managing the risk of deposit disintermediation and the optimal … · 2019-07-10 ·...
TRANSCRIPT
Managing the Risk of Deposit Disintermediation
and the Optimal Structure of Input Prices
Abstract
Most bank deposits contain an embedded withdrawal option which permits the depositor to sell the deposit back to the bank at will. Demand deposits generally allow costless withdrawal, while time deposits often require payment of an early withdrawal penalty. Managing the risk that depositors will exercise their withdrawal option is an important aspect of deposit pricing. This paper details the impact of the embedded withdrawal option upon the optimal time deposit rate.
1
At their most fundamental level banks are in the business of borrowing and
lending money. Bank managers are concerned with pricing their assets and their
liabilities in order to maximize their profits. While the academic literature on pricing
bank assets is vast and well developed, little attention has been given to pricing bank
liabilities.1 In standard banking models, the volume of deposits equate the expected
Lerner index of the input price with the inverse of the elasticity of supply to determine
the rate paid for deposits.2 The models have no explicit time dimension so that deposits
cannot have any “meaningful” length. Their implicit time to maturity is homogenous
across accounts and, without modeling the intertemporal behavior of interest rates,
deposits must be assumed to be held until maturity. Commercial banking models clearly
ignore two important aspects of input pricing: 1) deposit accounts have different times to
maturity, and 2) depositors can withdraw funds before maturity.
Most bank deposits contain an embedded withdrawal option which permits the
depositor to sell the deposit back to the bank at will. Demand deposits such as checking,
savings, and money market accounts generally allow costless withdrawal, while time
deposits often require payment of a pre-specified early withdrawal penalty.3 Managing
the risk that depositors will exercise their withdrawal option is an aspect of deposit
pricing.4 The purpose of this paper is to detail the impact of the embedded withdrawal
option upon the optimal time deposit rate.5 Our objective function is an intertemporal
expected cost of deposit funding that begins at 0t , the inception of the deposit D , and
ends at MT , the time to maturity of the deposit. The model has a benchmark rate of
interest whose continuous changes across time are given by trended brownian motion. If
the benchmark rate remains below Dr throughout the time horizon, then the bank’s cost
2
of funding is simply 0( )D Mr D T t⋅ ⋅ − . If the benchmark rate rises above Dr at IT then
disintermediation occurs. The bank suffers transaction costs τ and will have to replace
the funds at a rate which is higher than the original input price. The determination of the
expected cost of deposit funding depends upon the mean time to disintermediation. Using
the statistical properties of the benchmark rate of interest we find that time to
disintermediation T satisfies Kolmogorov’s diffusion equation.6 The solution to this
differential equation allows us to determine the probability density of the time to
disintermediation and its mean.7
We take the derivative of the expected cost of deposit funding with respect to Dr
and set it equal to zero. Unfortunately, the first order condition for *Dr is an integral
differential equation which refuses to yield an explicit solution for the optimal deposit
rate. Not being able to normalize on *Dr mutes any immediate intuition we might
otherwise gain. In addition, analytics provide very little direct insight into the
comparative static behavior of deposit pricing. Consequently, our solution for the bank’s
optimal input price was simulated over a host of alternative illustrative parameters. Our
solution for the input price depends upon the time to maturity, so that *Dr also yields the
optimal structure of time deposit rates.
I. The Time to Disintermediation
Today’s commercial bankers struggle to appropriately price their time deposits.
Bankers are convinced that as the economy grows and prospers, interest rates are going to
rise. Though anxious to “lock in” deposits at today’s low rates of interest, bankers are
aware of the threat of deposit disintermediation in an expansionary economic climate. In
the face of this dilemma we propose the following approach to deposit pricing.
3
Suppose that ( )R t is “the” rate of interest, a kind of average for all rates of
interest available on financial assets. Furthermore, we assume that ( )R t will be our
benchmark rate of interest, if ( )R t is less than Dr for all t over the interval 0[ , ]Mt T then
no disintermediation will occur and the cost of deposit funding is given by
0( )D Mr D T t⋅ ⋅ − . (1)
However, if the benchmark rate of interest is greater than Dr at some point IT then
disintermediation occurs and the cost of funding is given by
0( ) 2 ( )D I I M Ir D T t r D T Tτ⋅ ⋅ − + + ⋅ ⋅ − . (2)
τ is the fixed administrative and decision making costs of providing funds to fleeing
depositors and then securing new funds. The variable Ir is the emergency borrowing rate
the bank must pay to replace the disintermediated funds ( )I Dr r> at time IT .
Though interest rates in general have no long run trend, we believe it is
reasonable to assume that interest rates vary systematically with aggregate economic
activity. In particular, interest rates increase over the expansionary phase of the business
cycle and fall over the cycle’s contractionary phase.8 For our model, we will assume that
( )R t follows a brownian motion process with a drift of µ where µ >0 since current
expectations are that the economy will recover and interest rates will increase.9
Examining the expressions (1) and (2) it is clear that determining the expected
cost of deposit funding will involve the partial means of two random variables; the cost
of emergency borrowing and the time to disintermediation T . The expected cost of
emergency borrowing will be no more than the conditional partial expectation of the
benchmark rate over the range of [ , ]Dr ∞ . The evaluation of the partial mean of the time
4
to disintermediation is a little more subtle. However, a convenient statistical relationship
exists between our benchmark rate of interest and the time to disintermediation. In
particular, since
{ ( ) DP R t r< for 0 0 0| (0) } ( , ; ) ( , ; ).Dr
Dt T R r p r r t dr P r r t−∞
< = = ≡∫
is the probability that the time to disintermediation has not occurred by t , we have
0( , ; )DP r r t = ( ).prob T t≥
So that 0 0 0( , ; ) ( , ; ) 1 ( ; , )Dr
D DP r r t p r r t dr G t r r−∞
= = −∫
where 0( ; , )DG t r r is the cumulative probability of the time to disintermediation.
Consequently,
00
( , ; ) ( ; , )DD
P r r t G t r rt
∂ ′= −∂
or
00
( , ; ) ( | , )DD
P r r t g t r rt
∂− =
∂
which yields the marginal probability of T .
To solve for 0( | , )Dg t r r we simply take advantage of the fact that since 0( , ; )p r r t is
brownian motion it satisfies Kolmogorov’s diffusion equation.10 Since 0( , ; )DP r r t , like
0( , ; )p r r t , satisfies the diffusion equation then 0( | , )Dg t r r must satisfy Kolmogorov’s
equation, so we have11
2
20 0 0
0 0 0
( | , ) ( | , ) ( | , )12
D D Dg t r r g t r r g t r rt r r r
µ σ∂ ∂ ∂= +
∂ ∂ ∂ ∂. (3)
5
It is far more convenient to solve for 0( | , )Dg t r r using Laplace transforms with respect to
time.
The Laplace transformation changes the partial differential equation above involving the
interest rate coordinate r and the time coordinate t into an ordinary differential equation
in r . Thus we have12
* * 2 *2
0 0 0
12
g g gt r r r
µ σ∂ ∂ ∂= +
∂ ∂ ∂ ∂ where 0
0
* ( | , )stDg e g t r r dt
∞−= ∫ (4)
* 2 *2
0 0 0
1*2
g gsgr r r
µ σ∂ ∂= +
∂ ∂ ∂
22
0 0 0
12
sr r rφ φφ µ σ∂ ∂
= +∂ ∂ ∂
where *gφ = .
The expression above is a second order linear differential equation with constant
coefficients.
Adopting the trial solution of (4) as 00( ) rr Aeθφ = yields 0
0' ( ) rr Aeθφ θ= and
020'' ( ) rr Aeθφ θ= so that (4) becomes
0 0 0
0
0
2 2
2 2
2 2
12
1( ) 02
1( ) 0.2
r r r
r
r
sAe Ae Ae
Ae s
Ae s
θ θ θ
θ
θ
µθ σ θ
µθ σ θ
σ θ µθ
= +
+ − =
+ − =
6
Solving 2 21( ) 02
sσ θ µθ+ − = for θ using the quadratic formula yields
1( )sθ and 2 2
2 2
2( )
ss
µ µ σθ
σ− +
=m
. (5)
The general solution of (4) is
1 0 2 00 1 2( ) r rr A e A eθ θφ = + (6)
Note that if we take the positive square root in (5), for s real we have
1 2( ) 0 ( ) , ( 0)s s sθ θ< < > .
In (6), it is clear that 1A must be equal to zero because otherwise 1 01
rA eθ would be
unbounded as 0 approaches r −∞ with 1( ) less than 0sθ .13 Furthermore 2 02
rA eθ should be
written as 2 0( )Dr reθ − so that when 0 Dr r= , ( )Drφ equals one.14 With ( ) 1,Drφ = the inverse
function 0( | , )Dg s r r will be zero.15 This documents a zero likelihood of T>0 when
0 Dr r= and establishes the fact that disintermediation will take place immediately.
Finally then, we have
0 2( ) ( )0 0( ) *( | , ) Dr r s
Dr g s r r e θφ −= =
2 2 1/ 2 20( )( ( 2 ) ) /
0 0( ) *( | , ) Dr r sDr g s r r e µ µ σ σφ − − − += = for 0 Dr r< . (7)
We conjecture that
2 2 1/ 2 20( ){ ( 2 ) }/
0[g(t; , ); ] Dr r sDr r s e µ µ σ σ− − − +=L[ (8a)
where L represents the Laplace operator and 2
0 00 23
( ) ( )( ; , ) exp[ ]22
D DD
r r r r tg t r rttµ
σσ π
− − − −= .
7
0[g(t; , ); ]Dr r sL[ could be written as
2 2 20 0
2
2 2 2 2 20 0
2 2
0
[( ) 2( ) ]0 2
0 30
2( ) [( ) 2 ]0 2 2
0 30
( )0
0
( ) 1[g(t; , ); ] (8b)2
or as
( ) 1[g(t; , ); ] (8c)2
or as
( )[g(t; , ); ]2
D D
D D
D
r r t r r tstD t
D
r r r r t s tD t
D
r rD
D
r rr r s e e dtt
r rr r s e e dtt
r rr r s e
µ µσ
µ µ σσ σ
σ π
σ π
σ π
− − + − −∞−
− − − + +∞
−
−=
−=
−=
∫
∫
L[
L[
L[2 2 2
102 2 2
( ) 2[ ]2 2
30
1 (8d)
or, finally, as
Dr r st te dt
t
µ µ σσ σ σ
−− −∞ +−
∫
012
( )0
0 30
( ) 1[g(t; , ); ]2
Dr rct tD
Dr rr r s e e dt
t
µρσ
σ π−
− ∞− −−
= ∫L[ (8e)
where 2 2 2
02 2
( ) 2, ( )2 2
Dr r sc µ σρσ σ− +
= = .
Using the mathematical fact that (see Appendix H)
2 2002 22
2-3/2 /
0
2( )( ) 20 2 2
002
t
allows us to rewrite the expression above as
( )[g(t; , ); ] (8f)( )22
DD
cc t t
sr rr rD
DD
e e dt ec
r rr r s e er r
ρρ
µ σµσ σσ
π
πσ π
σ
∞−− −
+−− − ⋅
=
−= ⋅ ⋅
−
∫
L[
8
2 2 1/ 20 0
2 2
2 2 1/ 202
( ) ( )( 2 )[ ]
0
1 ( )[ ( 2 ) ]
0
[g(t; , ); ] (8g)
[g(t; , ); ] . (8h)
D D
D
r r r r s
D
r r s
D
r r s e e
r r s e
µ µ σσ σ
µ µ σσ
− − +−
− − +
=
=
L[
L[
So then indeed our conjecture is correct and the Laplace transform of
0g(t; , )Dr r =2
0 023
( ) ( )exp[ ]22
D Dr r r r tttµ
σσ π
− − − − is the right hand side of (8a).
