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Applied Business StatisticsApplied Business StatisticsCase studiesCase studies

A statistical model for real estate marketA statistical model for real estate market

Mauro BufanoMauro Bufano

Risk Management – Risk Management – BancaBanca Mediolanum Spa Mediolanum Spa

Real estate marketReal estate market Recently, a statistical analysis on the real estate market Recently, a statistical analysis on the real estate market

has grown its importance, due to several reasons:has grown its importance, due to several reasons: Many asset management companies have “real estate Many asset management companies have “real estate

portfolios” to manage, therefore buildings have to be treated as portfolios” to manage, therefore buildings have to be treated as stocks or bondsstocks or bonds

Among the assets of banks and insurances we find real estate Among the assets of banks and insurances we find real estate asset, whose yields and income have to be properly estimated asset, whose yields and income have to be properly estimated for for asset & liabilities management asset & liabilities management (ALM) purposes(ALM) purposes

The guarantee of many loans (especially mortgage loans) is The guarantee of many loans (especially mortgage loans) is constituted by real estate, whose depreciation could impact constituted by real estate, whose depreciation could impact heavily on bank profits (because LGD grows)heavily on bank profits (because LGD grows)

The origin of many economic crises (e.g. 2007-2009) has been The origin of many economic crises (e.g. 2007-2009) has been the excessive overvaluation of real estate goods!the excessive overvaluation of real estate goods!

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Anyway, there are many differences between traditional Anyway, there are many differences between traditional financial assets and real estate assets:financial assets and real estate assets:

Real estate assets are more “illiquid”, i.e. it’s generally difficult to Real estate assets are more “illiquid”, i.e. it’s generally difficult to buy and sell them quicklybuy and sell them quickly

Sometimes we don’t find a “fair” market price (especially for Sometimes we don’t find a “fair” market price (especially for institutional buildings)institutional buildings)

Negotiation times are very long (months, or years...)Negotiation times are very long (months, or years...) It’s difficult to estimate a real “yield” of a real estate asset, It’s difficult to estimate a real “yield” of a real estate asset,

because of hidden costs:because of hidden costs:• Maintenance costsMaintenance costs

• VacanciesVacancies

• High transaction costsHigh transaction costs

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Real estate marketReal estate market

Real estate market – international focusReal estate market – international focus

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We can see how in the period 2000-2008 in some countries (e.g. Great Britain and Spain) house prices have doubled, causing a bubble explosion in 2008-09 that have seriously damaged their economies. In Italy the growth has been more constant, while Germany has experienced a house price deflation

A statistical model for Italian real estate marketA statistical model for Italian real estate market

In next slides we will show a statistical model for italian In next slides we will show a statistical model for italian real estate market, used for risk management purposes real estate market, used for risk management purposes in a real estate portfolioin a real estate portfolio

Following market best practice, real estate market has Following market best practice, real estate market has been divided into 4 segmentsbeen divided into 4 segments1.1. ResidentialResidential

2.2. OfficesOffices

3.3. CommercialCommercial

4.4. Logistic Logistic

The model covers segments 1-3The model covers segments 1-3

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The variables analyzedThe variables analyzed Italian real estate prices (source: Nomisma) with semi-annual Italian real estate prices (source: Nomisma) with semi-annual

frequencyfrequency Macro-economic variables, also with semi-annual frequency. Macro-Macro-economic variables, also with semi-annual frequency. Macro-

economic variables have been lagged 6 months and 12 months (t-1 economic variables have been lagged 6 months and 12 months (t-1 and t-2, respectively), to take into account the rigidity of real estate and t-2, respectively), to take into account the rigidity of real estate market:market:

• GDP GDP growthgrowth rate rate

• Inflation rateInflation rate

• Stock market returnsStock market returns

• M1 growth rateM1 growth rate

• Real estate investments’ growth rateReal estate investments’ growth rate

• Private consumption growth ratePrivate consumption growth rate

• Euribor 3 monthsEuribor 3 months

• 10 year government bonds10 year government bonds

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Descriptive analysisDescriptive analysis The descriptive analysis of variables is preliminary to the The descriptive analysis of variables is preliminary to the

construction of the model, and it has been conducted on construction of the model, and it has been conducted on ““raw” dataraw” data percentage returnspercentage returns log-returnslog-returns

