Revenue Forecasting: Tips and Techniques for All Sources

Municipal Finance Institute December 4-5 Revenue Forecasting: Tips and Techniques for All Sources by Daniel Jordan, Ph.D., CGFM Director of Finance City of La Cañada Flintridge Adjunct Assistant Professor Price School of Public Policy University of Southern California LLC Municipal Finance Institute CLIENT LOGO HERE

Topics Introduction Elements of the revenue forecasting process Overview of forecasting techniques The choice of forecasting method it should be driven by the nature ( behavior may be a better term) of the revenue data, but remember that simple methods tend to perform best Criteria for measuring forecast accuracy Summing Up & Concluding Thoughts 2

A General Session pairing Revenue Forecasting & Council/Manager Relations??!! We can t compete with No. 1 on the Top 10 List of things that don t go together, but we certainly make the List 10 Things That Don't Go Together 1. Dennis Rodman and North Korea 3

The revenue data in this presentation is taken from the City of La Canada Flintridge 4

One reason you may have heard of La Canada Flintridge disasters Station Fire 8-28-09 Mudslides 2-6-10 General Fund Projection: FY 2013-14 to FY 2017-18 June 18, 2013 5

And perhaps another reason peafowl 6

Some vital statistics about La Canada Flintridge La Canada Flintridge (incorporated 1976) is a small city (21,000 population) with an even smaller City government (currently 24 full-time employees & about 20 part-time employees) A contract city contract for police, fire, public works, building and safety, etc. etc.) Using 21 funds currently (all governmental) For FY 2013-14 Budget, about $23 million in total expenditures General Fund expenditures: $11.3 million (another $600K being transferred out, mostly for capital projects All Other Funds: $11.7 million (mostly capital projects and debt service related to three sewer assessment districts) 7

The City carefully watches its reserves The City Council policy is that General Fund reserves should remain between 100%-150% of the annual operating budget $18 $16 $14 Ending Fund Balance Metrics: Historical and Projected Current FY 160% 140% 120% Millions $ $12 $10 $8 $6 $4 $2 100% 80% 60% 40% 20% $0 FY 2006-07 Act FY 2007-08 Act FY 2008-09 Act FY 2009-10 Act FY 2010-11 Act FY 2011-12 Act FY 2012-13 Est FY 2013-14 Proj FY 2014-15 Proj FY 2015-16 Proj FY 2016-17 Proj As a % of Revenues 123% 117% 119% 126% 119% 127% 118% 118% 118% 118% 119% 121% As a % of Expenditures 142% 142% 123% 123% 140% 138% 120% LCC Municipal 129% Finance 126% Institute 128% 130% 8134% FY 2017-18 Proj Fund Balance ($ millions) $15.25 $15.46 $13.85 $13.77 $14.75 $14.54 $13.55 $14.04 $14.37 $14.82 $15.34 $15.99 0%

Elements of the Revenue Forecasting Process LLC Municipal Finance Institute CLIENT LOGO HERE

A comprehensive forecasting process will always involve at least these elements Define Goal Collect and Clean Data Explore & Visualize Series Apply Forecasting Method(s) Evaluate & Compare Performance Implement Forecast(s) 10

You probably think about the goals for your forecast already, at least implicitly: Here s just a few with implications for choosing a forecast method Forecasting horizon one month, one quarter, 6 months, one year, 2+ years? A forecast for a period less than one year may be have to consider seasonality in the time series For long-range forecasting (2+ years), you should attempt to be broadly correct, not to be precise (because you will end up being precisely wrong) Costs associated with forecast errors is a revenue forecast that is too low better than one that is too high? (perhaps add a conservative tweak to your forecast process) Predictive vs. descriptive goals are you merely trying to forecast (predict) as accurately as possible, or are you aiming at some level cause-and-effect- understanding? Who is the audience? A technical department? The City Council? Etc. 11

Collecting and cleaning your revenue data: nothing sexy about this One thing to watch when compiling historical data: Be aware of any significant changes in revenue accrual practices 12

And which revenue data to clean? Focus on the big game (i.e., your largest revenue sources) Among the 37-41 General Fund revenue line items each year, relatively few the Big 8 account for most (>90%) of recurring operating revenue $4,500,000 $4,000,000 $3,500,000 $3,000,000 $2,500,000 $2,000,000 $1,500,000 $1,000,000 $500,000 General Fund Revenue, "The Big 8" & All Others, FY 2012-13 FY 2012-13 Total: $12.37 million 100.0% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% $0 Property Tax Sales Tax Prop Tax In- Lieu Bld Per & Pln Chk Franchise Fees Interest Income Zoning Fees Property Trans. Tax All Others Amount $3,992,151 $2,463,681 $1,926,365 $1,434,352 $608,238 $361,006 $255,995 $234,145 $1,100,680 % of Total 32.3% 19.9% 15.6% 11.6% 4.9% 2.9% 2.1% 1.9% 8.9% Cumul. % 32.3% 52.2% 67.7% 79.3% 84.2% 87.1% 89.2% 91.1% 100.0% 13 0.0%

