# Introduction

The psychologist Abraham Maslow proposed in the mid-20th century that humans have not just a set of needs or even a list of needs, but rather an ordered list or hierarchy that must be satisfied in sequence without skipping ahead. Only once the most basic needs of health, food, water and shelter have been fulfilled can we move to tier 2 of the 5-level hierarchy, this being safety. The presence of shelter in the top tier underlines the primacy of residential property as one of the most fundamental assets to our well-being, matching as it does the liability which Maslow put his finger on: everyone needs a home. In an age of ‘disruption’ this asset has a qualitatively different value relative to the conventional commercial property sectors, where for example retail is under heavy pressure from technology-driven change, and workplaces of the future may possibly differ from the offices we know.

The UK residential market is substantial in value. Taking in aggregate the properties sold since 1995 and revaluing them to today, we arrive at the sum of £4.5Tn as compared to £2.5Tn for the All-share. The full value including un-transacted property is estimated to exceed £7Tn. Back in 1995 the same. properties (including those only built subsequently) were worth 0.95Tn, a return of 6.5% per annum. This ignores the yield, which in several pleasant cities can quite easily be around 5% net, and the ‘imputed rent’ which the homeowner pays to themselves is of course not taxed, and neither is the capital gain. The annual volatility of capital gain is approximately 8-10%, and even in the GFC which was of course driven by property speculation the drawdowns across the country were remarkably consistent in the range 15-22% and so it only just qualifies as a bear market and is hardly a meltdown. To put it in context, Prime Central London rebounded entirely in 30 months, and within another 2 years was 16% above its pre-crisis high. In summary we have a fundamentally attractive asset class that deserves detailed study for reasons of return, risk, and taxation.

### Indices

Indices provide the basis on which to properly analyse this important market in a more systematic and quantitative way than for example the ‘average house price’. The challenge is to derive an accurate representation of the returns in an area, somehow overcoming the fact that many properties are unique and may be heterogeneous in size and quality. One way to eliminate the asset-specific, idiosyncratic features of each property is through using repeat sales of the same property, applying the assumption that the individual property has not materially changed and thus cancelling the unique features out. In some instances it will indeed have changed through improvement, dilapidation, or lease extension but on average it has simply ‘kept up with the Joneses’ and the required maintenance capex must be estimated and deducted from the total return.

In logs the known holding period return $$R$$ is a sum according to the days accrued $$A$$ in each period and an unknown index daily log rate of return $$x$$, a uniform rate of capital gain that applies to all residential properties. The relation is now linear in log returns: $$R=A.x+e$$ (Bailey 1963). Both $$R$$ and $$A$$ are known without error from the two sale dates and prices, the error term absorbing over/under payment, improvement/dilapidation and leasehold rundown/extension, which can plausibly be assumed uncorrelated with the regressors, so we have a well-posed problem for regression. Degrees of freedom are not so much a problem either, but there is an issue with collinearity: many observations have equal columns, simply because holding periods tend to be measured in years not months. The estimated index returns $$x$$ may have a lot of estimation variance, and especially so at the edges in time due to the lack of data, and inevitably we are very interested precisely in the leading edge which corresponds to the current date. A ‘smoothness prior’ (Shiller 1973) addresses the collinearity problem and is essentially a shrinkage method similar to ‘ridge regression’, but whereas ridge regression shrinks toward a ‘prior’ of zero the smoothness prior shrinks towards a second difference of zero. This, as the name suggests, directly reduces the volatility of the estimate. The prior strength is tuned via cross-validation to optimally condition the estimate for out of sample accuracy.

Seasonality is an important component so a 2-pass estimation can estimate and remove it - this matters because it is relatively high-frequency and yet it is a real systematic effect with significant amplitudes of up to 2%, as in the example illustrated for Prime Central London where there is a fairly strong summer peak. Some outlier rejection is also helpful. Following this procedure we arrive at a set of clean regularly sampled indices which show substantial trends superimposed with cycles and punctuated by inflection points.

It turns out that it is quite feasible to estimate the 104 postcode areas of England and Wales (the initial letter or two, such as W for west London) at a monthly frequency, with a little augmentation for some smaller areas. The use of postcodes is a convenience and also defines geo-locations according to real-world communities on a four-level hierarchical tree, something that geometric tessellations are less suited to. In doing this we have achieved a dimension reduction from 24 million transactions to an underlying model of 295 months and 104 areas. Joining this with floor area data from the Energy Performance Certificate we can reprice any of the 14m transacted properties to any date since 1995, and aggregating both value and floor area get histories of price/m2. Importantly we can do this for any aggregation with sufficient observations to attenuate noise in the numerator and the denominator - say over 1000 observations. In the following sections ‘price’ means $$£/m^2$$.

