Funding Liquidity Risk
Advanced Methods of Risk Management
Umberto Cherubini
Learning Objectives
• In this lecture you will learn
1. To evaluate and hedge funding
liquidity risk
2. To understand concepts, measures
and effects of market liquidity risk.
The credit crisis and liquidity risk
• If you do not trust your neighbour and do not trust
your assets, you are in liquidity trouble
• Funding liquidity risk: you must come up with
funding for your assets, but the market is dry.
Solutions: i) chase retail investors ii) rely on
quantitative easing (won’t last long)
• Market liquidity risk: you are forced to unwind
positions in periods of market stress, and you may
not be able to find counterparts for the deal,
unless at deep discount. Solution: quantitative
easing (place illiquid bonds as collateral)
Classical immunization: flows
• Maturity gap: banks lending on different
(longer) repricing periods than liabilities are
exposed to reduction of the spread earned
when interest rate rises.
• Cash flow immunization would call for
maturity matching. Assets should be have the
same repricing period of liability, or, deposits
should be hedged by being rolled over at the
short term rate.
Classical immunization: value
• Fisher – Weil: close the duration gap
– Immunization against parallel shifts
– Zero-coupon liability
• Reddington: keep an eye on convexity
– Immunization against parallel shifts
– Convexity of liabilities lower than that of assets
• Fong – Vasicek: the kind of shift matters
– Immunization against whatever shift
– Lower bound to losses positive or negative given
convexity of the shift
IRRM = ALM  risk management
• Asset-Liability-Management is about sensitivity of
balance sheet income and value to changes in the
economic scenario (ALM requires scenarios)
• Value-at-Risk is a matter of (i) time and (ii) chance. It
may be traced back to the system of margins in
derivatives markets.
• Stress-testing is a matter of information. We evaluate
the effect of a set of scenarios on a portfolio and the
amount of capital.
• Notice: ALM and risk management have in common
scenarios. Integration of the two (that we call interest
rate risk management requires to work on this
intersection)
Hedging by swaps
• Classical immunisation was non-stochastic and it was
not based on a model of the banking system.
• Jarrow and Van Deventer (1998) devised a model
with stochastic interest rates, market segmentation
and limited competition among banks, so that the
interest rate spread between the risk free rate and
the rate of deposits was allowed to be positive.
• In this case the present value of the spread adds to
the value of deposits, and may be read as the net
present value of a swap contract. In this case
hedging would require shorting this swap, and perfect
mathching would not work.
Extensions
• Return from maturity transformation. Assume
deposits are invested in long term (risk free)
assets. Then, the value of deposit would turn
into a CMS and would exploit a convexity
adjustment bonus.
• Swaptions. One could conceive contingent
hedging, triggered by market conditions, in
which case one should resort to receiver
swaptions (put options on swaps)
Basis risk
• In the standard model, it is often assumed that deposits are perfectly
correlated with the risk free rate, so that the hedging resolves in a
replication of a swap contract by positions in the risk-free bond market.
• Basis risk. An extension that seems mandatory in face of the recent
banking crisis is to allow for other elements determining the wedge
between risk free rates and rates on deposits. Following the same line
of Jarrow and Van Deventer model one should include other market
variables, first of all an indicator of the credit worthiness of the banking
system as a whole.
• A possible financial engineering could be buying insurance against the
increase in CDS spread in the banking system, or making the swap
contract “hybrid”.
Quantity risk
• What makes demand deposit hedging quite peculiar is quantity
risk. Since deposits can be withdrawn with no notice, returns on
assets and liabilities may fluctuate not only because of changes
in market rates, but also changes in the amount of deposits on
which this spread is computed. For this reason the swap
contract in the Jarrow-Van Deventer approach has a stochastic
amortizing structure.
• The problem is to model: i) the distribution of demand deposit in
each period of time; ii) the dependence structure between the
amount of deposits and interest rates.
• In a sense, it is the old problem of liquidity trading vs informed
trading.
Modelling deposit demand
• Structural models: these models should be based on the
micro-economic structure of demand deposits at the
individual level, followed by aggregation at the industry
level
• Reduced form models: these models should be based
on statistical regularities observed on the distribution and
the dynamics of the aggregate demand deposits.
• Notice. This distinction is new, but is motivated by the
similarity between quantity risk and credit risk
Structural models
Example from the literature
• A structural model coming from the academia is
Nystrom (2008).