Visualizing time series data begins with time series plots A basic time series plot graphs the revenue series as a line chart with time on the horizontal axis, but you can add relevant series and/or information to make it more useful analytically Note the recent (3-year) trend in Sales Tax 3,000,000 Annual Sales Tax Revenue 15% 2,500,000 10% 2,000,000 1,500,000 5% 1,000,000 0% 500,000 0-5% (500,000) FY 2003-04 FY 2004-05 FY 2005-06 FY 2006-07 FY 2007-08 FY 2008-09 FY 2009-10 FY 2010-11 FY 2011-12 FY 2012-13 $ Change 44,831 25,410 50,290 139,066 (123,806) (144,125) 204,079 141,470 217,786 $ Total 1,908,680 1,953,511 1,978,921 2,029,211 2,168,277 2,044,471 1,900,346 2,104,425 2,245,895 2,463,681 % Change 2.3% 1.3% 2.5% 6.9% -5.7% -7.0% 10.7% 6.7% 9.7% 14-10%

Note the HUGE drop, recovery, followed by a largely horizontal time series Annual Building Permit & Plan Check Revenue 2,000,000 40% 30% 1,500,000 20% 1,000,000 10% 0% 500,000-10% -20% 0-30% (500,000) -40% -50% (1,000,000) FY 2003-04 FY 2004-05 FY 2005-06 FY 2006-07 FY 2007-08 FY 2008-09 FY 2009-10 FY 2010-11 FY 2011-12 FY 2012-13 $ Change (26,970) 53,190 401,882 (13,327) (856,625) 134,711 378,314 (58,164) 14,977 $ Total 1,406,364 1,379,394 1,432,584 1,834,466 1,821,139 964,514 1,099,225 1,477,539 1,419,375 1,434,352 % Change -1.9% 3.9% 28.1% -0.7% -47.0% 14.0% 34.4% -3.9% 1.1% -60% 15

Basic metrics are also useful in conjunction with time plots May be useful in establishing likely upper and lower bounds for a forecast 3,000,000 Annual Sales Tax Revenue 15% 2,500,000 2,000,000 1,500,000 1,000,000 500,000 0 (500,000) FY03-04 to Measure FY12-13 Ave % Change 3.05% CAGR 2.88% Min. -7.05% Max. 10.74% Stan. Dev. 6.26% FY 2003-04 FY 2004-05 FY 2005-06 FY 2006-07 FY 2007-08 FY 2008-09 FY 2009-10 FY 2010-11 FY 2011-12 FY 2012-13 $ Change 44,831 25,410 50,290 139,066 (123,806) (144,125) 204,079 141,470 217,786 $ Total 1,908,680 1,953,511 1,978,921 2,029,211 2,168,277 2,044,471 1,900,346 2,104,425 2,245,895 2,463,681 % Change 2.3% 1.3% 2.5% 6.9% -5.7% -7.0% 10.7% 6.7% 9.7% 16 10% 5% 0% -5% -10%

$600,000 Visualizing the time series at different time intervals (e.g., monthly, quarterly) Any (consistent) seasonality? Not really Sales Tax Revenue by Quarter - Last 4 Years $500,000 $400,000 $300,000 $200,000 $100,000 $0 Revenue 17

Some other ways of visualizing the time series - create an index of your major revenue sources to compare their relative performance over time 18

Overview of Revenue Forecasting Techniques LLC Municipal Finance Institute CLIENT LOGO HERE

Forecasting methods (revenue and otherwise) fall into three broad categories Qualitative methods Time Series (a category of quantitative methods, sometimes referred to as extrapolation ) Causal (another category of quantitative methods) 20

And within those three categories, there are dozens of specific techniques Among the categories, time series methods are the true bread and butter, meat and potatoes of forecasting (i.e., the most frequently used by experts in forecasting competitions, among private sector forecasting professionals, the military, and local government). I ll spend most of today on time series methods Qualitative Methods Time Series Methods Causal Methods * Judgment/Intuition * Naïve * Simple Regression * Structured Group Research * Moving Averages * Multiple Regression * Etc. * Exponential Smoothing * Econometric * Trend Analysis * Input-Output * Simple Regression (against time) * Etc. * Advanced Smoothing techniques 21

All qualitative methods attempt to tap domain knowledge Common methods Expert judgment Ask a single (or group) of subject-area experts to make a projection Intuitive approaches Seek consensus on a projection through group interaction, brainstorming, gut feelings, crystal balls Delphi method Ask a bunch of experts forecast something through a series of sequenced questionnaires Beware of the obvious problems with qualitative techniques: they are subjective/biased, difficult to replicate, and often difficult to justify to the consumers of your forecast Used when little or no quantitative data available Qualitative information should ALWAYS be used in conjunction with quantitative forecast methods when possible 22

