To manage an asset class as effectively as possible we need a disciplined quantitative framework as decision support tool. One of the early steps in this process is to identify the risk drivers: those numbers that will determine the performance of our assets at the investment horizon. Here we list them out.

**Rental yield** - this is simply outside the scope of the model as it stands. The good news for homeowners is that it is totally irrelevant once the decision to buy has been made - you’ve left the realm of rents and landlords.

**Market risk** is the primary factor, more or less related to GDP and interest rates. However because it affects all areas, it’s of secondary importance for area choice.

**Cyclical risk** is quantified by factors 2 and 3, also known as **‘Twist’** and **‘Butterfly’** as described in the factors section. These measure the ripple effect which often dominates short-term relative performance. In sequence and by descending price each area/district/sector in turn receives buying pressure as the next tier up becomes less affordable and buyers lower their sights.

Within this model it is crucial to know three things: (1) the factor sensitivity of candidates for investment (2) the current phase in the cycle (3) the path of the factors between now and the investment horizon.

In this paper we focus on phase. In particular we aim to render the cycle as regular as possible, paving the way for a cyclical forecast.

Here we get a bit more technical - and to lighten up, a bit more metaphorical - about how the phase is measured.

First we need to deal with an unfortunate fact of cycles: that the phase angle is by its nature non-unique because we can add on multiples of a cycle and get the same coordinates.

The phase angle \(\theta=arctan(f_3/f_2)\) is a ratio of two single-period (one month) estimates with a trig function applied. As such it can be fairly noisy and subject to revision, but there is a strong motivation to reduce the noise because it tells us quite literally the current phase in the cycle. This is the focus in what follows.

So we do what is known to signal engineers as phase unwrapping. A slightly flippant analogy here may help: in the movie ‘Groundhog Day’ Julia Roberts experiences a ‘wrapped’ day where each time it is fresh, whilst Bill Murray experiences it ‘unwrapped’, recognising that it lies as just one in a sequential series. Dropping the analogy - which has its limitations - the good news is that an algorithm can do this unwrapping for us, placing the previous cycle ‘up where it belongs’ on the phase scale.

the two plots may make it clear what’s going on, if the slightly tortuous analogy did not - it’s exactly the same data, just displayed differently with some points displaced 360 degrees (and some 720 degrees).

`## Warning: Removed 86 rows containing missing values (geom_point).`

Having now unwrapped the phase the linear trend is visually pretty clear. However for forecasting purposes it’s a bit noisy and distorted. We would like to capture the trend and smooth out the short term reversals without over-smoothing it.

The motivation for this is that *if* the phase follows a linear trend *then* the cycle will be highly apparent. If a minor change in an arbitrary parameter can make the cycle more apparent - for example pooling time-periods - then this will make the forecasting easier and more accurate.

We borrow a method from a popular and recognised technique known as isotonic regression. The algorithm’s method is ‘pool adjacent violators’ (PAVA) ie to pool data until it always moves in one direction - in our case, down. However this is a very stringent requirement and actually goes beyond what we need. We are happy to accept points that simply lie close to a linear trend. So a simple adaptation of the algorithm pools adjacent periods recursively to maximise correlation with a linear trend.

To justify this, consider the fact that the one month period we start from is totally arbitrary - there is no reason to slice time regularly on any particular interval unless it serves a purpose to do so, and holding periods on property obviously tend to be many months. Stated another way, we can consider intra-period variations or meanderings to be essentially of no interest, especially if they are promptly retraced.

As it turns out, PAVA is quite successful. The left plot shows that we’ve achieved a pretty linear relation between phase and ‘flexitime’. Flexitime varies from one month (small dot) or up to nine months (bigger dots). The right plot re-stretches time into a more familiar scale for comparison.

To make it clearer when most of the compression occurs, it’s instructive to relate the two time scales. In the following plot it’s clearer what is occuring, with more pooling flattening the line. From when the market peaked in late 2007 the algo starts pooling, right up to 2014 when the need for pooling drops and we revert to ‘normal time’. What this means is that phase rotation did not stop when the market fell in 2007, but it went into ‘pause’ and did a little micro-gyration dithering around. It then reverted to rotation as before, just more slowly for a while before resuming normal speed.

Having completed this slightly abstract process of making the phase linear, we can revert back to cartesian coordinates and see if the factors 2 and 3 - **‘Twist’** and **‘Butterfly’** are even more recognisably cyclical.

Indeed the two factors now show their cyclical character rather clearly, and all we have done is to pool time-periods.

A few more observations are in order - it may help to refer to the section about these factors to see how they are constructed

‘Butterfly’ factor 3 in particular is somewhat more angular than a sine wave - the reversals are abrupt

‘Butterfly’ has apparently just reversed sign in 2019, pretty much right on cue given the amplitude of the upswing since 2014

‘Twist’ factor 2 has been deficient in amplitude over the last year at least - the upswing is feeble.

The last point is quite important because many places have big loadings on Twist. It is called twist because high-priced areas always have strong negative sensitivity to it, and lowest-priced areas have the most positive sensitivity, so London < Birmingham < Newcastle. It is pretty much ‘North versus South’ - refer back to the sensitivities scatterplot.

It’s instructive to see the two cumulative factors together as a scatterplot, with dates to show how the factors move. We rotate around clockwise from the 4 o’clock position in 1996, and 1999 is close to 2015, then the last three years have really tracked the last cycle as the strong Butterfly propelled the mid-priced Midlands and Wales versus London and the North. The latest point has however dropped a bit off the trajectory, reflecting a lack of punch in Twist, which last cycle really took off in 2003.

The implication of ‘deficient Twist’ is that we have not seen as strongly as expected the late phase of the cycle - where mid-priced cities such as Birmingham and Manchester lose pace to Bolton, Cleveland and their ilk. Instead we are seeing modest moves in this direction, but muted in magnitude.

Here are just two contrasting scenarios for what happens next:

the far North is left behind and so fails to rebound or ‘catch-up’ as the last remaining sunset industries are shuttered and it falls ever deeper into regional recession until perhaps something dramatic happens in the future

Some political initiative, natural phenomenon or economic circumstance revitalises these left-behind regions once more and the rebound picks up rapidly into 2020-21

The punchy late-cycle far-North returns last time were fuelled by a speculative lending frenzy to borrowers in the lowest-priced regions. Many were subprime borrowers. We know how that ended, so we should be careful what we wish for.

Taking a step back, consider what this tells us. There is a cycle in relative value/momentum, somewhat like a wave motion. The cycle is clearer when time is resampled, compressing or ‘fast-forwarding’ those phases where the action takes place a bit slowly. If we accept this, then the forecasting method for relative performance becomes clear.

Identify the current phase position (to be quantitative, the current phase angle)

Project the cycle forward to the investment horizon

Bearing in mind that the rate of rotation is somewhat random, assign error bars to the forecast

In step 3 we could even think about it another way: at least in part, it’s not so much the relative performance that is random, but the timing (horizon) at which it will come to fruition. Successful modelling breaks a seemingly complex phenomenon into small, regular parts, and we have taken a step down that road.

It’s not so much about *what* will happen next, but *by when*.