Supplemental Appendix to
“Liquidity Regulation and the Implementation of Monetary Policy”
Morten Bech
Todd Keister
Bank for International Settlements
[email protected]
Rutgers University
Paris School of Economics
[email protected]
December 28, 2015
Abstract
In this appendix, we do four things: (A) provide a proof of Proposition 1; (B)
show that the effects of liquidity regulation are amplified when the runoff rate for
loans from the central bank,   is larger than the runoff rate for deposits,  ; (C)
show that the equilibrium regulatory premium, ∗ − ∗  is (weakly) increasing in  ;
and (D) show how the effects of liquidity regulation on open market operations are
amplified when the central bank uses repurchase agreements rather than outright
transactions.
A. Proof of Proposition 1
Proof: To deal with the indicator function in the objective (14), we examine the first-order
conditions in the regions  ≥  and    separately. If  ≥  holds, the indicator
ª
©
function is zero and we have max   ̂ =  . The marginal benefit of funding in each market
for this case is given by
[  ]
= − +  + ( −  )
∆
Z
∞

¡ ¢
   
(23)
In the region where    holds, the value of the indicator function is one, we have
©
ª
max   ̂ = ̂  and the marginal benefits are
Ã
!
Z ∞
Z ̂
¡ ¢ 
¡ ¢ 
1 − 
[  ]
= − +  + ( −  )
   +
   
(24)
1 −  
∆
̂
1
For the first part of the proposition, suppose that    holds and consider the region
where  ≥   In this case, comparing (23) for  =  with  =  implies
[  ]
[  ]


∆
∆
meaning that the bank could increase its expected profit by borrowing one more dollar in market
 and one less dollar in market . It follows that the solution to the problem cannot lie in this
region and, hence, must satisfy     This solution is characterized by setting the derivatives
in (24) to zero, which directly yields equation (15) from the statement of the proposition.
Now suppose that  =  holds and consider the region where    holds. In this case,
the derivatives in (24) and    imply
[  ]
[  ]


∆
∆
meaning that the bank could increase its expected profit by borrowing one less dollar in market
 and one more dollar in market . It follows that the solution to the problem in this case cannot
lie in this region and must instead satisfy  ≥   The problem is solved in this case by any
{∆ } that equates (23) to zero for both  =  , which requires that  satisfy equation (16)
¥
from the statement of the proposition.
B. The model with   
The analysis in the text assumes that the runoff rate on loans from the central bank,  ,
is set below the runoff rate on deposits,  . In this section, we extend the analysis to the case
where    holds and show how the effects of the LCR regulation on equilibrium interest
rates are amplified in this case.
The demand for interbank loans. When    holds, the amount bank  must borrow
from the central bank’s lending facility to meet all of its regulatory requirements is still given
by (8), but the pattern of borrowing depicted in Figure 4 in the main text changes to that in
Figure 8. The principal difference is that the slope of the line associated with borrowing for
LCR purposes is now steeper than the line associated with the reserve requirement. As a result,
the LCR requirement now determines the amount borrowed for sufficiently large values of the
payment shock in panel () and for all values of the payment shock in panel ().
2
slope
slope
Figure 8: Bank ’s borrowing from the central bank (with    )
Bank ’s maximization problem (14) changes to reflect these new patterns:
max Ψ −
∆
(  ∆ )
³
P ´





∆
+

−

+

  
 ∆
P
⎧
⎪
⎨
¢ P  ´ ¡ ¢ 
R ̂ ³  ¡ 

I(  )   −  −  −  ∆   



− ( −  )
³
¡ 
¢ P 1−  ´ ¡  ¢ 
R
∞
⎪
1−
 −
⎩ +
−


 
 1− ∆   
max{̂  } 1−

To characterize the solution to this problem, we first define the constant
Λ ≡ (1 −  )  − (1 −  )  − ( −  )  
⎫
⎪
⎬
⎪
⎭

