence interval (0.1% or smaller), this has a VAR of zero. Expected shortfall corrects for that. The Expected shortfall is the expected value of the investment, conditioning on the fact that we are below the VAR cutoff. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 10 / 11 Statistical Decision Theory This combines the statistical and decision-making into one step. Widely used in machine learning. You view chosing the parameters as a decision problem. We will see more about this later. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 11 / 11