Hedging Demand Deposits Interest Rate Margins Risk Management and Financial Crisis Forum March 19th 2009. Mohamed HOUKARI
[email protected]
Alexandre ADAM, BNP Paribas Asset and Liability Management Mohamed HOUKARI, ISFA, Université de Lyon, Université Lyon 1 and BNP Paribas ALM Jean-Paul LAURENT, ISFA, Université de Lyon, Université Lyon 1
PRESENTATION OUTLOOK
Modeling Framework, Objective and Optimal Strategy
Empirical Results
Conclusions
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Demand Deposit Interest Rate Margin – Definition
Demand Deposit Interest Rate Margin for a given quarter:
Income generated by the investment of Demand Deposit Amount on interbank markets while paying a deposit rate to customers
Risks in Interest Rate Margins:
Interest Rate Risk:
1. Investment on interbank markets
2. Paying an interest rate to customers (possibly correlated to market rates)
3. Demand Deposit amount is subject to transfer effects from customers, due to market rate variations
Non hedgeable Risk Factors on the Deposit Amount:
Business Risk: Competition between banks, customer behavior independent from market conditions, etc.
Model Risk
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Setting the Objective Interest Rate Margin
IRM g (K T , LT ) = K T (LT − g (LT )) ⋅ ∆T
Deposit Amount at T Investment Market Rate during time interval [T,T+∆T] Customer rate at T
Mean-variance framework:
Including a return constraint – due to the interest rate risk premium
[
min E IRM g (K T , LT ) − S S
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]
2
under constraint
[
]
E IRM g (K T , LT ) − S ≥ r
Hedging Demand Deposits Interest Rate Margins
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Dynamics for Market Rate
Lt = L(t , T , T + ∆T )
Libor Market Model for Investment Market Rate
dLt = µ L dt + σ L dWL (t ) Lt
µL ≠ 0
Ex.: Brace, Gatarek, Musiela (1997)
Long-Term Investment Risk Premium
Coefficient specification assumptions:
Our model: µ L , σ L constant (and can be easily extended to time-dependent framework)
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Deposit Amount Dynamics
Diffusion process for Deposit Amount
[
]
dK t = K t µ K dt + σ K d WK (t )
(US marketplace)
Sensitivity of deposit amount to market rates
Money transfers between deposits and other accounts
Interest Rate partial contingence.
680
4
660
3,5
640
3
620 2,5 600 2 580 1,5 560 1
540
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juil-07
avr-07
janv-07
juil-06
oct-06
avr-06
janv-05
juil-04
oct-04
avr-04
janv-04
juil-03
oct-03
avr-03
janv-03
juil-02
dWK (t ) = ρdWL (t ) + 1 − ρ 2 dWK (t )
oct-02
avr-02
janv-02
juil-01
oct-01
avr-01
oct-00
0 janv-01
500 janv-06
Incomplete market framework
juil-05
0,5
US Demand Deposit Amount US M2 Own Rate
520
oct-05
Business risk, …
avr-05
−1 < ρ < 0
Hedging Demand Deposits Interest Rate Margins
6
97
Euro Overnight Deposits
EuroZone − µˆ K = 10.19%, σˆ K = 6.56%
Thursday, March 19th 2009 Turkey - M1-M0
Hedging Demand Deposits Interest Rate Margins sept-07
mai-07
janv-07
sept-06
mai-06
janv-06
sept-05
mai-05
janv-05
sept-04
mai-04
janv-04
sept-03
mai-03
janv-03
sept-02
mai-02
janv-02
0
mai-01
0
sept-01
100
janv-01
500
mai-00
1000
sept-00
300
janv-00
400
mai-99
1500
sept-99
30000
2500 600
15000 200
200
0
UAH Bln.
500
janv-99
2000
mai-98
US and Euro Zone
sept-98
700
TRY Bln.
3000
USD bln.