(25)
We then have the following result.
Proposition 3 Suppose     If Λ  0 the bank will choose {∆ } so that the critical values
¡   ¢
    ̂ defined in (9) — (11) satisfy
µ
¶
1 − 
 − 


 =  + ( −  )
+
[̂ ] − [ ] 
1 − 
1 − 
(26)
If Λ = 0 the bank will choose {∆ } so that these values satisfy
 ≥ 
and
 =  + ( −  )
£ ¤¢
1 −  ¡
1 −   
1 − 
(27)
Proof. The general structure of the proof follows that of Proposition 1 To deal with the
indicator function in the objective, we examine the first-order conditions in the regions  ≤
 and    separately. If  ≤  holds, the indicator function is zero and we have
3
ª
©
max ̂   =  . The marginal benefit of funding in each market for this case is given by
1 − 
[  ]
= − +  + ( −  )
1 − 
∆
Z
∞

¡ ¢
   
(28)
In the region where    holds, the value of the indicator function is one, we have
©
ª
max ̂   = ̂  and the marginal benefits are
ÃZ 
!
Z ∞
̂
¡ ¢ 
¡ ¢ 
1 − 
[  ]
= − +  + ( −  )
   +
   
(29)
1 −  ̂
∆

For the first part of the proposition, suppose that Λ  0 holds and consider the region where
 ≤   Starting from any point in this region, suppose the bank lends an additional (1 −  )
dollars in market  and borrows (1 −  ) more dollars in market  The change in expected profit
from this plan is
− (1 −  )
[  ]
[  ]
+
(1
−

)
= Λ  0

∆
∆
where the equality follows from using (28) for  =   and simplifying. The fact that this plan
raises expected profit from any starting point where  ≤  holds shows that the solution to
the bank’s problem cannot lie in this region. Instead, the solution in this case has    and
is characterized by setting (29) to zero for  =   which yields equation (26) in the statement
of the proposition.
Now suppose that Λ = 0 holds and consider the region where    holds. Starting from
any point in this region, suppose the bank borrows an additional (1 −  ) dollars in market 
and lends (1 −  ) more dollars in market  The change in expected profit from this plan is
(1 −  )
£  ¤¢
¡ £ ¤
[  ]
[  ]
−

 
−
(1
−

)
=
(
−

)
(
−

)