The expression 0( ; , )Dg t r r is then the probability density function of the time to disintermediation. The probability density function integrates to one and the likelihood of disintermediation has a mean of 0( ) /Dr r µ− . 16
II. The Expected Cost of Deposit Funding Using equations (1) and (2) we can easily determine the discounted expected cost of deposit funding, ( )E CDF . Recall from equation (1) that if the time to disintermediation is greater than the time to maturity, MT T< , then the cost of deposit funding is simply 0( )D Mr D T t⋅ ⋅ − so the present value contribution of this possibility to
( )E CDF would be
( )M
M
TD M
T
e r D T g t dtλ∞
− ⋅ ⋅ ⋅ ⋅ ∫ for 0t =0.
If MT T< then the fixed costs of both disintermediation and deposit refunding are incurred, 2τ , their discounted inclusion in ( )E CDF would be written as
0
2 ( )MT
te g t dtλτ −⋅ ∫ .
9
If the disintermediation occurs at IT then the cost of deposit funding across the time horizon to MT would be
( )D I I M Ir D T r D T T⋅ ⋅ + ⋅ ⋅ − . Weighted by the likelihood of occurrence and discounted to present value, the expression above would be
0
( )MT
tDr D e t g t dtλ−⋅ ⋅ ∫ ( )M
D
TM
r
e D T r f r drλ∞
−+ ⋅ ⋅ ⋅ ∫0
( , )M
D
Tt
r
D t r e z t r dt drλ∞
−− ⋅ ∫ ∫
where ( , )z t r is the joint likelihood function of t and r .
Resolving the joint likelihood of ( , )z t r , and collecting the terms above yields the
expected cost of deposit funding as17
2
2
2
2
0 0
( )2
2
( )2
20
( ) 2 ( ) ( )
1( )2
1( ) (9)2
M M
D
M M
M D
DM
M
D
T Tt t
D
r rT T
D M MT r
r rTT
r
E CDF e g t dt D r t e g t dt
e r D T g t dt D T e r e dr
D e t g t dt r e dr
λ λ
µλ λ σ
µλ σ
τ
πσ
πσ
− −
− − − ∆∞ ∞− − ∆
− − − ∆∞− ∆
= ⋅ + ⋅ ⋅
+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅∆
− ⋅ ⋅ ⋅∆
∫ ∫
∫ ∫
∫ ∫
whereλ is the intertemporal discount factor of the bank.
10
III. The Optimal Input Price Minimizing the expected cost of deposit funding with respect to Dr yields:
*0* *
0
* *0* *
0
** *
[ ( ) ]2 ( )
[ ( ) ]( ) ( ) ( )
[ ( ) ]
M
M
M
M
M M
M M
MM M
Tt
Tt
DD D
Tt
TT Tt
D M M DD DT T
TT TM D M
D D
e g t dtD r t e g t dt
r r
t e g t dtDD t e g t dt D r e T D g t dt e T r g t dt
r r
g t dtDD T e r e T
r r
λ
λ
λ
λ λλ
λ λ
τ
−
−
−∞ ∞
− −−
∞
− −
∂∂
⋅ + ⋅ ⋅∂ ∂
∂∂
+ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅∂ ∂
∂∂
+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅∂ ∂
∫∫
∫∫ ∫ ∫
∫ * 2
2
*
* 2
2
* 2
* 2
*
* 2
2
*
( )2
2
( )2
( )22
* * 20
( )2 0
*2
1[ ]2
1[ ]2 1( ) [ ]
2
[ ( ) ]1
2
D
D
D
DM
DM M
D
M
D
M
D
r r
r
r r
r rTrT T
MD D r
T
r rT
Dr
r e dr
r e drDe T D e t g t dt r e dr
r r
t g t dtD e r e dr
r
µσ
µσ
µλ λ σ
µλ σ
πσ
πσ
πσ
πσ
− − − ∆∞∆
− − − ∆∞∆
− − − ∆∞− − ∆
− − − ∆∞− ∆
∆
∂∆ ∂
+ ⋅ ⋅ − ⋅ ⋅ ⋅∂ ∂ ∆
∂− ⋅ ⋅ ⋅
∂∆
∫
∫∫ ∫
∫∫
* 2
2
*
( )2
2
*0
1[ ]2
( ) 0 (10)
D
M
DM
r r
TrT
D
r e dr
D e t g t dtr
µσ
λ πσ
− − − ∆∞∆
−
∂∆
− ⋅ ⋅ ⋅ =∂
∫∫
The integral differential equation above refuses to yield an explicit solution for
the optimal deposit rate.18 Examining (10), it is clear that the decision variable
arithmetically influences the first order condition in a number of ways. The *Dr appears in
the limit of five integrals, in ten integrands, in the power to which nearly every
exponential function in (10) is raised, and in the determination of the slope of demand for
deposits schedule. Given the disparate appearance of the optimal input price in (10),
normalizing on Dr is impossible. Not being able to isolate *Dr on the left hand side of such
11
a complex FOC mutes any immediate intuition. In addition, the complexity of the FOC
nearly guarantees that using the implicit function theorem for comparative static analysis
will be fruitless. However, it is clear that simulating the bank’s optimal input price would
provide us with a good deal of insight.
IV. Some Simulations
In regard to preparing equation (10) for simulation, we first chose a deposit supply
equation which is widely used in the literature,
DrD eα= . (11a) The exponential function above yields a positive relation between and DD r
Dr
D
D er
αα∂=
∂>0 (11b)
suggesting that to increase funds on deposit the bank merely needs to increase Dr .19 The second derivative is positive and helps insure that the second order condition for *
Dr is satisfied. The elasticity of supply of bank deposits is given by
DD
D
rD rr D
α∂⋅ =
∂ , (11c)
and rewriting (11c) we have
1
D
Dr D
α∂⋅ =
∂. (11d)
Consequently, dividing the LHS and the RHS of (10) by the level of deposits D , we can
12
replace the term 1
D
Dr D∂
⋅∂
with α wherever it appears in (10). Borrowing from an
intermediate level microeconomic textbook, we assume that the RHS of (11c) is less than one. Scrutinizing (10) it is clear that we still need to specify , , , and .MTµ σ τ λ In order
to employ realistic characterizations of the parameters µ and σ , we reviewed the
behavior of domestic interest rates over the last twenty years. We discovered five
intervals of time where short term interest rates increased by at least 200 basis points. The
following chart details the rise in the rate of return µ̂ on one year treasury securities and
the standard deviation σ̂ of the rates in each of the five epochs.
(place Table I here)
We annualize the percentage change in interest rates for each epoch, then weight each
µ̂ and σ̂ according to each epoch’s number of observations relative to the total. Our
overall average µ̂ and σ̂ provides us with the magnitudes for the parameters µ and σ .
Obtaining a characterization of the time to maturity MT for the base case is a little
more problematic. We initially computed an estimate of the average time deposit
maturity by using data from the 2004 10-K and 10-Q SEC filings for a large sample of
commercial banks, which reveal the gross amount of the CD liabilities per year. We
derived an estimate of MT by assigning a value of 1 to all CDs of one year and less, a
value of 2 for CDs two years to one year in maturity, etc. We then computed a weighted
average time to maturity for the banks in our sample of 1.36 years. However, due to the
fact that we do not know whether the CDs mature in 1 day or 365 days within each time
interval, our estimate is plagued with what amounts to be an “errors in variables”
13
problem. Consequently, we have selected four alternative times to maturity that cover the
spectrum of CD lengths which are popular at the retail level: 0.5, 1.0, 1.5, and 2.0 years.20
Serendipitously, this means that instead of having one base case we have four base cases,
one for each time to maturity. Our base case solution for *Dr provides the maturity
structure of the optimal input price and, addresses a major shortcoming of traditional
liability pricing models, it recognizes differences in the time to deposit maturity.
Using current (as of November 17, 2004) market data from Bloomberg on the cost
of equity, debt, and preferred stock for a large sample of publicly traded banks in the
United States, we find a weighted average cost of capital (WACC) of approximately 6%.
We then used this WACC as the bank’s intertemporal discount rate, λ .21
The integral differential equation given by (10) was solved numerically using Newton’s
search technique for first order conditions. In particular, equation (10) can be written in
general as
*, 1( ) 0D nFOC r + =
where n is the number of iterations. Using a first order Taylor expansion on the
expression above yields
* *, ,
* * *, , , 1( ) | ( ) | 0
D n D n
D n D n D nr r
FOC r FOC r dr +′+ = . (12a)
Solving for *, 1D ndr +
*
,
*,( ) |
D n
D nr
FOC r = *
,
*,( ) |
D n
D nr
FOC r′− * *, 1 ,( )D n D nr r+⋅ − (12b)
* *
, ,
* * * *, , , 1 ,( ) | ( ) | ( )
D n D n
D n D n D n D nr r
FOC r FOC r r r+′− = ⋅ − (12c)
14
*
,
*,* *
, 1 , *,
( )|
( ) D n
D nD n D n
rD n
FOC rr r
FOC r+ = −′
. (12d)
Newton’s approach to the numeric solution of (10) was chosen for its simplicity and its
convergence properties. A tolerance level of 710− was used in the iterations.
For the parameters specified above, the expected cost of deposit funding and the optimal
Dr is provided below for all four times to maturity associated with the base case.
(insert Figures 1, 2, 3, 4 here)
All four base cases have local minimums and the optimal input prices are all reasonable.
Given a 2.37% expected annual appreciation in the benchmark rate and MT =1 with
0r =0.25%, *Dr =3.997% seems like a pretty intuitive setting for a decision variable that
seeks to balance the probability of suffering the fixed costs of deposit disintermediation
with the cost of secure funding from 0t to MT . The fact that *Dr is increasing with respect
to the time to maturity is interesting. An increase in MT impacts all five RHS terms in (9)
and, consequently, its net impact on ( )E CDF is impossible to sign analytically. However,
insight into the positive relationship between MT and Dr can be gained by considering the
risk of disintermediation, “R of D”. The risk of disintermediation, 0
( )MT
g t dt∫ , appears in
four of the five RHS terms of ( )E CDF . While ( )g t is weighted in 3 of the 4 cases,
understanding how 0
( )MT
g t dt∫ behaves in the face of parametric changes is useful in
understanding how ( )E CDF behaves. For example, the partial derivative of the risk of
15
disintermediation with respect to MT is easily found to be [2 2
0( ) / 2032
D M Mr r T TD
M
r r eT
µ σ
σ π− − −− ], a
positive number. While the partial derivative of 0
( )MT
g t dt∫ with respect to Dr is
20 0
2 20 0
( ) ( )1[ ] ( ) [1 ]MT
D D
D
r r r rg t dtr r t
µσ σ− −
− +− ∫ , a negative number for the parameters used
in our simulations. If preserving the existing risk of disintermediation were the bank’s
objective function then the implicit function theorem would suggest that Dr will increase
with an increase in MT .
( 1) 0.D M
M
D
R of Ddr T
R of DdTr
∂∂
= − >∂∂
We believe this kind of intuition can be extended to Diagrams A, B, C, and D. Increasing
MT increases the risk of deposit disintermediation; this increase impacts ( )E CDF and,
consequently, *Dr increases to mitigate the impact of increasing the time to maturity.
In order to gain a better understanding of our solution for the optimal structure of
input prices, a comparative static analysis was performed. The original parameter settings
were alternatively increased by 10% and the results are provided immediately below.
(insert Table II here)
16
Examining the grid above it is clear that if the expected change in the rate of interest
increases, *Dr increases throughout the maturity structure. Rather than examining
( )E CDF directly, again, we think intuition is well served by considering the risk of
deposit disintermediation. The partial derivative of “ R of D ” with respect to µ is given
by
2 20( ) / 20 0
230
( )[ ]2
M
D
Tr r t tD Dr r r r te dt
tµ σ µ
σσ π− − −− − −
⋅∫ , a positive number . While the D
R of Dr
∂∂
is
negative, the implicit function theorem yields Ddrdµ
>0 for ( ) 0.d R of D = An increase
inµ increases the risk of disintermediation and, consequently, Dr increases to preserve
the existing level of risk of disintermediation. An increase in µ increases the risk of
disintermediation and adversely impacts ( )E CDF consequently, as documented in the
grid, *Dr increases across the term to maturity.
