1. Collateral Re-Hypothecation in a Networked Economy
Eric Tham
tham@nus.edu.sg
Date: December 28, 2013
1 Abstract
In September 2013, the Bank of International Settlements (BIS) recommended the re-use (re-hypothecation
or re-pledge) of collateral at the suggestions of the industry players. Whilst this can lower the costs of col-
lateral in trading amongst different counterparties, it may result in de-stabilising and greater contagion in
the financial systems. In this paper, the network model of firms in Elliot et al. [2013] is adapted to model
hypothecated collateral amongst a network of firms. A two stage model to model this re-hypothecated
collateral effects is also introduced with the second stage following a Google pagerank matrix. Model
inferences show that in the hypothecated model network, firms with high leverage actually decrease the
network firm of values as the system is ’pushed’ closer to default. In the re-hypothecated collateral
model, it is inferred that firms that do not re-hyothecate nor accept hypothecated collateral are actually
subsidising other firms that do. Further work needs to be done on how the contagion differs between the
two models.
2 Background
In the aftermath of the financial crisis, a series of financial regulations were instituted to
strengthen the financial system. One aspect of these regulations the EMIR in Europe and Dodd
Frank in the USA promotes mandatory clearing through central clearing parties (CCPs) for sufficiently
standardised OTC derivatives. By the end of 2012, between 40% − 50% of interest rate derivatives are
cleared through a CCP -BOE [2013]. CCPs promotes market transparency and the cross netting of
risks. However, the number of CCPs dominant in the market is few in particular LCH, ICE and CME1
accounts for most volume and range of OTC derivatives being cleared. This gives rise to a systematic
risk that of the too-big to fail CCPs. CCPs encourage financial stability through initial margining
of trades and a default fund where counterparty losses are mutualised. These funds are collateralised
through normally low risk government bonds, a list of which is publicly available on the CCPs websites.
Collateral is not only used amongst CCPs and their trading counterparties, but also for non-standard
OTC trades not cleared by CCPs.
It is postulated this increased use of collateral will result in a shortage of high quality col-
lateral for the functioning of the $630 trillion derivatives markets. This shortage of collateral in turn
potentially stifles hedging activity and economic growth. For this reason, financial players have suggested
the hypothecation of collateral or the re-use of collateral amongst trading counterparties. This rule was
finally published in the latest reform of the banking regulation by the Basel Committee and the central
bankers in September 2013 - BIS [2013], which allowed the hypothecation of collateral once. Whilst
re-hypothecation of collateral increases the availability of quality collateral, it may increase contagion,
as counterparties hold the ’same’ assets twice.
In this paper, we modify the model in Elliot et al. [2013] for two cases. The first case is similar
to the original model cited and is being used for a hypothecated collateral model. In the second case,
1These stand for the London Clearing House, Inter Continental Exchange and the Chicago Mercentile Exchange.
1