800
janv-98
EUR bln. 3500
sept-97
19 -0 9 9 19 8-0 98 1 19 -0 9 5 19 8-0 99 9 19 -0 9 1 19 9-0 99 5 20 -0 0 9 20 0-0 00 1 20 -0 0 5 20 0-0 01 9 20 -0 0 1 20 1-0 01 5 20 -0 0 9 20 2-0 02 1 20 -0 0 5 20 2-0 03 9 20 -0 0 1 20 3-0 03 5 20 -0 0 9 20 4-0 0 1 20 4-0 04 5 20 -0 0 9 20 5-0 05 1 20 -0 0 5 20 5-0 06 9 20 -0 0 1 20 6-0 0 5 20 6-0 07 9 20 -0 0 1 20 7-0 07 5 -0 9
19
Deposit Amount Dynamics – Examples
dK t = K t (µ K dt + σ K dWK (t )) Emerging Markets (Turkey, Ukraine) 400
25000 350
20000 300
250
10000 150
5000 100
50
0
US Demand Deposits
Ukraine - M1-M0
Turkey − µˆ K = 51.74%, σˆ K = 37.38% 7
Modeling Deposit Rate – Examples
We assume the customer rate to be a function of the market rate.
Affine in general (US) / Sometimes more complex (Japan)
g (LT ) = α + β ⋅ LT
g (LT ) = (α + β ⋅ LT ) ⋅ 1{LT ≥ R}
United States
Japan
3.00% M2 Own Rate
0,9 JPY Libor 3M
0,8
2.50%
Japanese M2 Own Rate
0,7 2.00%
0,6
Affine Dependance
0,5
1.50%
0,4 1.00%
0,3
Quasi Zero Rates !
0,2 0.50%
0,1 USD 3M Libor Rate
Hedging Demand Deposits Interest Rate Margins
mars-07
sept-06
mars-06
sept-05
mars-05
sept-04
mars-04
sept-03
mars-03
sept-02
mars-02
sept-01
mars-01
sept-00
mars-00
sept-99
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
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mars-99
0
0.00%
8
Sets of Hedging Strategies
1st case: Investment in FRAs contracted at t=0
… is contained in …
H S 1 = {S = θ (LT − L0 ) ; θ ∈ R}
2nd case: Dynamic self-financed strategies taking into account the evolution of market rates only
HS2
… is contained in …
T ⎧ ⎫ L L L = ⎨S = ∫ θ t dLt ; θ ∈ Θ ⎬ 0 ⎩ ⎭
Set of admissible investment strategies adapted to
F WL
3rd case: Dynamic strategies taking into account the evolution of the deposit amount T ⎧ ⎫ H D = ⎨S = ∫ θ t dLt ; θ ∈ Θ⎬ 0 ⎩ ⎭
z
Set of admissible investment strategies adapted to
F WL ∨ F WK
‘Admissible strategies’ are such that each of the sets above are closed
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Variance-Minimal Measure
Martingale Minimal Measure / Variance Minimal Measure T ⎛ 1T 2 ⎞ dP ⎜ = exp⎜ − ∫ λ dt − ∫ λdWL (t )⎟⎟
Martingale Minimal Measure: dP 0 ⎝ 20 ⎠ Föllmer, Schweizer (1990)
In ‘almost complete models’, it coincides with the variance minimal measure:
⎡ dQ ⎤ P ∈ Arg min E P ⎢ Q∈Π RN ⎣ dP ⎥⎦
2
Delbaen, Schachermayer (1996)
N.B.: In our case, the Variance Minimal Measure density is a power λ − function of the Libor rate. dP ⎛ LT ⎞ σ ⎛1 ⎞ = ⎜ ⎟ exp ⎜ ( λ 2 − λσ L ) T ⎟ L
dP
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⎝ L0 ⎠
⎝2
Hedging Demand Deposits Interest Rate Margins
⎠
10
Optimal Dynamic Hedging Strategy – Case #2 T ⎡ ⎤ P minL E ⎢ IRM g (K T , LT ) − ∫ θ t dLt ⎥ θ ∈Θ 0 ⎣ ⎦
In Case #2, we determine:
The projection theorem applies
Delbaen, Monat, Schachermayer, Schweizer, Stricker (1997)
In case #2, the solution consists in replicating
where
2
ϕ S 2 (LT )
ϕ S 2 ( x ) = E P [IRM g (K T , LT ) LT = x ]− E P [IRM g (K T , LT )]
This payoff can be replicated on interest rate markets.
