'Contagion is best defined as a significant increase in cross-market linkages after a shock to an individual country (or group of countries), as measured by the degree to which asset prices or financial flows move together across markets relative to this co-movement in tranquil times.' — Rudiger Dornbusch, Yung Chul Park, Stijn Claessens in their research paper titled Contagion: Understanding How It Spreads
A lot has been written on the 'contagion effect' in financial markets, but to my disappointment there hasn't been much written on how to model the dynamics of the financial market contagion effect. Therefore, I rummaged through most of the reputable biological journals I could find, in search of key insights about modeling the 'contagion effect' of diseases that afflict biological organisms. My initial hope was that if I found something 'good', I would draw analogies from it, to assist in the calibration of the contagion effect in financial markets.
As luck would have it, I found a deterministic model for calibrating the potential effects of a contagious virus on a human population in a paper, by Jim Caldwell and Kei Shing Ng, titled:
Deterministic Model in Contagious Disease, which this post heavily draws analogies from. The original logical format and the notation of their work is preserved, with of course slight adjustments made, to make the model relevant to the matter of discussion, which in this case is the contagion effect in financial markets. Therefore, I must point-out from the onset that I am just a translator, not an originator of this model - I'm just adapting it to make it relevant to financial markets.
The model in this post is the ensemble of equations which describe and interrelate the variables and parameters of a physical system - which in this particular case, is a hedge-fund sub-sector under conditions of distress that originate from a few individual funds within the sub-sector, and eventually cascade throughout the whole sub-sector through the contagion effect.
Whilst this model may lack robustness, it is my belief that it is a concrete starting point, for formulation of a robust risk management tool that explains the dynamics of the contagion effect, and will hopefully help industry players to mitigate systemic risks that stem from their individual activities.
...The ModelFrom the outset, I have to mention that I designed the model from the perspective of a fund-manager, who wants to know how vulnerable he is to being wiped-out because of the effects of stresses that may be affecting other funds with an operational strategy similar to his.
I consider here a (relatively) homogeneously mixed group of hedge-funds, i.e hedge-funds with a common operational strategy and overlapping portfolios, of size
n+a under the assumptions that initially
a individuals individuals are under conditions of stress with the remaining
n individuals all being susceptible, but not yet 'infected' by the stress. This leads to the following classical basic model:
Let time
t be the independent variable,
I(t) and
S(t) be continuous, where:
S(t) = number of susceptible funds at time
t, and:
I(t) = number of funds under stress at time
tThe key assumption of this model is that the rate of occurrence of new infections (i.e the rate at which the stress spreads the other funds) is proportional to both the number of infectives (i.e funds already under stress) and the number of susceptibles (i.e funds with a high likelihood of experiencing the stress through the contagion effect owing to a portfolio overlap with the infected funds and the use of leverage), we can write:
I(t+Δt) = I(t) + βI(t)S(t)Δt (A1.1)
Where: β = infection rate (or contact rate). β is directly proportional to the average percentage of leverage (in relation to NAV) used by the affected hedge-fund sub-population, which is denoted by the letter q, and, β is also directly proportional to the level of portfolio overlap between the hedge-funds, which in this case is measured by computing the average level of correlation between the funds' portfolios and the index that tracks the performance of funds in the specified strategy(ies), denoted by the letter v. β is also directly proportional to the average volatility of the overlapping portfolio, or the most common basket of securities in the portfolios of different funds operating in susceptible hedge fund sub-community, which in this case is denoted by the letter e. Furthermore, β is also directly proportional to the aggregate amount of redemption requests as a percentage of aggregate AUM of all affected and susceptible funds, which in this case is denoted by the letter j. Therefore Mathematically expressed:
β = k.qvej, where
k is a constant
.Therefore, the equation (A1.1) reads; the cumulative number of funds under stress during the current time period I(t+Δt) is equal to the sum of the cumulative number of funds under stress during the last time period I(t) AND the product of: the infection rate β; the cumulative number of funds under stress during the time period before the current one I(t); the cumulative number of susceptible funds during the time period before the current one S(t), and the time that has elapsed between the previous time period and the current time period Δt.
In the limit as
Δt→0 , this yields:
With initial conditions
S(0) =n, I (0) = a
In addition, since the total population size is always
n + a, and all individuals are either susceptible or infected, it is clear that
S(t) + I(t) = n + a for all
t , which means that:
S(t) = n + a - I(t) and when you put
n+a - I(t) in
(A1.2) in the place of
S(t) it follows that:
Whenever there is a shock in financial markets, there are bound to be casualties (blow-ups), that reduce the population of hedge funds in general. To make our original model more realistic, we can extend it by including a third variable
R(t) to represent the number of hedge funds that are removed from the affected population at a given time
t.
Therefore, the following assumptions are made for this model: The removals include infectives who are dead or recovered and immune; The immune or recovered removals enter a new class which is no longer susceptible to the stressing condition.
Hence, let
R(t) = the number of removals at time
t and γ is the removal rate, which in this case is partially comprised of the observed rate of bankruptcy of affected hedge funds owing to the prevailing condition of stress - such that we now have:
I(t) + S(t) + R(t) = n,
Where
n is the total size of the affected hedge fund sub-community (i.e the strategy to which affected funds belong to).
Stay tuned for part 2.
Please Note: This model is inspired by a paper co-authored by Jim Caldwell and Kei Shing Ng, titled: Deterministic Model in Contagious Disease, and is going through a continual overhaul to make it more robust. So far I've had this model critiqued by Rick Bookstaber, author of A Demon Of Our Own Design: Markets, Hedge Funds And The Perils of Financial Innovation, and he highlighted a lot of flaws, in the model and its presentation, that I'm currently addressing. Dr. Ernest Chan, the author of a quantitative text titled Quantitative Trading: How to Build Your Own Algorithmic Trading Business is also in the process of critiquing the model, and I'll also take his opinions on board. Furthermore, Dr. Jim Caldwell, the originator of the model I drew analogies from, also gave me a few pointers on how to make the model more realistic, and I'll also be taking his pointers on board. I also benefited from Eric Falkenstein's critique of the model, and I will be taking his suggestions on board as well. Please stay tuned!
Where:
And also:
When you graph (A1.8), you get a curve that looks something like this:
Computing the maximum casualty count is achieved by taking the limit of the contagion effect as t →∞ in equation (A1.8) which gives the equation for approximating the maximum casualty count as:
In the Hypothetical graphical illustration of equation (A1.8), the maximum casualty count is shown by the 'leveled-out' section of the trajectory between points b and c
Integration yields:
Which can be expressed as:
When t = 0:
And thus,
As:
The right-hand side of the equation above can be expanded as far as the term in R squared to give:
However, the removals (i.e the blow-ups) should be considered in the differential equation for the number of infected hedge funds i.e equation (A1.2), which should be modified to:
The equation for the number of hedge funds removed from the infectives with removal rate γ then becomes:
At the start of the crisis, when t = 0 , we assume that there are no removals, a very small number of infectives, Io , and the remaining population is susceptible, S, which is approximately equal to n. Thus, at t = 0, (S,I,R) take the values (So,Io,0). For convenience, we make use of
