quantitative methods applied to taxes forecasting

TERM PROJECT TEAM 5 SOPHIE MICHELOT MICKENSON PIERRE RYAN GARRINGER

QMB6603 QUANTITATIVE METHODS IN BUSINESS

SUMMARY

Introduction p.3 Data & modeling p.5 Formulation & results – Forecating p.9 Formulation & results – Decision analysis p.17 Conclusion p.25

3

INTRODUCTION

Explain the problem undertaken its significance

The problem we are undertaking is estimating more accurate future property tax expenses for

Barron Collier Company (BCC). Currently, at BCC property tax expenses are projected simply using

last year’s data. Since the company owns several hundred properties we will use the forecasting OR

method to try and forecast more accurate short term property tax figures. Once we have conducted

a forecasting projection we will be constructing a decision analysis to help identify a

recommendation for future property investments, through a property tax expense approach.

Explain the benefits of the project to the business where data is collected.

1. Breakdown in property category and in depth look at each property owned.

2. Forecasting future values for most properties.

3. Analysis to provide recommendations for future transactions.

Breakdown in Property Category

We will be focusing our analysis in residential and commercial type properties for our project. Once

we have allocated a property to a category we then further break down the category by market

value. Since we are analyzing a significant amount of properties we will be able to identify if any

outliers are present. If we see that a property is over or under taxed by a significant margin we can

recommend the company to take a further look at property to make sure they are being tax

accurately.

Forecasting future values for most properties We will be analyzing a lot of BCC property portfolio within the residential and commercial category

and it will be able to identify trends with individual properties and categories. Since we are

conducting a full scale analysis of the portfolio we hope we will be able to forecast values for most

of the properties within these categories.

Analysis to Provide Recommendations for Future Transactions

Once we have concluded a forecasting model we will be conducting a decision analysis. We will be utilizing current Collier County figures to produce a decision tree and hope that we provide BCC

4

with a recommendation to for future purchases if the company is strictly considering property tax

expenses for the purchase of a property.

Explain the management science (operations research) method used to formulate the problem such as

(linear programming, transportation/assignment, project management, waiting line, simulation,

decision analysis, etc.).

First method used : Forecasting analysis

Time series methods : moving average (3 periods selected : 3 months, 5 months, and

12 months), exponential smoothing (2 smoothing constants selected)

Linear trend : least squares calculations

Forecast accuracy analysis : MAD and MAPD

Second method : decision analysis

Determining factors influencing real estate investments

Establishing probabilities and payoffs

Building the decision tree

Fill out the below table showing the tasks you went thru to accomplish this project, and then write the

name of the group member who worked on the task, and tome spent. –Add more rows or change the

names of specific tasks as applies to your project.

PROJECT TASKS GROUP MEMBER

ASSIGNED/WORKED

%TIME/EFFORT

SPENT

Finding the problem Ryan 5%

Contact w/ Industry Ryan 5%

Data Collection Ryan, Mickenson, Sophie 15%

Data Analysis Ryan, Mickenson, Sophie 5%

Forecasting calculations Mickenson, Sophie 25%

Decision analysis Sophie 10%

Interpretation of results Ryan, Mickenson, Sophie 10%

Report Writing Ryan, Mickenson 20%

Power Point Presentation Sophie 5%

5

DATA $ MODELING

Describe the business you worked with. Their product/service, size, number of employees, sales volume,

location.

The business we worked with was Barron Collier Companies located in Naples, Florida. They are a

private family company that has investment interest in real estate, agriculture, minerals and other

resources. Along with its major focuses in different investment fields the company provides other

services such as but not limited to: engineering, title insurance company, environmental services

property management, etc. The total number of employees is within the 50-100 range at the

corporate office.

Describe clearly the data, assumptions, terms used, and how the data is collected. Describe your

data collection methodology.

