This session was the first ever math-based session for advanced revenue management practitioners to discuss computational challenges in revenue management and customer data analytics. Pricing isn’t just about looking at data on spreadsheets. You actually have to do math. Complex math. Well, you have to do complex math if your revenue management software system doesn’t bring the science and analytics together.

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In-depth Analytics of
Pricing Discovery
Donald Davidoff, D2 Demand Solutions
Annie Laurie McCulloh, Rainmaker LRO
Rich Hughes, RealPage

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Agenda
1. Forecasting
• Forecasting Model Options
• Principles of Forecasting
• Forecasting Methods
• Time Series Models
• Forecast Accuracy
2. Assessing Amenity Values
3. Procedurally Generated Content
4. Analyzing Performance
• Methodology
• Revenue Performance
• Intangible Benefits

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Forecast Model Options and Design
Theoretical Probability:
Coin:
P(heads) = 1 head on a 2 sided coin
= 1 out of 2
=
1
2
Dice:
P(6) = 1 side out of 6 sides of a die
(1,2,3,4,5,6)
= 1 out of 6
=
1
6
Both Heads and a 6 together:
= P(heads) * P(6)
=
1
2
*
1
6
=
1
12
or 8.3%
Experimental Probability:
Identify a trial:
• One trial consists of flipping a coin once and
rolling a die once
• Conduct 25 trials and record your data in the
table below:
Question: You are handed one die and one quarter. What’s the probability of rolling a 6
and getting a heads at the same time?
Legend:
Coin: H = Heads, T = Tails
Die: 1,2,3,4,5,6 = number rolled on the die
Head & 6: Y : Heads & 6 occurred, N: All other results
Results:
1 trial out of 25 resulted in a heads and a 6
= 1/25
Therefore,
P(heads,6) = 4%
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy
Trial 1 2 3 4 5 6 7 8 9 10 11 12 13
Coin T T T H T T T T H T T T T
Die 4 1 1 6 2 5 5 6 5 1 1 5 6
Head & 6 N N N Y N N N N N N N N N
Trial 14 15 16 17 18 19 20 21 22 23 24 25
Coin H H H H T H H T T H H H
Die 2 1 2 1 5 1 2 3 2 1 4 2
Head & 6 N N N N N N N N N N N N
ResultsResults

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Principles of Forecasting
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy
Grouping
of Data
Forecast
Accuracy
Quantity
of Data
Forecast
Accuracy
Recent
Data
Forecast
Accuracy
• Forecasts contain risk and uncertainty - they are rarely perfect
• Some characteristics of the data used to forecast can improve accuracy
• Forecasts should be systematically evaluated over time for accuracy

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Principle of Aggregating Data
• Since many times we must forecast off of sparse data, what are
some of the ways we aggregate data in our revenue management
forecasts?
- Lease type – Conventional New & Renewal, Affordable, Student, etc.
- Lead Source – ILS Vendor, Craig’s List, Property Website, Outdoor, etc.
- Unit types
- Lease terms
- Week types
- Move-in weeks
- Clustered communities
- Market
• Need “enough” observations/transactions to have predictive
capabilities
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Forecasting Methods
• Qualitative Methods
- Educated guesses based
on human judgement and
opinion
- Subjective and non-
mathematical
 Executive Opinion
 Market Research
 Delphi Method
• Quantitative Methods
- Based on mathematics
- Consistent and objective
- Only as good as the data
on which they are based
 Time Series Models
 Causal Models
 Associative Models
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Time Series Model
• Many of the forecasts used in revenue management
leverage time series models
• Time series models use historical data as the basis for
estimating future outcomes
- Moving average
- Weighted moving average
- Kalman filtering
- Exponential smoothing
- Autoregressive moving average (ARMA)
- Autoregressive integrated moving average (ARIMA)
- Extrapolation
- Linear prediction
- Trend estimation
- Growth curve
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Time Series Examples
Uniform distribution
between 1 and 2
Increasing trend
Quadratic growth
trend
Seasonal Model
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Time Series Problem - Seasonality
• A community manager must develop forecasts for the next
year’s quarterly or seasonal leads.
• The community has collected quarterly lead data for the
past two years.
• She has forecast total leads for next year to be 9000.
• What is the forecast for each quarter or season of next
year?
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Time Series Problem
2-period Moving Average
Quarter 2014 ‘14 Index 2015 ’15 Index Avg.
Index
2016
Fall 1900 ? 1900 ? ? ?
Winter 1400 ? 1700 ? ? ?
Spring 2300 ? 2200 ? ? ?
Summer 2400 ? 2600 ? ? ?
Total 8000 8400 9000
Average ? ? ?
=8000/42000
=1900/20000.95 =1900/21000.90
=8400/4 =9000/422502100
=(0.95+0.90)/20.925 =2250*.9252081
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy
1. Calculate the average leads per season for each of the past two years
2. Calculate a seasonal index for each season of the year
3. Average the indices by season
4. Calculate the average leads per season for next year by using total
forecast leads for the next year divided by the number of seasons
5. Multiply next year’s average seasonal leads by each average seasonal
index to get forecasted leads per season

