Models of Patterns. Lecture 3, SMMD 2005 Bob Stine

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1 Models of Patterns Lecture 3, SMMD 2005 Bob Stine

2 Review Speculative investing and portfolios Risk and variance Volatility adjusted return Volatility drag Dependence Covariance

3 Review Example Stock and index EBAY S&P 500 Index Question If you already own the market, would you want to own EBAY as well? Focus on relationship between returns of these two series Covariance measures the dependence

4 Daily Returns in 2004 Typical simple variation in returns, with EBAY showing much larger variation over the year 0 Return Day EBAY Return S&P 500 Return

5 Histograms Familiar bell-shaped distributions, particularly for returns on the S&P 500 EBAY Return S&P 500 Return Moments Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N

6 Normal Model The normal model (or normal distribution or Gaussian distribution) describes an idealized bell-shaped distribution Source of the Empirical Rule Stock returns typically follow this behavior, approximately Normal models become more and more important going forward, so useful to have a diagnostic for normality.

7 Normal Quantile Plot Shows deviations from the frequencies predicted by the idealized bell-shaped curve For the distribution of returns on SP500

8 Normal Quantile Plot Points should stay within dotted bands of the diagonal line Dotted bands are 95% confidence interval assuming normal model holds Normal Quantile Plot

9 Quantile Plot for EBAY Outliers deviate from normality, but otherwise close to normal as well Normal Quantile Plot

10 Volatility Adjusted Returns mean return - half variance of return Even after adjusting for its higher volatility, EBAY did better than the market as a whole mean var vol drag vol adj ret EBAY SP

11 Portfolio Might it help, in sense of reducing anticipated long-run return (aka, volatility adjusted return) to own some of the market in addition to EBAY? Need covariance to find variance of a portfolio that has market & EBAY EBAY Return S&P 500 Return

12 Optimal Mix Covariance between the two investments is positive, Cov(EBAY,SP500) = Plug this into formula for the variance of a portfolio Var(aX+bY) = a 2 Var(X)+b 2 Var(Y)+2abCov(X,Y) Because the covariance is positive, the variance of the portfolio is larger than it would be if the two stocks were independent What would be the ideal pair to combine? What s this called?

13 Closer Look at Covariance Is this a large covariance? Cov(EBAY, SP500) = Seems tiny, but plays an important role in forming portfolio Problem with covariance Difficult to interpret Units perfect for adjusting variance when mix stocks in a portfolio These same units mean that covariance depends on the way we measure data

14 Scale Dependence Would the covariance change if we used data on a percentage scale rather than return scale? Yes! Just like a variance Cov(%ChgEBAY,%ChgSP) = ( ) = 0.73 Does this mean that percentage changes are more dependent than raw returns? No, of course not. Just reflects scaledependence of the covariance Correlation removes this dependence on scaling

15 Correlation Scale-free version of the covariance Corr(X, Y ) = Cov(X, Y ) SD(X) SD(Y ) Definition implies that -1 Corr(X,Y) +1 Positive correlation implies data along / diagonal, negative correlation implies data pack along \ diagonal Corr (EBAY,SP500) = 0.54

16 Visual Correlations Ellipse shows the strength of the correlation Tighter the ellipse, strong correlation Plots are an important diagnostic when using correlation to measure dependence because of the effects of outliers EBAY Return 0.0 Correlation Variable S&P 500 Return EBAY Return Mean Std Dev Correlation Signif. Prob Number S&P 500 Return

17 Describing Dependence How would you describe the dependence between SP500 and EBAY? Direction: Strength: positive moderate More precise? Correlation describes the direction and strength, but often need something more predictive If you knew that SP increased by 1% today, what would you anticipate for EBAY?

18 Predict EBAY? What would you expect to be the return for EBAY if you knew (somehow) that the SP went up 1%? EBAY Return S&P 500 Return

19 Use an Average? Take an average of the returns on EBAY for those days on which SP increased about 1%? See any problems with this approach? EBAY Return S&P 500 Return

20 Regression Model If we accept some assumptions, we can use a model to describe the dependence The model will also let us use ALL of the data to answer the previous question Linear equation describes how the average value of one variable (Y) depends on the value of another (X) Avg(Y X) = a + b X Y = response, dependent variable X = predictor, covariate, independent var Never forget: model reality

21 What line? Where should the line go? Need a criterion for picking best line EBAY Return S&P 500 Return

22 Least Squares Find the single line that minimizes the sum of the squared vertical deviations Slope and intercept = least-squares estimates EBAY Return S&P 500 Return

23 Interpreting the Fit Avg(EBAY SP500) = SP500 Intercept = Slope = 1.5 Prediction = (0.01) = EBAY Return S&P 500 Return

24 Diamonds Price and weight are related! But which goes on the vertical axis (Y) and which goes on the horizontal axis (X) Not as obvious as it might seem Price (Singapore dollars) Weight (carats)

25 Fitted Line Least squares line Avg(price weight) = weight Units for slope and intercept? Price (Singapore dollars) Weight (carats)

26 Strength of Association R-squared summary statistic describes proportion of variation in response captured by fitted line Percentage of variation modeled by line R 2 = Corr(X,Y) 2 RMSE = SD(residuals) also measures amount of variation left over RMSE 2 (1-R 2 ) Var(Y) Has units of the response

27 Examples R 2 = RMSE = r R 2 = RMSE = 31.8 S$ EBAY Return S&P 500 Return Price (Singapore dollars) Weight (carats)

28 Nonlinear Models Linear means On average, changes of a fixed size in X are associated with constant changes in Y, regardless of the size of X Make sense Diamonds? Effects of advertising? Effects of price changes on sales? Returns to scale in manufacturing?

29 Pricing Models How do customers respond to changes in the price of a canned pet food? Do you expect the relationship to be linear? Sales Volume Price ($)

30 Linear Model Fails Residual plot zooms in to show the hidden detail. Large R 2 does not imply good fit. Sales Volume Price ($) Residual Price ($)

31 Economics Typical approach to this type of data used in economic models converts data to a log scale Why? 12 Log Sales Volume Log Avg Price Residual Log Avg Price

32 Interpreting log-log Nonlinear model Effects of changes in price do not have same effect. Depends on the price. Slope is known as an elasticity Avg(Log Volume Log Price) = Log(Price) On average, For each 1% increase in price, the volume sold decreases by 2.44 percent Sell more at a low price, but at less profit per item sold What do you need to know in order to find the right price for this item?

33 Different Look at Fit Original Coordinates Transformed Coordinates Sales Volume Log Sales Volume Price ($) Log Avg Price Different views of the same model!

34 Key Concepts Normal model and normal quantile plot Covariance and correlation Least squares regression Residual plots Interpretation of slope and intercept R 2 and RMSE Nonlinear models Elasticity

35 Software Notes Normal quantile plot diagnostic Analyze > Distribution > Normal quantile Density ellipse & correlation Analyze > Fit Y by X > Density ellipse Linear regression Analyze > Fit Y by X > Fit Line Use > Plot residuals option for fit Log-Log regression Analyze > Fit Y by X > Fit Special

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