The data we collected was through BCC and the Collier County Property Appraisers/Collier County

Tax Collector website. BCC provided a selective amount from their vast portfolio, in which we used

all the residential and commercial properties for our project. Once we received the ID numbers

associated with BCC owned properties we were able to start our property tax search. On the county

websites we received the following data:

Market value- The value assessed to the property from 2008-2012. This is a summation of

land value and improved land value for the property. The property tax amount is derived

from the market value.

Property Taxes- We also found the previous five years of tax values (2008-2012) for five

different time periods (Nov.-March) per each year. The total taxes paid are determined by

the payment date. For our project we will be taking an average of the five pay periods per

each year.

Below is an example of our constructed data :

6

Below is a sample of a property card with tax information:

7

Describe the objective function, decision variables, constraints-. If forecasting or decision analysis,

explain how the methods are implemented, probability figures, and so on.

Our objective function is to utilize past property tax information to help forecast future property

tax values for residential and commercial properties. We hope to take our forecasting findings

along with Collier County commercial/residential property information and preform a decision

analysis to provide recommendations to BCC for future property investments.

Forecasting

To implement forecasting we utilized several different methods including:

a. Moving Average

b. Exponential Smoothing

c. Least Square Calculations

d. MAD and MAPD

Moving Average

We decided to calculate first the moving average. We used three different time periods: 3 months, 5

months, and 12 months. The aim is to determine short-term forecast of the amount of tax expenses

due to the next payment periods. For more relevance, we decided to split our data into several

categories. We differentiated residential properties and commercial properties. Then, we built sub-

categories based on the value of the property. Indeed, tax expenses are directly linked to the value

of the taxed property. Here is a summary of the categories used:

Residential properties Commercial properties

$0 – $99,999 $0 - $199,999

$100,000 & more $200,000 - $999,999

- $1,000,000 & more

Thus, for each category, we calculated the average of tax expenses per payment periods. The

payment periods are November, December, January, February, and March for the accounting years

2008 to 2012.

8

Exponential Smoothing

The Second method of time series analysis we used is exponential smoothing. Here, the most recent

data are weighted more than the older one. We calculate the exponential smoothing with two

smoothing constants, 0.3 and 0.5. We used the average amount of tax expenses per pay period.

Least Square Calculations

The third method is different as we present forecasting based on linear trend line. We used least squares

calculations to determine the regression model where tax expenses are related to time.

MAD and MAPD

We close the forecasting analysis by determining the accuracy of our calculations. We calculated the

mean absolute deviation (MAD) and the mean absolute percent deviation (MAPD) for our exponential

smoothing and linear trend line.

Decision analysis

The purpose of decision analysis is to provide efficient tools to help the decision making process.

We developed a decision tree with probabilities in order to understand which action on the real

estate market BCC could make so as to maximize its profit.

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FORMULATION & RESULTS – FORECASTING

After having calculated the different average of tax expenses per payment periods, we start the

calculations of the 3-month moving average. Here is the formula for moving average :

MAn = ( 𝐷𝑖𝑛𝑖=1 )/n

Here the details of calculations for the average amount of tax expenses per payment periods:

This is an example of the details of calculations of moving average for commercial properties ($0–

$99,999):

10

We use the same principle to calculate the 5-month and 12-month moving averages:

11

Moving average results

Property category Nov 12 tax expenses

3-month moving average

Nov 12 tax expenses 5-month moving

average

Commercial $0 - $199,999

$7,005.44 ↑ $6,934.68 ↓

Commercial $200,000 - $999,999

$14,673.05 ↑ $14,524.83 ↓

Commercial $1,000,000 & more

$71,244.08 ↑ $70,532.34 ↑

Residential $0 - $99,999

$1,007.93 ↑ $997.75 ↑

Residential $100,000 & more

$5,383.44 ↑ $5,310.33 ↑

12

Exponential smoothing

Here again we used the average amount of tax expenses per payment period. . Here is the

formula for exponential smoothing :

