Applied Business Statistics Case studies A statistical model for real estate market

1. Applied Business StatisticsCase studiesA statistical model for real estate market Mauro Bufano Risk Management – BancaMediolanum Spa

2. Real estate market • Recently, a statistical analysis on the real estate market has grown its importance, due to several reasons: • Many asset management companies have “real estate portfolios” to manage, therefore buildings have to be treated as stocks or bonds • Among the assets of banks and insurances we find real estate asset, whose yields and income have to be properly estimated for asset & liabilities management (ALM) purposes • The guarantee of many loans (especially mortgage loans) is constituted by real estate, whose depreciation could impact heavily on bank profits (because LGD grows) • The origin of many economic crises (e.g. 2007-2009) has been the excessive overvaluation of real estate goods!

3. Real estate market • Anyway, there are many differences between traditional financial assets and real estate assets: • Real estate assets are more “illiquid”, i.e. it’s generally difficult to buy and sell them quickly • Sometimes we don’t find a “fair” market price (especially for institutional buildings) • Negotiation times are very long (months, or years...) • It’s difficult to estimate a real “yield” of a real estate asset, because of hidden costs: • Maintenance costs • Vacancies • High transaction costs

4. Real estate market – international focus 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

5. A statistical model for Italian real estate market • In next slides we will show a statistical model for italian real estate market, used for risk management purposes in a real estate portfolio • Following market best practice, real estate market has been divided into 4 segments • Residential • Offices • Commercial • Logistic • The model covers segments 1-3

6. The variables analyzed • Italian real estate prices (source: Nomisma) with semi-annual frequency • Macro-economic variables, also with semi-annual frequency. Macro-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 market: • GDP growth rate • Inflation rate • Stock market returns • M1 growth rate • Real estate investments’ growth rate • Private consumption growth rate • Euribor 3 months • 10 year government bonds

7. Descriptive analysis • The descriptive analysis of variables is preliminary to the construction of the model, and it has been conducted on • “raw” data • percentage returns • log-returns • Several tests have been conducted • Stationarity analysis (Dickey-Fuller test) • Normality tests (Kolmogorov Smirnov test) • Analysis of correlations (in order to avoid multicollinearity) In the construction of the model, variables have been transformed in percentage returns, due to the presence of trends (non-stationarity) in row variables

8. Principal component analysis • In order to choose the variables to include in the model, a principal component analysis (PCA) has been conducted: • We compute the correlation matrix of the time-series of real estate returns • The eigenvectors of the correlation matrix represents the principal components • The percentage of variance explained for each eigenvectors is given by the corresponding diagonal values of the eigenvalues’ matrix • In our model, the first 4 components explain 84.3% of the variance of the real estate market • Then, we select the macro-economic variables that are more correlated with the principal components

9. Models’ development – stepwise regression • The statistical model used here is a linear regression, whose parameters have been estimated with ordinary least squares (OLS) method. • In building the model, we have followed a stepwise regression (backward elimination): • Initially, all the potential explicative variables are included in the model • Then, the explicative variables are eliminated one-by one • For each step, we test (with a F-test) the difference in the model performance (with the residual sum of squares, RSS). If the difference is significant, the variable is hold in the model, otherwise is deleted

10. Models’ development – stepwise regression • The F test is • With p1 and p2 being, respectively, the number of parameters of the two models (with and without the variable) and n 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(p2-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 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 tests

11. Tests and model selection • “Absolute” goodness-of-fit: • Analysis of residuals: it has been conducted a series of test in order to evaluate the absence of auto-correlation in the residuals and in the squared-residuals (this condition is necessary otherwise the OLS is not anymore the most efficient estimator) • Correlogram analysis • LM test • White test (to test the presence of cross-correlations among residuals) • In the presence of auto-correlation, we have added to the regression some error-correction terms that “depurate” the model from autocorrelation (the model “learns” from previous errors)

12. Tests and model selection • Wald test (tests coefficients’ stability) • Jackniffe test (tests model robustness, by eliminating each observations one by one and testing the stability of betas) • “Relative” goodness-of-fit: • Fit analysis (with R-square) • Back test “out of time”: re-estimate the models eliminating the last observations and tests the “backward forecast” with the actual values, evaluating: • The forecast error • The directionality of forecasts

13. Model 1: Residential • 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%

14. Model 2: Offices • 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%

15. Model 3: Commercial • 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%

16. Portfolio analysis: expected returns and VaR • The models presented can be used, for instance, to work out expected returns and VaR of a real estate portfolio • Expected returns can be calculated as a weighted average of the 3 segments’ expected returns (in which the weights are given by the asset allocation of the portfolio) • Var can be obtained considering the standard errors of the regression (that represent forecasts’ uncertainty), either with parametric or Monte Carlo methodologies (in this case, by simulating the evolution of macro-economic variables)

17. Portfolio analysis: expected returns and VaR • Here is reported an example of a risk report with expected return and VaR of a real estate portfolio

18. Stress tests • In risk management practice, periodical reports are generally accompanied by stress tests and scenario analysis, in order to estimate the impact of unexpected events on portfolio returns. • Example of stress test: the inflation rate (positively correlated with the real estate returns) reduce by 2%, in a time horizon of 6 months. Using Monte Carlo methodology (with a simulated portfolio), we can see how with a drop in the inflation rate portfolio returns reduce a lot, with a bigger variance due to the uncertainty of the shock (given by the standard error of the relative coefficient).

19. Scenario analysis • Let’s now consider two historical scenarios and their potential impact on portfolio returns • Japan 1998-2005: in this period Japanese economy experienced deflation, with very low interest rate (in some days, below zero!!) and a negative GDP growth. This scenario would reduce portfolio returns with a high probability of having a loss, above all due to deflation. Simulated portfolio

20. Scenario analysis • 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 by keeping low interest rates and favouring monetary expansion. In this scenario, even if real economic conditions worsens, portfolio returns would improve, due to monetary expansion (M1 growth). 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. Simulated portfolio

21. Next steps • This analysis has been conducted by considering real estate segments, therefore ignoring the specificity of the single building • The aim of a specific statistical analysis would be the estimation of expected returns and VaR for each building, considering, i.e.: • The temporal distribution of (estimated) maintenance costs • The vacancy rate (i.e. the quote of the building that will not be rent, therefore not producing an income) • The default rate of the tenants

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