Residential factors
Coordinates
The initial factors are derived from the covariance of rolling annual returns on postcode areas. Polarity is defined by the highly liquid London postcode SW, whose value is dominated by prime property.
Factors are rotated such that factors 2 and 3 have zero return over a cycle.
| factor 1 | factor 2 | factor 3 |
|---|---|---|
| 0.91 | -0.11 | -0.39 |
| 0.10 | 0.99 | -0.04 |
| 0.40 | 0.00 | 0.92 |
This is a postmultiplier, so for example the rotated factor 2 has 0.11 of raw factor 1 deducted, and zero of raw factor 3.
The usual properties of a rotation apply:
\(\Sigma_j(r_{ij}^2)=\Sigma_i(r_{ij}^2)=1\) normalised rows and columns
\(R.R^T=I\) self-inverse
Changing to cylindrical coordinates. Radius r becomes a metric of quantity of cyclical risk, and radial angle \(\theta\) is the ‘phase angle’ which defines when in the cycle an area outperforms/underperforms.
\(\theta = arctan(\lambda_2/\lambda_3) . 180/\pi\)
\(r = \sqrt(\lambda_2^2 + \lambda_3^2)\)
This is non-standard in sign, origin, and units. Conventionally it would be \(arctan(y/x)\), and the units are degrees not radians.
Origin: prefer this origin since \(\theta=0\) corresponds to the point in the cycle when London starts to outperform Teesside.
Sign: prefer that angle increases with time, ie that the angular velocity is positive.
Units: radians are not widely used in the surveying / property domain, and engineering ‘grads’ are declining in popularity
Fundamentals
\(\theta\)
Notes
The range of thetad is 181 degrees, so TS has \(\theta=253\) and WC has \(\theta=72\)
The two metrics come from distinct ‘domains’ - \(£/m^2\) is entirely fundamental information from the raw datasets, whereas theta is wholly derived from price behaviour, specifically from covariance.
Regressing theta on \(log(£/m^2)\) :
\(\theta = a + b.(£/m^2)\)
| Dependent variable: | |
| thetad | |
| log(ppm2) | -85.748 |
| t = -39.420*** | |
| Constant | 878.614 |
| t = 50.058*** | |
| Observations | 103 |
| R2 | 0.939 |
| Note: | p<0.1; p<0.05; p<0.01 |
Where ‘thetad’ is \(\theta\) in degrees and ‘ppm2’ is \(£/m^2\), and logs are to base e (throughout this report).
Graphically therer are no obvious problems with the linear relation. There are clearly more points at the low-value end, but this need not be interpreted as higher residual variance. There is some hint of a ‘break’ around theta=200 - some of the above-trend prices relate to tourist and retirement destinations.
r
The original factor 2/3 scatterplot suggests that for the same theta, areas with low r may in general be lower rent. Therefore we test the hypothesis that r and rental yield are closely related.
The data source is problematic for several reasons
Geographic divisions are ‘Broad Rental Market Areas’ which need then to be mapped onto postcodes
History is short going back to 2014 and yield varies through time through both the numerator and the denominator
Reference rents are supplied for different numbers of rooms but not for a size metric such as \(m^2\)
The method of geographic mapping is to use (1) the standard postcode area names (2) where this fails make some custom mappings as tabulated below. Some are more satisfactory than others. In total 62 areas are mapped including a number in and around London, these last being somewhat unsatisfactory.
The history is simply snapshot as of the final datapoint - end June 2018 at the time of writing.
Valuation corresponding to a ‘standard n-bedroom dwelling’ is produced by taking \((N_{bedrooms+1})=N_{habitable}\) and averaging the size of a dwelling of that number of rooms, and pricing it by the \(£/m^2\) for the area. The resulting yields by number of rooms are averaged, unweighted.
The correlation across all 62 is 0.58. Arguably it makes sense to exclude the London area and include other major cities where the overlap between data sources is likely to be large. For example Lancaster is a well-defined city but the postcode LA indludes Kendal, Windermere and Ambleside which are very different. After this editing to 21 areas the correlation is 0.9, and regression results are as follows.
| Dependent variable: | |
| yield | |
| r | 0.829 |
| t = 8.823*** | |
| Constant | 0.024 |
| t = 6.458*** | |
| Observations | 20 |
| R2 | 0.812 |
| Note: | p<0.1; p<0.05; p<0.01 |
All this could be improved upon with more attention to the assumptions noted earlier, and the geographic mappings.
factor 1
Factor 1 has not been modified since the original rotation - it is the coordinate that is unchanged when going to cylindrical. One hypothesis about factor 1 would be as follows:
factor 1 is directional risk, cannot be hedged, and should be rewarded
factor 1 sensitivity and yield should be negatively correlated if gross total return is constant across areas
in an idealised world they would sum to a constant
The data does not support this hypothesis - there is a negative correlation, but it is weak and Brighton, Bristol and Oxford stand out as relatively large total return outliers. However the relatively low dispersion of returns at 2.5% p.a. suggests that it might be worth refining the model to remove some approximations in the join, and consider ex-ante not ex-post returns.
