University of Zürich, Switzerland

University of Zürich, Switzerland

RE - general asset features The inclusion of real estate assets in a portfolio has proven to bring diversification benefits both for homeowners [Mahieu, Van Bussel 1996] and for institutional investors [Hoesli, Hamelink 1997] These assets tend to have relatively stable cash flows over time. Is a good hedge for both expected and unexpected inflation. Low volatility in capital values (as compared to equity) and good diversification properties (low correlation to financial assets). Therefore: Direct real estate investments are present in many institutional portfolios as it offers a good way to match CF needs on the liability side: Swiss pension funds and insurance companies hold on average 19% of their portfolio in property [Anderson et al., 1993] But: The usual techniques for portfolio immunization tend to fail as the real estate asset is not behaving like a bond. The presence of multiple options of different types will invalidate the traditional duration concept as a measure of sensitivity to interest rates.

Academic Research Up to now very little work was found which tries to amend the concept of duration for direct real estate investments. Nevertheless, duration is a highly regarded tool in the industry, albeit improperly applied when used with real estate (due to the presence of options). The absolute value of direct RE investment and the reduced number of studies on this topic motivate the present work.

The Macaulay duration represents a measure of the maturity profile of the promised cash flows of an income security with a predefined cash-flow stream. The Macaulay s duration is defined as: D = P N t=1 P 0 t C(t) (1+r) t also represents the elasticity of an asset price with respect to the discount factor. As an elasticity measure,duration can be written as: D = dp (1 + r) dr P For bonds that have embedded options, such as puttable and callable bonds, Macauley duration and modified duration will not correctly approximate the price move for a change in yield.

Effective duration Bonds that have embedded options are usually analyzed using effective duration. Effective duration is a discrete approximation of the slope of the bond s value as a function of the interest rate. This is necessary because as interest rates change so do the cash flows of the bond. The effective duration is given as D e = V δy V +δy 2V 0δy with δy= the yield change, V δy the bond value for a fall in y For these assets interest-rate changes are the only causes for changes in cash-flows.

The Swiss Market - Identifying the options A survey of the existing Swiss rental contractual provisions identifies several options bundled together in the average rental agreement the interest rate option - rents increase/decrease when mortgage rates increase/decrease the inflation option - operating costs increase/decrease when inflation rates increase/decrease the renovation-option - rents on existing contracts can be below with possibility to reach after complete renovation lessee s reset option - tenant can ask for a rental level close to that one of the previous tenant if no major improvements were made to the dwelling. On top of these, forces also impact the changes in rents (thus cash-flows depend also on factors other than the change in interest rates/discount rates) Real estate is thus a claim to a stream of cash-flows depending on conditions and on the options embedded

Options impact Given the number, the structure and the interaction of the options, a yielding a closed form solution looks a priori involved. For the renovation option Merton proposes a closed form-solution for the option assuming rents follow a GBM and renovation costs are constant The value of the rent-revising options depend on the future path of mortgage rates and inflation In a DCF framework, how do changes in the discount rate and in the cash-flow relate to changes in the asset s price?

idea Let r t+1 be the log return at time t + 1 and p t be the log price at time t: r t+1 log(p t+1 + D t+1) log(p t) = p t+1 p t + log(1 + exp(d t+1 p t+1)) r t+1 k + ρp t+1 + (1 ρ)d t+1 p t where k and ρ are parameters of the linearization. The approximation can be solved forward (subject to a terminal condition) to obtain a formula for the log-price: p t = X k 1 ρ + (1 ρ) ρ j E t[d t+1+j ] j=0 X j=0 ρ j E t[r t+1+j ]

implementation The previous price approximation relates today s price to future cash-flows and discount-rates in a linear fashion. A change in the log price is then related to a revision in the expectations of future CF and discount rates. The next step is to get a measure for the expected CF and discount rates. C.S. advocate the use of a structural VAR in levels to capture the dynamic of the state variables used by the participants in forming estimates. x t+1 = Ax t + ɛ t+1 E t[x t+1+j ] = A j+1 x t All this works fine when you have enough data to properly estimate the VAR [For the US equity yearly values are used over a time span reaching from 1900 to 1987 (Campbell-Shiller 1987, J. of Finance)].