Causal methods all attempt to forecast based on posited cause-and-effect relationships Causal methods attempt to use a set of explanatory ( predictor, independent ) variables believed to influence the forecasted ( dependent ) variable (i.e., the revenue source you re forecasting). Assuming you have a good model, causal techniques can be useful if large changes in the revenue source are anticipated. BUT, there are myriad difficulties in using causal models to forecast: Is your model good (does it have the correct predictor variables) Do you have to predict your predictors, or can you used lagged predictor variables for which you have actual data to forecast Is data available (particularly at the right scale) Are your statistical skills up-to-date (i.e., you no doubt remember what an F-statistic is and how to interpret it ) 23

Time Series methods simply extrapolate (although many of these methods are not that simple) Time series methods use only the time series itself i.e., your past revenue data to build forecasting models and make forecasts. No attempt to forecast revenues on the basis of known, or supposedly known, cause-and-effect relationships. (This a both a strength and a weakness of time series techniques) Instead, you attempt to isolate and measure certain components within the time series itself, and then use a forecasting method that best accommodates the nature of your data. Relative simplicity, more modest data requirements & ease of implementation are among the strengths of time series methods 24

Pretty close to what we ve covered so far with respect to choosing a forecast method From Kavanagh & Iglehart, Government Finance Review, October 2012 (other published sources have similar matrices) 25

The Nature of Your Revenue Data, Choosing a Forecasting Method, and Criteria for Measuring Forecast Accuracy LLC Municipal Finance Institute CLIENT LOGO HERE

All time series (i.e., revenue data tracked over time) will contain one or more of the following elements 1. Level the average value of the series over a number of periods (i.e., the horizontal component what is left over in the absence of the all other time series components: trend, seasonality, cyclicality, and noise.) 2. Trend the continuing long-term pattern of increase or decrease in a time series over some number of periods. 3. Seasonality the repeating pattern of increases and decreases that occurs within a one-year period or less (i.e., hourly, daily, weekly, monthly, bi-monthly, quarterly, or even 4 or 6-month patterns) 4. Cyclicality the pattern of long-term increase or decrease in the time series, often in connection with the business cycle (i.e., recession and recovery); this causes a series to stay above or below the trend line for long periods of time. 5. Randomness (aka noise) Random variation in a time series due to measurement error, abrupt events (political, economic, natural); all time series have some level of random variation 27

What s worth trying to forecast versus what isn t For the most part, even the best of the best forecasters do not spend much(or any) time trying to forecast either: Cyclicality (aka cycles): Given the failure (up-to-now, anyway) of anyone to accurately forecast the timing and/or magnitude of future business cycles, most analysts try to incorporate cyclicality into the forecast of long-term trend. (An occasional exception is for very short-term forecasts) Randomness (noise): By its very nature, the random fluctuation in a time series does not repeat itself over time, therefore there is no historical pattern to model into the future So, to the extent that you use a formal, quantitative time series method forecasting, spend your time attempting to forecast: Level, Trend, and Seasonality. 28

The key question when choosing a forecasting technique The key question when choosing a forecasting technique to use with your particular stream of revenue data is to determine which of the factors Level, Trend, or Seasonality predominate 29

Some time series have only a Level and some degree of Randomness ( noise ) Time series with only a Level and some amount of Noise are said to be stationary. This means the noise fluctuates randomly around a fairly constant mean throughout the time series (or at least through that portion of the time series your deem relevant for forecasting the future) 30

What a relatively stationary time series looks like $25.00 Stationary Revenue Stream $20.00 $15.00 $10.00 $5.00 $0.00 Revenue ($000) Average 31

Jul-11 Aug-11 Sep-11 Oct-11 Nov-11 Dec-11 Jan-12 Feb-12 Mar-12 Apr-12 May-12 Jun-12 Jul-12 Aug-12 Sep-12 Oct-12 Nov-12 Dec-12 Jan-13 Feb-13 Mar-13 Apr-13 This too is a stationary time series, but at a new Level after a step-change at July 2012 (e.g., from a major increase in demand or price) $40.00 Stationary time series, but at a new Level $35.00 $30.00 $25.00 $20.00 $15.00 $10.00 $5.00 $0.00 Revenue ($000) Average1 Average2 32

With relatively stationary data (i.e., mostly Level and Noise, and little or no Trend or Seasonality), three primary forecasting techniques 1.Naïve forecasting 2.Moving average 3.Exponential smoothing Note that with each technique (except 1), you re simply trying to smooth out the Noise so you can forecast the Level 33

With a naïve forecast, your best guess of next period s revenue is last period s revenue Naïve forecast formula this actually eliminates none of the Noise, but reacts very quickly to a change in the Level Forecast = The most recent time series value where: F t+1 = forecast of the time series for period t+1 Y t = actual value of the time series at period t 34