The partial derivative of 0
( )MT
g t dt∫ with respect to σ is
20
30
( ) 1( ) [ ]MT
Dr r tg t dttµ
σ σ− −
−∫ , a positive number for all realistic values of σ , so that an
increase in σ increases the risk of disintermediation. With D
R of Dr
∂∂
< 0 then
( 1) 0.D
D
R of Ddr
R of Ddr
σσ
∂∂= − >
∂∂
An exogenous increase in σ occasions an increase in the
deposit rate if the existing risk of disintermediation is to be maintained. Extending this
logic to explain the results in the grid leads us to think an increase in σ increases the role
17
of " "R of D in equation (10) and this occasions the increase in *Dr , as detailed in the grid
for all four terms to maturity.
An increase in α essentially amounts to an increase in the elasticity of deposit
supply. Consequently, a reduction in the optimal input price across the maturity horizon
is not a surprise. Given the nearly identical arithmetic role of λ in all five right hand side
terms of ( )E CDF , it is hard to imagine the bank’s intertemporal discount rate having
much of an impact upon *Dr . In fact, though an increase in λ increases *
Dr in the grid; its
impact is nearly imperceivable.
The impact of an increase in τ , the fixed administration and decision-making costs of
disintermediation and re-intermediation, cannot be understood by considering its impact
on " "R of D . The risk of disintermediation is independent of the parameter τ ;
consequently, we must use the implicit function theorem upon the first order condition
for the input price and hope that the analytics do not become unmanageable.
In particular,
*
*
*
*
( )
( )
D
D
D
D
FOC rdr
FOC rdr
ττ
∂∂= −
∂∂
.
Fortuitously the partial derivative of *( )DFOC r with respect to τ is simply
*0
*
( )( ) 2
MTt
D
D
e g t dtFOC r
r
λ
τ
−∂∂
=∂ ∂
∫ .
The RHS of the expression above is the partial derivative of the discounted risk of
disintermediation with respect to *Dr which, according to our earlier analysis must be
18
negative. Given the convex appearance of ( )E CDF in the diagrams above, clearly the
second order condition for *Dr must be satisfied and an optimal input price exists. Since
* *( ) /D DFOC r r∂ ∂ is the second order condition, it must be positive. The partial derivatives
at hand lead us to conclude that * /Ddr dτ is positive which is what we have documented
for all four optimal deposit prices in Table 2.
V. The Impact of Early Withdrawal Penalties on Deposit Pricing
In pricing their deposit services banks can attach a penalty γ for early withdrawal of
funds. Though in practice γ is often nominal and is frequently waived, if used, it clearly
impacts the optimal structure of bank deposit prices22. Disintermediation now takes place
at benchmark rates that are greater than Dr γ+ since the benchmark’s yield must now
compensate the depositor for the penalty. Consequently, the probability density function
of the time to disintermediation is now given by
20
2( )
0 22 3
( )2
Dr r tD tr rg t e
t
γ µσγ
πσ
− + − −+ −=
and the conditional density of the benchmark rate at the time of disintermediation is
2 2( ( ) ) / 2
2
1 .2
Dr re γ µ σ
πσ− − + − ∆ ∆
∆
The new expected cost of deposit funding is distinguished from (9) in predictable ways
and is given as
19
2
2
2
2
0 0
( ( ) )2
2
( ( ) )2
20
( ) 2 ( ) ( ) ( )
1( )2
1( ) . (13)2
M M
D
M M
M D
DM
M
D
T Tt t
D
r rT T
D M MT r
r rTT
r
E CDF e g t dt D r t e g t dt
e r D T g t dt D T e r e dr
D e t g t dt r e dr
λ λ
γ µλ λ σ
γ
γ µλ σ
γ
τ γ
πσ
πσ
− −
− − + − ∆∞ ∞− − ∆
+
− − + − ∆∞− ∆
+
= ⋅ + ⋅ − ⋅
+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅∆
− ⋅ ⋅ ⋅∆
∫ ∫
∫ ∫
∫ ∫
Minimizing (13) with respect to Dr and then using our base case parameters, we solved
for the optimal structure of deposit prices with a prepayment penalty.23
(Insert Table III here)
The result above can be understood if we think in terms of managing the risk of
disintermediation. The partial derivative of 0
( )MT
g t dt∫ with respect to γ is
2 20 0
2 2( ) ( )
0 02 222 3 2 3
0
( )1{[ ] [ ]}2 2
M D DT r r t r r tD Dt tr r r r te e dt
tt t
γ µ γ µσ σγ γ µ
σπσ πσ
− + − − − + − −+ − − + − −+∫
a negative number. So that if ( )d R of D is to be equal to zero in the face of an increase in
γ , Dr must fall.
1 0.D
D
R of Ddr
R of Ddr
γγ
∂∂= − <
∂∂
An increase in γ reduces the risk of disintermediation and allows the bank to pay less for
deposits.
VI. Summary
Today’s commercial bankers are struggling to appropriately price their time deposits.
Bankers are convinced that as the economy grows and prospers, interest rates are going to
20
rise. Though anxious to lock in deposits at today’s low rates of interest, bankers are
aware of the threat of deposit disintermediation in an expansionary economic climate.
This paper has outlined a solution to deposit pricing problem faced by bankers. We first
proposed a stochastic process to describe the behavior of short term interest rates. The
process represented a benchmark rate of interest that signaled alternative returns available
to current bank depositors across time. If the benchmark rate moved above Dr , funds
would flow out of the bank and into other investment opportunities. Banks would suffer
fixed administrative costs and would have to refund its deposits at a rate of interest higher
than Dr . If the benchmark rate remained far below Dr for the deposit horizon, the bank
would have probably paid too much for its inputs. We used the prescribed behavior of the
benchmark rate to derive the probability density function of the time to disintermediation.
From here, we determined the expected cost of deposit funding and minimized it with
respect to Dr to yield the optimal input price. Unable to normalize on the optimal deposit
price, we solved for *Dr numerically. Furthermore, since the optimal input price is an
implicit function of the deposit’s time to maturity we were able to determine the optimal
maturity structure of time deposit rates. The comparative static behavior of the structure
of input prices was then studied under a host of parametric schemes. In the face of
alternative increases in , , , and µ σ τ λ the optimal structure of deposit prices shifted
upward. An increase in either α or γ shifted the maturity structure downward.
Squeezing any intuition out of the first order condition for the optimal input price was
nearly impossible. However, the comparative static behavior of the optimal input prices
was easily understood if we thought of * 'sDr behavior as trying to maintain the existing
21
likelihood of deposit disintermediation. Alternative changes in the model’s parameters
impact the risk of disintermediation. The subsequent change in the optimal input price
should be thought of as trying to preserve the original level of disintermediation risk, that
is , of trying to “manage” the risk of deposit disintermediation.
22
References
Adams, Robert M., Lars-Hendrick Röller, and Robin C. Sickles, 2002, Market Power in Outputs and Inputs: An Empirical Application to Banking, FEDS Working Paper No. 2002-52.
Ahn , Dong-Hyun, Jacob Boudouk, Matthew Richardson, Robert Whitelaw, 1999, Optimal Risk
Management Using Options, Journal of Finance 54, 369-375. Allen, Linda, 1988. The Determinants of Bank Interest Margins: A Note, Journal of Financial
and Quantitative Analysis, 23, 231-235. Amel, Dean, and Timothy Hannan, 1999, Establishing Banking Market Definitions Through
Estimation of Residual Deposit Supply Equations. Journal of Banking and Finance 23, 1667-1690
Angbazo, Lazarus, 1997, Commercial Bank Net Interest Margins, Default Risk, Interest-Rate
Risk, and Off-Balance Sheet Banking. Journal of Banking and Finance 21, 55-87. Bailey, Norman T. J. , 1964. An Introduction to Stochastic Processes (Cambridge University
Press, New York, NY). Baltensperger, Ernst, 1972. Economies of Scale, Firm Size, and Concentration in Banking.
Journal of Money, Credit, and Banking 4, 2009-2029. Baltensperger, Ernst, 1980, Alternative Approaches to the Theory of the Banking Firm, Journal
of Monetary Economics 6, 1-37. Barro, Robert, and Anthony Santomero, 1972, Household Money Holdings and the Demand
Deposit Rate, Journal of Money, Banking, and Credit, 4, 397-413. Bartlett, Maurice, 1955. Stochastic Processes. (Cambridge University Press, London). Berger, Allen and Timothy Hannan, 1998, The Efficiency Cost of Market Power in the Banking
Industry: A Test of the ‘Quiet Life’ and Related Hypothesis, Review of Economics and Statistics 80, 454-65.
Bharucha-Reid, Anthony T., 1960. Elements of the Theory of Markov Processes and Their
Applications.(McGraw-Hill, New York, NY). Darling, D. and A.J.F. Siegert, 1953, The First Passage Problem for a Continuous Markov
Process. Annals of Mathematical Statistics 24, 624-639. Dermine, Jean, 1986, Deposit rates, Credit rates and Bank capital: The Klein-Monti Model
Revisited, Journal of Banking and Finance 10, 99-114. Doob, Joseph L., 1953. Stochastic Processes. (Wiley, New York, NY). Edmister, Robert O., and Harry Merriken, 1989, Measuring Interest Rate Sensitivity of
Consumer Deposits, Journal of Financial Services Research 2, 133-145.
23
Federal Reserve Functional Cost Analysis Report (1992). Feller, William, 1957. An Introduction to Probability Theory and its Applications. 2nd edition.
(Wiley, New York, NY). Flannery, Mark, 1982. Retail bank deposits as quasi-fixed factors of production, American
Economic Review. June, 527-536. Freixas, Xavier, and Jean-Charles Rochet, 1997. Microeconomics of Banking. (MIT Press,
Cambridge, MA). Gilkeson, Jim, and Craig Ruff, 1996, Valuing the Withdrawal Option in Retail CD Portfolios.
Journal of Financial Services Research 10, 333-358. Gilkeson, Jim, Gary Porter, and Stanley Smith, 2000, The Impact of the Early Withdrawal Option
on Time Deposit Pricing, The Quarterly Review of Economics and Finance, 40, 107-120. Hannan, Timothy, and Allen Berger, 1991, The Rigidity of Prices: Evidence from the Banking
Industry, The American Economic Review 81, 938-945. Heffernan, Shelagh, 1992, A Computation of Interest Equivalences for Nonprice Characteristics
of Bank Products, Journal of Money, Credit and Banking, 24, 162-174. Heffernan , Shelagh, 2002, How Do UK Financial Institutions Really Price Their Banking
Products?, Journal of Banking and Finance 26,1997-2016. Ho, Thomas, and Anthony Saunders, 1981, The Determinants of Bank Interest Margins: Theory
and Evidence, Journal of Financial and Quantitative Analysis 16, 581-600. Hutchison, David, and George Pennacchi, 1996, Measuring Rents and Interest Rate Risk in
Imperfect Financial Markets: The Case of Retail Bank Deposits, Journal of Financial and Quantitative Analysis 31, 399-417.
Klein, Michael , 1971, A Theory of the Banking Firm, Journal of Money, Credit, and Banking 3,
205-218. Koch , Timothy, 1995. Bank Management, 3rd Edition, (Dryden Press, Hinsdale, IL). Mitchell, Douglas, 1979, Implicit Interest on Demand Deposits, Journal of Monetary Economics
5, 343-364. Monti, Mario, 1972, Deposit, Credit, and Interest Rate Determination Under Alternative Bank
Objective, in G.P. Szego and K. Shell, eds.: Mathematical Methods in Investment and Finance (Amsterdam: North-Holland, Amsterdam).