This is a function of
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LT
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Optimal Dynamic Hedging Strategy – Case #3 2 T ⎡ ⎤ P We recall the related problem: min E ⎢ IRM g (K T , LT ) − ∫ θ t dLt ⎥ θ ∈Θ 0 ⎣ ⎦
The solution is dynamically determined as follows: P ∂ E λ ** t [IRM (K T , LT )] θt = + EtP IRM g (K T , LT ) − Vt x** , θ ** ∂Lt σ L Lt
[ [
Delta term
+
Hedging Numéraire
] (
×
)]
Feedback term -
Shift between the RN anticipation of the margin and the present value of the hedging portfolio
Investment in some Elementary Portfolio which verifies This portfolio aims at some fixed return while minimizing the final quadratic dispersion. Thursday, March 19th 2009
2
⎡ λ ⎤ ⎡ ⎤ EP ⎢∫ dLt − (− 1)⎥ = min E P ⎢ ∫ θ t dLt − (− 1)⎥ θ ∈Θ ⎣ 0 σ L Lt ⎦ ⎣0 ⎦ T
T
Hedging Demand Deposits Interest Rate Margins
2
12
Optimal Dynamic Hedging Strategy – Some Remarks
Case of No Deposit Rate: g (LT ) = 0
Explicit solution (Duffie and Richardson (1991)):
[
] , L )] ⎛ ρσ = ⎜1 +
EtP IRM g (K T , LT ) = K t Lt exp[(T − t )(µ K − ρσ K λ + ρσ K σ L )]
[
∂EtP IRM g (K T ∂Lt
T
⎜ ⎝
K
σL
⎞ ⎟⎟ K t exp[(T − t )(µ K − ρσ K λ + ρσ K σ L )] ⎠
The model works for ‘almost complete models’
The Hedging Numéraire remains the following: t
HN t = 1 + ∫ 0
λ σ L Lt
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2
dLt
or
T T ⎡ ⎤ ⎡ ⎤ λ P P E ⎢∫ dLt − (− 1)⎥ = min E ⎢ ∫ θ t dLt − (− 1)⎥ θ ∈Θ ⎣ 0 σ L Lt ⎦ ⎣0 ⎦
Hedging Demand Deposits Interest Rate Margins
2
13
PRESENTATION OUTLOOK
Modeling Framework, Objective and Optimal Strategy
Empirical Results
Conclusions
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Comparing Strategies in Mean-Variance Framework
Efficient Frontiers
Dynamic Efficient Frontier vs. Other Strategies at minimum variance point
More discrepancies between strategies when the deposit rate escapes from linearity Mean-Variance Framework - No Deposit Rate
Mean-Variance Framework - Barrier Deposit Rate Barrier Threshold = 3,00% - L(0) = 2,50% Deposit Rate = a. L(T) + b if L(T) > Threshold; a = 30% ; b = -0,50%
3,45
3,20
3,40 3,35 Expected Return
Expected Return
3,15
3,10
3,05
3,30 3,25 3,20 3,15
3,00 3,10 2,95 0,20
0,22
0,24
0,26
0,28
0,30
0,32
0,34
0,36
0,38
0,40
3,05 0,15
0,20
0,25
0,30
0,35
Standard Deviation
Blue: Unhedged Margin
Green: Delta-Hedging at t=0 only
Red: Optimal Dynamic Strategy following only market rates
Purple: Dynamic Delta-Hedging
0,40
0,45
0,50
0,55
0,60
0,65
Standard Deviation
The performances of other hedging strategies strongly depend upon the specification of the deposit rate.
