Hedging Interest Rate Margins on Demand Deposits - - PowerPoint PPT Presentation

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Hedging Interest Rate Margins on Demand Deposits

slide-1
SLIDE 1

Hedging Interest Rate Margins on Demand Deposits

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slide-2
SLIDE 2

Presentation related to:

Hedging Interest Rate Margins on Demand Deposits

Working paper available on SSRN (to be updated soon)

Presentation Outlook

Modeling framework

customer rates

2

customer rates deposit amounts Interest rate margins

Optimal strategies

The blinkered investor Integrated risk management

Conclusion

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SLIDE 3

Prolegomena

Demand Deposits involve huge amounts

Bank of America Annual Report – Dec. 2007

Average Balance (Dollars in millions) 2007 2006 Assets Federal funds sold and securities purchased under agreements to resell $ 155,828 $ 175,334 Trading account assets 187,287 145,321 Debt securities 186,466 225,219 Loans and leases, net of allowance for loan and lease losses 766,329 643,259 All other assets 306,163 277,548 Total assets $ 1,602,073 $ 1,466,681

3

Total assets $ 1,602,073 $ 1,466,681 Liabilities Deposits $ 717,182 $ 672,995 Federal funds purchased and securities sold under agreements to repurchase 253,481 286,903 Trading account liabilities 82,721 64,689 Commercial paper and other short-term borrowings 171,333 124,229 Long-term debt 169,855 130,124 All other liabilities 70,839 57,278 Total liabilities 1,465,411 1,336,218 Shareholders’ equity 136,662 130,463 Total liabilities and shareholders’ equity $ 1,602,073 $ 1,466,681

Demand deposits involve both interest rate and liquidity risks

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SLIDE 4

Prolegomena

Can we think of a market value or a “fair value” of demand

deposits?

Clearly not for practical, accounting and financial theory

reasons

No real market to compute “exit values” As for loans, affectio sociatatis effects : contractors identities (bank and

customer) are essential elements of demand deposits

4

customer) are essential elements of demand deposits

Money does smell! Not a “law invariant contract” One cannot only rely on cash-flows This is not only a matter of bank credit risk

Think of the simplest case

No interest bearing, one USD deposit at t=0, withdrawn at some random

time t

Unlike a term deposit where the repayment date is known

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SLIDE 5

Prolegomena

Standard mathematical finance approach

Value of the contract: Q stands for a “risk-neutral” probability Obvious problem : t is not adapted to the usual filtration related to

interest rates

1 exp ( )

Q

E r s ds

τ

  • −

−

  • 5

interest rates

Not a standard interest rate contract Assume that is a stopping time adapted to some larger filtration

with intensity

Risk-neutral withdrawal intensity Similar to the valuation of defaultable securities One has no idea of the risk premia associated with the

uncertainty of the closing date

Since unlike liquid defaultable bonds, there is no liquid reference

market

λ

slide-6
SLIDE 6

Prolegomena

Standard mathematical finance approach

We end-up with an arbitrary choice of risk-neutral probability (or risk

premia) due to market incompleteness

Mark-to-Model valuation with large model risk

Can this issue be mitigated at a portfolio level?

Large pool and diversification effects

6

Large pool and diversification effects

Insurance ideas related to the law of large numbers Infinitely granular portfolios (Gordy, 2003) Well diversified portfolios (Björk and Naslünd, 1998, De Donno, 2004) Along this view, the uncertainty on the amortizing scheme of demand

deposits as sampling fluctuation could be neglected for large portfolios

This contradicts empirical evidence Demand deposit amounts at a bank level are not fully correlated to interest

rates

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SLIDE 7

Prolegomena

There exists some extra risk whatever its name

Liquidity risk Business risk As stated in the regression model relating demand deposit amounts to

interest rates in Jarrow & Van Deventer, 1998

The computation of the “value” of demand deposits through the

7

The computation of the “value” of demand deposits through the expected discounted approach is flawed

