Stochastic Models. Statistics. Walt Pohl. February 28, Department of Business Administration

Transcription

1 Stochastic Models Statistics Walt Pohl Universität Zürich Department of Business Administration February 28, 2013

2 The Value of Statistics Business people tend to underestimate the value of statistics. Statistics allows us to learn from history. Finance people tend to overestimate the value of statistics. History rarely repeats itself exactly. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

3 Statistical Methods 1 Method of moments 2 Maximum likelihood 3 Regression 4 many more... Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

4 Method of moments Simplest (and oldest) method of doing statistics. Pick some moments (usually mean and variance), and match with the data. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

5 Example: Lognormal Model of Stock Prices 1 Changes in stock prices are unpredictable. 2 Modeled by random changes in the growth rate. 3 We assume these random changes can be modeled by a lognormal distribution. In other words, if S t is the stock price, then the log growth rate, log S t+1 /S t is normally distributed. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

6 Example: Lognormal Model of Stock Prices The model can be written as log S t+1 = µ + log S t + ɛ t+1, where ɛ t is normally distributed Normal(0, σ 2 ). In financial circles, the variance or standard deviation is known as the volatility. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

7 Example: Lognormal Model of Stock Prices, cont d This model has two parameters, the mean µ and variance σ 2, both of which can be computed from moments. (We usually work with the standard deviation, since the variances are usually tiny.) Example: S&P 500 index. Mean = Standard Deviation = Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

8 Example: Lognormal Model of Stock Prices, cont d Is this a good model? No! 1 Distribution is fatter tails than the normal: more extreme events than predicted by the normal distribution. 2 Variances themselves vary over time. We consider the second. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

9 Example: The EWMA Volatility Model Forecasting is the art of predicting the future from today. The easiest, and frequently most successful, method is to predict that the future will be like the recent past. The exponentially-weighted moving average (EWMA) model for forecasting volatility is a simple example. We forecast tomorrow s volatility by a weighted sum of today s forecast and today s observation Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

10 Example: The EWMA Volatility Model, cont d Let σ 2 t be the variance of ɛ t for period t. Then the EWMA forecast is But how do we find λ? σ 2 t = λσ 2 t 1 + (1 λ)ɛ 2 t. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

11 Maximum Likelihood Estimation Maximum likelihood estimation is a general-purpose method for estimating parameters in models. One limitation: you must completely specify your probability distributions. (Here we must specify that ɛ t is normally distributed). Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

12 From PDF to Likelihood A probability density function (PDF) depends on 1 parameters 2 data Once we plug in the data, we get a function of the parameters alone the likelihood. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

13 Example: the Normal Distribution Recall the PDF of the normal distribution, Normal(µ, σ): 1 e (x µ)2 2σ 2 2πσ 2 If we have N independent draws, of a normal random variable with unknown mean µ and variance σ. ( ) N 1 e (x i µ) 2 2σ 2 2πσ 2 We choose as our estimates the µ and σ to maximize this expression. This is the maximum likelihood estimate (MLE). Walt Pohl (UZH QBA) Stochastic Models February 28, / 23 i

14 Example: the Normal Distribution, cont d. We can actually compute the MLE for the normal distribution explicitly: µ MLE = σ 2 MLE = N x i /N i N (x i µ MLE ) 2 /N i These are the sample mean and standard deviation. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

15 Likelihood for EWMA The likelihood for the EWMA is similar, but now σ i depends on the data. Let ɛ i = log S i+1 log S i α (where α is the sample mean). Then, given lambda, we know that σ 2 i = λσ 2 i 1 + (1 λ)ɛ 2 i. The (log) likelihood is then just ( ɛi 2σ 2 1 ) i i 2 log(2πσ2 i ) We must resort to numerical methods to maximize this expression. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

16 Linear Regression Linear regression is a technique for finding linear relationships between variables, Y t = α + βx t + ɛ Requires no distributional assumptions about Y t or X t. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

17 Example: Capital Asset Pricing Model Model to explain stock returns. Investors make money on the stock market from: 1 Dividends, D t actual case payments to investors. 2 Capital gains, S t S t 1 price changes in the stock. CAPM models total returns R t R t = S t + D t S t 1 S t 1. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

18 Example: Capital Asset Pricing Model, cont d 1 Investors always have an alternative investment available: risk-free government bonds. 2 The important quantity to explain, then, is excess returns, the return over the risk-free rate R f. 3 The Capital Asset Pricing Model claims that excess returns for a stock are (up to an error term), a linear function of the excess returns on the total market. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

19 Example: Capital Asset Pricing Model, cont d Let R M be return on the market. Then the CAPM says R R f = β (R M R f ) + ɛ CAPM is usually estimated using linear regression: R R f = α + β (R M R f ) + ɛ. Note the extra α. CAPM predicts α = 0. For market return we use the return on a broad-based index, such as the S&P 500 for US stocks. In the example, I use the CRSP value-weighted index, which includes all stock traded on the major US exchanges. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

20 Failures of the CAPM The CAPM is wildly popular, but doesn t have a good track record. Well-known failures include: Stocks for small firms do better than expected, large firms do worse than expected. Stocks for value firms firms with lots of assets for their size do better than growth firms firms few assets for their size. Here size means total market capitalization the cost of buying up all of the firms s stock. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

21 Example: Fama-French 3 Factor Model The main academic alternative to CAPM is factor models. Define several additional return differences, and compute several betas using multivariate regression. The Fama-French 3 factor model uses Market factor return on market minus risk-free rate (just like CAPM). SMB return on small stocks minus return on big stocks. HML return on value stocks minus return on growth stocks. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

22 Beyond Classical Econometrics Traditional statistics depends on stringent assumptions. Maximum likelihood requires specifying a distribution. Linear regression requires positing a linear relationship. There s essentially no reason to believe in these assumptions. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23

23 Big Data There are newer techniques that go under the generic name of machine learning. They require Lots of data. Computer power. We will discuss these techniques more later. Walt Pohl (UZH QBA) Stochastic Models February 28, / 23