Neumark, David, and Steven Sharpe, 1992, Market Structure and the Nature of Price Rigidity:
Evidence from the Market for Consumer Deposits. The Quarterly Journal of Economics 107, 657-680.
24
Parzen, Emanuel, 1960. Modern Probability Theory and its Applications. (Wiley, New York, NY).
Prisman, Eliezer Z., Myron B. Slovin, Marie E. Sushka, 1986, A General Model of the Banking
Firm Under Conditions of Monopoly, Uncertainty and Recourse, Journal of Monetary Economics 17, 293-304.
Rossiter, Rosemary, 1993. Deposit Rate Deregulation and Demand for Money in the 1980’s.
Quarterly Review of Economics and Finance. 33, 247-59. Rossiter, Rosemary, and T.H.Lee, 1987, Implicit Returns on Conventional Demand Deposit: An
Empirical Comparison, Journal of Macroeconomics 9, 613-624. Santomero, Anthony, 1984, Modeling the Banking Firm: A Survey, Journal of Money, Credit,
and Banking 16, 576-602. Sealey, C.W., 1980, Deposit Rate-Setting, Risk Aversion, and the Theory of Depository Financial
Intermediaries, Journal of Finance 35, 1139-1154. Sealey, C.W. and James T. Lindley, 1977, Inputs, Outputs, and a Theory of Production and Cost
at Depository Financial Institutions, Journal of Finance 32, 1251-1266. Stanhouse, Bryan and Duane Stock, 2004 .The Impact of Loan Prepayment Risk and Deposit
Withdrawal Risk on the Optimal Intermediation Margin, Journal of Banking and Finance. v.28 , 1825-1843
Uhlenbeck, George E., and L. S. Ornstein, 1930, On the Theory of Brownian Motion. Physics
Review 36, 823-41. Wong, Kit Pong, 1997, On the Determinants of Bank Interest Margins under Credit and Interest
Rate Risks. Journal of Banking and Finance 21, 251-571. Zarruk, Emilio R., 1989, Bank Spread with Uncertain Deposit Level and Risk Aversion. Journal
of Banking and Finance 13, 797-810. Zarruk, Emilio, and Jeff Madura,1992, Optimal Bank Interest Margin Under Capital Regulation
and Deposit Insurance, Journal of Financial and Quantitative Analysis 27, 143-149.
25
Footnotes 1 The literature on loan rate determination is extensive and many authors have considered
the deposit rate setting behavior of banks including the classic efforts of Klein (1971),
Monti (1972), Sealey (1980) and Flannery (1982). Baltensperger (1980), Santomero
(1984) and Freixas and Rochet (1997) provide excellent reviews of these two segments of
research on commercial banking. Often in the banking literature, pricing of bank outputs
and inputs is considered simultaneously in the determination of the optimal
intermediation margin. In Ho and Saunders (1981), banks are dealers in loan and deposit
markets where the major difficulty in managing the process is that loans and deposits
arrive stochastically and not at the same time. The authors show that a bank will charge
an intermediation fee for the immediate provision of loan and deposit accounts to its
customers. This intermediation margin is shown to be dependent upon management’s
degree of risk aversion, the bank’s market structure, the average size of bank transaction,
and the variance of interest rates. Subsequent papers have also focused on the optimal
intermediation margin as it relates to various types of uncertainty which are common to
the banking environment. Wong (1997) analyzes the intermediation margin under interest
rate risk and finds it positively related to market power, operating costs, and risk
aversion. Wong contrasts his model with others such as Zarruk (1989) where the sole
source of uncertainty is funding risk and Zarruk and Madura (1992) where credit is risky.
Zarruk (1989) finds that increases in bank capital typically increase the intermediation
margin while deposit volatility reduces the margin. Zarruk and Madura (1992) find that
26
increases in bank capital requirements and deposit insurance premiums reduce borrowing
and lending margins. Allen (1988) has analyzed the impact of cross elasticities of bank
products upon the intermediation margin; the margin is shown to be dependent on
monopoly power, a risk premium and multi-product diversification. Angbazo (1997) has
empirically confirmed that banks with riskier loans and higher interest rate risk exposure
enjoy larger intermediation margins. Stanhouse and Stock (2004) derive the optimal
intermediation margin in a model where loans can be prepaid and deposits can be
withdrawn, the simplifying assumptions of their analysis undermines the usefulness of
their results. For example, in order to obtain *Dr , Stanhouse and Stock (2004) assume
that all deposits have the same implicit time to maturity. Furthermore, they assume that
the undeclared time to maturity of deposits coincides with that of bank loans. Finally,
they allow the benchmark rate to change just once between recalibrations of the optimal
output price and input price.
2 Please see Appendix A for an illustration of the standard approach to deposit pricing.
3 Insured time deposits, such as certificates of deposit, provide banks with a significant
source of funds. Although banks face increasing competition in all product areas from
nonbanks such as finance, investment and insurance companies, the federally insured
retail deposit franchise remains unique to the bank charter, thereby offering a potential
comparative advantage. In fact as of August 1, 2004, retail CD balances (time deposits
less than $100,000) totaled 794.4 billion or 10.1% of assets for commercial banks in the
United States. Wholesale CDs (Jumbo CDs, or deposits greater than $100,000) added
27
another 1042.3 billion or 13.3% of bank assets. Thrift institutions seem to rely even more
heavily on CDs. In a recent (6/30/2004) survey of savings institutions by FDIC, total time
deposits totaled 343.0 billion dollars or 35.7% of assets.
4 Ahn, Boudoukh, Richardson, and Whitelaw (1999) minimize the firm’s risk by
determining the optimal exercise price on options the firm issues. Similarly, this paper
could be considered a risk management paper where the bank seeks to manage the risks
of early deposit withdrawal. Optimal deposit rates are the optimal striking prices set by
the bank as the issuer of the implicit options on deposits.
5 In an already complex model the bank is assumed to face only one supply schedule,
therefore only one set of supply elasticities. The bank’s supply schedule is, of course, the
consumer’s demand schedule for deposit services. Heffernan (2002), Rossiter (1993), and
Hannan (1994) all insist that consumer price elasticities for bank products are different;
in particular the interest rate elasticities of depositors for demand deposits and time
deposit services are distinct. Having to accommodate the model and use only one input
supply schedule, our analysis will focus on time deposits. We chose time deposits partly
due to the fact that they are six times larger than demand deposits ($1.818 trillion
compared to $310 billion according to current Federal Reserve estimates). But, in
addition, the solution yielded by our model would not be directly applicable to demand
deposit pricing by commercial banks. In particular, as documented by Heffernan (1992),
Barro and Santomero (1972), Rossiter and Lee (1987), and Mitchell (1979), banks “pay”
non-pecuniary returns for demand deposits which can take the form of gifts, free
28
checking, discounted brokerage fees, and a variety of other services. If our model was
stylized to compute the optimal input price for demand deposits, the solution would be
gross of these implicit payments and would not be immediately useful to a commercial
bank.
6 Please see appendix B for a heuristic derivation of the Kolmogorov diffusion equation.
7Even the most casual reader will recognize that what we call “time to disintermediation”
is what statisticians call the “time to first passage”. Please see Bailey (1964), Bartlett
(1955), Bharucha-Reid (1960), Darling and Siegert (1953), Doob (1953), Feller (1957),
Parzen (1962), and Uhlenbeck and Ornstein (1930) for the background analysis that
enabled us to derive the probability density for the time to disintermediation.
8 The notion that interest rates rise during periods of economic recovery and growth is
probably justified. Over the last twenty years, we find five periods of time where short
term interest rates have risen at least 200 basis points (these epochs are also discussed in
the text on page 12 and in footnote #9). The appreciation of the one year rate of return on
treasuries during the relevant intervals can be seen in Table I.
If the correlation coefficient ρ between the return on treasuries and real gross domestic
product is computed for the epochs at hand , the results dramatically endorse the
existence of positive relationship between the two variables.
(insert Table IV here)
29
9 In order to gain support for characterizing the behavior of short term interest rates as
trended brownian motion with 0µ > , we reviewed domestic interest rates over the last
twenty years. We discovered five intervals of time where short term rates increased by at
least 200 basis points (see Table I) and then tested the appropriateness of a linear trend
over the five epochs.
In particular, the mean squared errors (MSE) of linear, quadratic, and exponential trends
were computed for one year treasury rates of returns across the length of each interval.
For example, during the 2/12/1988-1/27/1989 interval, the three specifications yield the
following MSEs:
linear trend quadratic trend exponential trend
.027486 .06709 .02926
Clearly the linear entries of the RHS variable fit the data best, which supports our trended
brownian motion characterization of interest rates. Similar results hold for the other
epochs and for two year treasury rates.
10 Please see Appendix D for a demonstration of the fact that trended brownian motion
satisfies the Kolmogorov diffusion equation.
11 Please see Appendix E and F for demonstrations that 0( , ; )DP r r t and 0( | , )Dg t r r ,
respectively, satisfy Kolmogorov’s diffusion equation.
30
12 Please see Appendix G for a brief tutorial on Laplace transforms.
13The benchmark rate of interest has unbounded support in this paper. Consequently,
problems of negative interest rates can arise. Throughout the paper, the non-negative
constraint is ignored for simplicity. The probability of negative interest rates can be made
arbitrarily small by the appropriate choice of the underlying statistical parameters that
characterize the density function of ( )R t .
14 That is 2DrA e−= .
15 Clearly when 0 Dr r= , then ( ) 1Drφ = . The inverse of ( )Drφ is
20 0
0 23
( )( ; , ) exp[ ].22
D DD
r r r r tg t r rttµ
σσ π
− − − −=
If 0 Dr r= the starting value for 0r that made ( )Drφ equal to 1, we have 0( ; , )Dg t r r =0.
16 Please see appendix I and then appendix J for confirmation that 0
( )g t dt∞
∫ equals 1 and
that 0
0
( ) Dr rt g t dtµ
∞ −=∫ .
17 Please see appendix K for the resolution of the joint likelihood ( , )z t r into a marginal
and a conditional probability density function.
31
18 Please see appendix L for the arithmetic details associated with equation (10).
19 The supply of time deposits schedule is positively sloped so that the bank can secure
more deposit funds only by increasing Dr ( the bank is assumed to a monopsonist). Our
assumption that a bank’s input market is not perfectly competitive is consistent with a
large number of published articles. Hannan and Berger (1991) as well as Neumark and
Sharpe (1992) report that deposit input prices demonstrate significantly more price
rigidity when the stimulus for a price change is upward rather than downward. Adams,
Roller and Sickles (2003) find that banks display non-competitive behavior in the deposit
market. Sealey and Lindley (1977), Prisman, Slovin and Sushka (1986), Dermine (1986)
and Hutchison and Pennacchi (1996) all presume non-competitive bank deposit markets
as a point of departure in their respective analyses.
20For CDs with terms to maturity longer that the anticipated expansionary phase of the
business cycle, our results will be of only limited usefulness.
21 Using data provided by the Federal Reserve’s Functional Cost Analysis Report (1992)
and Koch (1995), τ , administrative costs for small and large CDs, can range between
0.65 and eight cents per dollar.
22 See Gilkeson, Porter, and Smith (2000) for an alternative analysis of the early withdrawal option.
32
23 Please see Appendix M for the arithmetic details associated with the determination of
*Dr when prepayment penalties are involved.
33
Appendices
Appendix A
Consider the following commercial banking model where the expected profit is given as
( ) ( ) ( )K
L L D Dk
E Lr Lc r D c D L E D x f x dxπ = − − − − − − ∫
and where L =loan size Lc =administrative cost of loan portfolio D =deposit size Dc =administrative cost of deposit liabilities E =equity Lr =rate of return of loan Dr =rate of return on deposits, that is, the input price x is the bank’s cost of borrowing. Evaluating the expected cost of borrowing yields
( ) ( )L L D D xE Lr Lc r D c D L E Dπ µ= − − − − − − . Taking the derivative of ( )E π with respect to Dr
( )D D x
D D D D
E D D DD r cr r r rπ µ∂ ∂ ∂ ∂
= − − − +∂ ∂ ∂ ∂
.
