Statistics Review

1. Statistics Review 1

2. Why We Care • Analysts must be able to evaluate quantitative information to make predictions. • Although statistical tools my not be sharp, they are necessary. • The point is for you to be able to • use the tools, • Understand their limitations. 2

3. Statistical Background • Two Steps • Review of the basics • Application to estimating growth • Which will illustrate different ways of estimating growth using past data on different types of securities. • Not surprisingly, the methods work better with certain types of securities. 3

4. Estimating the mean and variance • We will begin by computing a sample mean and a sample variance. • These are estimates. • That means that there is error associated with the estimates. • The error can be large especially • when the time series used to estimate is short, • when you’re estimating the mean. • This can be a problem when estimating mean growth rates. 4

5. Means and Variances • Start with T numbers for earnings, Et, for t=1,…T. • For example, T could be 20, where we have quarterly earnings numbers for the past five years. • Then the samplemean, or average earnings is: M = (1/T)(Σt Et) • The sample variance for the same set is: S2 = (1/(T-1))(Σt (Et-M)2) The T-1 is just to make the estimator unbiased. • The sample standard deviation, S, is the square root of the sample variance. 5

6. Samples and populations • We estimated a sample mean M = Sample Mean = (1/T)(Σt Et) • and sample variance S2 = Sample variance = (1/(T-1))(Σt (Et-M)2) • These sample parameters are estimates of “true” population parameters (let’s call them μ and σ2. • What we want are “good” estimates. • What makes for a good estimate? 6

7. Confidence • Answer: A good estimate is accurate. • We need to to estimate the variance, or standard deviation, of your estimate of the sample mean. • This is NOT the same as the estimate of the variance. • An estimate of the variance of earnings may change as you add more data, but it will not shrink necessarily. • But the variance in an estimate of the sample mean goes down as you get more data. • If you had an infinite amount of data, this sampling error goes to ZERO under certain conditions. • You need stationarity. • e.g. there can’t be a trend with time. 7

8. Confidence – Standard Errors • Usually, the more data, the smaller the standard error (the more accurate your estimate). • Computationally, the variance of your estimate of the sample mean is proportional to the number of observations you have: Variance of estimate of sample mean = σ2/T In sample, that’s S2/T • The standard error is S/sqrt(T) • This square root is going to cause problems, because it means you need a lot more data to get small improvements in precision. 8

9. Numerical example of confidence • Suppose you have 20 observations of earnings drawn from a normal distribution with mean of $1 per share with variance of ¼ so that the standard deviation was $0.50 per share. • If you simulated data from this distribution and the std. error would be something like • S/sqrt(20) = .50/4.47 = .11 • You’d be confident at the 5% level from looking at the data that the mean was between $0.78 and $1.22. • Your estimate of the standard error is ¼ as much as your estimate of the std. deviation. 9

10. Numerical example of confidence • If you had 100 observations (25 yrs), std error S/sqrt(100) = .50/10 = .05 • You’d be confident at the 5% level from looking at the data that the earnings would be between $0.90 and $1.10 • Your estimate of the standard error is 1/10 as much as your estimate of the std. deviation. • 5 times as much data and your precision only doubles. 10

11. Where do we go from here? • Let’s see if we can apply this tool + our knowledge of regression to a prediction problem. • We want to apply these tools to estimate the growth rate for a firm’s earnings. • We could apply this to dividends and use a DDM • Or to free cash flow and price using a FCFE model. 11

12. Example with real data • You have 10 years of earnings data and you want to forecast earnings out for another few years. • The reason is that you want to do valuation using either a free cash flow to equity approach or a dividend discount approach. • Suppose you have no access to analyst forecasts. How would you begin? • We’ll talk about analyst forecasts later. 12

13. Easiest Case • The past is like the future. • This means that you don’t have to worry about innovations of any kind that can increase (or reduce) growth rates. • Innovations can be important, because then past data may not be a good guide to future performance. 13

14. Easiest Case • When might the past not be like the future? • new innovations affect future performance. • In the past, the company had a different product mix. • what would you do then? • In the past, the company had made numerous acquisitions. • essentially, whenever the business going forward looks different from the business in the past. 14

15. Forecasting from past earnings • Suppose we have quarterly earnings data for 10 years. Potential problems. • Seasonality. • For example, 4th quarter earnings for retailers need not be the same as 2nd quarter earnings. • Seasonality should NOT affect stock prices. Everyone knows that retailers earn the most in December – it’s no surprise to anyone. • One fix is to use annual earnings only. 15