Several tests have been conductedSeveral tests have been conducted Stationarity analysis (Dickey-Fuller test)Stationarity analysis (Dickey-Fuller test) Normality tests (Kolmogorov Smirnov test)Normality tests (Kolmogorov Smirnov test) Analysis of correlations (in order to avoid multicollinearity)Analysis of correlations (in order to avoid multicollinearity)

In the construction of the model, variables have been transformed in In the construction of the model, variables have been transformed in percentage returns, due to the presence of trends (non-percentage returns, due to the presence of trends (non-stationarity) in row variablesstationarity) in row variables

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Principal component analysisPrincipal component analysis In order to choose the variables to include in the model, In order to choose the variables to include in the model,

a principal component analysis (PCA) has been a principal component analysis (PCA) has been conducted:conducted:

We compute the correlation matrix of the time-series of real We compute the correlation matrix of the time-series of real estate returnsestate returns

The eigenvectors of the correlation matrix represents the The eigenvectors of the correlation matrix represents the principal componentsprincipal components

The percentage of variance explained for each eigenvectors is The percentage of variance explained for each eigenvectors is given by the corresponding diagonal values of the eigenvalues’ given by the corresponding diagonal values of the eigenvalues’ matrixmatrix

In our model, the first 4 components explain 84.3% of the In our model, the first 4 components explain 84.3% of the variance of the real estate marketvariance of the real estate market

Then, we select the macro-economic variables that are more Then, we select the macro-economic variables that are more correlated with the principal componentscorrelated with the principal components

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Models’ development – stepwise Models’ development – stepwise regressionregression

The statistical model used here is a linear regression, The statistical model used here is a linear regression, whose parameters have been estimated with ordinary whose parameters have been estimated with ordinary least squares (OLS) method.least squares (OLS) method.

In building the model, we have followed a stepwise In building the model, we have followed a stepwise regression (backward elimination):regression (backward elimination):

Initially, all the potential explicative variables are included in the Initially, all the potential explicative variables are included in the modelmodel

Then, the explicative variables are eliminated one-by oneThen, the explicative variables are eliminated one-by one For each step, we test (with a F-test) the difference in the model For each step, we test (with a F-test) the difference in the model

performance (with the residual sum of squares, RSS). If the performance (with the residual sum of squares, RSS). If the difference is significant, the variable is hold in the model, difference is significant, the variable is hold in the model, otherwise is deletedotherwise is deleted

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The F test isThe F test is

With With p1p1 and and p2p2 being, respectively, the number of parameters of the two models (with and without the variable) and being, respectively, the number of parameters of the two models (with and without the variable) and nn the number of observations the number of observations Under the null hypothesis (the two models aren’t different, i.e. the variables can be excluded), F has a distribution of a Fisher’s F(Under the null hypothesis (the two models aren’t different, i.e. the variables can be excluded), F has a distribution of a Fisher’s F(p2-p1,n-p1p2-p1,n-p1) ) If F is bigger than its critical value (given a confidence level, e.g. 95%), the null hypothesis is rejected and the variable is hold in the model, otherwise it’s excludedIf F is bigger than its critical value (given a confidence level, e.g. 95%), the null hypothesis is rejected and the variable is hold in the model, otherwise it’s excluded At the end of the process we don’t get the final model, but the variables selected have been used to build several regression models, that have to be compared with goodness-of-fit testsAt the end of the process we don’t get the final model, but the variables selected have been used to build several regression models, that have to be compared with goodness-of-fit tests

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Models’ development – stepwise Models’ development – stepwise regressionregression

Tests and model selectionTests and model selection ““Absolute” goodness-of-fit:Absolute” goodness-of-fit:

Analysis of residuals: it has been conducted a series of test in Analysis of residuals: it has been conducted a series of test in order to evaluate the absence of auto-correlation in the residuals order to evaluate the absence of auto-correlation in the residuals and in the squared-residuals (this condition is necessary and in the squared-residuals (this condition is necessary otherwise the OLS is not anymore the most efficient estimator)otherwise the OLS is not anymore the most efficient estimator) Correlogram analysisCorrelogram analysis LM testLM test White test (to test the presence of cross-correlations among White test (to test the presence of cross-correlations among

residuals)residuals) In the presence of auto-correlation, we have added to the In the presence of auto-correlation, we have added to the

regression some error-correction terms that “depurate” the regression some error-correction terms that “depurate” the model from autocorrelation (the model “learns” from previous model from autocorrelation (the model “learns” from previous errors)errors)