̂





∆
∆
where the equality follows from using (29) for  =   and simplifying. In this region, ̂  
holds (see Figure 8) and, therefore, the expression above is strictly positive; this plan raises
expected profit from any starting point in the region. It follows that the solution to the bank’s
problem must satisfy  ≤  . Any {∆ } that equates (28) to zero for  =   solves the
problem, which yields equation (27) in the statement of the proposition.
4
¥
Intuition. When    holds, the LCR requirement will always determine the bank’s need
to borrow from the central bank for some realizations of   while the reserve requirement may
or may not bind for some   (See Figure 8.) The parameter Λ measures the direct cost to the
bank of borrowing more in market  and lending more in market  in a way that leaves the
cutoff  associated with the LCR requirement unchanged. The fact that    holds implies
that the amount lent in market  under this plan is less than the amount borrowed in market 
so the bank gains reserves in net. In other words, Λ measures the cost to the bank of improving
its reserve position by an amount  −  while leaving its LCR position unchanged.
When the cost Λ is positive, the bank will choose {∆ } so that     ̂ holds, as
depicted in panel () of Figure 8, and the reserve requirement will indeed bind for some   One
way to understand the first part of the result is to multiply both sides of (26) by (1 − − ) and
subtract the result for  =  from that for  =  to get
¡ £ ¤
£ ¤¢
(1 −  )  − (1 −  )  = ( −  )  + ( −  ) ( −  )  ̂ −  
or, using the definition in (25),
¡ £ ¤
£ ¤¢
Λ = ( −  ) ( −  )  ̂ −   
The right-hand side of this equation is the expected marginal benefit to the bank of improving
its reserve position by ( −  ) dollars: it saves the net cost of borrowing from the central bank,
 −   for each of these dollars whenever the realization of  falls between  and ̂ . (See
Figure 8.) The first-order conditions in (26) thus say that the bank equates this benefit to the
marginal increase in net interest costs from the plan.
When Λ is zero, in contrast, there is no cost to the bank of using this plan to improve its
reserve position. In this case, the solution to the bank’s problem must have the property that
the reserve requirement does not bind for any realization of   as in panel () of Figure 8 The
second component of (27) says that the bank’s marginal cost of funds in each market must equal
the marginal value of loans in that market for meeting the LCR requirement. Many pairs {∆ }
will satisfy the condition in (27) and the bank will be indifferent between any of these actions.
Finally, note that if Λ  0 the bank has a pure arbitrage opportunity and hence its problem
has no solution. Equilibrium interest rates obviously cannot lie in this case.
5
Equilibrium interest rates. The process of aggregating banks’ demands for loans in each
market is identical to that in the main text, as are the equilibrium conditions ∆ = 0 for
 =   Substituting the equilibrium critical values from (20) into the demand functions from
Proposition 3 yields the equilibrium pricing relationships for this case.
Proposition 4 When     the unique equilibrium interest rate in market  =   is given
by
∗
=  + ( −  )
µ
¶
1 − 
∗ ∗
∗
∗
(1 −  [max {̂   }]) + max{[̂ ] − [ ] 0} 
1 − 
This result provides the counterpart to Proposition 2 in the main text. It follows immediately
from the assumption    that ∗  ∗ always holds in this case, meaning that the regulatory
premium associated with the LCR requirement is always strictly positive.
Corollary 4 When     ∗  ∗ always holds.
Effects of open market operations. The impact of an open market operation on the balance
sheet of the banking system is again as depicted in Figure 5 in the main text. The relationship
between these changes and equilibrium interest rates, however, is now governed by Proposition
4. Figure 9 illustrates the results for this case by plotting the equilibrium interest rates {∗ }
associated with two different values of  in each panel:  = 025 and  = 0 These runoff
rates correspond to the minimum standards under the original and revised LCR rules, respectively. The two panels of the figure corresponds to the case where the central bank conducts
outright purchases/sales of HQLA (panel ) and of non-HQLA (panel ), both with banks as
counterparties. The initial value of the liquidity surplus of the banking system is set to 0 = 0
which means that the interest rates associated with  = 0 are the same those in the first
two graphs in the top row of Figure 7 in the main text. The new curves in each panel are the
equilibrium interest rates associated with  = 025
For the case of operations with HQLA, when  is sufficiently negative, the LCR is only a
binding concern for extreme values of the payment shock  and the overnight rate is virtually
the same as in the standard model regardless of the value of   For sufficiently positive values
of  the overnight rate is at the floor of the corridor for both values of   For intermediate
values of  however, we see that the higher value of  pushes the overnight rate lower. The
difference can be significant: when  = 0 the overnight rate is at the midpoint of the corridor
6
HQLA with banks
-
non-HQLA with banks
-
-
-
:
(with
)
(with
)
:
(with
)
(with
)
Figure 9: Effects of an OMO with    vs.  = 0
with  = 0 but at the floor of the corridor with  = 025 In addition, the higher value of 
pushes the term rate up substantially for both moderate and sufficiently positive values of 
Overall, panel () of Figure 9 shows how a higher runoff rate  tends to increase the regulatory
premium and amplify the effect of the LCR requirement on equilibrium interest rates.
Panel () presents the corresponding figure for the case where the central bank conducts
outright purchases/sales of non-HQLA ( = 0). The same general effects are present here, and
the differences between the two cases are even larger. With  = 0 the overnight rate lies at
the midpoint of the corridor for all   0 With  = 025 in contrast, the overnight rate lies
on the floor of the corridor for all   0. In addition, the term interest rate can now rise above
the ceiling of the corridor,   since a dollar of term borrowing in this case will save the bank
from having to borrow (1 −  )−1  1 dollars from the central bank’s lending facility in some
states. Together, these two panels clearly illustrate how the effects of liquidity regulation on
equilibrium interest rates identified in the main text for the case of    are amplified when
 is instead set larger than  
C. The effect of increasing 
In this section, we establish that the equilibrium regulatory premium is always a (weakly)
increasing function of the runoff rate  for loans from the central bank. This result spans both
the case of    studied in the main text and that of    studied in the previous
section of this appendix.
7
Proposition 5 The equilibrium regulatory premium ∗ −∗ is a continuous, (weakly) increasing
function of  .
Proof: To begin, note that the equilibrium critical values ∗ and ∗ in (20) do not depend on
  whereas ̂∗ does. It is clear from the expression that ̂∗ is a continuous function of  in
the regions    and    separately, but is discontinuous at  =   Differentiating
yields
̂∗
(1 −  ) ( −  − )
=