Fₓ₊₁ = αDₓ + ( 1 – α ) Fₓ

D corresponds to the average amount of tax expenses per payment periods and F relates to

the forecast. Here are the details of calculations :

13

Exponential smoothing (α = 0.5) results


Exponential smoothing


$7,051.32 ↑

Commercial $200,000 - $999,999

$14,712.55 ↑


$71,293.78 ↑


$1,009.56 ↑


$5,385.73 ↑

Least squares calculations

We present now other methods of forecasting based on linear trend line. We used least squares

calculations to determine the regression model where tax expenses are related to time. We

determined the different factors composing the least squares calculations :

b = 𝑥𝑦−𝑛𝑥𝑦

𝑥2−𝑛𝑥 2

a = 𝑦 - b𝑥

Here are our detailed calculations :

15

Linear trend results


Linear trend


$5,233.39 ↓

Commercial $200,000 - $999,999

$13,433.81 ↓


$74,490.36 ↑


$983.54 ↓


$3,834.76 ↓

Forecast accuracy

We close the forecasting analysis by determining the accuracy of our calculations. We calculated the

mean absolute deviation (MAD) and the mean absolute percent deviation (MAPD) for our

exponential smoothing and linear trend line. Here are the formulas we used to determine both MAD

and MAPD :

MAD = 𝐷−𝐹

𝑛

MAPD = 𝐷−𝐹

𝐷

Here are our detailed calculations :

16

Category Method MAD MAPD


Exponential smoothing α = 0.5

3,222.32 27.37%


2,831.02 24.05%

Commercial $100,000 - $999,999


616.71 3.44%


852.62 4.75%



2,358.39 3.31%


4,534.31 6.37%

Residential $0 – $99,999


18.82 1.77%


18.65 1.75%



489.24 6.42%


709.2 9.30%

17

ecision analysis helps, by developing tools like decision tree, to make decision in a situation

characterized by uncertainty. It consists in determining the decision alternatives, the states

of nature likely to occur under some probabilities, and the different payoffs resulting from

this succession of events. The aim of the decision analysis is to help BCC in its investments decisions

in the local real estate market. By building decision tree, we aim at providing efficient tools to make

the best decision at the highest profit for the company.

Investing in real estate is almost a science. A large number of factors can influence the

demand of properties for sale. We start our analysis with the principle that if the demand increases,

the prices on the real estate market will increase too, by a phenomenon of scarcity. Thus, the

situation of the market is favorable for the seller, but not for the buyer. On the contrary, if the

demand is low, the prices will tend to decrease by a phenomenon of the scarcity of demand, and an

abundance of offer. Here, the situation of the market is favorable for the buyer, and unfavorable for

the seller. Based on that, we get our two decision alternatives : should BCC invest or not invest ?

Now, we have to determine the states of nature.

According to the economic press and related sources of information, there are three major

factors influencing the demand on the real estate market. The first one in terms of importance is the

tax conditions. Thanks to the previous forecasting part, we have an accurate idea of what the tax

expenses will represent for BCC in the short-term future. If the tax expenses are likely to increase,

because of new tax rates or new fiscal laws for instance, the price and cost for the buyer are likely

to increase too. On the contrary, it can be a real advantage for the seller, who could be willing to

decrease its costs. The second factor is the mortgage conditions. It refers to the level of mortgage

rates, home loans, etc. If the mortgage rates increase, the situation becomes unfavorable for the

buyer, thus decreasing the demand. If the mortgage rates decrease, the demand will increase. The

third factor gathers the different elements defining the conditions of economic environment. We

will see these elements in a deeper analysis in the coming paragraphs.

We decided to conduct the decision analysis on one specific category defined in the

forecasting part : the residential properties with an estimated value between $0 and $99,999. We

will not tackle the commercial estate market as we consider that investing in that kind of property

corresponds to specific projects between BCC and external partners, and not necessarily to a pure

answer to market opportunity, as it is more the case with residential properties.