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111 ...

tkk xxy

Wald test (tests coefficients’ stability)Wald test (tests coefficients’ stability) Jackniffe test (tests model robustness, by eliminating each Jackniffe test (tests model robustness, by eliminating each

observations one by one and testing the stability of betas)observations one by one and testing the stability of betas)

““Relative” goodness-of-fit:Relative” goodness-of-fit: Fit analysis (with R-square)Fit analysis (with R-square) Back test “out of time”: re-estimate the models eliminating the Back test “out of time”: re-estimate the models eliminating the

last observations and tests the “backward forecast” with the last observations and tests the “backward forecast” with the actual values, evaluating:actual values, evaluating: The forecast errorThe forecast error The directionality of forecastsThe directionality of forecasts

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Tests and model selectionTests and model selection

Model 1: ResidentialModel 1: Residential

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Explicative variables:• GDP growth rate(t - 1)• Inflation rate (t - 1)• Private consumption index (t - 1 and t - 2)• Autoregressive coefficient (t - 2)• M1 growth rate (t - 2)• “Error correction” coefficient (t - 1) The resultant R-square is 92.44%

Model 2: OfficesModel 2: Offices

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Explicative variables:• Inflation rate (t - 2)• Private consumption index (t - 1 and t - 2)• Autoregressive coefficient (t - 2)• “Error correction” coefficient (t -1 and t - 3)

The resultant R-square is 86,66%

Model 3: CommercialModel 3: Commercial

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Explicative variables:• Inflation rate (t - 2)• Private consumption index (t - 1 and t - 2)• Autoregressive coefficient (t - 2)• “Error correction” coefficient (t -1)

The resultant R-square is 79,92%

Portfolio analysis: expected Portfolio analysis: expected returns and VaRreturns and VaR

The models presented can be used, for instance, to work The models presented can be used, for instance, to work out expected returns and VaR of a real estate portfolioout expected returns and VaR of a real estate portfolio

Expected returns can be calculated as a weighted Expected returns can be calculated as a weighted average of the 3 segments’ expected returns (in which average of the 3 segments’ expected returns (in which the weights are given by the asset allocation of the the weights are given by the asset allocation of the portfolio)portfolio)

Var can be obtained considering the standard errors of Var can be obtained considering the standard errors of the regression (that represent forecasts’ uncertainty), the regression (that represent forecasts’ uncertainty), either with parametric or Monte Carlo methodologies (in either with parametric or Monte Carlo methodologies (in this case, by simulating the evolution of macro-economic this case, by simulating the evolution of macro-economic variables)variables)

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Portfolio analysis: expected Portfolio analysis: expected returns and VaRreturns and VaR

Here is reported an example of a risk report with expected Here is reported an example of a risk report with expected return and VaR of a real estate portfolioreturn and VaR of a real estate portfolio

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Segment Expected return*Residential 2,00%Commercial 2,50%Offices 3,50%

Building Market Value Rent Segment Expected return* Value at Risk (99%)*1 10.000.000 € 400.000 € Residential 600.000 € 400.000 € 2 5.000.000 € 250.000 € Commercial 375.000 € 250.000 € 3 12.000.000 € 540.000 € Offi ces 960.000 € 420.000 € 4 7.500.000 € 262.500 € Commercial 450.000 € 375.000 € 5 10.500.000 € 630.000 € Commercial 892.500 € 525.000 € 6 11.000.000 € 352.000 € Offi ces 737.000 € 385.000 € 7 3.500.000 € 175.000 € Residential 245.000 € 140.000 € 8 4.000.000 € 176.000 € Offi ces 316.000 € 140.000 € 9 8.000.000 € 240.000 € Offi ces 520.000 € 280.000 €

10 2.500.000 € 50.000 € Commercial 112.500 € 125.000 € Portfolio 74.000.000 € 3.075.500 € 5.208.000 € 2.220.000 €

7,04% 3,00%* Horizon: 6 months

Stress testsStress tests In risk management practice, periodical reports are generally In risk management practice, periodical reports are generally

accompanied by stress tests and scenario analysis, in order to accompanied by stress tests and scenario analysis, in order to estimate the impact of unexpected events on portfolio returns.estimate the impact of unexpected events on portfolio returns.