( −  )2
(30)
for any  6=   The remainder of the proof is divided into three parts.
Part () :    . In this region, we can use Proposition 2 to write the equilibrium regulatory
premium as
∗ − ∗ = ( −  )
µ
¶
 − 
max { [̂∗ ] −  [∗ ]  0} 
1 − 
(31)
Since ̂∗ is a continuous function of  in this region (and ∗ is independent of  ), it follows
that the premium is a continuous function of  in this region. To see that it is also a (weakly)
increasing function, first note that if ̂∗ ≤ ∗  then the premium is equal to zero for any   In
the reverse case, one can use (20) and (30) to show that the following inequalities are equivalent:
̂∗  ∗
⇔ ∗  ∗
⇔ −  ⇔
̂∗
 0

(These relationships can also be seen in Figure 4 where the slope of the flatter (green) line
is increasing in  ) Since the distribution function  is strictly increasing, it follows that the
expression for the premium in (31) is increasing in  whenever ̂∗  ∗ holds as well.
Part () :    . The equilibrium regulatory premium is actually strictly increasing in  in
this region. To see why, note that we can use Proposition 4 to write the premium in this region
as
∗ − ∗ = ( −  )
 − 
(1 −  [max {̂∗  ∗ }]) 
1 − 
(32)
If ̂∗ ≤ ∗  the argument of  is independent of  and the overall expression is clearly strictly
increasing in   For the reverse case, using (20) and (30) shows that the following inequalities
are equivalent when    :
̂∗  ∗
⇔ ∗  ∗
⇔ −  ⇔
8
̂∗
 0