Establishing probabilities and payoffs

There are three decisions opened to BCC : invest, sell, or do nothing. We will refer to these decisions

in the decision tree as Invest, Sell, Status quo.

The states of nature are the conditions of the market, the conditions of mortgage, and the

conditions of tax expenses. We will refer to these states of nature in the decision tree as : Good/Bad

market conditions, Good/Bad mortgage conditions, and Tax increase/Tax decrease. We will first

describe how to establish the probabilities for each state of nature.

D FORMULATION & RESULTS – DECISION ANALYSIS

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Market conditions

We established earlier that the market conditions are determined by several factors. We

select four of them to forecast the fluctuation of the demand on the real estate market. The first

factor is the prices evolution of residential properties in Collier County. After a strong decrease

from 2007 to 2009, the prices increase slowly, and we thus bet on an increase of prices (source :

movoto.com). This situation is favorable for the seller, and unfavorable for the buyer. The second

factor is the demographic evolution. The number of inhabitants in Collier County is going to

increase at regular pace (source : US census bureau). This will have a positive impact on demand,

and thus contribute to an increase of price. The situation is favorable for the seller, and unfavorable

for the buyer. The third factor is the vacancy rate. The actual rate amounts 32.5% (source :

marconews.com). We bet on a decrease on this rate, which is quite high today. This will have an

impact on the scarcity of properties, thus increasing prices. The situation is favorable for the seller,

and unfavorable for the buyer. The fourth factor is the housing starts rate. The actual rate is 52.9%

(source : metrostudyreport.com). Analysts predict that this rate is going to increase. This means

that the number of available and new houses will increase. This situation is more favorable for the

buyer than for the seller. To conclude, we estimate that the market conditions will sustain the offer

of properties. We define the probabilities of this state of nature as follows :

Market conditions Probability

Good 0.6

Bad 0.4

Mortgage conditions

To establish the probabilities regarding mortgage conditions, we look at the mortgage rates. After a

certain decrease, the rate slightly increases. Nonetheless, we bet on a stabilization of the rate for the

coming months. We define the probabilities of this state of nature as follows :

Mortgage conditions Probability

Good 0.5

Bad 0.5

Tax conditions

To determine the probability of increase or decrease of the tax expenses, we use our results from

the forecasting part. The forecasts for the tax expenses linked to the residential properties with an

estimated land value between $0 and $99,999, based on the least squares calculations methods,

follows this scheme :

19

Period Tax expenses

1 983.5417941

2 980.6746253

3 977.8074566

4 974.9402878

5 972.073119

6 969.2059502

7 966.3387814

8 963.4716127

9 960.6044439

10 957.7372751

9706.395346

920

940

960

980

1000

1020

1040

1060

1080

1100

No

v 0

7

Jan

08

Mar

08

De

c 0

8

Feb

09

No

v 0

9

Jan

10

Mar

10

De

c 1

0

Feb

11

No

v 1

1

Jan

12

Mar

12

Actual

Linear trend

For the coming two years, the average tax expenses will amount approximately $9,706. The linear

trend seems to forecast a decrease in tax expenses.

We define the probabilities of this state of nature as follows :

Taxes Probability

Increase 0.3

Decrease 0.7

We can now give a first overview of the decision tree with the different probabilities defined.

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Now let’s determine the different payoffs. We start with the following assumptions. Let’s

consider that BCC owns one property of the category chosen (40-$99,999), and that it wants to

analyze the payoffs of the different decisions taken after two years. Thus, BCC has three decision

choices : invest and buy a new property with the same characteristics that the one it already owns,

sell this property, or do nothing. Notice that the property currently owned yields an average rent of

$886 per month. To determine the payoffs, we attribute an increase or decrease in dollar terms to

each states of nature that occurs. The following table describes the calculations of payoffs :

Decision Market conditions Mortgage conditions

Tax variation Payoff

Invest Good Average price of the property : $20,000 If the market conditions are good, we estimate that the demand is high, thus the prices are high too. We add $6,000 to the starting price.