2. we build upon the first model for a two stage model of re-hypothecated collateral. In the second stage
of this model, we introduce a Google pagerank-like matrix to compute f values. As far as the authors
are aware, this is new in the literature. The merits and implications of the model are substantiated. We
conduct some simulation and compare some results from these two models. To conclude, further areas
of work are proposed. A literature review is first outlined.
3 Literature Review
The literature for financial networks modelling has grown in recent years. In an early paper
by Eisensenger and Golub [2001], the existance of a unique payment vector was proven via fixed point
theorem. This payment vector is one that clears the financial network of its obligations. Later papers
examine the different network structures and its effect on contagion. These include an independent
study by Tahbaz-Salehi et al. and related studies by Gourieroux et al. [2012]. These study the effects
of different networks structures on how the shocks propagate through a network based on debt holdings
or inter-bank lending. Main results from these papers indicate that for moderate shocks, a perfectly
diversified holdings is optimal while for large shocks, a diversified network is the worst in terms of
contagion. The results from Elliot et al. [2013] for random shocks also indicate that up to an optimal
point, diversification increases contagion as mutual assets are ’spread across’. Beyond this optimal point,
the financial network increases stability as shocks are absorbed by the network.
The intial focus of our paper is to examine how firms with re-hypothecated collateral affects
their value and other firm values. As the concept of re-hypothecated collateral is still new, it is expected
that initial cross holdings of re-hypothecation to be sparse. Many firms may not accept re-hypothecated
collateral or allow their collateral to be re-hypothecated.
4 Model for Hypothecated Collateral
The model derived in Elliot et al. [2013] is:
V = ˆC(I − C)−1
(Dp − βIυ<υ) (1)
where:
• V = n × 1 matrix of firm values
• ˆC = n × 1 matrix of firm ’safe’ assets not hypothecated
• C = n × n cross holdings matrix of hypothecated collateral
• I = n × n identity matrix
• D = n × m matrix of n firm holdings of m assets not pledgeable as collateral
• p = m × 1 vector of primitive assets not pledgeable as collateral
• β = n × 1 vector of firm bankruptcy costs
• Iυ<υ = n × 1 indicator variables if firm values υ breach thresholds υ for bankruptcy.
Both ˆC and C are in fractional forms expressed on a vector Q of Tier 1 capital (cash, high quality bonds)
the firms have.
The derivation of this equation is similar as in Elliot et al. [2013] and will not be repeated here.
Whilst in the original paper, C indicates firm’s cross holdings matrix of shares in one another, here
the matrix C represents the network of collateral firms have against one another. C and p represent
the full ’asset’ value of a firm, where C represents the ’safer’ assets allowed for colateral use. This can
refer to cash, investment grade bonds and gold. The p refers to the trading book values including the
interest rate swaps, FX swaps, commodity products that a bank typically trades either on the exchange
or with OTC counterparties. Notice that p can be potentially be very large with the different number of
assets and trades a firm holds. However, p can be understood as the ’risk deltas’. These deltas could be
the interest rates risks, different commodity risks, equity risks or credit risks through the asset matrix D.
The bankruptcy vector β indicates the liquidation and sell-off costs if the firm declares bankrupt.
2

3. This happens when its value υ goes below υ. There is thence a discontinuity in value with a firm
bankruptcy.
To illustrate a raw cross holdings matrix, suppose:
M =




1 2 3 4
1 0 5 0 3
2 2 0 2 4
3 3 0 0 4
4 6 6 3 0




Firm 1 has pledged 8 units of collateral (5 unitds to firm 2 and 3 units to firm 4) and received 11
units (2 units from firm 2, 3 units from firm 3 and 6 units from firm 4) of collateral. The vector receipts
for firms 1 to 4 is (11, 11, 5, 16). The C matrix is fractionalised on collateral pledged to the amount of
collateralisable assets - Q = (20.0, 15.0, 9.0, 16.0) 2
From vector Q, firm 1 has the most resilient capital
position. C=




1 2 3 4
1 0 0.25 0 0.15
2 0.13 0 0.13 0.27
3 0.33 0 0 0.44
4 0.38 0.38 0.19 0




The ˆC of collateral refers to the fractional amount of cash/ quality bonds that is not pledged and
retained within the firm. This can be set aside for ’internal buffer’ as capital requirements. 3
. For this
case, ˆC = (0.6, 0.467, 0.222, 0.063). High values of ˆC will necessarily make firms more stable and less
likely to bankrupt.
4.1 Algorithm for contagion cascade
The algorithm for contagion cascade via the hypothecated model is proposed as below:
1. Apply shock(s) to the vector p of asset prices sufficiently for firm(s) value υ to decrease below υ.
2. Apply bankruptcy costs β to bankrupted firm(s) values.
3. Re-compute V by equation (1).
4. If no new firm(s) is bankrupted, the algorithm stops. Otherwise apply bankrupcty costs and return
to step 2.
Note that D indicates the different exposure risks firms have to the asset vector p. For example, a
shock in the short term interest rates can have different impacts on firm values compared to emerging
markets interet rates. A firm value Vi not only depends on its own exposure to the different asset classes,
but also through the collateral (debt) pledged with it, and that it pledged.
4.2 Model Inferences
Some simulations are done to test the model parameters on the firm values. Aside from intuitive impli-
cations that a firm’s capital and asset position increase its value, there are also interesting implications
from the network cross matrix C.
1. A firm impact on a network of firm values depends on its leverage.
(a) The reduction in debt of a firm that has a negative collateral position and weak capital position
increases its value and makes an overall positive value gain on the financial network. Vice
versa is true if the heavily indebted firm increases its debt.
(b) The reduction in debt of a firm that has a positive collateral position and strong capital
position increases its value and makes an overall negative value loss on the financial network
up to a certain optimal point. Vice versa is true if the well capitalised firm increases its debt.
2This is a subtle difference from Elliot et al. [2013]. In Elliot et al. [2013], the assets are summed and fractionalised over
the column total, whilst in our paper, the row sums are fractionalised over total amount of cash/ capital. Doing so reflects
the capital position of the firms, whilst summing up by columns would reflect the amount of collateral debt it receives.
3Under the Basle Accord requirements, banks typically retained 8% of risk weighted assets for capital requirements.
3

