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Funding Liquidity Risk Advanced Methods of Risk Management Umberto Cherubini

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In this lecture you will learn

To evaluate and hedge funding liquidity risk

To understand concepts, measures and effects of market liquidity risk.

Learning ObjectivesIf 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)

The credit crisis and liquidity riskMaturity 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: flowsFisher – 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

Classical immunization: valueAsset-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)

IRRM = ALM risk managementClassical 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.

Hedging by swapsReturn 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)

ExtensionsIn 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”.

Basis riskWhat 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.

Quantity riskStructural 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

Modelling deposit demandA 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 literatureA 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.

Structural models Example from the industrySpecification 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.

Reduced form modelsA 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 proposalAssume 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 structural modelEstimate 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 copula based algorithmAssume 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.

A liquidity modelMarket 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

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

Market liquidity measures

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.

Prudent valuation and AVA
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