Central Bank Liquidity Swaps and Optimal
Swap Policy
April 22, 2013
Kalok Chu
Department of Economics, Indiana University
Third Year Paper
Abstract
After the financial crisis in 2007, there was a shortage of dollars in the overseas financial
market. To provide liquidity, the Federal Reserve Bank established liquidity swap lines with
other central banks. This paper provides a theoretical model of liquidity swap of central
banks. We show that i) as the default risk of local banks increases, the centrals banks should
extend the amount of swap; and ii) there exists an optimal level of swap line as a portion of
the total loan market.
JEL Classification: E50, G10
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1.1
Introduction
Background
After the global financial crisis in 2007, the supply of US dollar overseas was very low
which leaded to a shortage in dollars. (Ivashina and Scharfstein, 2010) The Federal Open
Market Committee (FOMC) announced that it had authorized temporary currency arrangements (central bank liquidity swap lines) with the European Central Bank and the Swiss
National Bank, in order to improve liquidity conditions in U.S. and foreign financial markets
by providing foreign central banks with the capacity to deliver U.S. dollar funding to institutions in their jurisdiction during times of market stress. In 2009, the FOMC extended the
central bank liquidity swap lines to other twelve central banks. Figure 1.1 shows the amount
of liquidity swap outstanding of the federal reserve.
[Insert figure 1.1 here]
The goal of this paper is to provide a theoretical model of central bank swap in a banking sector model, to find out the optimal swap policy and to analysis the policy implication
of the optimal policy.
1.2
Procedure of Swap
A central bank liquidity swap consists of two transactions.
If two central banks agrees to establish a liquidity swap line, they will first swap their own
currency at the market exchange rate for a certain period of time. The maturity date can
range from overnight to three months. After the two central banks acquire the fund of
foreign currency, they can loan out the foreign currency to local banks in their jurisdiction.
At maturity, the two central banks swap the currency back using the same exchange rate
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as the first transaction, so there is no exchange rate risk involved. And at the end of the
second transaction, the borrower pays a market interest rate to the lender at the end of the
transaction. For example, the European Central Bank will pay a market interest rate to the
Federal Reserve at the conclusion of the liquidity swap arrangement, while the FED does
not have to pay any interest to the ECB.
The transactions can be summarized in figure 1.2.
[Insert figure 1.2 here]
1.3
Review of Literature
Obstfeld, Shambaugh and Taylor (2009) gives a very detailed empirical results regard-
ing financial stability and foreign currency reserve. In their paper, looking at a country’s
reserve holdings and the predicted reserve holdings after the 2007 crisis can significantly
predict exchange rate movements of both emerging and advanced countries in 2008. They
also find that the ratio of the amount of swap to the total foreign exchange reserve for a
country is a very good indicator to predict GDP movement. They also find that if a country
has a higher foreign exchange reserve, then the currency of this country tends to appreciate
in the financial crisis.
The remainder of the paper is organized as follows. Section 2 presents the model
and the solutions. Section 3 includes data and the application of the model. Section 4 is
conclusion.
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Model
I will introduce the role of central bank in Ivashina, Scharfstein and Stein (2012) global
bank model.
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2.1
Model Setup
A global bank that lends in both the U.S. and Europe, in both dollars LD and euros LE ,
and the return of the loans are g(LD ) and h(LE ) respectively, where g( ) and h( ) are both
increasing and concave. The bank also faces an aggregate lending constraint LD + LE ≤ K,
where K is an exogenous given lending limit.
Assume that if the bank wants to lend, it has to be funded in the respective currency, that
means it cannot take on any unhedged FX risk. For simplicity, further assume the risk free
rate in both the U.S. and Europe veto be equal to r, and the spot dollar euro exchange rate
X S is equal to 1.
The bank can default with probability p. We assume that euro borrowing is insured by the
government, while dollar borrowing is only partially insured. Then the interest rate on the
borrowings are rE = r and rD = r + αp respectively, where α is the portion of the uninsured
dollar borrowing.
As stated in Ivashina, Scharfstein and Stein (2012), the covered interest parity does not hold
thus there is a limited arbitrage opportunity for investors to participate in the swap market,
where investors can convert appropriate amount of euro funding into dollar borrowing and
make a profit.
A representative investor with wealth W can engage in swap activity (S) and invest in other
projects (I). If they choose to engage in swap, they have to set aside a haircut H as collateral,
which we assume to be linear in S, i.e. H = γS. Then the investor’s budget constraint can
be written as I = W − H = W − γS. Denote the net return from investment be f(I), which
is an increasing concave function. In equilibrium, the marginal return on both swap and
investment must be equal. Denote the forward exchange rate be X F = 1 + ∆, then we have
∆ = γf 0 (W − γS).