Implicitly assumes no risk premia for liquidity or business risk Minimal martingale measure Which seems rather unrealistic, given relationships between monetary

aggregates, stock markets and real economy

Let us remark that the same issue holds for the valuation of the

mortgage prepayment option

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SLIDE 8

Prolegomena

Accounting framework of deposit accounts involve mainly:

IASB (International Accounting Standards Board) : IAS 39 FASB (US Financial Accounting Standards Board) FASB mention that demand deposits “involves consideration of non

financial components” and propose to postpone recognition of those liabilities at fair value

Approaches differ but the two boards are connected

8

Financial Crisis Advisory Group, G20, European Commission,

Financial Stability Forum also involved

Amortized cost versus Fair value

Fair value measurement is likely to be updated Exposure draft, May 2009 Accounting boards, SEC, Basel Committee on Banking Supervision,

European Banking Federation favour the amortizing cost approach for demand deposits

One should thus focus on interest rate margins rather than fair value

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SLIDE 9

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)

( )

T T

L L g ⋅ + = β α

&.

( ) ( ) { }

R L L L g

T T T

≥ ⋅ ⋅ + = 1 β α

9

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

USD 3M Libor Rate M2 Own Rate

56 ' &

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 mars-99 sept-99 mars-00 sept-00 mars-01 sept-01 mars-02 sept-02 mars-03 sept-03 mars-04 sept-04 mars-05 sept-05 mars-06 sept-06 mars-07

JPY Libor 3M Japanese M2 Own Rate

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SLIDE 10

Dynamics for Market Rate : forward Libor rate

Market Model for forward Libor rate(s)

( )

t L L L t

dL dt dW t L µ σ = +

t

L ≠ µ

,

10 10

≠

L

µ

,

Coefficient specification assumptions:

Our model: Assumptions can be relaxed: Time dependent coefficients CEV type Libor models

L L σ

µ ,

constant

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SLIDE 11

Deposit Amount Dynamics

( )

t t K K K

dK K dt dW t µ σ

  • =

+

  • Diffusion process for Deposit Amount

Sensitivity of deposit amount to

market rates

Money transfers between deposits

620 640 660 680 2,5 3 3,5 4

  • 11

Money transfers between deposits

and other accounts

Interest Rate partial contingence.

Business risk, … Incomplete market framework

( ) ( ) ( )

2

1

K L K

dW t dW t dW t ρ ρ = + − 1 < < − ρ

500 520 540 560 580 600

  • ct-00

janv-01 avr-01 juil-01

  • ct-01

janv-02 avr-02 juil-02

  • ct-02

janv-03 avr-03 juil-03

  • ct-03

janv-04 avr-04 juil-04

  • ct-04

janv-05 avr-05 juil-05

  • ct-05

janv-06 avr-06 juil-06

  • ct-06

janv-07 avr-07 juil-07 0,5 1 1,5 2 2,5 US Demand Deposit Amount US M2 Own Rate

slide-12
SLIDE 12

Deposit Amount Dynamics – Examples

( )

( )

t W d dt K dK

K K K t t

σ µ + =

.&5

  • 2500

3000 3500 600 700 800 25000 30000 300 350 400

12 12

500 1000 1500 2000 1 9 9 7

  • 9

1 9 9 8

  • 1

1 9 9 8

  • 5

1 9 9 8

  • 9

1 9 9 9

  • 1

1 9 9 9

  • 5

1 9 9 9

  • 9

2

  • 1

2

  • 5

2

  • 9

2 1

  • 1

2 1

  • 5

2 1

  • 9

2 2

  • 1

2 2

  • 5

2 2

  • 9

2 3

  • 1

2 3

  • 5

2 3

  • 9

2 4

  • 1

2 4

  • 5

2 4

  • 9

2 5

  • 1

2 5

  • 5

2 5

  • 9

2 6

  • 1

2 6

  • 5

2 6

  • 9

2 7

  • 1

2 7

  • 5

2 7

  • 9

EUR bln.