Setting the expression above equal to zero to solve for the optimal price of inputs yields
**
*
*
*
**
*
*
*
( ) 0
1( )
1
1
D D xD
D D x
D
D D x
DD
D
D D x
D
DD r cr
r c D Dr
r crDr
r D
r cr
µ
µ
µ
µε
∂+ + − =
∂
+ − = −∂∂
+ − −=∂∂
+ −=
34
where ε is equal to the elasticity of supply and the LHS is the familiar Lerner index. Please note that *
Dr has no explicit time dimension and that the model cannot accommodate the possibility of early withdrawal by depositors. Appendix B
Suppose that the random variable R representing the process in question takes the value
0r at time 0t and 1r at time 1t , i.e., 0 0( )R t r= and 1 1( )R t r= . Let us write 0 0 1 1( , ; , )p r t r t for
the conditional frequency function of the variable 1r at time 1t given the value 0r at the
previous time 0t . Notice that the order of the pairs 0 0( , )r t and 1 1( , )r t represent the
direction of the transition. We now consider the three epochs of time 0 1 2t t t< < , with the
corresponding variable values 0 1,r r , and 2r where
0 0 2 2 0 0 1 1 1 1 2 2 1( , ; , ) ( , ; , ) ( , ; , )p r t r t p r t r t p r t r t dr∞
−∞
= ∫ (i)
And we are assuming the usual Markov property that, given the present state of the
system, the future behavior does not depend on the past. The equation above can be
easily proved by considering first a path from 0 0( , )r t to 2 2( , )r t through a particular
intermediate point 1 1( , )r t . The probability of this specific path for a Markov process is
0 0 1 1 1 1 2 2( , ; , ) ( , ; , ).p r t r t p r t r t Thus the total probability for a transition from 0 0( , )r t to
2 2( , )r t is obtained by integrating over all possible intermediate points, i.e., integrating
with respect to 1r as shown above.
Suppose we consider, for 0 1t t< , the difference
35
0 0 1 1 0 0 1 1( , ; , ) ( , ; , )p r t t r t p r t r t−∆ − (ii)
From the expression (i) above, we have
0 0 1 1 0 0 0 0 1 1( , ; , ) ( , ; , ) ( , ; , )p r t t r t p r t t z t p z t r t dz∞
−∞
− ∆ = −∆∫
Furthermore, we can write
0 0 1 1( , ; , )p r t r t = 0 0 1 1 0 0 0 0( , ; , ) ( , ; , )p r t r t p r t t z t dz∞
−∞
− ∆∫
because clearly 0 0 0 0( , ; , ) 1p r t t z t dz∞
−∞
− ∆ =∫ .
From here we have
0 0 0 1 1 0 0 1 1( , ; , ) ( , ; , )p r t t r t p r t r t−∆ − = 0 0 0 0 0 1 1 0 0 1 1( , ; , )[ ( , ; , ) ( , ; , )] .p r t t z t p z t r t p r t r t dz∞
−∞
− ∆ −∫ (iii)
Rewriting the second term in the integrand in terms of a Taylor expansion, yields
220 0 1 1 0 0 1 1
0 0 0 0 0 00 0 0
( , ; , ) ( , ; , )1( , ; , )[( ) ( ) ]2
p r t r t p r t r tp r t t z t z r z r dzr r r
∞
−∞
∂ ∂−∆ − + −
∂ ∂ ∂∫ .
So that we have
0 0 1 10 0 0 1 1 0 0 1 1 0 0 0 0 0
0
220 0 1 1
0 0 0 0 00 0
( , ; , )( , ; , ) ( , ; , ) ( ) ( , ; , ) (iv)
( , ; , )1 ( ) ( , ; , ) .2
p r t r tp r t t r t p r t r t z r p r t t z t dzr
p r t r t z r p r t t z t dzr r
∞
−∞
∞
−∞
∂− ∆ − = − −∆
∂
∂+ − −∆
∂ ∂
∫
∫
Let us now suppose that there exist infinitesimal means and variances for changes in the
36
basic random variable ( )R t defined by
00 0 0 0 0 0 00
0
1( , ) lim ( ) ( , ; , )t
r t z r p r t t z t dzt
µ∞
∆ →−∞
= − −∆∆ ∫ (v)
0
2 20 0 0 0 0 0 00
0
1( , ) lim ( ) ( , ; , )t
r t z r p r t t z t dzt
σ∞
∆ →−∞
= − −∆∆ ∫ .
We may have to restrict the ranges of integration on the right hand side of (v) to ensure
convergence, but it is unnecessary to pursue this point in our present non-rigorous
discussion.
Finally, dividing both sides of (iv) by t∆ and letting 0t∆ → yields:
220 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1
0 0 0 00 0 0 0
( , ; , ) ( , ; , ) ( , ; , ) ( , ; , )1[ ] ( , ) ( , )2
p r t r t p r t t r t p r t r t p r t r tr t r tt r r r
µ σ− −∆ ∂ ∂− = +
∆ ∂ ∂ ∂
or
220 0 1 1 0 0 1 1 0 0 1 1
0 0 0 00 0 0 0
( , ; , ) ( , ; , ) ( , ; , )1[ ] ( , ) ( , )2
p r t r t p r t r t p r t r tr t r tt r r r
µ σ∂ ∂ ∂− = +
∂ ∂ ∂ ∂.
Writing 0( , ; )p r r t as the probability density function of r at time t for the time invariant
case, we have
220 0 0
0 0 0 0
( , ; ) ( , ; ) ( , ; )1[ ]2
p r r t p r r t p r r tt r r r
µ σ∂ ∂ ∂− = +
∂ ∂ ∂ ∂ (vi)
This then is the backward Kolmogorov diffusion equation for a continuous variable,
continuous time stochastic process.
37
Appendix C
2( ) 3f x x= for 0 1x≤ ≤ ; 3( )F x x= for 0 1x≤ ≤
2 3 3 30
0
{0 } 3 | 0a
aP x a x dx x a a≤ ≤ = = = − =∫
2( ) 3cdf F x x pdf
x x∂ ∂
= = =∂ ∂
Appendix D
Consider 2
02
( )2
01( , ; )2
r r ttp r r t e
t
µσ
σ π
− − −
=
( )2 2
0 02 2
( ) ( ) 2 2 23/ 22 2 0 0
4 22
2( ) [2 ] 2 ( )1 1 [ ]2 422
r r t r r tt tp r r t t r r tt e e
t tt
µ µσ σ µ µ σ σ µ
σσ πσ π
− − − − − −−∂ ⋅ − − + − −= − +
∂
( ) 00 2
0
2( )( , ; ) ( 1)2
p r r tp r r tr t
µσ
∂ ⋅ − − −= −
∂
220
0 04 2 20 0
( )( ) 1( , ; ) ( , ; )[ ]r r tp p r r t p r r tr r t t
µσ σ
− −∂ ⋅ −= +
∂ ∂ Substituting into the backward Kolmogorov equation yields (of course since 0t has been
suppressed we use t where ( )pt
∂ ⋅∂
= ( )0
( 1)pt
∂ ⋅−
∂)
2 2 21 0 0
4 2
4( ) 2 ( )1 ( ) ( )[ ]2 4
r r t t r r tt p pt
µ µσ σ µσ
− − − + − −− ⋅ + ⋅
=2
20 02 4 2 2
( ) ( )1 1( ) ( )[ ]2
r r t r r tp pt t tµ µ µ σ
σ σ σ− − − −
⋅ + ⋅ −
38
2
0 01
2 2
1( ) ( )1 2[ ]2
r r t t r r tt
t
µ µ µ
σ−
− − + − −− + =
20 0
2 2 2
( ) ( ) 12 2
r r t r r tt t tµ µ µ
σ σ− − − −
+ −
220 0
0 02 2 2 2 2
1( ) ( ) ( ) ( )22
t r r t r r t r r t r r tt t t
µ µ µ µ µ µσ σ σ
− − + − − − − − −= +
2 20 0
2 2 2 2
( ) ( )2 2
r r t r r tt tµ µ
σ σ− − − −
= ,
so ( )p ⋅ =2
02
( )21
2
r r tte
t
µσ
σ π
− − −
satisfies the Kolmogorov diffusion equation.
Appendix E
220 0 0
0 0 0
( , ; ) ( , ; ) ( , ; )12
D D DP r r t P r r t P r r tt r r r
µ σ∂ ∂ ∂= +
∂ ∂ ∂ ∂
because if
20 0 0
0 0 0 0
( , ; ) ( , ; ) ( , ; )12
p r r t p r r t p r r tt r r r
µ σ∂ ∂ ∂− = +
∂ ∂ ∂ ∂
then
20 0 0
0 0 0
( , ; ) ( , ; ) ( , ; )12
p r r t p r r t p r r tt r r r
µ σ∂ ∂ ∂= +
∂ ∂ ∂ ∂
then
39
0 0 02
0 0 0
( , ; ) ( , ; ) ( , ; )12
D D Dr r r
p r r t dr p r r t dr p r r t dr
t r r rµ σ−∞ −∞ −∞
∂ ∂ ∂= +
∂ ∂ ∂ ∂
∫ ∫ ∫
220 0 0
0 0
( , ; ) ( , ; ) ( , ; )12
D D DP r r t P r r t P r r tt t r r
µ σ∂ ∂ ∂= +
∂ ∂ ∂ ∂.
Appendix F
We use 0( | , )Dg t r rt
∂−
∂instead of 0
0
( | , )Dg t r rt
∂∂
because we suppressed 0t in the notation.
2 20 02 2
22
0 0 0
( ) ( )3/ 22 20 0
3
12
( )22
D Dr r t r r tt tD D
g g gt r r r
r r t r rg e et
µ µσ σ
µ σ
σ πσ π
− − − − − −−
∂ ∂ ∂= +
∂ ∂ ∂ ∂
− −= =
202
2 20 02 2
202
5/ 2 ( ) 2 2 202 0 0
2 2 2
( ) ( )2 20 0
23 30
( )22 0
30 0
3( ) ( ) 2 ( )(2 ) ( ) 22 [ ]42
( )1 [ ]2 2
1 [2
D
D D
D
r r tD
t D D
r r t r r tt tD D
r r tt D
t r r r r t t r r tg e gt t
r r r r tg e er tt t
r rg er r t
µσ
µ µσ σ
µσ
µ µ σ µ σσ σσ π
µσσ π σ π
σ π
− − − −
− − − − − −
− − −
− − − − + − −∂= +
∂
− − −∂ −= +
∂
−∂ −= ⋅
∂ ∂
202
2 20 02 2
( )2 0
2 23
( ) ( )22 20 0 0
2 2 2 23 3
1] [ ]2
( ) ( ) 1[ ] [ ]2 2
D
D D
r r tt D
r r t r r tt tD D D
t r r tet tt
r r r r t r re et tt t
µσ
µ µσ σ
µ µσ σσ π
µσ σ σσ π σ π
− − −
− − − − − −
− − −−+ ⋅
− − − − −+ +
We can see that Kolmogorov’s diffusion equation holds because
40
52
2( )0 0 0
2 2 2
( ) ( )0 023 3
0
( ) ( ) ( )3( ) [ ] [ ]2 22
( ) [ ]2 2
D D D
D D
r r r r t r r tg e g gt t tt
r r r r tg e er tt t
µ µ µσ σσ π
µµµ µσσ π σ π
⋅
⋅ ⋅
− − − − −∂= − + +
∂
− − −∂= − +
∂
22
2 ( ) ( ) ( )0 0 0 02 23 3 3
0 0
( ) ( )1 1 1 1 1[ ] [ ] [ ]2 2 22 2 2
D D D Dr r t r r r r t r rg e e er r t t tt t t
µ µσσσ π σ π σ π
⋅ ⋅ ⋅− − − − − −∂ −= ⋅ + −
∂ ∂ Plugging the right hand side of the three expressions above into the diffusion equation yields:
52
2( )0 0 0
2 2 2
( ) ( ) ( )3( ) [ ] [ ]2 22
D D Dr r r r t r r te g gt tt
µ µ µσ σσ π
⋅− − − − −− + + =
( ) ( )0 0
23 3
( ) [ ]2 2
D Dr r r r te ett tµµ µ
σσ π σ π⋅ ⋅− − −
− +
2
( ) ( )0 0 02 23 3
( ) ( )1 1 1 1[ ] [ ] [ ]2 22 2
D D Dr r t r r t r re g et t tt t
µ µσσ π σ π
⋅ ⋅− − − − −− ⋅ + − .
