Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 74 What Drives the Price-Rent Ratio in the UK Housing Market? An Unobserved Component Approach Jan R. Kim Department of International Economics and Law Hankuk University of Foreign Studies, Korea kjryoul@hotmail.com (preferred), kjryoul@hufs.ac.kr, Abstract The Campbell-Shiller present value formula decomposes the price-rent ratio into the sum of present discounted values of housing market fundamentals (i.e., rent growth, real interest rates, and excess returns from investing in housing). Treating the expectations of market fundamentals as unobserved components, we cast the Campbell-Shiller formula into a state-space model, and apply it to explain what has driven the UK housing market since the 1970s. We find that the variations in the price-rent ratio are mostly explained by the expected housing premium, whereas the real interest rate and expected rent growth account for only small fractions of the variation in the price-rent ratio. It is also found the correlations among the fundamental factors considerably dampen fluctuations in the price-rent ratio. Keywords: price-rent ratio, present value model, unobserved components.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 75 Introduction Since the mid-1990s, many developed countries have experienced booms and busts in house prices and housing returns on an unprecedented scale, and the UK is no exception. The goal of this paper is to examine the sources of variation in the price-rent ratio in the UK housing market. One key practical issue in the literature, including this paper, is how to estimate the market fundamentals. Following Binsbergen and Koijen (2009)), we acknowledge that expected market fundamentals are unobserved per se and specify them as latent factors in a parsimonious autoregressive form. We then cast the Campbell-Shiller formula for the price-rent ratio in a state-space model and estimate the deep parameters of the model using the Kalman filter. Three main findings emerge from the estimated present-value model for the U.K. housing market. First, the expected housing premium turns out to be the main source of volatility in the price-rent ratio. Second, the correlation structure among the three factors considerably dampens the total volatility of the price-rent ratio. The expected real interest rate and the housing premium turn out to be positively correlated, and the expected rent growth and the housing premium are negatively correlated. In combination, there correlations help reduce the volatility by more than 400%. Model and Data Let Ht+1 denote the gross real return on housing Ht+1=(Pt+1+Rt+1)/Pt, where P t is the real house price at the end of period t and R t+1 is the real rent payment during period t. Applying the log-linear approximation as in Campbell and Shiller (1988a, 1988b), we have
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 76 pr t = K + ρpr t+1 + Δr t+1 h t+1 (1) where pr t = log (P t R t ), Δr t+1 = log (R t+1 R t ), h t+1 = log(h t+1 ), ρ = e pr /(1 + e pr ), and pr is the average of the log price-rent ratio over the sample 1. Following Balke and Wohar (2002) and Kishor and Morely (2010), we break down the log gross real return, h t, into the real interest rate, i t (corresponding to the risk-free rate of return), and the excess rate of return, π t, (reflecting the risk premium for investing in housing): h t = i t +π t (2) Ruling out explosive behavior of the log price-rent ratio, we recursively substitute the pr t+i terms for i 1 in equation (2). Taking the conditional expectation E t ( ) of the resulting equation, we obtain pr t = K + E 1 ρ t{ j=0 ρ j ( Δr t+j+1 i t+j+1 π t+j+1 )} (3) which shows that that the log price-rent ratio is a weighted average of the expected future housing market fundamentals, i.e., rent growth, real interest rate, and excess returns. We treat the expected rent growth, g t = E t [Δr t+1 ], expected real interest rate, μ t = E t [i t+1 ], and the expected housing premium, λ t = E t [π t+1 ], as unobserved components. We 1 In our sample, the size of pr is 2.966, implying that the value of ρ is 0.951.