Hedgind Demand Deposit Liabilities: Market
Conditions, Term Structure of Interest Rates
and Risk Management
Umberto Cherubini
University of Bologna
Workshop: Interest Rate Risk Management
21 March 2011, Bank of Italy, Rome
Outline
• Interest rate risk: from ALM to risk mgmt
– Classical non-stochastic immunization theory
– Interest rate risk management
– Immunization with stochatic interest rates
• Quantity risk: modelling deposit demand
– Structural models (?) vs reduced form
– Dependence between demand deposit and interest rates:
a copula approach proposal
• Stress-testing and VaR measures
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 shif 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 JarrowVan 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
16
Vasicek density function
14
12
10
R h o = 0 .2
8
R h o = 0 .6
R h o = 0 .8
6
4
2
0
0
0 ,1
0 ,2
0 ,3
0 ,4
0 ,5
0 ,6
0 ,7
0 ,8
0 ,9
1
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 bayesian specification of the
interest rate law
• Cherubini and Della Lunga (1999) (but see also Kupiec,
1998), propose a bayesian model for risk evaluation, based
on: i) historical info; ii) scenarios
• They use Black-Litterman approach from asset
management to yield the posterior distribution of profits
and losses of a levered portfolio conditional on a set of
scenarios
• It turns out that stress-testing and VaR are the extreme
outcomes of the same from the same procedure: stresstesting is obtained when perfect confidence is assigned to
the precision of the scenario and VaR is obtained when no
confidence is given. Scenario based risk management is in
the middle.
An example of scenario
• Assume a model in which interest rates are the only state
variable (and determine deposit volumes). Suppose you
have historical information on the 1, 10 and 30 year
maturity of the Italian yield curve, summarized in the
statistics
é5.04ù
é0.24ù
m º ê5.39ú s º ê 0.11ú
ê
ú
ê
ú
êë 6.21úû
êë 0.11úû
0.86 0.81ù
é 1
R º ê0.86
1
0.97 ú
ê
ú
1 úû
êë 0.81 0.97
• A possible scenario is described by the statement:
– “The 1 year rate will rise to 6% (± 10 bp s.e.)”
The Black & Litterman approach
• In mathematical terms:
e1' r = q + 1 = 6% + 1 , 1 ~ N(0,0.001)
• …and from the joint distribution
V e1
ér ù  é m ù é V
ù

,ê
ú
ê ú ~ N  ê
ú
ë q û  ë e 1 ' m û ë e 1 ' V ' e 1 ' Ve 1    û 
• where V is the covariance matrix...
The conditional scenario
…we may compute the conditional scenario
(
r q ~ N  m  Ve1 s 12   12

)
-1
(6% - e1 ' m )
(
; V - Ve1 s 12
)
2 -1
  1 e1 ' V 
and new (posterior) statistics...
é6.00ù
é 0.01ù
m º ê5.77 ú s º ê0.06ú
ê
ú
ê
ú
êë6.58úû
êë0.07 úû
0.04 0.03ù
é 1
R º ê0.04
1
0.92ú
ê
ú
1 úû
êë0.03 0.92

The effects on the yield curve...
Figure 1
7
6.5
Yields
6
Historic
Scenario
5.5
5
4.5
1
6
11
16
Maturities
21
26
…and on the value of positions
Table 1. Stress testing report
Scenario: the short term rate increases to 6% (0.1 s.d.)
Maturity Nominal Cur. MtM Scen.MtM Mean Loss Scen. VaR
1 Year
100
95.08
94.18
0.90
0.92
10 Years
100
58.35
56.16
2.17
2.93
30 Years
100
15.54
13.88
1.66
2.30
MtM = Marking-to-Market Value
Mean Loss = Expected values of the difference between current and future MtM
values
Scen. VaR = 1% percentile of the difference between current and future MtM values
References (some…)
•
•
•
•
•
•
•
•
Cetin U. – R.A. Jarrow – P. Protter (2004): “Liquidity Risk and Arbitrage Pricing
Theory”, Finance and Stochastics, 8, 311-341.
Cherubini U. – G. Della Lunga (1999): “Stress-Testing Techniques and VaR Measures: A
Unified Approach”, Rivista di Matematica per le Scienze Economiche e Sociali, 22, 7799.
Dewachter H – M. Lyrio – K. Maes (2008): “A Multifactor Model for the Evaluation
and Risk Management of Demand Deposits”, working paper n. 83, National Bank of
Belgium
Jarrow R.A. – D.R. Van Deventer (1998): “The Arbitrage-Free Valuation and Hedging
of Demand Deposits and Credit Card Loans”, Journal of Banking and Finance, 22, 249272
Kalbrener M. – J. Willing (2004): “Risk Management of Non-Maturing Securities”,
Journal of Banking and Finance, 28, 1547-1568
Kupiec P.H. (1998): “Stress-Testing in a Value-at-Risk Framework”, The Journal of
Derivatives, 6, 7-24.
Nystrom K. (2008): “On Deposit Volumes and the Valuation of Non-Maturing
Liabilities”, Journal of Economic Dynamics and Control, 32, 709-756
Nawalka S.K. – G.M. Soto (2009): “Managing Interest Rate Risk: The Next
Challenge?”, working paper, www.ssrn.com/abstract=1392543