4. Observe that in matrix M, firms 1 and 2 have a net positive collateral position while firms 3 and 4 have
net negative position. When firms 1 and 2 increases its leverage (debt position), its net values decrease.
The collateral pledged however has an overall positive effect on the network firm values, compared to
firms 3 and 4 having a negative effect.
Figure 1: Well capitalised firm impact on network values
The figure shows that for a well capitalised firm, increasing its leverage increases total network value
beyond a certain point when net debt position increases. Decreasing its leverage have the opposite effect
of ascribing less ’value’ to other netowrk firms.
Figure 2: Under capitalised firm impact on network values
In a firm that is already highly leveraged, reducing or increasing its leverage has a higher absolute
impact on network of firms values. Increasing its leverage brings down the network firm values, while
decreasing its leverage has an overall positive impact on network firm values.
5 Two stage re-hypothecated Collateral Model
In the two stage re-hypothecated model, there are two equations with an additional sub-equation that
models the secondary interactions among the re-hypothecated collateral.
V = ˆC(I − C)−1
(Dp − ∆V − βIυ<υ) (2)
∆V = ˆC + (I − ˆC )C ∆V (3)
The equation (2) is similar to equation (1) except for a ∆ V term. The other terms are:
1. ˆC = n by 1 matrix of proportion of pledged collateral that is not re-hypothecated. It is surmised
that there will be regulatory rules on this proportion.
2. I = n by 1 ones matrix
4

5. 3. C = n by n cross holdings matrix of re-hypothecated collateral that is being re-hypothecated by a
firm.
In this two stage model, a firm derives value from the network of re-hypothecated collateral from the
second stage as well as its asset values and hypothecated collateral values at the first stage. Observing
equation (3), economics intution is that re-hypothecation can potentially change the firm value through
its cross-holdings of re-hypothecated collateral (the right hand side of the equation) and the portion of
the hypothecated collateral that is not kept within the firm.
Assuming the case when there is re-hypothecation allowed, C = 0 and ˆC = I such that ∆V = 0. The
second equation then collapses to the original first stage equation 1 with no re-hypothecation.
5.1 Relation to Google PageRank
In our previous discussion on the hypothecated collateral, the model infers that the network value
increases for well capitalised firms that increases its collateral pledges. Firms ’more integrated’ to this
well-capitalised firm increases greater in value than other second degree firms, through a direct ’link’.
The form of this secondary matrix is also similar to the weighted Pagerank equation, which is damped
by a damping factor.
5.2 Algorithm for two-stage contagion cascade
The algorithm for a two-stage contagion cascade follows economic intuition in that the rehypothecated
collateral is first unwound in the event of a default. The unwounding of the re-hypothecated collateral
then changes the ∆V which is substituted back in the first stage equation. Except for this unwounding,
the two stage cascade algorithm is similar to the original algortihm. Note that the asset values of the
firm Dp does not come into the second stage equation. This is firstly to avoid double-counting and at the
second stage, the firm has only an indirect exposure from the original debtor from the re-hypothecated
collateral .
1. Compute firm values V from equations in the two stages.
2. Apply shock(s) to the vector p of asset prices sufficiently for firm(s) value υ to decrease below υ.
3. Apply bankruptcy costs β to bankrupted firm values.
4. In the secondary cross-holdings matrix, set the collateral that originates from the bankrupted firm
to = 0 Suppose firm 3 hypothecates collateral to firm 2, which in turn re-hypothecates the collateral
to Firm 1. Firm 3 goes bankrupt. Set the re-hypothecated collateral from firm 2 to firm 1 to = 0.
5. Re-compute ∆V by equation (3) by an iterative process, and substitute back into equation (2) to
calculate new firm V .
6. If no new firm(s) is bankrupted, the algorithm stops. Otherwise apply bankrupcty costs and return
to step 3.
5.3 Model Results
The model is used to study the impact on firm that do not re-hypothecate its collateral nor accept
re-hypothecated collateral. A firm also may not allow its collateral to be re-hypothecated. Although
the Basel Committee has permitted re-hypothecation of collateral, these risk-adverse firms are likely to
curb this re-hypothecation in its portfolio, in a perception that it increases credit risk for the firms. The
equation (3) is solved in Python through an linalg.solve in the numpy library.
1. Continuing from the earlier section on hypothecated collateral, suppose firm 1 does not re-hypothecate
nor accept re-hypothecated collateral. A hypothetical secondary cross-holdings matrix is then:
M =