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 77 further assume they follow AR(2) processes 2 : g t = γ 0 + γ 1 g t 1 + γ 2 g t 2 + ε t g, (4a) μ t = δ 0 + δ 1 μ t 1 + δ 2 μ t 2 + ε t μ, (4b) λ t = θ 0 + θ 1 λ t 1 + θ 2 λ t 2 + ε t λ, (4c) where the shocks to the expectations, ε t = (ε t g, ε t μ, ε t λ ), are specified as a Gaussian i.i.d. process with a covariance matrix Σ ε. The realized series of rent growth and real interest rate are equal to their respective expectations plus idiosyncratic innovations 3 : Δr t = g t 1 + u t r, i t = μ t 1 + u t i (5) where the unexpected innovations, u t = (u t r, u t i ), follow a Gaussian i.i.d. distribution with a diagonal covariance matrix, Σ u = diag(σ r 2, σ i 2 ). We further assume that ε t and u t are mutually uncorrelated at any leads and lags. Using equation (5) and the laws of motion (4a)-(4c) in the expectation part of equation (3), we 2 In an earlier version of the paper, we also used the AR(1) specification, as in van Binsbergen and Koijen (2010), but the AR(2) specification turned out to better fit the data. 3 By considering the observations of the rent growth and real interest rate only, we are treating housing risk premium as residuals given data on price-rent ratio.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 78 represent the log price-rent ratio (up to a measurement error u t pr ) as follows: G t pr t = pr t + u pr t = A + [B 1 B 2 B 3 ] [ Μ t ] + u t Λ t (6) where G t = [ g t γ 0 g t 1 γ 0 ], M t = [ μ t δ 0 μ t 1 δ 0 ], Λ t = [ λ t θ 0 λ t 1 θ 0 ], and (A, B 1, B 2, B 3 ) are factor loading coefficients determined by deep parameters of the model. In equation (6), pr t is the fundamental log price-rent ratio determined by equation (3). Given the estimated parameters, we can evaluate the contributions of the variance- covariance terms to the overall variations in the price-rent ratio. Equation (6) allows us to decompose the variance of pr t as: var(pr t ) = B 1 var(g t )B 1 + B 2 var(m t )B 2 + B 3 var(λ t )B 3 2B 1 cov(g t, M t )B 2 2B 1 cov(g t, Λ t )B 3 + 2B 2 cov(m t, Λ t )B 3. (7) The above decomposition implies that var(pr t ) depends on the variance-covariance of the unobserved components and their persistence measured by the AR(2) coefficients. The raw data used in this paper are the quarterly UK series of real house prices, real rent, pricerent ratios, nominal interest rates, and the core CPI, all spanning 1971:Q1 to 2013:Q4. The series of real house prices and price-rent ratios are obtained from the OECD statistical warehouse as seasonally adjusted indices. Because the price-rent ratio is available as an index with 2010 as the base year, we rescale the original series to match the national account version of the rental yields
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 79 (i.e., the inverse of the ratio) in Ward (2011) 4. We then obtain the real rent series by dividing the real house price index with the rescaled price-rent ratio. The nominal interest rate is the 20-year government bond yield rate obtained from the FRED of St. Louis Fed 5. The core CPI is also obtained from the FRED and transformed into year-on-year inflation rates, which are then subtracted from the nominal interest rate to yield the real interest rate series. Empirical Results Table 1 presents the maximum likelihood estimates of the model parameters. To save on space, we report and discuss what is relevant to the main theme of the paper: the relative importance of market factors in driving price-rent ratio. In the first panel, we find that three components of the housing market fundamentals are highly persistent. More specifically, the long-run AR coefficients of the expected interest rate (μ t ) and expected excess return (λ t ) are δ 1 + δ 2 = 0.9517 and θ 1 + θ 2 = 0.8909, respectively. The long-run AR coefficient of the expected rent growth is γ 1 + γ 2 = 0.7047. The second panel reports the standard deviations of the innovations (u t s) to the realized rent growth and interest rate, the shocks (ε t s) to agents expectations on market fundamentals, and the measurement error (u pr t ). The estimated standard deviations of the expected housing premium (ε λ t ) and the expected rent growth (ε g t ) are of similar size and larger than that of the real 4 The sum of actual and imputed rents divided by the value of housing stock was 3.56% over the year 2010, which is equal to a price-rent ratio of 112.36 in annual terms. 