Naïve indeed (but admit it, you ve gone with the naïve forecast) Given the data s relatively narrow & consistent variability around the mean (i.e., its not just a random walk ), and assuming you anticipate no sudden change in the Level, you could do better than the naïve forecast. (This is the data for the first 12 periods of the stationary series shown earlier.) Period Notation Y t-11 Y t-10 Y t-9 Y t-8 Y t-7 Y t-6 Y t-5 Y t-4 Y t-3 Y t-2 Y t-1 Y t Period Count 1 2 3 4 5 6 7 8 9 10 11 12 Time Period Jul-11 Aug-11 Sep-11 Oct-11 Nov-11 Dec-11 Jan-12 Feb-12 Mar-12 Apr-12 May-12 Jun-12 Revenue ($000) $17.00 $21.00 $19.00 $23.00 $18.00 $16.00 $20.00 $18.00 $22.00 $20.00 $15.00 $22.00 $25.00 Stationary Revenue Stream $20.00 F t+1 = F 13 = Y t = Y 12 = $22.00 $15.00 $10.00 $5.00 $0.00 Revenue ($000) Average 35

A moving average is another technique for relatively stationary data, focusing on the more recent periods Moving average formula Forecast = Average of the most recent k time series values where: F t+1 = forecast of the time series for period t+1 Y t = actual value of the time series at period t k = number of periods of time series data used to generate the forecast 36

Calculating moving averages for July 2012 (period 13) Period Count Period Y 1 Jul-11 $17.00 2 Aug-11 $21.00 3 Sep-11 $19.00 4 Oct-11 $23.00 5 Nov-11 $18.00 6 Dec-11 $16.00 7 Jan-12 $20.00 8 Feb-12 $18.00 9 Mar-12 $22.00 10 Apr-12 $20.00 11 May-12 $15.00 12 Jun-12 $22.00 F Revenue ($000) Forecast 13 Jul-12?????? 3-period moving average the three most recent periods 3 22 15 20 3 19.00 4-period moving average the four most recent periods 3 22 15 20 22 4 19.75 5-period moving average the five most recent periods 3 22 15 20 22 18 5 19.40 37

Period Count Calculating moving averages (cont.) The average is moving because as new periods come to pass, the oldest periods are dropped from the calculation of the average. The assumption is that newer periods are more likely to accurately forecast a future period (below is a 3-period MA) Period 1 Jul-11 $17.00 2 Aug-11 $21.00 3 Sep-11 $19.00 Y F Revenue ($000) Forecast 4 Oct-11 $23.00 $19.00 5 Nov-11 $18.00 $21.00 6 Dec-11 $16.00 $20.00 7 Jan-12 $20.00 $19.00 8 Feb-12 $18.00 $18.00 9 Mar-12 $22.00 $18.00 10 Apr-12 $20.00 $20.00 11 May-12 $15.00 $20.00 12 Jun-12 $22.00 $19.00 13 Jul-12 $19.00 38

Notice how the moving average tends to smooth out the random fluctuation in the time series $25.00 Forecast with 3-period moving average $20.00 $15.00 $10.00 $5.00 $0.00 Revenue ($000) 3-period MA 39

Jul-11 Aug-11 Sep-11 Oct-11 Nov-11 Dec-11 Jan-12 Feb-12 Mar-12 Apr-12 May-12 Jun-12 Jul-12 Aug-12 Sep-12 Oct-12 Nov-12 Dec-12 Jan-13 Feb-13 Mar-13 Apr-13 May-13 Different degrees of smoothing with different length moving averages In general, a shorter moving average provides a better forecast for data with lots of Randomness around the mean (or for a sudden change in the Level), while using more data is better for time series with relatively little variation and a steady Level $35.00 Forecast with naïve, 3-period & 6-period moving averages $30.00 $25.00 $20.00 $15.00 $10.00 Revenue ($000) Naïve 3-period MA 6-period MA 40

A weighted moving average allows you to vary the weights given to each period in calculating the average The standard (or simple ) moving average assigned the same weight to each value used in the calculation; a weighted moving average lets you decide on the weights. Note that the sum of the weights must always equal 1.0 Forecast = Weighted average of the most recent k time series values where: F t+1 = forecast of the time series for period t+1 Y t w t = actual value of the time series at period t = weight applied to the actual value of the time series for period t k = number of periods of time series data used to generate the forecast 41

Calculating a 3-period moving average In general, the more recent periods used in the calculation receive more weight the numbers below used for the period 4 forecast are from the earlier data, with period 3 from that data getting a 50% weight, and earlier periods given progressively less weight. 42

Exponential smoothing a special case of the weighted moving average method (and the basis for nearly all of the more complex fixed model time series techniques) With exponential smoothing, only one weight the weight for the most recent actual value in the series is chosen That weight is known as a smoothing constant, and is referred to as Alpha (the first letter of the Greek alphabet) The weights of all previous values are computed automatically based on the weight for Alpha Forecast = Exponentially smoothed weighted average of the time series values 1 where: F t+1 = forecast of the time series for period t+1 Y t F t = actual value of the time series at period t = forecast of the time series for period t α = smoothing constant 0 1 43