### Factors

A principal components type of factor model shows a steep eigenvalue spectrum, with the first 3 factors accounting for 98% of annual covariance, compared with 58% for 100 SPX index constituents present 1995-2019, so very dominated by low-dimensional systematic risk. No surprise that as usual in asset markets the first factor is a positive-weights directional average that broadly represents ‘the market’ and is highly dominant, so attention turns to the factors 2 and 3 which account for 12% of variance and represent systematic relative performance.

A principal components factor model has high explanatory power but can be difficult to interpret. To assist interpretation, we can rotate the factors whilst preserving their orthogonality, only sacrificing the property that each factor in turn maximises the $$R^2$$ available. Specifically we apply a small rotation such that all the return premium is in factor 1, so factors 2 and 3 are pure risk cyclical systematic risk showing long periods of trending punctuated with more-or-less abrupt reversals, with insignificant net premium between reversals. The rotation is small because factors 2 and 3 are already very close to ‘pure cyclical’. Their last cycle has been about 16 years, and corresponds to a known cycle of 16-18 years in property development - two beats on the Juglar cycle - which has a deep and lengthy history in economic literature using historical datasets such as brick production (Barras 2009) and also extends internationally. So some kind of cyclical behaviour is leaving its footprints very clearly in factors which are by construction orthogonal and resemble the familiar sine and cosine ‘factors’ of a cycle represented in cartesian coordinates. We will return to this using polar coordinates, which can be a more useful basis for models of cyclical phenomena.

The factor timeseries capture all the time-variation in systematic return, while the loadings define each area’s sensitivities. It turns out that the latter fall on an arc around the origin. This observation has a couple of implications: firstly that all areas have non-zero cyclical risk, and secondly that judicious portfolio construction can eliminate quite a lot of cyclical risk. This does nothing to reduce directional risk inherent to the asset class which is represented in factor 1, rather the management of factors 2 and 3 relates to timing the long cycle, which matters if your investment horizon differs from 16-18 years.

There is always an ambiguity in these models about definitions of polarity and phase angle from which axis, so some convention is needed. Here the factors are defined such that for assets loading zero factor 2, the rate of change of factor 3 with respect to price is negative (and vice-versa), so for example around London the loading $$\lambda_3$$ is close to zero and $$\partial \lambda_3/\partial p<0$$, and around Birmingham $$\lambda_2$$ is close to zero $$\partial \lambda_2/\partial p<0$$. The net effect is that Prime London has negative loadings on both factors, while Birmingham is pure positive factor 3, and Cleveland is pure positive factor 2. Other conventions could be used. Adopting the convention that factor 2 is the x-axis and factor 3 the y-axis, define phase angle from the loadings in the usual way: $$\theta=arctan(\lambda_3/\lambda_2)$$.

As it turns out the angular coordinate is a close correlate of price, or more precisely log(£/m2). The intuition then is that the factor model has identified the ‘ripple effect’ whereby early in the cycle prices rise in the most prime areas first, and only when these are getting unaffordable does price momentum gradually ‘ripple out’. At any one time the factors are rewarding maximally a particular price-point, with those above it on the wane and those below it picking up. It is convenient henceforth do define x=-log(£/m2), the sign being a convenience because of the direction of travel from high to low price, so x is here flipped for a more convenient positive direction of travel.

### Relative value and mean-reversion.

Almost exactly the same procedure applied to prices enables us to factor-analyse the covariance of x rather than returns, the motivation being to get a mean-reverting series for price versus an equilibrium value, in the style of a statistical arbitrage model. It turns out that factor 1 ‘the market’ is the only component with a unit root, and the others are indeed mean-reverting but not in a classic AR(1) type of sense, instead following the cyclical pattern already noted in returns which implies a continual overshooting of fair value. By construction there is always a balance of over and under-valued areas in cross-section. The benefit of modelling in levels is that the model is now in units of over/under price in £/m2, which is an unambiguous metric of value suitable for further quantitative applications.