5 We tried to us the long-term mortgage rate available from the Bank of England, but the data series was too short.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 80 interest rate (ε t μ ). The third panel reports the estimated correlation coefficients among the expectation shocks, ε t s. From the estimated covariance structure, we observe that the shocks (ε t g, ε t μ ) to the expected rent Table 1: Maximum Likelihood Estimates of Parameters Parameter Estimate Standard Error AR coefficients γ 1 0.7138* 0.0122 γ 2-0.0091 0.0085 δ 1 1.5450* 0.0011 δ 2-0.5933* 0.009 θ 1 0.4744* 0.0038 θ 2 0.4105* 0.0035 standard dev. SD(u r t ) 0.0018 0.0021 SD(u i t ) 0.0004 0.0003 SD(ε g t ) 0.0265* 0.0054 SD(ε μ t ) 0.0057* 0.0014 SD(ε λ t ) 0.0243* 0.0053 SD(η pr t ) 0.0015 0.0011 corr. coeffs. ρ gμ -0.2739* 0.0066 ρ gλ 0.7709* 0.0007 ρ μλ -0.7596* 0.0007 Note: estimates with * are significant at the 5% critical level. growth and interest rate are negatively correlated via ρ gμ = 0.2739. Our interpretation of this
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 81 negative correlation is as follows: to the extent that the house price and rents are related via a stable price-rent ratio, shocks that induce agents to expect a higher real interest rate would dampen the purchase demand for housing units and thereby lead to slower expected rent increases. We also find a negative correlation of ρ μλ = -0.7595 between the expected interest rate and the housing premium and a positive correlation of ρ gλ = 0.7709 between the expected rent growth and housing premium. These two findings have a straightforward explanation if house prices increase slowly in response to good news. Regarding the positive correlation of ρ gλ = 0.7709, the house prices do not increase enough during the periods of rising rent growth, and therefore, there follows a contemporaneous increase in the housing premium. Similarly, if house prices do not rise enough when the interest rate is expected to fall, agents expect a contemporaneous increase in the housing premium. Thus, we observe a negative correlation of ρ μλ = 0.7595 between the interest rate and housing premium. In sum, the two latter findings imply that house prices have not moved quickly enough to fully capitalize on the changes in expected future real interest rates and rent growth. Using the estimated version of equation (7), we can now assess the contributions of the three expectation factors (g t, μ t, λ t ) more formally. Figure 2 display the loadings of the three market factors onto the price rent ratio. Strikingly, the contribution of expected housing premium moves very closely with the price-rent ratio throughout the sample period and nearly in tandem since the mid-1980s. Compared with other factors, the contribution of the housing premium is far more substantial and dominant. Such dominance of the risk premium is reminiscent of Campbell and
1971 Q3 1973 Q3 1975 Q3 1977 Q3 1979 Q3 1981 Q3 1983 Q3 1985 Q3 1987 Q3 1989 Q3 1991 Q3 1993 Q3 1995 Q3 1997 Q3 1999 Q3 2001 Q3 2003 Q3 2005 Q3 2007 Q3 2009 Q3 2011 Q3 2013 Q3 1971 Q3 1973 Q3 1975 Q3 1977 Q3 1979 Q3 1981 Q3 1983 Q3 1985 Q3 1987 Q3 1989 Q3 1991 Q3 1993 Q3 1995 Q3 1997 Q3 1999 Q3 2001 Q3 2003 Q3 2005 Q3 2007 Q3 2009 Q3 2011 Q3 2013 Q3 1971 Q3 1973 Q3 1975 Q3 1977 Q3 1979 Q3 1981 Q3 1983 Q3 1985 Q3 1987 Q3 1989 Q3 1991 Q3 1993 Q3 1995 Q3 1997 Q3 1999 Q3 2001 Q3 2003 Q3 2005 Q3 2007 Q3 2009 Q3 2011 Q3 2013 Q3 Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 82 Figure 2: Decomposition of Log Price-Rent Ratio logptr Loading of G(t) 0.6 0.4 0.2 0-0.2-0.4-0.6 log(ptr) Loading of MU(t) 0.6 0.4 0.2 0-0.2-0.4-0.6 log(ptr) Loading of Lambda(t) 0.6 0.4 0.2 0-0.2-0.4-0.6
