A hot topic in UK news over the last 12 months has been the so-called “cost of living”, i.e. how expensive it has become just to live (and survive) in the UK. This cost has changed significantly in recent years. Whilst economists and politicians can endlessly debate the reasons for this, the following three factors are often part of the conversation:
Brexit – how various infrastructure within the UK now operates differently post-Brexit compared to pre-Brexit, and how this has added extra costs to industry and production.
Covid – how the lockdown periods resulting from the Covid outbreak have permanently disrupted the employment market, supply chains, and consumer demand patterns.
Energy – how the wholesale cost of energy generation has increased, either due to forces of nature or international conflict, which impacts almost everything else in the economy.
The purpose of this article isn’t to measure the relative significance of these factors. Rather it is to use this situation as an example, to describe some more abstract principles behind the nature of cause and effect.
Observables and risk factors
The cost of living is an example of an observable (i.e. measurable) variable, which is dependent on multiple factors, each of which are themselves variable (as described above).Since nobody could anticipate ahead of time how those multiple factors would turnout, there’s no way anyone could have anticipated how our chosen observable would turn out.
But the cost of living is just one such example: everyday life is full of many situations where a measurable observation is dependent on multiple uncertain factors (which in business language we call “risk factors”). Other examples include:
· The arrival time of a familiar journey (e.g. a commute to work). This is dependent on both the departure time(which may be later than expected), and the traffic level on the transport network (which may be greater than expected).
· The waiting time for a doctor’s appointment. This is dependent on both the number of patients also seeking an appointment at that time, and the number of doctors employed or available at that time.
· The incremental growth achieved by a living plant in a certain time frame, dependent on both how much sunlight it receives, and how much rainfall it receives.
Components of change
Now for the quantitative angle. In each of these situations we might ask ourselves the following question: out of the total change that we can measure in our observable output, between any two instances or time points, how much of this change is driven by each of the changes in our various input factors? Are we able to decompose that overall change into distinct quantifiable components, which can be described or explained separately, by attributing it to each of the factors which contribute to it?
Why might we care about this? Sometimes it is simply about apportioning credit or blame; e.g. am I late for work mainly because I left home late, or mainly because of extra traffic, or equally due to both? At other times it is about wanting to anticipate what future observations of the same variable might be, if the same input factors were to change again, perhaps in different combinations, or by much larger amounts than have been seen previously.
For example, suppose we have observed the wait time for a doctor’s appointment on several different days, and we know what the state of each input factor was on each day (i.e. both the number of patients and the number of doctors). Compared to our most recent observation, how much longer than this would the wait time hypothetically be, if the number of patients increased by a further 10% and the number of doctors decreased by a further 5%. Can we reliably estimate this from the data we already have?
To help answer these questions, we need a quantitative “model” for our observable. A model is a scientific theory, a formula, or a computational algorithm, which describes how our observable behaves in response to different inputs. In particular, we can plug different combinations of inputs into our model, and simulate the hypothetical outputs. This allows us to consider alternative scenarios for our observable to the one that we have experienced in reality.
In many cases, such an algorithm can be quite complex, meaning that it is practical to compute it for only a small number of scenarios, but impractical to compute it for a much larger number of scenarios. Therefore it helps if we are able draw wider conclusions about the behaviour of an observable from computing a relatively small number of scenarios.
First order risks
The first thing we might try to do is examine the impact of each input in isolation, in the absence of any other changes. So we might use our model to ask the following:
· How much would the cost of living situation have differed from normal, if only Brexit had happened (but neither Covid nor increased energy costs)?
· How much would the cost of living have changed, if only Covid had happened (but neither Brexit nor increased energy costs)?
· What would the cost of living be, if only energy costs had increased (but neither Brexit nor Covid had occurred)?
What we typically find is that the overall impact is greater than the sum of these individual impacts. We might quantify “cost of living” in terms of average consumer cost per year. And we may be able to quantify the answers to the preceding three questions, i.e. the additional cost of living in each case. But adding up these three additional costs, will still give us a significantly lower value than the impact we would experience with all three factors occurring simultaneously. We say that there remains a “residual impact”, which is not explained by these “first-order effects”. This exists even if there are no other causation factors outside of the ones which we’re looking at.
We might reasonably ask: why is this, and how can we further account for this residual impact?
Second order risks
To proceed further we have to try and measure “second-order effects”, i.e. the impact that one factor has on the impact of another factor. This is more complex but is often quite significant!
If we measure the extent to which the cost of living is worse under Brexit than in the absence of Brexit, we will find that the size of this impact is itself greater in the presence of Covid than in the absence of Covid. Likewise, the extent to which the cost of living is worse with increased energy costs than with normal energy costs, is greater in the presence of Brexit than it would have been in the absence of Brexit. And finally, the extent to which Covid has impacted the cost of living, is greater with increased energy costs than it would be with normal energy costs. The sensitivities to each of the three factors are all dependent on each other!