### Shift, twist and butterfly

For an analyst familiar with the yield curve it is instructive to consider the loadings on the first 3 factors across the price spectrum. To avoid clutter the graphic shows 9 areas which are reasonably evenly spaced in log price, and factors 1, 2, and 3 correspond to shift, twist, and butterfly. This means that when factor 2 is strong as now, it favours the cheapest versus the most expensive (‘twist’), and when factor 3 is strong as it was 2014-2019 it favoured the mid-band mid-priced areas on the periphery of the London commuter zone versus both London and the far North (‘butterfly’). Compared to the yield curve the evolution of these factors through time is a lot more predictable, implying that this market is not weak-form efficient. This is not surprising because much activity in the housing sector is not driven solely by investment considerations.

### Granularity

Armed with the insight relating price to the factor model in polar coordinates we can go a step further to make it relevant to a very important use case: district and sector selection. District and sector are the second and third tiers of the four-level postcode definition, so additional model constraints are required to make this feasible and not overfit. There is more than one way to do this.

The first is to take the residuals of the observed repeat sales for the area-level index estimate, and in a binary way, rank sectors on price and recursively partition, fitting accruals within increasingly longer periods, thus trading off reduced temporal sampling frequency against increased spatial granularity. This can usefully be done 5 or 6 times, for 32 or 64 quantiles and pushing out the time sampling to 3-5 years. A second method is applied locally on touching areas to regress factor loadings on price, and for those factors that are significant, to infer loadings for sectors within the target area from their price. Another method takes price quintiles on the sectors in the touching areas, and directly estimates the indices similarly to the original estimation. Each method has pros and cons. They can be tested for overfitting using cross-validation methods so that only the higher granularities that fit well out of sample are then used.

The higher granularity indices are important decision support information for those homebuyers or investors who are constrained to one or two postcode areas through their life choices, yet seek guidance at their particular point in the cycle about whether from an investment perspective to buy a larger property in a cheap district or a smaller one at a higher price-point. For a Londoner: is now the time to sell up my Knightsbridge studio and buy a house in Tooting, and how much urgency is there?

### Waves

This section is a tentative formalisation of the idea that any given area has slightly upmarket and downmarket ‘neighbours’, and if the upmarket differential has expanded relative to downmarket (positive curvature in price), this constitutes a force that will accelerate price.

If what we have identified is the ripple effect then it’s reasonable to ask whether a relation can be unearthed whose fundamental underpinnings can reasonably be supposed to be universal and timeless. This is important because the cycle is long and our dataseries is only 25 years, so if more theory can be injected into the model then some of the data shortcomings can be offset.

To proceed we use the relative value model on levels, as described earlier. The wave in our application propagates in a positive direction along the affordability axis we defined before as $$x=-ln(£/m^2)$$. Consider first the wave equation applied to cyclical component of price $$c$$, namely the component attributable to factors 2 and 3. This component is a function of price ranking $$i$$, starting with rank 1 the highest, and time $$t$$ : $$\partial^2 c/\partial t^2=v^2.\partial^2c/\partial i^2$$, then taking its finite difference form, rearranging to make a forecasting relation, and relaxing the constraints on the coefficients.

$$\Delta^2_t(c(t,i))=v^2.\Delta^2_i(c(t,i))$$ finite differences

$$\Delta_tc(t+1)=\Delta_tc(t)+v^2.(\Delta_ic(i+1)-\Delta_ic(i))$$

$$\Delta_tc(t+1)=a_0+a_1.c(t,i)+a_2.\Delta_tc(t,i)+a_3.\Delta_ic(t,i)+a_4.\Delta^2_ic(t,i))$$ general unrestricted forecasting form

The ‘finite difference wave equation’ form is a special case of the general unrestricted form with the appropriate restrictions applied. By relaxing the restrictions on the right-hand we get a finite difference version of a generalised wave equation which is highly significant with over 70% $$R^2$$ on 1-year forecasts. Whilst the momentum or continuation models have high predictive power over shorter periods of less than 1/4 cycle and value has high predictive power up to 1/2 cycle, the model based on cyclical wave motion captures both in an encompassing framework. The regression results are shown in the appendix.

### Risk

There was in the aftermath of Lehman a synchronised fall across all areas over the period December 2007 - April 2009. The magnitude and duration of this fall was remarkably uniform across areas - actually more uniform than under normal market conditions. Once the market bottomed, the cyclical pattern resumed from where it was in 2007. This was early cycle, so Prime Central London was good value on the relative value model, and the rebound here was consequently sharp, whilst low-priced areas had a long wait to recover pre-crash levels. Some areas have yet to do so in 2019.