16. Annual earnings: 10 observations. • This number, 10, is not atypical. • Strategies for estimating growth: • Naively assume that future earnings will be the same as earnings last year. • This is easy, but what are the shortcomings? • Take the 10-year average as an estimate • This assumes that all variation in the past is just noise. • This too has shortcomings. For example? • Assume that earnings are growing somehow. • Figure out how. 16

17. A simple example • Suppose you have a deterministic trend. • let E1=1, E2=2,...E10=10, you expect E11=11. • Plug into the data analysis package. Results: • Using E11=E10 = 10 works well. • TakingE11 = sample average of 5.5 does not because you’ve ignored the trend. • What about running a regression? 17

18. Regressions • You could run a linear regression here and that would give you a perfect fit. • Et = a + b Et-1 • a=1, b=1 so that • Et = 1 + Et-1 • You could also de-trend the data (usually good to do) and calculate average growth rates in % terms: • g = (Et+1/Et – 1) which is not quite constant. Here, growth is always $1 per year but growth is slowing in % terms (from 100% in year 1 to 11% in year 9) 18

19. Dealing with growth – summing up • Simple regressions: Earningst+1 = a + b earningst + et • A and b are regression coefficients, and et is the regression error term (which is different for each year). Earnings growtht,t+1 = a + b earnings growtht-1,t • Second, just calculate average earnings growth over the period and use that. • In either case, make sure your quantitative estimates are • Reasonably precise (high standard errors mean you have junk) • Pass the laugh test (weird results are probably wrong) • Remember, you have so few observations that anything you get is likely to be imprecise at best. Don’t think that just because you CAN run a regression, you should! 19

20. Earnings Growth • Why do we focus so much on growth for stocks? • The DDM: D1/(k-g) • How reasonable is a growth rate of 10% • Sears? • Home Depot? • Google? 20

21. Real data • From COMPUSTAT - annual earnings for firms from 1993-2002 for • Amgen • Colgate • Exxon • P&G 21

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23. What to do? • Predict returns next period. • Estimate average growth rates. • These are the two things we’ll start with. • The data are from wrds. • Past earnings are in the COMPUSTAT portion of the database. 23

24. Descriptive statistics first. • Q: for which companies are we likely to be able to predict best? Worst? Why? • Let’s focus on • Average annual earnings per share in dollars • The standard error for that • Average growth rates in percent • The standard error for that • Notice for growth how the means and medians diverge. • Also, what do you do for growth when earnings are low? Or even worse, negative? 24

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27. Sample statistics • the mean is always less than the most recent observation, because there’s a trend. • so, you wouldn’t use the mean to predict, necessarily. • the means are estimated pretty precisely. 27

28. Regressions • For Amgen, Et = .11 + .956 Et-1 • Our E11 guess: .11+.956(1.07) = 1.13 • Next slide, we look at % changes 28

29. Predictions • Here, the mean growth rate for AMGEN is 17%, so that our forecast would be $1.25. • Colgate and P&G have mean growth rates of over 35%. Why? • Outliers – and these are often driven by accounting items. • Look at the next page • 1996-7 for Colgate • 1995-4 for P&G 29

30. Outliers create problems • Look at Colgate in 1995, 1996, and 1997. • Earnings were .955, .26, and 1.0475 • % changes were -73% and + 303% • The avg percent change over the 95-6 and 96-7 was +115%. • This number is meaningless and it drives all the results 30

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32. Geometric Averages. • The geometric average growth rate for 10 years would be ((1+g)10)(1/10) • In this case, just take (E10/E1)(.1) • For Colgate, that’s (2.02/0.73)0.1 -1 = .107 or 10.7%, which is a lot less than the arithmetic average of over 36%. • The difference is driven by variance: • There is a sense in which the geometric mean ignores outliers – although you have to be careful – an outlier at either endpoint would cause problems. 32

33. Now, how did the professionals do? • IBES 2003 most recent growth forecasts for • Colgate 13% • P&G 9.5% • Exxon 6% • Amgen 23% • Actual 2003 earnings and growth: • AMGN 1.75 CL 2.60 XOM 3.16 PG 2.19 63% 28%, 43%, 16% 33