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Dealing with Deposits’ ‘Specific’ Risk
Comparing the optimal dynamic strategy following only market rates (blue) and the optimal dynamic strategy following both rates and deposits (pink):
At minimum variance point (risk minimization)
As expected, the deposits’ ‘specific’ risk is better assessed using a dynamic strategy following both rates and the deposit amount Risk Reduction and Correlation Total Deposit Volatility = 6.5% - K(0) = 100 0,35
Hedged Margin Standard Deviation
0,30 0,25 0,20 0,15 0,10 Optimal Dynamic Hedge (Rates)
0,05
Optimal Dynamic Hedge (rates + deposits)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Deposit / Rates Correlation Parameter
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Robustness towards Risk Criterion
The mean-variance optimal dynamic strategy (following deposits and rates) behaves quite well under other risk criteria
Example of Expected Shortfall (99.5%) and VaR (99.95%). ES (99.5%)
Standard Deviation Barrier Deposit Rate
Expected Return Level
Risk Reduction
Level
VaR (99.95%)
Risk Reduction
Level
Unhedged Margin
3.16
0.39
Static Hedge Case 1
3.04
0.28
-0.11
-2.34
-0.32
-2.26
-0.36
Static Hedge Case 2
3.01
0.23
-0.16
-2.26
-0.24
-2.04
-0.14
Jarrow and van Deventer
3.01
0.24
-0.15
-2.35
-0.33
-2.25
-0.35
Optimal Dynamic Hedge
3.01
0.22
-0.17
-2.38
-0.36
-2.29
-0.39
The optimal dynamic strategy features better tail distribution than for other strategies
Blue: Optimal Dynamic Strategy (following rates)
Pink: Optimal Dynamic Strategy (following both deposits and rates)
-1.90
Probability Densities Hedging Following Rates vs. Hedging Following Deposits and Rates 1,6 1,4
Hedging Following Rates
1,2
Hedging Following Rates and D it
1,0 Density
-2.02
Risk Reduction
0,8 0,6 0,4 0,2 -
-
0,50
1,00
1,50
2,00
2,50
3,00
3,50
4,00
4,50
5,00
Interest Rate Margin Level (incl. Hedge)
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Dealing with Massive Bank Run
Introducing a Poisson Jump component in the deposit amount:
[
]
dK t = K t µ K dt + σ K d WK (t ) − dN (t )
(N (t ))0≤t ≤T is assumed to be independent from WK and
WL
∂EtP [IRM (KT , LT )] λ + EtP IRM g (K T , LT ) − Vt x** , θ ** Then, we have: θ = ∂Lt σ L Lt ** t
[ [
] (
)]
EtP ⎣⎡ IRM g ( KT , LT ) ⎦⎤ = e −γ (T −t ) × (Previous conditional expectation term)
Due to independence, the jump element can be put out the conditional expectations
N.B.: When a bank run occurs, the manager keeps investing the current hedging portfolio’s value in the Hedging Numéraire
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PRESENTATION OUTLOOK
Modeling Framework, Objective and Optimal Strategy
Empirical Results
Conclusions
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Conclusions (1)
A dynamic strategy to assess risk in mean-variance framework
Results about Mean-variance hedging in incomplete markets yield explicit dynamic hedging strategies
Practical Conclusions:
Better assessment of deposits’ ‘specific’ risk with a dynamic strategy taking into account both deposits and rates;
Lack of stability for other strategies towards the deposit rate’s specification;
Robustness towards risk criterion
No negative consequences as for tail distribution
Additivity of Optimal Dynamic Strategies
Applicable to various balance sheet items
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Conclusions (2)
We use some mathematical finance concepts:
For Financial Engineering problems
with the aim of providing applicable strategies
And improve risk management processes
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Technical References
Duffie, D., Richardson, H. R., 1991. Mean-variance hedging in continuous time. Annals of Applied Probability 1(1).
Gouriéroux, C., Laurent, J.-P., Pham, H., 1998. Mean-variance hedging and numéraire. Mathematical Finance 8(3).
Hutchison, D., Pennacchi, G., 1996. Measuring Rents and Interest Rate Risk in Imperfect Financial Markets : The Case of Retail Bank Deposits. Journal of Financial and Quantitative Analysis 31(3).
Jarrow, R., van Deventer, D., 1998. The arbitrage-free valuation and hedging of demand deposits and credit card loans. Journal of Banking and Finance 22.
O’Brien, J., 2000. Estimating the value and interest risk of interest-bearing transactions deposits. Division of Research and Statistics / Board of Governors / Federal Reserve System.
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