Part III. International Liquidity, and the Cost of Currency Crises - Mateusz Szczurek
3.3. International Liquidity: Simple Model
reserves in any case, especially if the exchange rate regime is rigid.
Probably, the most often used measure of hot liabilities is Bank for International Settlements' (BIS) statistic of short-term debt in the foreign banking sector. This mea-sure is available for most of the emerging market counties in semi-annual frequency, which makes it the statistic of choice for most of the cross-country estimations of inter-national liquidity [Bussiére and Mulder, 1999; Tornell, 1999; Radelet and Sachs, 1998; Rodric and Velasco, 1999].
Even disregarding the domestic liabilities of the central bank, this short-term debt measure is obviously biased downwards. For example, a five year treasury bond held by a foreign fund is not included (the liability is more than one year, and moreover it is not versus the foreign bank-ing sector). Similarly, portfolio equity investments can flow out of the country in minutes, but they are not included in the BIS statistic as a short-term liability.
Money stock is clearly the upper limit for the short-term liabilities of the central bank, provided the M2 does not grow during the crisis as a result of sterilised foreign exchange interventions [10].
common with the liquidity models based on bank-run liter-ature [see e.g. Chang and Velasco, 1999]. Secondly, because the function is so widely used in the empirical study, it can be claimed that it is believed by both policy-makers and the creditors to be true representation of the currency crisis risk. It suffices for the analysis of the policy-maker's optimi-sation that follows.
The functional form of the crisis probability function has powerful implications for the policy options faced by the government/central bank. In particular, regardless on how bad the fundamentals are, the central bank could come up with liquidity in t-1sufficiently high to prevent the crisis in time t, as shown in Figure 3-2.
A crisis in period t results in real depreciation in the same period. This is the only effect of the crisis on the fun-damental variables. The scale of the crisis-triggered depre-ciation is a function of the liquidity and the other country-specific variables in t-1.
(3.2) where µ, θ> 0; λ< 0
A crisis and devaluation/depreciation is possible even with undervalued real exchange rate, provided internation-al liquidity, or other fundamentinternation-als are bad enough. Formu-lation above ensures that the two consecutive crises in a country are possible, yet unlikely – the worse the funda-mentals leading to the first crisis, the bigger the deprecia-tion, and bigger the improvement in the external stability outlook in the following period.
A crisis in time tresults in certain costs χtto the econo-my and the policy makers. One could argue that the cost
should be some function of the severity of the crisis (or mis-alignment of the fundamental variables and insufficient liq-uidity), reflecting the adjustment costs (presumably higher with high current account deficit, high public debt growing in line with real exchange depreciation), distress to the banking system, etc. This type of cost can be observed empirically as, e.g., the deviation of the post-crisis GDP growth from its long-term trend.
The overall crisis cost to the policy-maker, however, includes a second type of cost: reputation loss. It is much more difficult to assess empirically in an explicit way, no explicit form of the total crisis cost function will therefore be considered here. Presumably, the reputation cost is the function of the degree of the rigidness of the foreign exchange regime, and the length for which the regime was maintained, past inflation experience, but also personality of the central banker, etc.
The actual expected gain from the additional unit of reserves in t-1is therefore:
where χt<0, (3.3)
which is the decrease in expected value of the policy maker's crisis cost in tas a result of the higher international liquidity in t-1. Apart from the liquidity's influence on the financing costs (considered in the following section), this is the only benefit from international reserves in the model.
The marginal gain reflects the shape of the probability surface, and looks as in Figure 3-3 (assuming cost indepen-dent of REER).
The peak of the marginal return to reserves is reached for higher levels of liquidity as the fundamentals get worse.
Figure 3-2. Probability of a crisis vs. international liquidity and a budget deficit
0 5
5 5
10 0
2 4
6 0
0.25 0.5 0.75
1
2.
7.