With this in mind, there is a slight puzzle about how best to measure and manage risk. We have a covariance matrix estimated on rolling overlapping 12-month returns, but this tells us quite little about the small or even negligible dispersion of losses in the crash, and equally little about the long trough in recovery for low-priced areas. Using what we know, the GFC moves can be modelled with technologies such as the panic copula. The slow recovery in the north is understood when conditioned on the relative valuation metric, since as we saw the GFC simply put the cycle briefly on pause. It was the position in the cycle that led to protracted recoveries in the north - the synchronised downturn did nothing to alter the value cycle, which resumed its path and held down the areas low in price but unattractive on value. The conclusion is that a simple beta metric is at best a partial reflection of the magnitude and duration of downside risk.

### Applications

The factor models for return and value open up a range of applications - these are some examples.

• Through cluster analysis in factor space we can identify natural groupings which are more meaningful, or at least complementary to traditional geographic regions, maximising the inter-cluster variance and minimising the intra-cluster variance. From the finding of the link with price, inevitable they tend to look a lot like price quantile groupings.

• The ability of a factor model to identify and quantify accurately in time and space the non-systematic local effects around infrastructure projects such as crossrail or HS2 stations can provide insight into the value-add from these projects.

• New build property tends to sell at a premium, and we can understand the decay of this premium through the residual return on the first resale. This has practical importance for homebuyers tempted by the government ‘Help to Buy’ scheme, amongst others.

• One insight is that cyclical risk cannot be diversified away by geographical diversification, needing instead price-point diversification. Granular indices and factor models allow residential property to be managed using the quantitative toolkit that is applied to security market investors, including cyclical forecasts, scenarios, and stress-tests.

### Status and outlook

The ripple effect is the dominant feature throughout the market once the directional factor has been removed. It is largely about affordability, but it is in part also about equity release as people move or retire from the capital to cheaper areas with different attractions. In these phases of divergence and later convergence the early cycle is all about London pulling ahead and expanding its price premium, while as the cycle progresses this slows down and finally reverses so that all price points converge once again. This is happening now late 2019. Convergence never totally recoups the ground lost in divergence due to ‘beta shortfall’ - or more simply perhaps the metropolisation process - but the upside can be brutally sharp as in 2004, driven by easy credit and strong value fundamentals. We are approaching or even entering that phase.

A glance at the factors suggests that based on the length and amplitude of its last trend, the factor 3 ‘butterfly’ has just now flapped its wings - flipping polarity after a 5-year up-trend. At the time of writing the latest data (September 2019) suggests the factor 3 flip may already have happened, but due to edge effects, data scarcity, and delayed reporting that plague the front month it awaits confirmation. In terms of observables, factor 3 changing sign corresponds to the observation that the far north (Bolton, Wigan, Cleveland) are delta-positive, Birmingham rolling over, and Prime Central London picking up. Some of these - Wigan is the poster child - are also plays on the HS2 project phase 2b, which must be considered at high risk even if the earlier HS2 phases come to fruition. The main implication of factor 3 reversing is that mid-priced, midlands areas including Birmingham and up to Manchester soon will face headwinds for some time as factor 3 swings down.

The upswing in Factor 2 should accelerate and have important negative impact on the entire London commuter belt, as well as the M4 corridor. This is the late cycle phase where the general rule is ‘the cheaper the better’. The only exception being Prime Central London, where as already noted, the factor 3 ‘butterfly’ will increasingly come to the rescue as we gradually approach the next cycle start.

One caveat: the model tells us a lot about relative performance, cyclical risk management, and timing area switches, but not a lot about factor 1, the market. In the case of factor 1 - the reward-bearing factor - the highest ‘betas’ have gone to in-sample-gentrified areas of London. But gentrification is a one-off, and factor 1 right now is hostage to the outcome of brexit and the path of the regional and global economy.

Even if it’s a challenge to comment on the future path of factor 1, it’s tempting to say a few things we do know about the capital. Some allocation to the primest London (£20,000/m2) could prove beneficial if brexit plays out without excessive harm to the capital, and especially if distressed developers offer attractive prices.

A few points on Prime Central London, which:

• has quite a lot of political risk - downside and upside - due to demographics and industry mix

• has low yields which offer not much cushion to capital loss, but better than sovereign bonds

• currently is entering ‘good value’ on a cyclical basis after 3.5 years of nominal price decline

• should start to get the benefits of the next cycle in the next 2-3 years, from where it looks good

• competes now against the late-cycle cheap areas which would need some nerve, pending better visibility on brexit and HS2, but have better yield to compensate

The ‘heat disk’ is quite an accurate representation of the heat gradient, because the colour intensity is proportional to the displacement along the cyclical factor vector. It rotates clockwise, the cycle being 16-18 years. The display here shows a critical moment in the cycle - not reached yet, but not very long off - when the most expensive and least expensive areas are both ‘lukewarm’, and everywhere else a bit chilled.