So then
52
( ) ( ) ( )0 0 0 02 23 3
( ) ( )0 03 3
( ) ( ) ( )3( ) [ ] [ ]2 2 2 2
( )1 1 1[ ] [ ],22 2
D D D D
D D
r r r r t r r r r te g e et tt t t
r r t r re et tt t
µ µ µµ µσ σσ π σ π σ π
µ
σ π σ π
⋅ ⋅ ⋅
⋅ ⋅
− − − − − −− + = − +
− − −− ⋅ −
then
41
52
( ) ( )0 0 02 23
( ) ( )0 03 3
( ) ( )3( ) [ ] [ ]2 2 2
( )1 1 1[ ] [ ]22 2
D D D
D D
r r r r t r r te g e gt tt t
r r t r re et tt t
µ µ µµ µσ σσ π σ π
µ
σ π σ π
⋅ ⋅
⋅ ⋅
− − − − −− + = − +
− − −− ⋅ −
and then
52
( ) ( ) ( )0 0 0 02 23 3
( ) ( ) 1[ ] [ ] [ ].2 2 2
D D D Dr r r r t r r t r r te g e g et t tt t t
µ µ µ µµ µσ σσ π σ π σ π
⋅ ⋅ ⋅− − − − − − −− + = − + − ⋅
After further cancellations we have
52
( ) ( ) ( )0 03 3
( ) 1 [ ]2 2 2
D Dr r r r te e ett t t
µµσ π σ π σ π
⋅ ⋅ ⋅− − −− = − − ⋅ .
And finally
2 20 02 2
5 52 2
( ) ( )( ) ( )2 20 0( ) ( )
2 2
D Dr r t r r tt tD Dr r r re e
t t
µ µσ σ
σ π σ π
− − − − − −− −
− = − .
So we have shown the pdf for the time to disintermediation satisfies the Kolmogorov diffusion equation. Appendix G The Laplace transform is a method for solving differential equations. For example:
42
( , ) ( ) 0
( , ) ( )[ ] [ ] 0
( , ) ( )[ ] [ ] 0
f x t fxx t
f x t fxx t
f x t fxx t
∂ ∂ ⋅+ =
∂ ∂
∂ ∂ ⋅+ =
∂ ∂
∂ ∂ ⋅+ =
∂ ∂
L L
L L
Then we change the order of the operators on the two terms appearing on the left hand side of the expression above.
( ) [ ( ) ( ,0)] 0
( ) ( ) 0.
f x s f f xx
f xs fx
∂+ − =
∂
∂+ =
∂
L L
L L
This then we may regard as an ordinary differential equation in x since derivatives with respect to time do not occur in the equation.
If ( )W f= L then we have
0W xsWx
∂+ =
∂ or we have
21
20( , )
sxW x s W e
−
= . From here we have ( , ) ( )W x s f= L , but to get f we need to eventually obtain the solution for the Laplace inverse.
Detailing the fourth equation above, by definition 0
( ') '( )stf e f t dt∞
−= ∫L .
Using integration by parts
( )d uv u dv v du= +
43
|a a
ab
b b
uv u dv v du− =∫ ∫
let’s say that
stv e−= '( )du f t dt= .
So that we have
( ).
stdv se dtu f t
−= −=
Plugging into our formula
00
0
( ) | ( )
( ) | ( )
( 0) ( ( )) ( '( )).
ast a st
bb
st st
f t e s f t e dt
f t e s f t e dt
f t e s f t f t
− −
∞− ∞ −
+
+
− = + =
∫
∫L L
Finally, we have
( '( )) ( ( )) ( 0)f t s f t f t= − =L L
Consider a Laplace transform
0
[ ( ); ] ( )tf t p e f t dtρ∞
−= ∫L
where ρ is a positive constant such that 0
[ ( ); ] ( )tf t p e f t dtρ∞
−= ∫L converges.
Consider an example:
44
0
0
0
[1; ] [1]
1 1|
Consider another example:
[ ; ]
t
t
t
p e dt
e
t p te dt
ρ
ρ
ρ
ρ ρ
∞−
− ∞
∞−
=
− =
=
∫
∫
L
L
( )
|
|
a aab
b b
a aab
b b
d uv u dv v du
uv v du u dv
uv v du u dv
= +
= +
− =
∫ ∫
∫ ∫
00
02 2
1, , , and
1 1|
1 1|
t t
t t
t
u t du dt dv e dt v e
te e dt
e
ρ ρ
ρ ρ
ρ
ρ
ρ ρ
ρ ρ
− −
∞− ∞ −
− ∞
= = = = −
− +
− =
∫
Appendix H
Consider
2 2 2 2
0
( , ) a u b uI a b e du−
∞− −= ∫ .
If v au= , then we can rewrite the expression above as
22 2
2( )
0
av bve du
∞ − −
∫ or as
45
2 2 2 1
0
1( , ) (1, )v b vI a b e dv a I aba
−∞
− − −= =∫ using v au= .
Consider
2 2 2 22
0
( , ) 2 a u b uI a b b u e dub
−∞
− − −∂= −
∂ ∫
Rewriting the derivative of ( , )I a b with respect tob in terms of v
where this time 1v bu−= , we have 2 2( )vub
− = , 2dv bu du−= − , yields
22 2 2
20 ( )2 2 12 ( ) ( )
b va bv bvb e u b dv
b− −
−
∞
− −∫
Keep in mind when 0,u v= = ∞ and when , 0u v= ∞ =
2 2 2 2
0
2 v a b ve dv−
∞− −− ∫ , so
( , ) 2 (1, )I a b I abb
∂= −
∂, where 1v bu−=
2 2 2 21
0
( , ) v a b vI a b a e dv−
∞− − −= ∫ , where v au=
1( , ) (1, )I a b a I ab−= , where v au=
( , ) 2 (1, )I a b I abb
∂= −
∂, where 1v bu−=
1( , )( ) 2
I a b aIb
−
=∂ ⋅ −∂
which implies ( , )2 ( , ) I a baI a bb
∂− =
∂
( , ) 2 (1, )I a b I abb
∂= −
∂
46
The solution to the differential equation immediately above is 2( , ) ( ,0) abI a b I a e−= , where
( ,0)I a is an initial condition.
2 2 1
0
1( ,0)2
a uI a e du aπ∞
− −= =∫
1 21( , )2
abI a b a eπ − −=
Finally, we have 2 2 2 2 2
0 2a u b u abe du e
aπ−
∞− − −=∫
Briefly,
2 2 2 2 2 2 2 21
0 0
a u b u v a b ve du a e dv− −
∞ ∞− − − − −=∫ ∫ , where v au=
2 2 2 2
2 2 2 20
0
[ ]2
a u b u
v a b v
e due dv
b
−
−
∞− −
∞− −
∂= −
∂
∫∫ , where 1v bu−=
2 2 2 2
2 2 2 2
0
1
0
[ ]
2
a u b u
a u b u
e du
ba
e du
−
−
∞− −
∞ −− −
∂
−∂ =
∫
∫
So we have
( ) 2 ( )f afb
∂ ⋅= − ⋅
∂
2( ) abf e−⋅ = , in particular.
In general, we have
47
2
2
( ) ( 0)
( ) .2
ab
ab
f f b e
f eaπ
−
−
⋅ = =
⋅ =
In order to demonstrate the usefulness of this result, let’s consider the
integral
12
1 2 22
2 2
2 2 2 2
1/ 2 /
0
2
0
0
0
.
Changing the variable of integration necessitates the following substituions: , , 2
2
2
2
c t t
cu u
u cu
a u b u
t e e dt
t u t u dt u du
t e e u du
e du
e du
ρ
ρ
ρ
−
−
−
∞− − −
∞− − −
∞− −
∞− −
= = =
∫
∫
∫
∫where 2 2, , , and .a b c a b cρ ρ= = = =
Using the result above
22[ ]2
abeaπ −
2 ce ρπρ
−
21/ 2 /[ ; ] cc tt e e ρπρρ
−− − =L
Consider another very important result. Take the derivative of the left hand side and the
right hand side of the previous expression with respect to c .
21/ 2 / 1/ 21 1[ ( ) ; ] ( 2)( )( )2
cc ttt e e cρπρ ρ ρ
ρ−− − −− = −L
48
Be sure to multiply the LHS and the RHS by (-1)
23/ 2 / 1[ ; ] cc tt e ec
ρπρ ρρ ρ
−− − =L
2 cec
ρπ −=
If it was conjectured that
2 2 1
0
12
a ue du aπ∞
− −=∫ would I believe this?
I know that
2
21 12
z
e dzπ
∞−
−∞
=∫
2
2 2z
e dz π∞
−
−∞
=∫
2
2
0
22
z
e dz π∞
−=∫
2 21[ ]2
0
1 22
ze dz π
∞ −=∫ , therefore
2 2 1
0
12
a ze dz aπ∞
− −=∫ , where 12
a =
so I would believe the conjecture.
49
Appendix I
Does 0
( )g t dt∞
∫ equal 1?
2 2 20 0
2
2 2 2002 2
20 0
230 0
[( ) 2 ( )]3/ 20 2
0 0
2( ) 1 1 [( ) ]3/ 20 22
0 0
2(0
0
( ) ( )( ) exp[ ]22
( )( )2
( )( )2
( )( )2
D D
DD
D D
r r t t r rD t
r r r r tD t
rD
r r r r tg t dt dttt
r rg t dt t e dt
r rg t dt e t e dt
r rg t dt e
µ µσ
µµ
σ σ
µσσ π
σ π
σ π
σ π
∞ ∞
− − + − −∞ ∞−
− ∞∞ − − +−
∞
− − − −=
−=
−=
−=
∫ ∫
∫ ∫
∫ ∫
∫2 2
10 02 2 2
012
) ( )1 13/ 2 2 22
0
( )3/ 20
0 0
( )( )2
D D
D
r r r tt
r rct tD
t e e dt
r rg t dt e t e e dt
µ µσ σ σ
µρσ
σ π
−
−
− −∞− −
−
− ∞∞− − −−
=
∫
∫ ∫
where c=2
02
( )2
Dr rσ− ,
2
22µρσ
= . According to Appendix H
1 23/ 2
0
cct tt e e dt ec
ρρ π−∞
−− − − =∫ so we have
2 21/ 20 0
2 2 2( ) ( )2[ ]
0 2 22
002
( )( )( )2
2
D Dr r r rD
D
r rg t dt e er r
µ µσ σ σπ
σ πσ
− −∞ − ⋅−= ⋅
−∫
0 02 21 12( ) [ ]( )
02
00
( ) 2( )( )2
D Dr r r rD
D
r rg t dt e er r
µ µσ σσ π
σ π
∞ − − −−=
−∫
0
( )g t dt∞
=∫ 1
50
Appendix J Let’s check out the mean:
2 2 20 0
2
2 210 0
2 2 2
20 0
20 0
[( ) 2 ( )]1/ 20 2
0 0
2( ) ( )1 11/ 20 2 22
0 0
0
0
( ) ( )( ) exp[ ]22
( )( )2
( )( )2
( )( )
D D
D D
D D
r r t t r rD t
r r r r ttD
D
r r r r ttg t dt dttt
r rtg t dt t e dt
r rtg t dt e t e e dt
r rtg t dt
µ µσ
µ µσ σ σ
µσσ π
σ π
σ π
σ
−
∞ ∞
− − + − −∞ ∞−
− −∞∞ − −−
∞
− − − −=
−=
−=
−=
∫ ∫
∫ ∫
∫ ∫
∫0
12( )
1/ 2
02
Dr rct te t e e dt
µρσ
π−
− ∞− − −∫
where c=2
02
( )2
Dr rσ− ,
2
22µρσ
= .