Example of stress test: the inflation rate (positively correlated with Example of stress test: the inflation rate (positively correlated with the real estate returns) reduce by 2%, in a time horizon of 6 months.the real estate returns) reduce by 2%, in a time horizon of 6 months.

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Shock sul Portafoglio

Previsione attuale

Using Monte Carlo Using Monte Carlo methodology (with a methodology (with a simulated portfolio), we simulated portfolio), we can see how with a drop can see how with a drop in the inflation rate in the inflation rate portfolio returns reduce a portfolio returns reduce a lot, with a bigger variance lot, with a bigger variance due to the uncertainty of due to the uncertainty of the shock (given by the the shock (given by the standard error of the standard error of the relative coefficient).relative coefficient).

Scenario analysisScenario analysis Let’s now consider two historical scenarios and their potential impact on portfolio Let’s now consider two historical scenarios and their potential impact on portfolio

returnsreturns

1.1. Japan 1998-2005: in this period Japanese economy experienced deflation, Japan 1998-2005: in this period Japanese economy experienced deflation, with very low interest rate (in some days, below zero!!) and a negative GDP with very low interest rate (in some days, below zero!!) and a negative GDP growth.growth.

This scenario would reduce portfolio returns with a high probability of having This scenario would reduce portfolio returns with a high probability of having a loss, above all due to deflation.a loss, above all due to deflation.

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Scenario Giappone

Previsione attuale

Simulated portfolio

2.2. USA after 9/11: the worsening of economic conditions (after the .com bubble of USA after 9/11: the worsening of economic conditions (after the .com bubble of 1999-2000) was exacerbated by the terroristic attack. The Fed reacted quickly 1999-2000) was exacerbated by the terroristic attack. The Fed reacted quickly by keeping low interest rates and favouring monetary expansion.by keeping low interest rates and favouring monetary expansion.

In this scenario, even if real economic conditions worsens, portfolio returns In this scenario, even if real economic conditions worsens, portfolio returns would improve, due to monetary expansion (M1 growth).would improve, due to monetary expansion (M1 growth).

It’s common opinion that the monetary expansion of the Fed has been the It’s common opinion that the monetary expansion of the Fed has been the ultimate cause of real estate bubble in USA and of the ultimate economic crises.ultimate cause of real estate bubble in USA and of the ultimate economic crises.

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Scenario analysisScenario analysis

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250

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Next stepsNext steps This analysis has been conducted by considering real This analysis has been conducted by considering real

estate segments, therefore ignoring the specificity of the estate segments, therefore ignoring the specificity of the single buildingsingle building

The aim of a specific statistical analysis would be the The aim of a specific statistical analysis would be the estimation of expected returns and VaR for each estimation of expected returns and VaR for each building, considering, i.e.:building, considering, i.e.:

The temporal distribution of (estimated) maintenance costsThe temporal distribution of (estimated) maintenance costs The vacancy rate (i.e. the quote of the building that will not be The vacancy rate (i.e. the quote of the building that will not be

rent, therefore not producing an income)rent, therefore not producing an income) The default rate of the tenantsThe default rate of the tenants

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ReferencesReferences1. Benjamin J. D., Guttery R. S. and Sirmans C. F. (2004), “An Introduction to

Multiple Regression Analysis for Real Estate Valuation”, The Journal of Real Estate Practice and Education, Volume 07, Number 1, pp. 65-78.

2. Geltner D. and Goetzmann W. (2000), “Two Decades of Commercial Property Returns: A Repeated-Measures Regression-Based Version of the NCREIF Index”, The Journal of Real Estate Finance and Economics, Volume 21, Number 1, pp. 5-21.

3. Hamilton, J. D., (1994), “Time Series Analysis”, Princeton University Press.

4. Lin Z. and Vandell K. D. (2007), “Illiquidity and Pricing Biases in the Real Estate Market”, Real Estate Economics, Volume 35, Number 3, pp. 291-330.

5. Ling D. C. and Naranjo A. (1997), “Economic Risk Factors and Commercial Real Estate Returns”, The Journal of Real Estate Finance and Economics, Volume 14, Number 3, pp. 283-307.

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