### Summary

The ‘stack’ of models has firm foundations in the high-quality Land Registry Price Paid dataset which records all residential transactions. The first link in the model toolchain is the repeat sales indices, time-series of performance on each geographic unit. Next a factor model partitions risk and return into orthogonal components, systematic and idiosyncratic. This in turn leads through the observation of a clear 16+ year factor cycle to a model for the ripple effect based on a modified wave equation, which dominates simple momentum and value models. This toolkit opens the door to a familiar landscape of quantitative methods for portfolio management and asset evaluation, backward-looking and forward-looking.

One of the biggest insights perhaps is the cyclical nature of the market. It was Isaac Newton’s question ‘does the moon also fall?’ that led to recognition of lunar orbital motion as a delicate poised balance between momentum and gravitational force. The precise counterparts in residential property are price continuation and a convexity-driven mean-reversion which naturally leads to overshooting and hence the property cycle. This should be of direct relevance not only to homeowners keen to get a step up on the property ladder, but also potentially to investors, developers, and lenders and securitisers. At the very least it can pep up dinner party conversations.

Likely the press will find other ways to ‘explain’ whatever moves that we may see, whether it be free ports, HS2, the decline of London as a regional power, or general lifestyle trends. The perspective described here shows that the likely relative price surge in the north is at least in part attributable to a persistent cycle with economically significant amplitude and considerable predictability.

Website: The models are maintained with the most recent monthly data from the Land Registry at www.anest.uk.

Giles Heywood 2019-09-27

### References

16-18 year cycle

Barras, R., 2009. Building cycles: growth and instability (Vol. 27). John Wiley & Sons.

ripple effect

Meen, G., 1999. Regional house prices and the ripple effect: a new interpretation. Housing studies, 14(6), pp.733-753.

Holly, S., Pesaran, M.H. and Yamagata, T., 2011. The spatial and temporal diffusion of house prices in the UK. Journal of Urban Economics, 69(1), pp.2-23.

repeat sales index

Bailey, M.J., Muth, R.F. and Nourse, H.O., 1963. A regression method for real estate price index construction. Journal of the American Statistical Association, 58(304), pp.933-942.

Nagaraja, C., Brown, L. and Wachter, S., 2014. Repeat sales house price index methodology. Journal of Real Estate Literature, 22(1), pp.23-46.

OECD, et al. (2013), “Repeat Sales Methods”, in Handbook on Residential Property Price Indices, Eurostat, Luxembourg. DOI: http://dx.doi.org/10.1787/9789264197183-8-en

smoothness prior

Kitagawa, G. and Gersch, W., 1996. Smoothness priors analysis of time series (Vol. 116). Springer Science & Business Media.

Shiller, R.J., 1973. A distributed lag estimator derived from smoothness priors. Econometrica (pre-1986), 41(4), p.775.

PCA factor models

Connor, G., Goldberg, L.R. and Korajczyk, R.A., 2010. Portfolio risk analysis. Princeton University Press.

Connor, G., 1995. The three types of factor models: A comparison of their explanatory power. Financial Analysts Journal, 51(3), pp.42-46.

 Dependent variable: d.c(t+1) = next year log return unrestricted heat equation momentum mean-reversion c -0.284*** -0.354*** -0.118*** (0.025) (0.029) (0.032) d.c 0.589*** 0.726*** (0.063) (0.050) D.c -0.697*** -1.601*** (0.132) (0.110) D.D.c -0.355*** -0.815*** (0.086) (0.085) Constant -0.0002 0.002 -0.0003 0.003 (0.002) (0.002) (0.002) (0.003) Observations 161 168 207 216 R2 0.738 0.593 0.510 0.060 Adjusted R2 0.731 0.585 0.508 0.056 Residual Std. Error 0.019 (df = 156) 0.024 (df = 164) 0.028 (df = 205) 0.038 (df = 214) F Statistic 109.847*** (df = 4; 156) 79.597*** (df = 3; 164) 213.285*** (df = 1; 205) 13.773*** (df = 1; 214) Note: p<0.1; p<0.05; p<0.01 c is the cyclical component (attributed to factors 2 and 3) of price D operator is difference in cross-section: D.x(i,t)=x(i,t)-x(i-1,t) d operator is difference in time: d.x(i,t)=x(i,t)-x(i,t-1)