available in Appendix H is the mathematical fact that
1 1/ 21/ 2 2( )
0
ct t ct e e dt eρ ρπρ
−∞
− − −=∫ so that
00
2( )( ) 2[ ]
20 222
0
( )( ) 22
DD r rr rDr rt g t dt e e
µµσσσ π σ
σ π µ
−−∞ −−=∫
0
0
( ) Dr rt g t dtµ
∞ −=∫
51
Appendix K The partial expectation of the joint probability density function
0
( , )M
D
Tt
r
e t r z t r dr dtλ∞
−∫ ∫
can be rewritten as
0
( ) ( )M
D
Tt
r
e t r g t f r dr dtλ∞
−∫ ∫
as soon as it is realized that at the time of disintermediation IT , when the benchmark rate
equals or exceeds Dr , another clock starts. This new clock is the reaction time∆ of
depositors to this new opportunity of disintermediation, let’s say 4 hours or
4/(24 ⋅7 ⋅52)=1/2184 of a year. Conditional on passage of the benchmark rate through
Dr and ∆ , the probability density function of the benchmark rate is given as
2
2( )
2
2
12
Dr r
eµ
σ
πσ
− − − ∆∆
∆.
So that we now have
2 202 2
( ) ( )2 20
3 20
( ) 12 2
D DM
D
r r t r rTtt D
r
r re t r e e dt drt
µ µσ σλ
σ π πσ
− − − − − − ∆∞∆− −
⋅∆
∫ ∫
2
2( )
2
20
1[ ( ) ]2
DM
D
r rTt
r
t e g t dt r e drµ
σλ
πσ
− − − ∆∞∆−
∆∫ ∫
2
2( )
2
20
1[ ( ) ][ ]2
DM
D
r rTt
r
t e g t dt r e drµ
σλ
πσ
− − − ∆∞∆−
∆∫ ∫ which is what we have in the paper.
52
Appendix L Now let’s explicate some of the partial moments for further differentiation, recall
02( )
220
3( )
2
Dr r t
tDr rg t e
t
µ
σ
σ π
− − −−= .
From here I can show that
0
( ) 1g t dt∞
=∫
and that
0
0
( ) Dr rt g t dtµ
∞ −=∫
2 2 2 2
0[ ( ) 2 ]/ 203 2
0 0
( )2
M M
D
T Tr r t t tt Dr re g t dt e dt
tµ λσ σλ
π σ− − − −− −
=∫ ∫
2 2 2 2
0[ ( ) 2 ]/ 202
0 0
( )2
M M
D
T Tr r t t tt Dr rt e g t dt e dt
tµ λσ σλ
π σ− − − −− −
=∫ ∫
2 2
0[ ( ) ]/ 20
0 0
( )2
M M
D
T Tr r t tDr rt g t dt e dt
tµ σ
σ π− − −−
=∫ ∫
Now the remaining derivatives:
2 2 2 20
2 2 2 20
[ ( ) 2 ]/ 203
0
[ ( ) 2 ]/ 20 023
0
[ ( ) ]12
( )[ ]2
M
M
D
M
D
Tt
Tr r t t t
D
Tr r t t tD D
e g t dte dt
r tr r r r te dt
tt
λ
µ λσ σ
µ λσ σ
σ π
µσσ π
−
− − − −
− − − −
∂=
∂
− − − −+
∫∫
∫
53
2 2 2 20 0( ) / 2 ( ) / 20 0
23 2 3 2
[ ( ) ]( ) ( )1 [ ]
2 2M D D
M M
T r r t t r r t tD D
D T T
g t dtr r r r te dt e dt
r tt tµ σ µ σ µ
σπ σ π σ
∞
∞ ∞− − − − − −
∂− − − −
= + ⋅∂
∫∫ ∫
2 2 2 20
2 2 2 20
[ ( ) 2 ]/ 20
0
[ ( ) 2 ]/ 20 022
0
[ ( ) ]12
( ) ( )[ ]2
M
M
D
M
D
Tt
Tr r t t t
D
Tr r t t tD D
t e g t dte dt
r tr r r r te dt
tt
λ
µ λσ σ
µ λσ σ
σ π
µσπ σ
−
− − − −
− − − −
∂=
∂
− − − −+ ⋅
∫∫
∫
2
2( )
2
2
1[ [ ] ]2
D
D
r r
r
D
r e dr
r
µσ
πσ
− − − ∆∞∆∂
∆∂
∫=
2 2( ) / 2
2( 1)
2Dr e µ σ
πσ− − ∆ ∆−
∆
2
2( )
222
( )1[ ] [ ]2
D
D
r r
D
r
r rr e drµ
σ µσπσ
− − − ∆∞∆ − − ∆
+ ⋅∆∆
∫
2 2 2 20 0[ ( ) ]/ 2 [ ( ) ]/ 20 0 0
220 0
[ ( ) ]( ) ( )1 [ ]
2 2
M
M M
D D
T
T Tr r t t r r t tD D
D
t g t dtr r r r te dt e dt
r tt tµ σ µ σ µ
σσ π π σ− − − − − −
∂− − − −
= + ⋅∂
∫∫ ∫
54
Operationalizing the first order condition for simulation, using DrD eα= and 1
D
Dr D
α∂⋅ =
∂
yields:
*
*0*
0 0
* 0*
0
* **
[ ( ) ]( ) ( )
[ ( ) ]( )
[ ( ) ]( ) ( )
1[2
M
M M
M
M
MM M M
M M
M
D
Tt
T Tt t
DD
Tt
Tt
DD
TT T TM M D M D
DT T
TM
r
e g t dtFC e g t dt FC r t e g t dt
r
t e g t dtt e g t dt r
r
g t dte T g t dt e T r g t dt T e r
r
e T r
λ
λ λ
λ
λ
λ λ λ
λ
α α
α
απσ
−
− −
−
−
∞
∞ ∞− − −
∞−
∂⋅ ⋅ + ⋅ + ⋅ ⋅
∂
∂+ + ⋅
∂
∂
+ + + ⋅ ⋅ ⋅∂
+
∫∫ ∫
∫∫
∫∫ ∫
∫* 2
2
* 2
2
*
* 2
2
*
* 2
2
*
*
( )2
2
( )2
2
*
( )2
20
( )2 0
*2
0
]
1[ ]2
1( ) [ ]2
[ ( ) ]1
2
[
( )
D
D
DM
DM
M
D
M
D
M
D
M
DM
r r
r r
rTM
D
r rTT
r
T
r rT
Dr
TrT
e dr
r e dr
e Tr
e t g t dt r e dr
t g t dte r e dr
r
r
e t g t dt
µσ
µσ
λ
µλ σ
µλ σ
λ
πσ
απσ
πσ
− − − ∆∆
− − − ∆∞∆
−
− − − ∆∞− ∆
− − − ∆∞− ∆
−
∆
∂∆
+∂
−∆
∂− ⋅
∂∆
∂
− ⋅ ⋅
∫
∫ ∫
∫∫
∫
* 2
2( )
2
2
*
1 ]2
0
Dr r
D
e dr
r
µσ
πσ
− − − ∆∞∆
∆=
∂
∫
55
Appendix M
Minimizing the expected cost of deposit funding with respect to Dr (under the
assumption of a prepayment penalty) yields:
*0* *
0
* *0* *
0
**
[ ( ) ]2 ( ) ( )
[ ( ) ]( ) ( ) ( ) ( )
[ ( ) ]
M
M
M
M
M M
M M
MM M
Tt
Tt
DD D
Tt
TT Tt
D M M DD DT T
TT TM D M
D
e g t dtD r t e g t dt
r r
t e g t dtDD t e g t dt D r e T D g t dt e T r g t dt
r r
g t dtDD T e r e T
r
λ
λ
λ
λ λλ
λ λ
τ γ
γ
−
−
−∞ ∞
− −−
∞
− −
∂∂
⋅ + −∂ ∂
∂∂
+ + ⋅ − + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅∂ ∂
∂∂
+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅∂ ∂
∫∫
∫∫ ∫ ∫
∫ * 2
2
*
* 2
2
* 2
* 2
*
* 2
*
( ( ) )2
* 2
( ( ) )2
( ( ) )22
* * 20
( ( ) )
2
1[ ]2
1[ ]2 1( ) [ ]
2
12
D
D
D
DM
DM M
D
D
M
D
r r
D r
r r
r rTrT T
MD D r
r rT
r
r e drr
r e drDe T D e t g t dt r e dr
r r
D e r e
γ µσ
γ
γ µσ
γ µγλ λ σ
γ
γ µλ
γ
πσ
πσ
πσ
πσ
− − + − ∆∞∆
+
− − + − ∆∞∆
− − + − ∆∞+− − ∆
+
− − + − ∆∞−
+
⋅∆
∂∆ ∂
+ ⋅ ⋅ − ⋅∂ ∂ ∆
− ⋅∆
∫
∫∫ ∫
∫2
* 2
2
*
2 0*
( ( ) )2
2
*0
[ ( ) ]
1[ ]2
( ) 0
M
D
M
DM
T
D
r r
TrT
D
t g t dtdr
r
r e dr
D e t g t dtr
σ
γ µσ
γλ πσ
∆
− − + − ∆∞∆
+−
∂⋅
∂
∂∆
− ⋅ ⋅ ⋅ =∂
∫
∫∫
Now let’s explicate some of the partial moments for further differentiation, keeping in mind that the evaluations will be impacted by γ .