Exponential smoothing why the funny name? Exponential smoothing was originally called an exponentially weighted moving average If you look at the formula (shown again below), you can see that the calculation i.e., your forecast is just the weighted average of two values the most recent actual value, and the model s forecast of that most recent actual value. BUT, due to the computation, the weights given to each prior period used in the calculation decrease at an exponential rate, yet they are still having an impact on the forecast for the next period. In other words, the forecast is a weighted average of ALL the previous values in the time series used in the calculation 44

Period Count An exponential smoothing calculation To initiate a exponential smoothing calculation, you need to fill in a number for the first forecast value (F t ). The first actual value in the series (Y 1 ) is typically used Period Y F Revenue ($000) Forecast 1 Jul-11 $17.00 n/a 2 Aug-11 $21.00 3 Sep-11 $19.00 4 Oct-11 $23.00 5 Nov-11 $18.00 6 Dec-11 $16.00 7 Jan-12 $20.00 8 Feb-12 $18.00 9 Mar-12 $22.00 10 Apr-12 $20.00 11 May-12 $15.00 12 Jun-12 $22.00 13 Jul-12 1 1 3 1 3 1 3 45

With alpha = 0.2, a forecast for period s 1 to 13, given actual revenue through period 12 Alpha (α) = -------------------------------> 0.2 Period Count Period Y F Revenue ($000) Forecast 1 Jul-11 $17.00 n/a 2 Aug-11 $21.00 $17.00 3 Sep-11 $19.00 $17.80 4 Oct-11 $23.00 $18.04 5 Nov-11 $18.00 $19.03 6 Dec-11 $16.00 $18.83 7 Jan-12 $20.00 $18.26 8 Feb-12 $18.00 $18.61 9 Mar-12 $22.00 $18.49 10 Apr-12 $20.00 $19.19 11 May-12 $15.00 $19.35 12 Jun-12 $22.00 $18.48 13 Jul-12 $19.18 46

Forecast performance with different Alpha values Note that higher Alpha values make the forecast very responsive to Noise (or a sudden Level change), with a lower Alpha smoothing the random fluctuation to a much larger degree $24.00 Forecast with exponential smoothing & different Alpha values $22.00 $20.00 $18.00 $16.00 $14.00 $12.00 $10.00 Revenue ($000) ES, Alpha = 0.2 LCC Municipal ES, Alpha Finance = Institute 0.8 47

The problems/questions facing you when using one of the previous smoothing techniques Moving average: How many historical periods should be included to calculate the average? Weighted moving average: Same question as above, plus: how much weight to give each period? Exponential smoothing: What should the value of Alpha be? (knowing that higher values of Alpha give more weight to more recent periods in determining the forecast, and that lower values give more importance to earlier periods) 48

A criteria for choosing one quantitative forecasting method (including the ones just discussed) over another You (generally) want to choose the forecasting method that will give you the best forecast i.e., minimize the amount of error between your forecast and the actual revenue value This will always involve qualitative considerations: judgment, trial and error, input from experts, detailed domain knowledge But in purely quantitative terms, you can also measure how well a particular method would have forecasted actual revenues in the past, and choose the one that minimized the forecast error. HUGE assumption when using the minimized past error criterion to pick a method: that past performance IS indicative of future results i.e., that a method that would have worked well in the past will work well in the future, and vice versa (i.e., methods that worked relatively poorly in the past will continue to perform poorly). 49

Measuring forecast error The forecast error for a particular time period (e t ) is simply the difference between the actual value (Y t ) and the forecasted value (F t ). Specifically: e t = Y t - F t There are three main measures of forecast error: 1. Mean Absolute Error (MAE) 2. Mean Square Error (MSE) 3. Mean Absolute Percentage Error (MAPE) The preferred forecasting method is the one that minimizes the amount of error as computed by one of these measures 50

Measuring forecast error for the previous exponential smoothing forecast Forecast (ES, Alpha = 0.2) Absolute Value of Forecast Error Squared Forecast Error Mean Absolute Percentage Error (MAPE) = 147.43 / 11 = ------------------------------------------------------------> 13.40 51 Absolute Value of the Percentage Error Period Count Period Revenue ($000) Forecast Error Percentage Error 1 Jul-11 $17.00 2 Aug-11 $21.00 $17.00 4.00 4.00 16.00 19.05 19.05 3 Sep-11 $19.00 $17.80 1.20 1.20 1.44 6.32 6.32 4 Oct-11 $23.00 $18.04 4.96 4.96 24.60 21.57 21.57 5 Nov-11 $18.00 $19.03-1.03 1.03 1.07-5.73 5.73 6 Dec-11 $16.00 $18.83-2.83 2.83 7.98-17.66 17.66 7 Jan-12 $20.00 $18.26 1.74 1.74 3.03 8.70 8.70 8 Feb-12 $18.00 $18.61-0.61 0.61 0.37-3.38 3.38 9 Mar-12 $22.00 $18.49 3.51 3.51 12.34 15.97 15.97 10 Apr-12 $20.00 $19.19 0.81 0.81 0.66 4.05 4.05 11 May-12 $15.00 $19.35-4.35 4.35 18.94-29.01 29.01 12 Jun-12 $22.00 $18.48 3.52 3.52 12.38 15.99 15.99 TOTALS 10.92 28.56 98.80 35.86 147.43 Mean Absolute Error (MAE) = 28.56 / 11 = ------------------> 2.60 Mean Square Error (MSE) = 98.80 / 11 = -------------------------------------------> 8.98