• Each individual demands transaction balances and
demand deposits as a function of:
– i) income dynamics
– ii) a target deposits/income ratio
• The key point is that the target ratio is a function of the
difference between the deposit rate and a reservation
(strike) price.
• Aggregation is obtained by averaging income dynamics
and dispersion around average behavior is modelled by
selecting a distribution function of the strikes.
Structural models
Example from the industry
• A major Italian bank is pursuing a policy of buying and selling its
bonds at the same credit spread as the placement day. This way, the
bonds issued by the bank are substitute of deposits from the point of
view of customers.
• In the evaluation of this policy, the bank relies on a behavioral model
according to which:
– the customer decision to sell and buy the bond is triggered by the
difference between the current spreads prevailing on the banking
system and the original spread (a real option model, like that of
Nystrom)
– customers are assumed to be sluggish to move in and out,
because of irrational exercize behavior or monitoring costs. This is
modelled by multiplying the spread difference times a participation
rate lower than one.
Reduced form models
• Specification of deposits demand is based on
statistical/econometric analysis.
• Typical specification:
– Linear/log-linear relationship with the interest rate
dynamics
– Autoregressive dynamics
• What is missing: would be interesting to
include a liquidity crisis scenario using the
same technology applied by Cetin, Jarrow
Protter (2004) to market liquidity risk.
A copula based proposal
• A natural idea stemming from the similarity
between the demand deposit problem and
large credit portfolio models is to resort to
copulas.
• Copula functions could provide:
– Flexible specification of the marginal distributions
of deposits and interest rates
– Flexible representation of the dependence
structure between deposits demand and interest
rates
– Flexible representation of deposits dynamics
A copula-based structural model
• Assume a homogeneous model in which all agents have the same
deposit income ratio and same correlation with an unobserved
common factor.
• Possible specifications are Vasicek model (gaussian dependence)
or Schonbucher (Archimedean dependence)
• These specifications would yield the probability law of the deposit
income ratio that could be used as the marginal distribution for
deposits.
• The dynamics would be finally recovered by applying the dynamics
of income to the ratio.
• Notice: this is conjecture. Everything should be proved in a model
built on micro-foundations, and probably different specifications
would come out
A copula based algorithm
• Estimate the dependence structure between deposit
volumes and interest rates (moment matching, IFM,
canonical ML) and select the best fit copula:
– Notice. The conditional distribution of deposit volumes is the
partial derivative of the copula function.
• Specify the marginal distribution of deposit volumes
(the structural model above or a non parametric
representation).
• Specify the marginal distribution of interest rates: the
distribution may be defined on the basis of historical
data and/or scenarios (we suggest a bayesian
approach).
A liquidity model
• Assume that an obligor issues a long term
bond for an amount D0. The bond expires in N
periods.
• The curve of the obligor is v(t0,ti)
• In every period, the obligors receives net cash
flows Si, and it pays interest rates on debt Ri =
1/v(ti,ti+1) – 1.
• The difference between Ri Di –1 and Si
increases or decreases the amount of debt Di.
Market liquidity
• Market liquidity impact on prices:
– Difficult to compare prices on different
markets (best execution)
– Illiquid markets reduce transparency of
prices
Illiquid markets  Noisy information
Market liquidity measures
Risk
Measure
Dimension
Breadth
Bid-ask spread
Price
Depth
Slippage
Quantity
Resiliency
Autocorrelation
Time
Market liquidity measures
– Bid-ask spread: difference btw the price at
which it is possible to buy or sell a security
(does not take into account the dimension of
transaction)
– Slippage: difference btw execution cost of a
deal and bid-ask average (mid price). Takes
into account dimension. Bigger orders “eat” a
bigger share of the order book.
– Resiliency: time needed to reconstruct the
book once that a big order has eaten part of it
Slippage example
140
120
100
Quantità
80
60
40
20
0
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
Slippage
0,2
0,3
0,4
0,5
0,6
Prudent valuation and AVA
• The most recent regulatory innovation is the
conservative analysis of pricing.
• Under the new accounting standard, banks are required
to evaluate at fair value the trading book. So every time
that losses are marked-to-market, they are deducted
from the economic balance.
• The new regulation requires that capital is allocated
against wrong valuation of the trading book. The
difference between fair value and conservative valuation
is called AVA (additional valuation adjustment) and
capital is allocated to hedge this evaluation risk.