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Time Series Problem
Solution
Quarter 2014 ‘14 Index 2015 ’15 Index Avg.
Index
2016
Fall 1900 0.95 1900 0.90 0.925 2081
Winter 1400 0.70 1700 0.81 0.755 1699
Spring 2300 1.15 2200 1.05 1.100 2475
Summer 2400 1.20 2600 1.24 1.220 2745
Total 8000 8400 9000
Average 2000 2100 2250
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
SeasonalityFactor
Week
1-Bedroom Seasonality Factors
1X1
How this applies?
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Measuring Forecasting Accuracy
• Forecasts are never perfect
• The forecast error is the difference between the actual value and the forecast value for
the corresponding period
Et = At - Ft
where E is the forecast error at period t, A is the actual value at period t, and F is the
forecast for period t.
• Measures of aggregate error:
- Mean Absolute Error (MAE) or Mean Absolute Deviation (MAD)
- Mean Absolute Percentage Error (MAPE) or Mean Absolute Percentage Deviation
(MAPD)
- Mean Squared Error (MSE) or Mean Squared Prediction Error (MSPE)
- Cumulative Forecast Error (CFE)
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Forecast Accuracy Problem
• An asset manager is measuring the accuracy of
her forecasts using data from the past 5
Thursdays.
• Average difference = (4+6-3-6-2)/5 = -0.2
• Is this an accurate forecast?
Forecast Actual Difference
43 39 4
40 34 6
34 37 -3
36 42 -6
38 40 -2
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Forecast Actual Difference
Absolute
Difference
43 39 4 4
40 34 6 6
34 37 -3 3
36 42 -6 6
38 40 -2 2
MAE 4.2
MAE: Mean Absolute Error
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Forecast Actual Difference
Absolute
Difference
% of Actual
43 39 4 4 10.3%
40 34 6 6 17.6%
34 37 -3 3 8.1%
36 42 -6 6 14.3%
38 40 -2 2 5.0%
MAPE 11.1%
MAPE: Mean Absolute Percent Error
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Week Type Unit Category
Lease Term
Category
Move-in Week Etc.
Level of Granularity
Number of Days Out
Measure accuracy where the forecast has the best potential for performing well
Measure accuracy with appropriate lead time so that your yielding decisions will have value
Too far out:
- Decisions mean little
- Typically less
accurate
Too close in:
- Decisions made
too late
Key Questions when
Measuring Accuracy
1. Forecasting Model Options
2. Principles of Forecasting
3. Forecasting Methods
4. Time Series Models
5. Forecast Accuracy

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Using T-tests to Assess Unit Amenity Values
• The Problem: how do we know whether our unit
amenities are priced too high or too low (or just right)?
• The Solution: Use Days on Market (DOM) as a proxy for
market response and assess how statistically significantly
different the average DOM is for leases with versus
without the amenity
• Application: Any individual or bundle of unit-level
amenities including renovations

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Example 1
T-test examines whether 2
samples are different;
commonly used with
small sample sizes
 First two parameters are the
ranges of the two samples
 Third parameter is set to 1
for one-tailed distribution
and 2 for two-tailed
 Fourth parameter is set to 1
for paired data, 2 for equal
variance and 3 for unequal
variance
Conclusion: PRICED RIGHT

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Example 2
 Only 3 bundles can be analyzed
 BA partial and Kitchen partial (26)
 BA full and Kitchen full (65)
 No renovations (12)
BA Minor BA Partial Kitchen Appliance Kitchen Partial BA Full Kitchen Full LseCount AvgDOM
50 75 150 175 No Amenity No Amenity 1 1.0
No Amenity 75 150 175 No Amenity No Amenity 1 33.0
No Amenity No Amenity No Amenity 1 30.0
No Amenity 175 No Amenity No Amenity 26 43.5
No Amenity 150 No Amenity No Amenity No Amenity 2 9.5
No Amenity 175 250 No Amenity 1 16.0
No Amenity No Amenity 2 44.5
No Amenity 250 450 65 78.4
No Amenity No Amenity 12 46.8
Grand Total 111 62.9

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Example 2
 Conclusion: PARTIALS PRICED OK; FULL RENO PRICED TOO HIGH

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0,1,1,2,3,5,8,13,21,34,55

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Old
Data
Rules
New
Data

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"the map is not the territory"
“...no matter how many instances of white
swans we may have observed, this does not
justify the conclusion that all swans are white.”