Excel s Solver tool can optimize Alpha (i.e., find the value of Alpha that minimizes the MAPE). So can dedicated forecasting software 52

Now to a Trend data example Revenue exhibit a trend when the data shows a distinctive direction (either increasing or decreasing) over time For such data, you generally want to straight-line your projection in a way consistent with the trend (assuming you don t anticipate any big changes), but different methods give you somewhat different straight lines Key question: Is the trend you re trying to forecast more global (e.g., a trend exhibited over all your historical data) or more local (do more recent periods show a different trend, and is this more instructive for your forecast)? No perfect answer here judgment and context are critical 53

Trend data the global trend and several local trends Year (Period) 1 2 3 4 5 6 7 8 9 10 Year 02-03 03-04 04-05 05-06 06-07 07-08 08-09 09-10 10-11 11-12 Revenue ($Mil.) $21.6 $22.9 $25.5 $21.9 $23.9 $27.5 $31.5 $29.7 $28.6 $31.4 $35.0 Revenue Data Series with Trend $30.0 $25.0 $20.0 $15.0 $10.0 $5.0 $0.0 02-03 03-04 04-05 05-06 06-07 07-08 08-09 09-10 10-11 11-12 Annual Revenue ($Mil.) 54

One straight line : the Average Incremental Percentage Change (APC) approach Simply the average percentage change using all (or some) of your historical time series data Year (Period) Year Annual Revenue ($Mil.) 1 02-03 $21.6 % Change 2 03-04 $22.9 6.02% 3 04-05 $25.5 11.35% 4 05-06 $21.9-14.12% 5 06-07 $23.9 9.13% 6 07-08 $27.5 15.06% 7 08-09 $31.5 14.55% 8 09-10 $29.7-5.71% 9 10-11 $28.6-3.70% F 11 = Ave. % Change = (6.02% + + 9.79%) / 9 = 4.71% then: Multiply the most recent period by 1 + Ave. % Change F 11 = $31.4 x (1.0471) = $32.9 10 11-12 $31.4 9.79% 11 $32.9 4.71% 55

Altering the previous straight line forecast based on judgment and context Based on answers to questions like: Where are we in the business cycle? What local information do I have that impacts the forecast? Are we making a conservative or aggressive forecast?, you can alter (either dampen or accelerate) the average growth rate for your forecast (4.71% is the average annual growth from previous slide) Historical Average % Growth Forecast Factor Forecasted % Growth 4.71% x 1.50 = 7.06% 4.71% x 1.25 = 5.88% 4.71% x 1.00 = 4.71% 4.71% x 0.75 = 3.53% 4.71% x 0.50 = 2.35% 4.71% x 0.25 = 1.18% 56

Another straight line the Compound Average Growth Rate (aka CAGR) approach Uses the compound growth rate formula - ending value divided by beginning value, all to the power of 1 divided by the number of growth periods, then minus 1 Year (Period) Year Annual Revenue ($Mil.) 1 02-03 $21.6 2 03-04 $22.9 3 04-05 $25.5 4 05-06 $21.9 5 06-07 $23.9 6 07-08 $27.5 7 08-09 $31.5 8 09-10 $29.7 9 10-11 $28.6 10 11-12 $31.4 CAGR 11 $32.7 4.24% F 11 = CAGR = [($31.4 / $21.6) (1/9) ]-1 = 4.24% then: Multiply the most recent period by 1 + CAGR F 11 = $31.4 x (1.0424) = $32.7 57

And yet another straight line least squares regression (using period count as the independent variable) You use the familiar equation of a straight line as a tool to forecast the Trend Forecast = Linear function of the independent varible t where: t = the time period F t = the linear trend forecast in period t (i.e., the estimate of the value of Y t in period t) b 0 b 1 = the Y-intercept of the linear trend line = the slope of the linear trend line 58

Excel (or any forecasting software) makes this very easy The forecast is that with each additional year, revenue will increase by $1.1million (the slope is 1.1) you are forecasting period 11 $35.0 Annual revenue with linear least squares regression (LSR) line $30.0 $25.0 y = 1.1x + 20.4 R² = 0.7648 $20.0 $15.0 F 11 = 20.4 + 1.1(11) = $32.5 $10.0 $5.0 $0.0 1 2 3 4 5 6 7 8 9 10 11 Annual Revenue ($Mil.) linear LSR December line 4, 2013 59