implementation II The standard is a square VAR x t+1 = β 11x t + β 12y t + β 13z t + ɛ x t+1 y t+1 = β 21x t + β 22y t + β 23z t + ɛ y t+1 z t+1 = β 31x t + β 32y t + β 33z t + ɛ z t+1 Alternatives are error-correcting VARs.

implementation III The problem arises when not enough data is available for the estimation of the VAR In real estate s the methodology has been used yet the original VAR contains trending variables (disregarding any question of stationarity of the time-series - CF are used in levels) [Geltner, Mei 1995] The parameter estimates become unreliable when many variables are needed in the VAR and the sample is not large enough (low t-values on non-trending variables, high t-values on trending)

A potential solution Use a different econometric specification for the expectation equation which preserves the linear structure and is consistent from a statistical view-point A good candidate is an ADL (autoregressive distributed lag) with exogenous variables x t = α + px i=1 β 1i x t i + qx j=0 β 2j y t j + mx k=0 β 3k z t k + ɛ t The lag-length selection procedure is dictated by the data and not imposed a priori (using some Information Selection Criteria)

Using the method for the Swiss RE For Switzerland the index measuring the performance of direct real estate is the IAZI Investment Index (available at quarterly values). The index is available since 1987 with roughly 6 years representing a generally-accepted bubble [Hoesli, Giaccotto 1997] It is a total return index used to compute the capital requirement of insurance companies investing in direct real estate [the reason why the analysis is done on this index]. An index for cash-flows is unfortunately unavailable; a rental index of the BFS is used instead For the discount rate the Swiss Confederation bond yield is used r IAZI t = α + px i=1 β 1i r IAZI t i + qx j=0 β 2j r rents t j + mx k=0 β 3k r SNB t k + ɛ t

The Swiss Direct Real Estate Market 100 120 140 160 180 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 IAZI Performance Index - Levels -0.04 0.00 0.04 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 IAZI Performance Index - Returns

The rental 100 120 140 160 180 1980 1985 1990 1995 2000 2005 Rental Index - levels -0.01 0.01 0.03 0.05 1980 1985 1990 1995 2000 2005 Rental Index - quarterly returns

The discount factor 0.006 0.010 0.013 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 CH 20y bond yields Series : All.Ret[, "SNB.20Y"] ACF -0.2 0.2 0.6 1.0 0 5 10 15 Lag

The discount factor - unit root test summary(adf.unit20y) Test for Unit Root: Augmented DF Test Null Hypothesis: there is a unit root Type of Test: t-test Test Statistic: -4.395 P-value: 0.004781 Coefficients: Value Std. Error t value Pr(> t ) lag1-0.3512 0.0799-4.3945 0.0001 lag2 0.4373 0.1176 3.7180 0.0005 constant 0.0046 0.0011 4.3699 0.0001 time -0.0024 0.0006-4.1726 0.0001 Regression Diagnostics: R-Squared 0.3249 Adjusted R-Squared 0.2860 Durbin-Watson Stat 2.0935 Residual standard error: 0.000485 on 52 degrees of freedom F-statistic: 8.342 on 3 and 52 degrees of freedom, the p-value is 0. 0001263

The ADL - first look Using a grid selection the better is given by rt IAZI = α + β 11rt 1 IAZI + β 14rt 4 IAZI + β 21rt 1 rents + β 22rt 2 rents + + β 30r SNB t + β 31r SNB t 1 + β 32r SNB t 2 + ɛ t But the bond yield is autocorrelated implying the estimates will have large standard errors