202
( )0 2
3( )
2
Dr r tD t
r rg t e
t
γ µσ
γ
σ π
+ − −−+ −
=
56
From here I can show that
0
( ) 1g t dt∞
=∫
and that
0
0
( ) Dr rt g t dt γµ
∞ + −=∫
2 2 2 2
0[ ( ) 2 ]/ 203 2
0 0
( )2
M M
D
T Tr r t t tt Dr re g t dt e dt
tγ µ λσ σλ γ
π σ− + − − −− + −
=∫ ∫
2 2 2 2
0[ ( ) 2 ]/ 202
0 0
( )2
M M
D
T Tr r t t tt Dr rt e g t dt e dt
tγ µ λσ σλ γ
π σ− + − − −− + −
=∫ ∫
2 2
0[ ( ) ]/ 20
0 0
( )2
M M
D
T Tr r t tDr rt g t dt e dt
tγ µ σγ
σ π− + − −+ −
=∫ ∫
Now the remaining derivatives:
2 2 2 20
2 2 2 20
[ ( ) 2 ]/ 203
0
[ ( ) 2 ]/ 20 023
0
[ ( ) ]12
( )[ ]2
M
M
D
M
D
Tt
Tr r t t t
D
Tr r t t tD D
e g t dte dt
r tr r r r te dt
tt
λ
γ µ λσ σ
γ µ λσ σ
σ π
γ γ µσσ π
−
− + − − −
− + − − −
∂=
∂
+ − − + − −+
∫∫
∫
2 2 2 20 0( ) / 2 ( ) / 20 0
23 2 3 2
[ ( ) ]( ) ( )1 [ ]
2 2M D D
M M
T r r t t r r t tD D
D T T
g t dtr r r r te dt e dt
r tt tγ µ σ γ µ σγ γ µ
σπ σ π σ
∞
∞ ∞− + − − − + − −
∂+ − − + − −
= + ⋅∂
∫∫ ∫
57
2 2 2 20
2 2 2 20
[ ( ) 2 ]/ 20
0
[ ( ) 2 ]/ 20 022
0
[ ( ) ]12
( ) ( )[ ]2
M
M
D
M
D
Tt
Tr r t t t
D
Tr r t t tD D
t e g t dte dt
r tr r r r te dt
tt
λ
γ µ λσ σ
γ µ λσ σ
σ π
γ γ µσπ σ
−
− + − − −
− + − − −
∂=
∂
+ − − + − −+ ⋅
∫∫
∫
2
2( ( ) )
2
2
1[ [ ] ]2
D
D
r r
r
D
r e dr
r
γ µσ
γ πσ
− − + − ∆∞∆
+
∂∆∂
∫=
2 2( ) / 2
2( 1)
2Dr e µ σγ
πσ− − ∆ ∆+
−∆
2
2( ( ) )
222
( ( ) )1[ ] [ ]2
D
D
r r
D
r
r rr e drγ µ
σ
γ
γ µσπσ
− − + − ∆∞∆
+
− + − ∆+ ⋅
∆∆∫
2 2 2 20 0[ ( ) ]/ 2 [ ( ) ]/ 20 0 0
220 0
[ ( ) ]( ) ( )1 [ ]
2 2
M
M M
D D
T
T Tr r t t r r t tD D
D
t g t dtr r r r te dt e dt
r tt tγ µ σ γ µ σγ γ µ
σσ π π σ− + − − − + − −
∂+ − − + − −
= + ⋅∂
∫∫ ∫
58
Operationalizing the FOC, using DrD eα= and 1
D
Dr D
α∂⋅ =
∂, yields:
*
*0*
0 0
* 0*
0
* **
[ ( ) ]2 ( ) ( ) ( )
[ ( ) ]( ) ( )
[ ( ) ]( ) ( )
M
M M
M
M
MM M M
M M
M
D
Tt
T Tt t
DD
Tt
Tt
DD
TT T TM M D M D
DT T
TM
r
e g t dte g t dt FC r t e g t dt
r
t e g t dtt e g t dt r
r
g t dte T g t dt e T r g t dt T e r
r
e T r
λ
λ λ
λ
λ
λ λ λ
λ
γ
α τ α γ
γ
α
α
−
− −
−
−
∞
∞ ∞− − −
−
+
∂⋅ ⋅ + ⋅ + −
∂
∂+ + −
∂
∂
+ + + ⋅ ⋅ ⋅∂
+
∫∫ ∫
∫∫
∫∫ ∫
* 2
2
* 2
2
*
* 2
2
*
* 2
2
*
( ( ) )2
2
( ( ) )2
2
*
( ( ) )2
20
( ( ) )2 0
2
1[ ]2
1[ ]2
1( ) [ ]2
[ ( ) ]1
2
D
D
DM
DM
M
D
M
D
M
D
r r
r r
rTM
D
r rTT
r
T
r rT
r
e dr
r e dr
e Tr
e t g t dt r e dr
t g t dte r e dr
γ µσ
γ µσ
γλ
γ µλ σ
γ
γ µλ σ
γ
πσ
πσ
απσ
πσ
− − + − ∆∞∆
− − + − ∆∞∆
+−
− − + − ∆∞− ∆
+
− − + − ∆∞− ∆
+
∆
∂∆
+∂
−∆
∂− ⋅
∆
∫
∫
∫ ∫
∫∫
* 2
2
*
*
( ( ) )2
2
*0
1[ ]2
( ) 0
D
M
DM
D
r r
TrT
D
r
r e dr
e t g t dtr
γ µσ
γλ πσ
− − + − ∆∞∆
+−
∂
∂∆
− ⋅ ⋅ =∂
∫∫
59
Appendix N Leibniz’s rule: Consider the derivative of an integral where the limits are a function of the variable of differentiation.
( )
( )
( )h x
g x
y f x dx= ∫
( )( )
( )
( ) ( )' ( ( )) ( ( ))h x
g x
y h x g xf x dx f h x f g xx x x∂ ∂ ∂
= + ⋅ − ⋅∂ ∂ ∂∫
Appendix O
For example, consider the probability density function for trended brownian motion
2 20 0 0( ( )) / 2 ( )
0 00
1( , ; , )2 ( )
r r t t t tp r r t t et t
µ σ
σ π− − − − −=
−
by observation 0t enters 0 0( , ; , )p r r t t as the mirror reflection of t and, consequently,
0
( ) ( )p pt t
∂ ⋅ ∂ ⋅= −
∂ ∂.
60
Appendix P
When most of us “hear” trended brownian motion, we think
0 0 0
00
00
0
2
0
2 2
0
.
However, integrating across the time horizon we have
( 0)
( )
( ) [ ]
( ) [ ] .
u u
t t t
u u
t
t u
t
t u
tt
t u
t
t u
dr du dW
dr du dW
r r t dW
r r t dW
E r r t
V r V dW
V r E dW
µ σ
µ σ
µ σ
µ σ
µ
σ
σ
= +
= +
− = − +
= + +
= +
=
=
∫ ∫ ∫
∫
∫
∫
∫
Ito’s isometry allows us to write
0 0
[[ ][ ]]t t
u uE dW dW∫ ∫ as 0
t
du∫ , so we have
2
02
20
( )
( )
~ ( , ).
t
t
t
t
V r du
V r t
r N r t t
σ
σ
µ σ
=
=
+
∫
with these characteristics the pdf for the left hand side variable tr must be as follows
2 20( ) / 2
01( , ; )2
r r t tp r r t et
µ σ
σ π− − −= .
61
Table I: Rises in one-year interest rates This table details recent increases in the one-year treasury interest rates over five different time periods. Also included are the annualized increases, µ̂ ,and their standard deviation, σ̂ , for each epoch. In each case the one-year rates increased more than 200 basis points. Date # of months bps∆ µ̂ σ̂ 8/22/86-10/16/87 14 248 .0213 .0061 2/12/88-1/27/89 12 238 .0238 .0078 10/8/93-5/13/94 7 214 .0367 .0102 6/10/94-12/30/94 6 205 .0410 .0127 10/23/98-5/19/00 17 239 .0151 .0042 Average µ̂ .0237 Average σ̂ .0070
62
TABLE II: Optimal Input Prices
At the top of the table are the base case optimal time deposit rates for a given time to
maturity, MT , using the following parameters:
0.0237, 0.007, 1.0, 0.10, 0.06.µ σ α τ λ= = = = =
Below these optimal rates are the changes in the *Dr given alternative ten percent
increases in the parameters at hand.
Base case
*Dr =0.02687
MT =0.5
*Dr =0.03997
MT =1.0
*Dr =0.05087
MT =1.5
*Dr =0.05922
MT =2.0
*Dr∆ *
Dr∆ *Dr∆ *
Dr∆ µ∆ +0.00111 +0.00227 +0.00433 +0.00529 σ∆ +0.00104 +0.00107 +0.00069 +0.00000 α∆ -0.00003 -0.00004 -0.00013 -0.00021 λ∆ +0.00000 +0.00001 +0.00000 +0.00005 τ∆ +0.00017 +0.00038 +0.00064 +0.00129
63
Table III Optimal Input Prices with a Prepayment Penalty*
This table details the optimal time deposit rates with and without a prepayment penalty of
0.5%, using the following base case parameters:
0.0237, 0.007, 1.0, 0.10, 0.06.andµ σ α τ λ= = = = =
MT =0.5 MT =1.0 MT =1.5 MT =2.0
Base Case (γ =0) *Dr =0.02687 *
Dr =0.03997 *Dr =0.05087 *
Dr =0.05922
γ =0.005 *Dr =0.02187 *
Dr =0.03497 *Dr =0.04572 *
Dr =0.05383
* The typical commercial bank structures their early withdrawal penalties based on the original time to
maturity of time deposits. For example, a CD that is issued with 12 months or less to maturity will have 1
to 3 months worth of interest as an early withdrawal penalty. A CD with 1 year or greater to maturity will
have 3 to 6 months interest as a penalty. These penalties vary from bank to bank. Using these guidelines, a
2.0%, 1 year CD will have a 0.5% penalty on average. A 4%, 2 yr. CD will have between 0.5% and 1.0%
penalty on average. The overall withdrawal penalty as a percentage of earned interest decreases the longer
the customer waits for early withdrawal. Based on current market rates and average bank penalties, it is
reasonable to choose γ =0.005 (0.5%) for our simulations.
64
Table IV: Relationship between interest rates and Real GDP
This table details the correlation between one year treasury interest rates and
Real Gross Domestic Product during five time intervals of rising short term rates. The
correlation coefficient and t-statistic for their significance are given for each interval.
Date Correlation coefficient ( ρ ) t-statistic* 8/22/86-10/16/87 .8634 3.423 2/12/88-1/27/89 .9384 4.704 10/8/93-5/13/94 .9476 4.195 6/10/94-12/30/94 .8658 1.730 10/23/98-5/19/00 .9613 9.869 *the computed value of t = ρ ⋅ sqrt((N-2)/(1- 2ρ ), with N-2 degrees of freedom
65
Figure 1: Base Case with MT =0.5 This figure shows the expected cost of deposit funding on the vertical axis and the time deposit rate on the horizontal axis. ( )E CDF as a function of Dr is plotted using the following base case parameters:
00.0237, 0.007, 0.10, 0.0025, 0.06, 0.50.Mr Tµ σ τ λ= = = = = = The optimal rate, the *Dr
that minimizes the expected cost of deposit funding, is listed below the graph. The figure is plotted using Mathematica.
*Dr =0.02687
0.02 0.04 0.06 0.08rD0.02
0.03
0.04
0.05
0.06
0.07
E(CDF)
E(CDF)=f(rD)
66
Figure 2 : Base Case with MT =1.0
This figure shows the expected cost of deposit funding on the vertical axis and the time deposit rate on the horizontal axis. ( )E CDF as a function of Dr is plotted using the following base case parameters:
00.0237, 0.007, 0.10, 0.0025, 0.06, 1.0.Mr Tµ σ τ λ= = = = = = The optimal rate, the *Dr
that minimizes the expected cost of deposit funding, is listed below the graph. The figure is plotted using Mathematica.
*Dr =0.03997
0.04 0.05 0.06 0.07 0.08rD0.065
0.07
0.075
0.08
0.085
0.09 E(CDF)
E(CDF)=f(rD)
67
Figure 3 : Base Case with MT =1.5
This figure shows the expected cost of deposit funding on the vertical axis and the time deposit rate on the horizontal axis. ( )E CDF as a function of Dr is plotted using the following base case parameters:
00.0237, 0.007, 0.10, 0.0025, 0.06, 1.50.Mr Tµ σ τ λ= = = = = = The optimal rate, the *Dr
that minimizes the expected cost of deposit funding, is listed below the graph. The figure is plotted using Mathematica.
*Dr =.05087
0.04 0.05 0.06 0.07 0.08rD0.12
0.14 0.16
0.18 0.2
0.22 0.24
E(CDF)
E(CDF)=f(rD)
68
Figure 4: Base Case with MT =2.0
This figure shows the expected cost of deposit funding on the vertical axis and the time deposit rate on the horizontal axis. ( )E CDF as a function of Dr is plotted using the following base case parameters:
00.0237, 0.007, 0.10, 0.0025, 0.06, 2.0.Mr Tµ σ τ λ= = = = = = The optimal rate, the *Dr
that minimizes the expected cost of deposit funding, is listed below the graph. The figure is plotted using Mathematica.
*Dr =.05922
0.05 0.06 0.07 0.08 0.09rD0.12
0.14 0.16
0.18 0.2
0.22 0.24 E (CDF)
E(CDF)=f(rD)