100 200 300 400 500

USD bln. Euro Overnight Deposits US Demand Deposits

5000 10000 15000 20000 sept-97 janv-98 mai-98 sept-98 janv-99 mai-99 sept-99 janv-00 mai-00 sept-00 janv-01 mai-01 sept-01 janv-02 mai-02 sept-02 janv-03 mai-03 sept-03 janv-04 mai-04 sept-04 janv-05 mai-05 sept-05 janv-06 mai-06 sept-06 janv-07 mai-07 sept-07

TRY Bln.

50 100 150 200 250

UAH Bln. Turkey - M1-M0 Ukraine - M1-M0

% 56 . 6 ˆ %, 19 . 10 ˆ = = −

K K

σ µ EuroZone % 38 . 37 ˆ %, 74 . 51 ˆ = = −

K K

σ µ Turkey

slide-13
SLIDE 13

Demand Deposit Interest Rate Margin

For a given quarter T Income generated by:

Investing Demand Deposit Amount on interbank markets while paying a deposit rate to customers

13

( ) ( ) ( )

T L g L K L K IRM

T T T T T g

∆ ⋅ − = ,

,

' ,& 78

slide-14
SLIDE 14

We need to focus on Interest Rate Margins

According to the IFRS (International accounting standards) :

The IFRS recommend the accounting of non maturing assets and

liabilities at Amortized Cost / Historical Cost

( ) ( ) ( )

T L g L K L K IRM

T T T T T g

∆ ⋅ − = ,

14

liabilities at Amortized Cost / Historical Cost

Recognition of related hedging strategies from the accounting

viewpoint

Interest Margin Hedge (IMH). The fair value approach does not apply to demand deposits

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SLIDE 15

Risks in interest rate margins Interest rate risk

Direct interest rate risk on unit margins Indirect interest rate risk due to correlation between and

Business risk

( )

T T

L g L −

T

K

( ) ( )

( )

,

g T T T T T

IRM K L K L g L T = − ⋅∆

T

L

15

Business risk

Deposit amounts are not fully correlated to interest rates

Hedging tools

Interest rate swaps (FRA’s)

  • three months forward Libor at date t for quarter T
  • : incremental cash-flow at time T associated with a unit FRA

t

L

t

dL

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SLIDE 16

Sets of Hedging Strategies

  • Θ

∈ = =

  • L

L T t L t S

dL S H θ θ ;

2

Hedging strategies based on FRAs Amount held in FRA’s varies with available information 1st case: (myopic) self-financed strategies taking into account the evolution

  • f market rates only
  • 16
  • Θ

∈ = =

  • t

t S

dL S H θ θ ;

2

  • Θ

∈ = =

  • θ

θ ;

T t t D

dL S H

Management of interest rate risk achieved by trading desk far-off ALM 2nd case: self-financed strategies taking into account the evolution of

the deposit amount

Integrated risk management

  • L

W

F

K L

W W

F F ∨

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SLIDE 17

( ) ( )

( )

, = − ⋅∆

g T T T T T

IRM K L K L g L T

Mean-variance framework: ,

' ,& 78

  • 17

( )

[ ] 2

, min S L K IRM

T T g S

− E

( )

[ ]

r S L K IRM

T T g

≥ − , E

Mean-variance framework:

Including a return constraint – due to the interest rate risk premium. Incomplete markets: perfect hedge cannot be achieved with interest

rate swaps

&

slide-18
SLIDE 18

Useful mathematical finance concepts

Martingale Minimal Measure:

  • risk premium associated with holding long term assets

Risk-neutral with respect to traded risks (interest rates) Historical with respect to non hedgeable risks (business risk)

In our framework (almost complete markets), coincides with the variance

( )

exp

T T L

d dt dW t d λ λ

  • =

− −

  • P

P

2

1 2

/

L L

λ µ σ =

18

In our framework (almost complete markets), coincides with the variance

minimal measure:

Arg min

RN

d d

∈Π

  • =
  • P

Q

Q P E P

2 Here, the Variance Minimal Measure density is a power function of the

Libor rate :

( )