The ADL - estimates Call: OLS(formula = IAZI.TR ~ ar(1) + tslag(iazi.tr, k = 4) + tslag(miete.r) + tslag(miete.r, k = 2) + SNB.20Y.1 + tslag(snb.20y.1) + tslag( SNB.20Y.1, k = 2), data = All.Ret.S, na.rm = T) Residuals: Min 1Q Median 3Q Max -0.0509-0.0112-0.0009 0.0114 0.0365 Coefficients: Value Std. Error t value Pr(> t ) (Intercept) 0.0473 0.0162 2.9253 0.0054 tslag(iazi.tr, k = 4) -0.2581 0.1519-1.6985 0.0965 tslag(miete.r) -1.5850 0.8711-1.8195 0.0756 tslag(miete.r, k = 2) 1.1731 0.7890 1.4868 0.1442 SNB.20Y.1-8.5202 5.5983-1.5219 0.1352 tslag(snb.20y.1) 7.6549 8.1622 0.9378 0.3534 tslag(snb.20y.1, k = 2) -2.9263 5.2002-0.5627 0.5765 lag1 0.2991 0.1502 1.9911 0.0527 Regression Diagnostics: R-Squared 0.3132 Adjusted R-Squared 0.2039 Durbin-Watson Stat 1.8190 Residual Diagnostics: Stat P-Value Jarque-Bera 0.5686 0.7525 Ljung-Box 23.4378 0.1355 Residual standard error: 0.01898 on 44 degrees of freedom Time period: from Dec 1994 to Sep 2007 F-statistic: 2.866 on 7 and 44 degrees of freedom, the p-value is 0.01485

The ADL - transformation A transformation of variables is used in order to obtain a reliable estimate for the long-run impact of a change in bond yields. The transformation entails estimating the using as explanatory variables rt SNB, rt SNB rt 1 SNB, rt SNB rt 2 SNB where the parameter estimate of rt SNB now becomes the long-run impact or propensity of a change in the bond yield.

The ADL - transformed rt IAZI = α + β 11rt 1 IAZI + β 14rt 4 IAZI + β 21rt 1 rents + β 22rt 2 rents + + β30r t SNB + β31 r g t 1 SNB + g β 32rt 2 SNB + ɛt

The ADL - transformed estimates Call: OLS(formula = IAZI.TR ~ ar(1) + tslag(iazi.tr, k = 4) + tslag(miete.r) + tslag(miete.r, k = 2) + SNB.20Y.1 + S21.diff + S22.diff, data = All.Ret.S, na.rm = T) Residuals: Min 1Q Median 3Q Max -0.0503-0.0104-0.0005 0.0127 0.0341 Coefficients: Value Std. Error t value Pr(> t ) (Intercept) 0.0496 0.0159 3.1146 0.0032 tslag(iazi.tr, k = 4) -0.2666 0.1527-1.7463 0.0877 tslag(miete.r) -1.6315 0.8655-1.8849 0.0661 tslag(miete.r, k = 2) 1.1474 0.7890 1.4541 0.1530 SNB.20Y.1-3.9142 1.5007-2.6083 0.0124 S21.diff -4.9252 7.5380-0.6534 0.5169 S22.diff 3.7655 4.4762 0.8412 0.4048 lag1 0.2634 0.1503 1.7518 0.0868 Regression Diagnostics: R-Squared 0.3100 Adjusted R-Squared 0.2002 Durbin-Watson Stat 1.8223 Residual Diagnostics: Stat P-Value Jarque-Bera 0.5687 0.7525 Ljung-Box 23.6093 0.1304 Residual standard error: 0.01903 on 44 degrees of freedom Time period: from Dec 1994 to Sep 2007 F-statistic: 2.823 on 7 and 44 degrees of freedom, the p-value is 0.0161

The ADL - transformed estimates Normal Q-Q Plot 0.04 ADL36.fit Dec 1999 Mar 2000 0.02 Residuals 0.0-0.02-0.04 Dec 1996-2 -1 0 1 2 Quantile of Standard Normal

The ADL - transformed estimates Residuals versus Time ADL36.fit -0.05-0.03-0.01 0.01 0.03 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

The ADL - transformed estimates Residual Autocorrelation 1.0 0.8 ADL36.fit 0.6 ACF 0.4 0.2 0.0-0.2 5 10 15 Lag

The ADL - transformed estimates Residual^2 Autocorrelation 1.0 0.8 ADL36.fit 0.6 ACF 0.4 0.2 0.0-0.2 5 10 15 Lag

A few conclusions Only the long-run impact is obtained with sufficient reliability - given a 1% permanent increase in the bond yield one expects a roughly 4% drop in the IAZI performance index. Thus an interest-rate sensitivity is obtained - this is nevertheless not the same as a duration number. Given the normal errors and their lack of autocorrelation, the OLS residuals can be used for historical simulation in order to obtain a distribution of returns for the next period.