1 2 3 4
1 0 0 0 0
2 0 0 2 2
3 0 0 0 3
4 0 3 0 0




5

6. With the C : =




1 2 3 4
1 0 0 0 0
2 0 0 0.18 0.18
3 0 0 0 0.6
4 0 0.273 0 0




Hence,
ˆC = (1, 0.73, 0.82, 0.22) (4)
Whence, for firm 4, 0.73 = 1−0.273 = 3/11 where 11 is the total hypothecated collateral that firm
4 receives and 3 is the amount of collateral it re-hypothecates. The rest of the matrix is similarly
obtained. Solving for ∆V ,
∆V = (−16.5, 10.5, 6.6, 0.39) (5)
Notice that, compared to firms 2 to 4, firm 1 which does not re-hypothecate nor accept re-
hypothecated collateral has decreased in value. This is an intuitive result since a firm can ’gain
value’ through ’re-using’ its collateral. The greatest gain in value occurs for firms 2 and 3 which re-
hypothecates a substantial portion of its received collateral. This gain is offset by the loss in value
of the firm 1. The weighted values sum(∆V ) = 1 which is multiplied by each re-hypothecated col-
lateral value for the absolute value. Firms which do not re-hypothecate nor accept re-hypothecated
collateral thence ’subsidise’ the other firms.
6 Further Work and Conclusion
This paper has adapted Elliot et al. [2013] for a hypothecated collateral network model and
introduced a new model for re-hypothecated collateral in a networked economy. From our model, it is
inferred that net effect of a high leveraged firm on the network of firm values is negative, as the system is
’pushed closer’ to default. In addition, a well capitalised fim increasing its leverage has a ’positive effect
on other firm values’ as it imparts positive economic value to other firms.
We also develop models for contagion in the hypothecated and re-hypothecated collaterals. Fur-
ther work is needed especially to compare how contagion spreads through these two models and also the
probability of first default on the system. A shock on assets correlation matrix D instead of a singular
shock on an asset class is more appropriate since firms are usually exposed to certain key risk deltas. The
network re-hypothecated model also postulates that a firm that does not re-hypothecate its collateral is
in a disadvantage compared to other firm that do. It is effectively subsidizing other firms.
6

7. References
BIS. Margin requirements for non-centrally cleared derivatives. https://www.bis.org/publ/bcbs261.
pdf, September 2013. [Online; accessed Nov-2013].
BOE. Otc derivatives: new rules, new actors, new risks. bank of england financial stability review. April
2013. [Online; accessed Nov-2013].
Larry Eisensenger and Thomas Hoe Golub. Systemic risk in financial systems. Management Science, 47,
February 2001.
Matthew Elliot, Benjamin Golub, and Matthew O. Jackson. Financial networks and contagion. Septem-
ber 2013. [SSRN id 2175056].
Christian Gourieroux, Alain Montfort, and Heam JC. Bilateral exposures and systematic solvency risk.
Canadian journal of economics, 45, 2012.
A. Tahbaz-Salehi, Acemoglu D., and Ozdaglar A. Systematic risk and stability in financial networks.
mimeo.
7