(33)
(See also Figure 8.) Since the distribution function  is strictly increasing and bounded above by
1 it follows that the expression for the premium in (32) is strictly increasing whenever ̂∗  ∗
holds as well.
Part (): The boundary  =   What remains is to show that the equilibrium regulatory
premium is a continuous function of  as it crosses the boundary  that defines the two
regions studied separately above. We divide this analysis into two cases. First, suppose that
∗ ≤ ∗ holds. In this case, it follows from (31) and the discussion above that the regulatory
premium is zero for all     For the region where     note that the expression in
(20) for ̂∗ implies
lim ̂∗ ( ) = ∞
 ↓
(This fact can also be seen from Figure 8.) Using the expression for the premium in (32), we
then have
lim ∗ − ∗ = ( −  )
 ↓
 − 
(1 −  [∞]) = 0
1 − 
This equation establishes that the regulatory premium increases from zero as  rises above 
and, hence, is a continuous function of  when ∗ ≤ ∗ holds.
Now suppose ∗  ∗ holds. The expression for ̂∗ in (20) now implies
lim ̂∗ ( ) = ∞
 ↑
(see also Figure 4). Using this fact and (31), the limiting value of the premium as  approaches
 from below is given by
lim ∗ − ∗ = ( −  )
 ↑
 − 
(1 −  [∗ ]) 
1 − 
In the region where    holds, the inequalities in (33) show that ̂∗  ∗ always holds in
this case and, using (32), the limiting value of the premium as  approaches  from above is
given by
lim ∗ − ∗ = ( −  )
 ↓
 − 
(1 −  [∗ ]) 
1 − 
The fact that these last two expressions are equal shows that the equilibrium regulatory premium
is also a continuous function of  in this second case, where ∗  ∗ holds. We have, therefore,
¥
established the proposition.
9
D. Repurchase agreements
In this section, we show how the effects of an open market operation change when it is
structured as a repurchase agreement (repo) rather than as an outright purchase/sale by the
central bank. In a repo transaction, the exchange of reserves for assets between the central bank
and its counterparties is reversed at a pre-specified future date. The key difference between a
repo and an outright transaction in our framework is that a repo typically involves a haircut,
meaning that the value of the assets encumbered by the operation is larger than the quantity
of reserves created. We show here that this fact amplifies the regulatory premium that arises
when the central bank does operations using bonds with banks as counterparties.20
Figure 10 shows the balance sheet effects of an operation in which the central bank purchases
 units of bonds using a repurchase agreement with haircut  ≥ 0. As in the main text, we use
 to denote the fraction of these bonds sold by banks. Taking into account the haircut, the total
quantity of bonds in the banking system that are encumbered by this operation is    (1 − ) 
As the LCR rules only allow unencumbered assets to be included in the calculation of a bank’s
high-quality liquid assets, an operation with  = 1 now decreases the LCR position of the
banking system rather than leaving it unchanged.21
Central Bank
Assets
Liabilities
Loans
Reserves
Bonds
Equity
Banking System
Assets
Liabilities
Loans
Deposits
Bonds
- encumbered
Reserves
Equity
Figure 10: Balance sheet effects of a repo operation
20
The results in this section also apply to collateralized loans, which in the context of our model are identical to
a repo transaction with the same haircut. In practice, repurchase agreements generally require that the collateral
involved is marketable, while collateralized loans can be used with a much wider range of countervalue.
21
It is straightforward to see that the effects of a repo operation using only loans ( = 0) or with only non-bank
counterparties ( = 0) are exactly the same in our framework as the outright transactions studied in the main
text. While a repo transaction again encumbers additional assets, Propositions 2 and 4 show that the quantities
of unencumbered loans held by banks and unencumbered bonds held by non-banks have no effect on equilibrium
interest rates in our model.
10
Figure 11 presents the effects of an open market operation structured as a repurchase agreement on the equilibrium interest rates in our model. The parameter values are the same as those
used to generate Figure 7 in the main text, including  = 0 and  = 1. The initial LCR surplus
of the banking system is set to 0 = 0, so that this figure corresponds to the upper-left panel
in Figure 7. The dashed lines in the figure show the effects of an operation with zero haircut,
which is equivalent to an the outright purchase/sale depicted in Figure 7. The solid lines show
the equilibrium interest rates when the haircut is set to 25%. (This relatively large value for the
haircut is useful for illustrating the effects visually in the graph.)
-
:
(with
)
(with
)
:
(with
)
(with
)
Figure 11: Effects of a repo operation with banks using HQLA
As the figure shows, the regulatory premium is larger when the haircut is higher and the
difference increases with the size of the operation. In particular, the higher haircut causes the
overnight interest rate to fall slightly faster as reserve supply increases, while it causes the term
interest rate to rise. The fact that the term interest rate increases when the central bank adds
reserves is particularly striking; this result can be understood by looking back at Figure 4 When
the central bank adds reserves using this operation, the equilibrium cutoff ∗ moves to the right
as before. However, the quantity of unencumbered high-quality liquid assets held by the banking
system falls due to the haircut, which moves the equilibrium cutoff associated with the LCR
requirement, ∗  to the left in the figure. The result is that the LCR constraint is more likely to
be a binding concern for each bank, which drives up the interest rate on term loans and widens
the regulatory premium, as seen in Figure 11 The figure thus illustrates how the effects studied
in the main text tend to be amplified when the central bank uses repurchase agreements.
11