Notice how the APC, CAGR, and LSR straight lines diverge in the forecast out-years (and note that the APC and CAGR lines are not truly straight ) You could average the three forecasts, you could try to compare their past accuracy (the MAPE), but there is no perfect technique for choosing. It is, again, judgment and context that matters $41 $39 $37 $35 $33 $31 $29 $27 $25 Three "straght line" forecasts of revenue trend ($ Mil.) 11-12 Act. 12-13 For. 13-14 For. 14-15 For. 15-16 For. 16-17 For. APC $31.4 $32.9 $34.4 $36.0 $37.7 $39.5 CAGR $31.4 $32.7 $34.1 $35.6 $37.1 $38.7 LSR $31.4 $32.5 $33.6 $34.7 $35.8 $36.9 60

Now to data with Seasonality (but no Trend) You can see a repeating seasonal pattern over the five years: highest in Quarter 2, lowest in Quarter 4, and in between in Quarters 1 and 3 $180.0 $160.0 $140.0 $120.0 $100.0 $80.0 $60.0 $40.0 $20.0 Quarterly revenue with seasonality (no trend) (Revenue in $000) $0.0 1-1 1-2 1-3 1-4 2-1 2-2 2-3 2-4 3-1 3-2 3-3 3-4 4-1 4-2 4-3 4-4 5-1 5-2 5-3 5-4 Revenue ($000) 61

Being a bit more precise in demonstrating Seasonality Revenue by Quarter Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 1 $125.0 $153.0 $106.0 $88.0 2 $118.0 $161.0 $133.0 $102.0 3 $138.0 $144.0 $113.0 $80.0 4 $109.0 $137.0 $125.0 $109.0 5 $130.0 $165.0 $128.0 $96.0 % of Annual Revenue by Quarter Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 1 26.48% 32.42% 22.46% 18.64% 2 22.96% 31.32% 25.88% 19.84% 3 29.05% 30.32% 23.79% 16.84% 4 22.71% 28.54% 26.04% 22.71% 5 25.05% 31.79% 24.66% 18.50% 5 Yr. Average 25.25% 30.88% 24.57% 19.31% Min 22.71% 28.54% 22.46% 16.84% Max 29.05% 32.42% 26.04% 22.71% Quarterly Index (Actual Qtr. Revenue / Ave. Quarterly Revenue) Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 1 1.059 1.297 0.898 0.746 2 0.918 1.253 1.035 0.794 3 1.162 1.213 0.952 0.674 4 0.908 1.142 1.042 0.908 5 1.002 1.272 0.987 0.740 5 Yr. Average 1.010 1.235 0.983 0.772 Min Max 0.908 1.162 1.142 1.297 LCC 0.898 Municipal Finance 0.674Institute 1.042 December 4, 0.908 2013 62

Quantify and forecast Seasonality through seasonal dummy variables With this technique, each season is treated as a categorical (aka 0-1) variable, sometimes known as dummy variables You always include one less dummy variable than there are seasons (so, 4 quarters -1 = 3 dummy variables); the fourth quarter is forecasted when the other 3 seasonal variables = 0) Use multiple linear regression to find the value of b 0, b 1, b 2, and b 3 Forecast = Function of the independent seasonal dummy variables where: 1 2 3 3 b 0 b 1, b 2, b 3 1 2 = the Y-intercept = the multiple regression slopes for each quarter 1 1 0 1 2 0 3 1 3 0 63

A multiple regression forecast model & forecast of quarterly revenue (again, data exhibits Seasonality, but no Trend) Regression Statistics Multiple R 0.89 R Square 0.80 Adjusted R Square 0.76 Standard Error 11.32 Observations 20.00 ANOVA df Regression 3 Residual 16 Total 19 Q1 revenue = 95.0 + 29.0(1) + 57.0(0) + 26.0(0) = 124.0 Q2 revenue = 95.0 + 29.0(0) + 57.0(1) + 26.0(0) = 152.0 Q3 revenue = 95.0 + 29.0(0) + 57.0(0) + 26.0(1) = 121.0 Q4 revenue = 95.0 + 29.0(0) + 57.0(0) + 26.0(0) = 95.0 Coefficients Intercept 95.00 Qtr1 29.00 Qtr2 57.00 Qtr3 26.00 64

Finally, revenue data that exhibits both a Seasonal and Trend component $9.0 Quarterly revenue data over 4 years (16 periods) with seasonality and trend $8.0 $7.0 $6.0 $5.0 $4.0 $3.0 $2.0 $1.0 $0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Revenue ($000) 65

A multiple regression model to forecast a revenue time series with both Seasonality and Trend Same dummy variable structure as the previous model (for the Seasonality), but now you add the time period as an additional independent variable to incorporate the Trend Use multiple linear regression to find the value of b 0, b 1, b 2, b 3 and b 4 Forecast = Function of the independent seasonal dummy variables & time 1 2 3 3 where: F t b 0 b 1, b 2, b 3, b 4 1 2 = forecast in period t = the Y-intercept = the multiple regression slopes for each quarter and for time 1 1 0 1 2 0 3 1 3 0 t = the time period 66