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0%
2%
4%
6%
8%
10%
12%
14%
16%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Probability
Sum of 3 Dice
Actual
Distribution
Mean = 10.5 Standard deviation = 2.96

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Distribution Kolmogorov Smirnov Statistic Chi-Squared Statistic Parameters
Dagum 0.03621 0.37197 k=0.34965 alpha=3.3322 beta=131.63
Burr 0.04574 0.8922 k=5.2827 a=1.4273 b=289.97
Weibull 0.13813 7.634 a=1.259 b=100.07
Perason6 0.17274 18.404 alpha1=1.553 alpha2=35.978 beta=2091.8
Average Days vacant

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Distribution Kolmogorov Smirnov Statistic Chi-Squared Statistic Parameters
Burr 0.06249 0.1269 k=146.87 alpha=15.96 beta=127.03
Weibull 0.08687 0.03283 alpha=13.597 beta=92.483
Gumbel Min 0.0707 0.08378 sigma=5.5687 mu=93.014
Pert 0.07139 0.16172 m=95.711 a=57.213 b=100.43
Occupancy

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Analyzing Performance:
Measurement Methodology
1. Methodology
2. Performance Results
3. Intangible Benefits
1. Measure “Rental Revenue”
• Account for both rent and occupancy
- Method 1 – Month End Financials
- Method 2 – RPU (Revenue per Unit)
2. Incorporate a Benchmark
• Before and After - Pre vs. Post Revenue Management
• 3rd party “market” data
• Test vs. Control Data Set
3. Measure over Time
• Revenue management is a marathon, not a sprint
4. Account for the Intangibles

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Method 1 - Month End Financials
1. Methodology
2. Performance Results
3. Intangible Benefits
• Measure the month end revenue line items that Rev Mgmt can directly
impact:
› Market Rent
› Vacancy Loss
› Loss & Gain to Lease
› Concessions – New & Renewal
› Month to Month and Short Term Lease Fees
• Don’t incorporate line items that Rev Mgmt cannot control like Bad Debt,
Write Offs, etc…
July Aug Sept Oct Nov Dec Jan Feb Mar Apr May June Baseline July Aug Sept
Market Rent $883,825 $884,575 $884,575 $884,575 $884,575 $884,635 $884,635 $885,850 $885,050 $885,050 $885,075 $878,940 $878,955 $878,980 $878,965
Vacancy Loss ($100,575) ($105,145) ($113,045) ($124,755) ($129,710) ($138,758) ($145,801) ($148,955) ($152,526) ($132,854) ($116,498) ($112,907) ($101,941) ($97,407) ($94,924)
Loss to Lease ($16,966) ($15,784) ($14,793) ($13,518) ($12,378) ($11,836) ($11,221) ($11,301) ($10,686) ($10,975) ($10,126) ($10,084) ($9,965) ($10,897) ($14,484)
Gain to Lease $110 $125 $105 $230 $100 $100 $110 $135 $135 $110 $110 $5,890 $5,885 $6,413 $6,250
Concessions - Renewals ($31,629) ($34,866) ($36,552) ($14,469) ($10,343) ($13,925) ($12,010) ($3,110) ($7,820) ($17,015) ($22,490) ($19,290) ($31,230) ($24,030) ($34,430)
Concessions ($11,412) ($12,225) ($18,875) ($11,826) ($19,769) ($22,280) ($19,241) ($4,880) ($6,440) ($21,082) ($15,620) ($19,947) ($22,206) ($19,699) ($15,447)
Month to Month Fee $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0
Short Term Monthly Fee $775 $1,115 $64 $701 $843 $835 $706 $590 $500 $400 $675 $770 $970 $990 $1,463
Total Rev $724,128 $717,795 $701,479 $720,938 $713,318 $698,771 $697,178 $718,329 $708,213 $703,634 $721,126 $723,372 $712,357 $720,468 $734,350 $727,393
YOY -0.5% 2.3% 3.7%

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Method 2 – Revenue per Unit (RPU)
1. Methodology
2. Performance Results
3. Intangible Benefits

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Analyzing Performance:
Incorporate a Benchmark
1. Methodology
2. Performance Results
3. Intangible Benefits
86%
88%
90%
92%
94%
96%
98%
100%
102%
Baseline July Aug Sept Oct
%ofIndex
Test (Rev Mgmt) vs. Control (No Rev Mgmt)
Avg Net Rental Income - Test (Rev Mgmt)
Avg Net Rental Income - Control (No Rev Mgmt)

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Analyzing Performance:
Account for the Intangibles
1. Methodology
2. Performance Results
3. Intangible Benefits
• Steady pricing with measured market
response
• Strategic approach to pricing with more
attention and visibility to amenity-based
pricing
• Better, more consistent insight into
competitive market space
• Movement away from market rent and
toward net effective pricing

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Thank you!
donald@d2demand.com
amcculloh@letitrain.com
Rich.Hughes@RealPage.com