Funding Liquidity Risk

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• Funding Liquidity Risk
• Advanced Methods of Risk Management
• Umberto Cherubini

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Learning Objectives

• In this lecture you will learn
• To evaluate and hedge funding liquidity risk
• To understand concepts, measures and effects of
  market liquidity risk.

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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 (wont 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)

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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.

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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

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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)

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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.

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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)

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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.

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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.

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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

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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.

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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.

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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.

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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

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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

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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).

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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,ti1) 1.
• The difference between Ri Di 1 and Si increases
  or decreases the amount of debt Di.

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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

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Market liquidity measures
Risk Measure Dimension
Breadth Bid-ask spread Price
Depth Slippage Quantity
Resiliency Autocorrelation Time
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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

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Slippage example
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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.