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 83 Ammer (1993): in the US stock market approximately 70% of the variance of excess stock returns is attributable to news about future risk premiums for holding stocks, whereas approximately 15% of the stock return variance is attributable to news about future dividends. Table 2 presents how the unconditional variance of pr t is split across the three unobserved house market factors (g t, μ t, λ t ) 6. The results suggest that a significantly larger portion of the total variance in the price-rent ratio is attributable to the expected housing premium than to the expected rent growth or interest rate. In fact, fluctuations in the expected future housing premium cause more than ten times the variations in the price-rent ratio, and the expected rent growth takes the smallest portion of the variations in the ratio. Table 2: Decomposition of Unconditional Variance estimate share var(pr t ) 0.0345 100% B 1 var(g t )B 1 0.01116 33.52% B 2 var(m t )B 2 0.0357 108.79% B 3 var(λ t )B 3 0.1159 333.55% -2B 1 cov(g t, M t )B 2 0.0098 28.43% -2B 1 cov(g t, Λ t )B 3-0.0439-127.03% 2B 2 cov(m t, Λ t )B 3-0.0964-279.26% The bottom panel of Table 2 shows that the covariance structure among the three factors 6 When calculating the variance-covariance terms, we abstract away uncertainty about the estimated parameters.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 84 substantially dampens the overall variations in the price-rent ratio. The highly positive correlation ρ gλ = 0.7709 between the expected rent growth and the housing premium is translated into the large negative contribution to the total variance, dampening the total variance of the price-rent ratio by more than 300%. Furthermore, the correlation ρ μλ = -0.7595 between the expected real interest rate and housing premium makes even larger negative contribution to the overall variability of the price-rent ratio by about 300%. Finally, the correlation ρ gμ = -0.2739 between the expected rent growth and real interest rate tends to render the price-rent ratio slightly more volatile. Conclusion The estimated parameters and the resulting variance decompositions reveal some important characteristics of the UK s housing market behavior. First, the expected housing premium turns out to be the principal source of variation in the price-rent ratio. Over the whole sample period, our results indicate that the variation in the expected housing premium accounts for more than 300% of the variation in the price-rent ratio, whereas the expected rent growth and real interest rate explain about 33% and 100%, respectively. Second, we find that the correlations among the three components of the rent-price ratio considerably dampen fluctuations in the rent-price ratio. In particular, the correlation between the expected future premium and rent growth is positive and lowers the variation in the ratio by almost 300%. It is also found that the negative correlation between the expected housing premium and the real interest rate is large enough to cancel out most of the volatility caused by the expected risk premium.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 85 References Balke, Nathan S. & Mark Wohar (2002) Low-Frequency Movements in Stock Prices: A State Space Decomposition," The Review of Economics and Statistics, MIT Press, vol. 84(4), 649-667. van Binsbergen, J. H. & R. S. J. Koijen (2010) Predictive Regressions: A Present-Value Approach, The Journal of Finance, vol. 65, 1439 1471. Campbell, John Y. & J. Ammer (1993) What Moves the Stock and Bond Markets? A Variance Decomposition for Long-Term Asset Returns, Journal of Finance, vol. 48, 3-48. Campbell, John Y. & Robert J. Shiller (1988a) The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies, vol. 1(3), 195-228. Campbell, John Y. & Robert J. Shiller (1988b) Stock Prices, Earnings, and Expected Dividends, Journal of Finance, vol. 43, 661-676. Kishor, Narayan & James Morley (2010) What Moves the Price-Rent Ratio: A Latent Variable Approach, working paper, University of Wisconsin, Milwaukee. Ward, Simon (2011) UK House Prices at fair value, Based on Rents, an article posted on Mindful Money. Acknowledgement This work is supported by the research grant from Hankuk University of Foreign Studies.
Journal of Marketing and Management, 6 (2), 74-86, Nov 2015 86 Biography Prof. Kim received a PhD degree from Yale University in 2003 in economics. He worked at Federal Reserve Bank of Minneapolis as an associate economist during 2001-2004. After working at Korea Development Institute, a leading government-funded think tank in Korea, he has been teaching at Hankuk University of Foreign Studies since 2005. His interest lies in dynamic stochastic general equilibrium models and empirical research in business cycles.