To quantify all of this, we have to use our model to compute the impact of one factor on our observable, under two different scenarios: i.e. both in the presence and in the absence of a second factor. Doing this requires evaluating our model in a total of 4 different scenarios:
a) without a change in either the first or second factor,
b) with a change in the first factor but not the second,
c) with a change in the second factor but not the first, and
d) with a change in both first and second factors.
We can then perform a sequence of subtractions on these four model outputs, to compute the difference in the impact of the first factor, between two scenarios of the second factor.
This calculation gives us a measure of the “co-dependence” of the observable on those two factors – in finance language we call this a “cross-gamma”, and in mathematics language we call it a “mixed partial derivative”. There is one of these measurements that could be made for each pair of factors that we consider, so in a situation with three input factors there are three distinct pairs on which we could make such a measurement.
This count of the number of these metrics increases quite rapidly (in mathematical language “quadratically”)with the number of factors, e.g. with just two factors there is only one possible pair and hence one possible cross-gamma metric; with five factors the number of cross-gamma metrics increases to 10.
For many practical situations, these cross-gammas exhibit a nice symmetry, meaning that we don’t need to measure both possible orientations within the same pair of factors. Using our example above, the impact of Brexit on the impact of Covid is exactly the same as the impact of Covid on the impact of Brexit (if measured in equivalent units). This symmetry can be far from intuitive: it is rarely obvious that these two measurements are even meaningfully comparable. However for most reasonably-behaved observables this equality holds true.
Reducing the residual
Going back to our three-factor example (the cost of living), we can now try adding up our separate impacts again, but including all these additional components – so we now have six components in total:
· The impact of Brexit alone
· The impact of Covid alone
· The impact of energy costs alone
· The impact of Brexit on the impact of Covid
· The impact of Brexit on the impact of energy costs
· The impact of Covid on the impact of energy costs
If we add up these six components, we will find that the total more closely approximates the observed change in cost of living when the three factors have changed simultaneously. This means we have done a better job than previously of accounting for the change in our observable. There is still a residual impact, which is not explained by any of these six factors, but it is typically much smaller and can often be neglected.
Pure gammas vs cross gammas
A special case of the above concept occurs when we consider the same factor twice within the same pair, i.e. computing the impact that changing a factor has on the sensitivity of the same factor. This isn’t meaningful for binary factors, which either occur or don’t occur, but is very relevant for continuous factors which can vary by different amounts, and where the dependency is nonlinear.
For example in the “doctor’s wait time” observable, the degree of sensitivity to the number of patients is small when the number of patients is much lower than the number of doctors (i.e. one additional patient will not impact the average waiting time)but the very same sensitivity metric is large if the number of patients exceeds the number of doctors (the same additional patient would then need to wait fora busy doctor to become available).
An example with numbers
We can illustrate all of this numerically using the plant growth example mentioned earlier, where the incremental growth is dependent on both sunlight and rainfall. Let’s suppose we have a very simple mathematical model that tells us the amount of growth for a given value of sun and rain, based on the following formula:
Growth increment = (3 x sunlight + 2 * rainfall) ^ 3
And further suppose that we make two observations, one with sun value 1.0 and rain value 1.0,yielding a plant size of 125, and another with sun value 2.0 and rain value 2.0,yielding a plant size of 1000. We can likewise also make some intermediate model calculations at the following combinations of inputs:
· Sun value 1.01, Rain value 1.00
· Sun value 1.00, Rain value 1.01
· Sun value 1.01, Rain value 1.01
Then by using these states of the model to consider the various sensitivities (both one-factor and two-factor), we can apportion the overall plant growth from 125 to 1000 in the following manner:
· Proportion of growth due to sunlight alone: 26%
· Proportion of growth due to rainfall alone: 17%
· Proportion of growth due to impact of sunlight on further sunlight: 15%
· Proportion of growth due to impact of rainfall on further rainfall: 7%
· Proportion of growth due to impact of sunlight on impact of rainfall: 21%
· Proportion of growth which remains unexplained by any of the above: 14%
Conclusion
There is an important application of all the above theory, within the field of finance. When financial investments either gain value or lose value, quantitative analysts try to apportion this gain or loss into the various market factors that drive it, such as changes in interest rates, changes in stock prices, or changes in volatilities & correlations. The principles used are exactly the same as described above, and often involve measuring second order effects, or cross-gammas, between different market factors. This process is known as P&L Explain (or P&L Attribution), because it seeks to explain (or attribute) in quantifiable terms where a profit or loss has arisen from.
This helps risk managers to understand past financial performance so that financial strategy (i.e. trading strategy) can be assessed and improved. It also enables them to estimate profit/loss in real-time as market movements occur, even for highly complex assets where there might not be time to calculate a computationally expensive full valuation.
If you would like help with managing market risk or measuring asset performance, you may consider seeking out specialist quantitative analytics skills. RocketFin is a leader in this field, offering advanced consulting and analytics services that can help maximise your quantitative capability.
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