Forecasting periods 17 (Quarter 1, Year 5) & 18 (Quarter 2, Year 5) Regression Statistics Multiple R 0.99 R Square 0.98 Adjusted R Square 0.97 Standard Error 0.22 Observations 16.00 F 17 = 6.07 1.36(1) 2.03(0) 0.30(0) + 0.15(17) = 7.19 F 18 = 6.07 1.36(0) 2.03(1) 0.30(0) + 0.15(18) = 6.67 ANOVA df Regression 4 Residual 11 Total 15 Coefficients Intercept 6.07 Period 0.15 Qtr1-1.36 Qtr2-2.03 Qtr3-0.30 67

And there are many (dozens) more time series models to play with, if you dare double exponential smoothing, triple exponential smoothing, etc. E.g., The Holt-Winters model is a form of triple exponential smoothing you create smoothed forecasts for the time series Level, Trend, and Seasonality (with three separate smoothing constants), and then combine those separate forecasts into a single forecasted value Forecast = Exponentially smoothed weighted average of the time series Level, Trend, and Seasonality 1 1 1 where: L = Level T = Trend SA t = Seasonal Adjustment for period t C = Cycle length of the seasonable pattern α = smoothing constant for Level β = smoothing constant for Trend γ = smoothing constant for Seasonaility 0 1 0 1 0 1 m = number of periods into the future to forecast 68

But before we move on Experts agree: Simple is better! (when it comes to choosing a forecasting technique). This always makes me feel better During the 1970s, one of the authors, a statistician who was working in a business school, realized that executives were deeply preoccupied with forecasting. The statistician was concerned that practitioners were making their forecasts without the benefit of the latest, most theoretically sophisticated methods. Instead, they seemed to prefer simpler techniques, which they could at least explain to their bosses. And so the statistician decided to teach them a lesson. He embarked on a research project that would demonstrate the superiority of the latest statistical techniques To the theoreticians chagrin, the practitioners simple techniques turned out to be more accurate than their own statistically sophisticated methods In the wake of his embarrassment, the statistician searched for a way to explain why that was so. His rationale: Complex models try to find nonexistent patterns in past data; simple models ignore such patterns and just extrapolate trends. Over the years,the same empirical truth came back each time: Simple statistical models are better at forecasting than complex ones. Spyros Makridakis, Why Forecasts Fail. What to Do Instead MIT Sloan Management Review, Winter 2010 69

Time series forecasting method selection guidelines A summary matrix of what I ve just put you through Revenue Time Series Characteristics Stationary (Level and Noise only, little/ no Trend or Seasonality) Time Series Forecasting Technique(s) * Moving Average * Weighted Moving Average * Exponential Smoothing Trend (no Seasonality) * Trend forecast (APC, CAGR) * Linear least squares regression (LSR) * Double Exponential Smoothing (Level and Trend) Seasonality (no Trend) * Regression w/ seasonal dummy variables * Seasonal indexes & averages Trend and Seasonality Long-Range Forecasting (2+ years) * Regression w/ seasonal dummy variables & time * Triple Exponential Smoothing (Level, Trend, Seasonality) * Trend forecast (APC, CAGR), dampened or accelerated 70

Summing Up & Concluding Thoughts LLC Municipal Finance Institute CLIENT LOGO HERE

Invest the time necessary to learn (or re-learn) the basic time series forecasting techniques While there are a few equations to understand and some notation to conquer, most quantitative time series techniques can be learned and applied relatively quickly & without too much pain Excel (or a dedicated forecasting software) does much of the heavy lifting 72

Create a forecasting sandbox to experiment with different forecasting techniques Maintain & update a clean data series for your major revenue sources, and compare forecast results (with MAPE ) with different forecasting methods & data time intervals $1,800,000 Property Tax Revenue by Quarter - Last Four Years - Talk about Seasonality (and a sprinkle of Trend)!! $1,600,000 $1,400,000 $1,200,000 $1,000,000 $800,000 $600,000 $400,000 $200,000 $0 35.00% Revenue Change in Assessed Value & Change in Property Tax - Last Nine Years - A Surprisingly Bad Fit!!! 30.00% 25.00% 20.00% y = 1.7856x - 0.0234 R² = 0.3234 % Change - Property Tax 15.00% 10.00% 5.00% LCC Municipal 0.00% Finance Institute -5.00% 73-2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% % Change - Assessed Value

Integrate a quantitative revenue forecast within your organization s overall forecasting process It is likely that you/ your organization has a process for implementing revenue forecasts (i.e., every quarter, at mid-year, as part of the annual budget) Scene from Tweak 3 the boss expresses his thoughts on my revenue forecast It may be that a quantitativebased forecasting method is just one part of that process (in my City, the quantitative forecast goes through several tweaks before going live ) Forecast w/ Quantitative Method(s) Tweak 1: Tap Domain Knowledge **********Qualitative Tweaks********** Tweak 2: Accuracy or Conservatism Tweak 3: Meeting with Boss Final Revenue Forecast 74

THANK YOU VERY MUCH! All questions, comments, and critiques welcome LLC Municipal Finance Institute CLIENT LOGO HERE