The European Central Bank will conduct liquidity swap with the Federal Reserve, and will
lend the dollar fund to the global bank. The ECB will swap an amount S C with the FED
and lend to the global bank. The global bank then will pay an interest equal to the market
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interest rate to the ECB at the end of the transaction. There is an operational cost C(S C )
for the ECB to conduct the swap. This cost can be thought as the social cost of holding
foreign reserve as described in Rodrik (2006). This cost also includes the interest cost that
the ECB has to pay to the Federal Reserve. We further assume that C( ) is increasing and
convex. This cost will be funded by the net interest paid. Therefore we can write the ECB
budget constraint as (r + αp)S C = C(S C ).
One may notice now the total amount of lending in dollars can be written as LD = B D +
S + S C where B D is the direct dollar borrowing, and all the lending in euro is funded by
direct borrowing where LE = B E .
2.2
Global Bank’s Optimization Problem
Now we can write down the global bank’s optimization problem:
max g(LD ) − LD (1 + r) + h(LE ) − LE (1 + r) − αp(B D + S C ) − S∆
LD ,LE ,S
subjected to
LD = B D + S + S C
LD + LE ≤ K
and the global bank will take S C , K, r, ∆ as exogenously given.
The first order conditions yield
g 0 (LD ) = h0 (LE ) + αp
∆ = αp = γf 0 (W − γS)
The main result in Ivashina, Scharfstein and Stein (2012) shows that the total amount
of dollar lending decreases as the default risk increases, and there is a threshold ∆∗ such that
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if the default risk is higher than this threshold, then the direct borrowing in dollar B D will
be zero and the entire lending in dollar will be funded by swap activity from the investor.
This result can be summarized by figure 2.1.
[Insert figure 2.1 here]
When the central bank engage in the liquidity swap line, S C > 0. Since the global
bank takes this policy as given, the total lending in dollar increases by the amount of S C no
matter what the default risk is, since the central bank swap is risk free. Therefore the LD
curve will shift upward by the amount of S C . This result can be illustrated by figure 2.2.
[Insert figure 2.2 here]
2.3
Social Planner’s Problem
Now we will look at the social planner’s problem. The social planner has to choose an
optimal level of swap. We can write the social planner’s problem as
max
LD ,LE ,S,S C
g(LD ) − LD (1 + r) + h(LE ) − LE (1 + r) − αpB D − S∆ − C(S C ) + rS C
subjected to
LD = B D + S + S C
LD + LE ≤ K
The social planner’s optimality condition yields
g 0 (LD ) = h0 (LE ) + αp
∆ = αp = γf 0 (W − γS)
C 0 (S C ) − r = αp
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It is easy to see that as the default risk, measured by αp, increases, the optimal amount of
central bank swap increases as well.
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3.1
Result
Optimal Policy
In this section, we will show that an optimal swap policy exists.
Assumption 1
The marginal cost for the ECB to replace the entire lending market by central bank swap
is higher than the marginal return to the investor when the investor does not engage in any
swap activity such that the investor will invest all his wealth into capital investment. That
is, C 0 (LD∗ ) − r ≥ γf 0 (W ).
Proposition 1 (Optimal Swap Policy)
Under assumption 1, there exists a unique optimal level of central bank swap.
The unique optimal level S C∗ is the solution of C 0 (S C ) − r = γf 0 (W − γ(LD∗ − S C )), which
is illustrated by figure 3.1.
[Insert figure 3.1 here]
The social optimal solution can be shown in figure 3.2.
[Insert figure 3.2 here]
3.2
Data and Implication
(to be added)
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Conclusion
(to be added)
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References
Correa, R., Sapriza, H. and Zlate, A. (2012) ”Liquidity Shocks, Dollar Funding Costs,
and the Bank Lending Channel During the European Sovereign Crisis,” Working Paper.
Goldberg, L., Kennedy, C. and Miu, J. (2010) ”Central Bank Dollar Swap Lines and
Overseas Dollar Funding Costs,” Federal Reserve Bank of New York Staff Reports, no. 429
Ivashina, V. and Scharfstein, D. (2010). ”Bank Lending During the Financial Crisis of
2008.” Journal of Financial Economics, Vol. 97, No. 3, 319338.
Ivashina, V., Scharfstein, D. and Stein, J. (2012). Dollar Funding and the Lending
Behavior of Global Banks,” Harvard Business School Working Paper.
Obstefeld, M., Shambaugh, J. and Taylor, A. (2009). ”Financial instability, Reserves,
and Central Bank Swap Lines in the Panic of 2008,” The American Economic Review: Papers & Proceedings, Vol. 99, No. 2, 480-486.
Rodrik, D. (2006) ”The Social Cost of Foreign Exchange Reserves,” International Economic Journal Vol. 20, No. 3, 253266.
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Appendix
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Figure 1.1: Total Amount of Swap Line Outstanding
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Figure 1.2: Liquidity Swap Procedure
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Figure 2.1: Equilibrium when S C = 0 as a function of αp
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Figure 2.2: Equilibrium when LD > S C > 0 as a function of αp
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Figure 3.1: Find the optimal level of swap
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Figure 3.2: Optimal solution as a function of αp
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