=$26,000

Good If the mortgage conditions are good, we estimate that the demand is high, thus the prices are high too. We add $2,000 to the starting price.

=$28,000

Increase If the tax increases, we estimate that the amount of tax expenses for the two coming years will be $10,000.

= $38,000

The cost of purchase a new property is estimated to $38,000. However, BCC will get $21,264 ($886x12) of rent revenue from its current possession, plus another $21,264 from its new acquisition. BCC will also have to pay for the tax of its current property ($10,00). Thus, the final payoff will be $42,528 - $38,000 - $10,000

=-$5,472

Invest Good =$26,000

Good =$28,000

Decrease If the tax decreases, we estimate that the amount of tax expenses for the two coming years will be $9,706.

= $37,706

$42,528 - $37,706 - $9,706

=-$4,884


Bad If the mortgage conditions are bad, we estimate that the demand is low, thus the prices are lower. We substract $2,000 to the starting price.

=$24,000

Increase =$34,000

$42,528 - $34,000 - $10,000

=-$1,472


Bad =$24,000

Decrease =$33,706

$42,528 - $33,706 - $9,706

=$884

Invest Bad If the market conditions are bad, we estimate that the demand is low,

Good =$16,000

Increase =$26,000

$42,528 – $26,000 - $10,000

=$6,528

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Invest Bad If the market conditions are bad, we estimate that the demand is low, thus the prices are low too. We substract $6,000 to the starting price.

=$14,000

Good =$16,000

Increase =$26,000

$42,528 – $26,000 - $10,000

=$6,528

Invest Bad =$14,000

Good =$16,000

Decrease =$25,706

$42,528 - $25,706 - $9,706

=$7,116

Invest Bad =$14,000

Bad =$12,000

Increase =$22,000

$42,528 - $22,000 - $10,000

=$10,528

Invest Bad =$14,000

Bad =$12,000

Decrease =$21,706

$42,528 - $21,706 - $9,706

=$11,116

Sell Good =$26,000

Good =$28,000

Increase An increase in tax expenses can discourage the demand and thus decrease the prices. We estimate an impact of $2,000 on the selling price.

=$26,000

By selling its property, BCC gives up its rent revenues. Thus, the only got comes from the sale of the property.

=$26,000

Sell Good =$26,000

Good =$28,000

Decrease A decrease in tax expenses can encourage the demand and thus increase the prices. We estimate an impact of $2,000 on the selling price.

=$30,000

=$30,000

Sell Good =$26,000

Bad =$24,000

Increase =$22,000

=$22,000

Sell Good =$26,000

Bad =$24,000

Decrease =$26,000

=$26,000

Sell Bad =$14,000

Good =$16,000

Increase =$14,000

=$14,000

Sell Bad =$14,000

Good =$16,000

Decrease =$18,000

=$18,000

Sell Bad =$14,000

Bad =$12,000

Increase =$10,000

=$10,000

Sell Bad =$14,000

Bad =$12,000

Decrease =$14,000

=$14,000

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The expected value of the decision analysis is $22,000, which corresponds to the decision to sell the

property. This result refers to an analysis of revenue for the coming two years. The result could be

different for a longer period, but we can assume that it could also be less accurate as the conditions of

the market would have evolved. However, this analysis represents a good tool to help the decision

making.

Status quo Good By doing nothing, BCC gives up to act on the real estate market.

Thus, any changes in the market will not directly affect BCC.

Good By doing nothing, BCC gives up to act on the real estate

market. Thus, any changes in the market will not directly affect BCC.