2

1 exp 2

L

T L

L d T d L

λ σ

λ λσ

−

  • =

−

  • P

P

slide-19
SLIDE 19

Only depends upon terminal Libor! Involves a nonlinear regression of IRR on terminal Libor European option type payoff / Analytical computations

Optimal Hedging Strategy – blinkers’ case

( )

Payoff of optimal hedging strategy:

( ) ( ) ( )

2

, ,

S T g T T T g T T

L IRM K L L IRM K L ϕ

  • =

−

  • P

P

E E

19

Optimal hedging strategy in FRAs consists in replicating

( )

T S

L

2

ϕ

slide-20
SLIDE 20

Optimal Hedging Strategy – Integrated Risk Management

( ) ( )

( )

** ** **

, , ,

t g T T t t g T T t t L t

IRM K L IRM K L V x L L λ θ θ σ

  • ∂
  • =

+ −

  • ∂

P P

E E

The optimal investment in FRA’s is determined as follows:

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+ ×

20

( ) ( )

2 2

1 min 1

  • −

− =

  • −

−

  • Θ

∈ T t t T t t L

dL dL L θ σ λ

θ P P

E E

$% $

, ! :

  • Computations are fully explicit

Case of No Deposit Rate:

Optimal strategy reduces to earlier results of Duffie & Richardson

( )

=

T

L g

slide-21
SLIDE 21

Dealing with Massive Bank Run

( ) ( )

t t K K K

dK K dt dW t dN t µ σ

  • =

+ −

  • ( )

( )

T t

t N

≤ ≤ K

W

Introducing a Poisson Jump component in the deposit amount:

&/& &

L

W

&

Same hedging numéraire and variance minimal measure as before. Computations are still explicit :

21

( ) [ ] ( )

[ ] ( )

[ ]

* * * * * *

, , , θ σ λ θ x V L K IRM L L L K IRM

t T T g t t L t T T t t

− + ∂ ∂ =

P P

E E

Due to independence, the jump element can be put out the conditional

expectations:

( )

( )

,

T t t g T T

IRM K L e

γ − −

  • =

×

  • P

E

&'

Previous technique has a wide range of applications Changing the deposit amount dynamics Changing the customer rate specification

slide-22
SLIDE 22

Relevance of Integrated Risk Management ?

Risk Reduction and Correlation

  • Optimal strategy based on market rates only (blue) and the one knowing

both rates and deposits (pink):

At minimum variance point (risk minimization) Taking into account deposit amounts leads to higher accuracy when correlation

between interest rates and deposit amounts is low

  • corresponds to a complete market case

1 ρ =

22

Risk Reduction and Correlation Total Deposit Volatility = 6.5% - K(0) = 100

  • 0,05

0,10 0,15 0,20 0,25 0,30 0,35 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Deposit / Rates Correlation Parameter Hedged Margin Standard Deviation

Optimal Dynamic Hedge (Rates) Optimal Dynamic Hedge (rates + deposits)

slide-23
SLIDE 23

Choice of Risk Criterion

Level Risk Reduction Level Risk Reduction Level Risk Reduction ES (99.5%) VaR (99.95%) Barrier Deposit Rate Expected Return Standard Deviation

  • 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%).

23

Unhedged Margin 3.16 0.39

  • 2.02
  • 1.90

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
  • Mean-variance optimal dynamic strategy are additive with respect to

different items of the balance sheet

  • One can deal separately with demand deposits and mortgages (say)
  • Which is not the case with ES or VaR
slide-24
SLIDE 24

Conclusions

Abstract mathematical finance concepts lead to analytical and easy to

implement optimal hedging strategies for demand deposits:

Taking both into account interest rate risk and business risk Sheds new light on risk management architecture Consistent with standard accounting principles Robust with respect to choice of risk criteria

24

Can cope with a wide range of specifications Applicable to various balance sheet items That can be dealt with separately (additivity) But… Lack of stability towards deposit rate’s specification Growth and volatility of deposit amounts As usual, significant model risk