Increase BCC chooses to keep its current property and thus

has to pay taxes. =-$10,000

With its current property, BCC earns $21,264 of rent revenue over the period. =$11,264

Status quo Good Good Decrease =-$9,706

=$11,558

Status quo Good Bad Increase =-$10,000

=$11,264

Status quo Good Bad Decrease =-$9,707

=$11,558

Status quo Bad Good Increase =-$10,000

=$11,264

Status quo Bad Good Decrease =-$9,706

=$11,558

Status quo Bad Bad Increase =-$10,000

=$11,264

Status quo Bad Bad Decrease =-$9,706

=$11,558

Here is an overview of the decision tree with probabilities, payoffs, and expected values : (next

page)

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CONCLUSION

1. Commercial Properties

After calculating the 3-months, 5-months, and 12-months moving average of the commercial

properties, we found out that the all three moving average forecasts smooth out the variability in

the actual average. However the 3-months moving average period was the closest to the real

average. This is because the 5-months and 12-months moving average consider older data than the

3-months moving average. Moving average does not react well to variations that occur for a reason,

therefore, it will be best to use the moving average for short-term forecasting rather than

forecasting that is too far into the future. We also found out that there is a trend in the movement of

the taxes throughout the five years. Therefore, we can confidently predict that what happened in

the past will happen again.

The second calculation we performed was the exponential smoothing. We found out that alpha =

0.5 was actually closer to the actual average than alpha = 0.3. And also, we found out that there is a

linear trend in the forecast and actual average. This information is crucial to the client, because they

can predict that what happens in the past will eventually happen in the future. For example if tax

increases in a month last year, in the current year, one can expect taxes to increase throughout the

same period.

The last method we use is the linear trend line method. This method clearly shows that there is a

linear trend which closely reflects the actual data.

And finally, we determined the forecast accuracy of our data by computing the MAD and MAPD for

all our forecasts. For the exponential smoothing forecast, alpha = 0.5 has a MAPD of 27.37 %, while

alpha = 0.3 has an MAPD of 29.90%, and the linear trend has an MAPD of 24.05%. The smallest

MAPD of those three forecasts is the linear trend with an MAPD of 24.05%, therefore we would

recommend that BCC uses the linear trend method to forecast future expenses for the short term.

2. Residential Properties

We performed a moving average forecast for the residential properties and the same results as the

commercial results. The 3-months moving average was the closest to the actual average, which

means that the moving average will work best if it is performed in the short run as opposed to the

long run.

The exponential smoothing also yields the same result as the results for the commercial properties.

We found out that the forecast using alpha = 0.5 was closer to the actual average than the forecast

using alpha = 0.3, and also that there is a trend in the results meaning that what happened in one

period the previous year will also happen in the same period during the upcoming year.

26

The last forecasting method we used was the linear trend method. We found out that there is a

linear trend in the data which clearly reflects the actual data.

And finally, we also calculated the MAD and the MAPD of all our forecasts. We found out that the

MAPD for our exponential smoothing was 8.25 for alpha = 0.3 and 5.61 for alpha = 0.5, and the

MAPD for the linear trend line was 8.15%. The lowest MAPD of all the forecasts is the exponential

smoothing alpha = 0.5 with an MAPD of 5.61%. Therefore we would recommend BCC to use

exponential smoothing with alpha = 0.5 to forecast its future tax expenses for the short-term

period.

To validate the results, we would compare each one of them to the actual data and determine what

percentage did we deviate from the actual data. Currently, BCC uses the previous year’s data to

forecast future expenses, but in our methodology, we use data from the previous five years. In the

future, instead of using the data of previous year, BCC could use economic conditions and attempt

to predict their future expenses instead of just looking at the trends, and they need to analyze the

effects of certain conditions on the total value of the properties. If we had to redo this project, we

would definitely pick another topic. Forecasting and decision analysis are two extremely

complicated methods to use and comprehend. We definitely benefited greatly from doing this

project. It helped us understand the different methods of forecasting as well as how to apply

decision analysis to a real-world situation.

quantitative methods applied to taxes forecasting

Business

property category

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property tax search

future property investments

bcc property tax expenses

property tax expense

lot of bcc property

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