P(y=1)
Liquidity
Budget deficit
( )
= +
+ +
−
= =
−
−
−
−
−
1 ,
0 ,
1 1 1
1 1
t t t t
t
t t
t REER l REER G if y
y if REER REER
θ µ
λ
φ
( )
(
l REER G)
tG REER l t
t t t
t
t t t
t t t
e e l
y P l
Eχ χ β χ
δ γ β α
δ γ β α
) 2 (
) (
1
1 1 1 1
1 1 1
1 ) 1 (
−
−
−
−
−
−
+ + +
+ + +
−
− = +
∂
=
=∂
∂
∂
The reason is that when fundamentals are really bad, a mar-ginal increase in liquidity from zero will not markedly reduce the probability of the crisis. Alternatively, if the fundamentals are really good, an increase in already high liquidity will not reduce the probability of a crisis, because it is very close to nil anyway.
3.3.2. International Liquidity and its Cost
At each non-crisis point of time, the policy maker faces the following choice: he can either keep his foreign exchange reserves (receiving international yield i* and ben-efiting from the increased security it brings), or he can get rid of them using the cash to pay off its foreign debt, avoid-ing payavoid-ing interest i. We assume that when there is no run on the currency, the policy maker can both easily borrow on the international bond markets and get down with its inter-national debt up to the size of foreign exchange reserves (to allow a possibility to vary international liquidity between zero to infinity).
The real alternative cost of the foreign exchange reserves is thus the difference between the country's inter-national bond and US treasury yield, equal toi-i*, which is the country risk premium over the international borrowing rate. This is the only cost of holding foreign exchange reserves. The way the reserves are acquired does not
mat-ter, to the policy maker's choice – it can always run down or increase his reserves holding afterwards [11].
The risk premium faced by the policy makers depends on the credit assessment by the foreign investors, which is directly related to the probability of the crisis:
(3.4) Because an improvement of the credit assessment makes servicing the existing foreign debt cheaper, the mar-ginal, immediate cost of reserves πtis:
(3.5) where Dtis the amount of foreign debt to be rolled over in t. Because βis less than zero, it is possible that πtfalls below zero, given the existing debt is high enough.
We assume the level of taxation constant, so an increase in international liquidity in t-1increases Gtby πt.
The immediate cost of reserves is not the only cost faced by the policy maker. Second round effects also play a role. There are two dynamic problems to worry about.
[11] Somewhat more subtle point is how international liquidity is defined. If it is just foreign exchange reserves, or foreign exchange reserves scaled by M2, the annual cost of "a unit of liquidity" is i-i*(possibly scaled up by a constant). If it is the ratio of foreign exchange reserves to the foreign short-term debt (as in most of the recent empirical work), the cost is higher, because the bonds become a short short-term obligation one year ahead of the matu-rity. The average annual cost then becomes larger by , where m is the maturity of the benchmark international bond of the country. The effect becomes insignificant when the bonds are sufficiently long maturity, and does not change the overall results qualitatively.
Figure 3-3. Marginal return to international liquidity vs. liquidity and REER
0 0.2
0.4 0.6
0.8 1
0 0.05 0.1 0.15 0.2 US$ c
0
2
4
6 Liquidity
Overvaluation
−1 m
m
) 1 (
*= + =
−i P yt
i ε ξ
( )( )
[ ]
( )
[ ]
(
l REER G t)
21 tG REER l
G REER l 1
t t 1 t t
1 t 1 t 1 t
1 t 1 t 1 t
1 t 1 t 1 t
e 1
D l e
1
l e D l i i
−
−
−
−
−
−
−
−
−
+ + +
− +
+ +
+ + +
−
−
+
+ + +
+
∂ = +
−
=∂
δ γ β α
δ γ β α
δ γ β α
β ξ ε
π * ( )
First, is the obvious negative impact of debt servicing costs on budget deficit in the subsequent period [12]. This nega-tive effect can be counterbalanced by reserve borrowing higher by , at a cost of , where ris the poli-cy maker's discount rate.
Second cost is much less straightforward, and is related to the fact, that crises in t and t+1are not independent events. A crisis in t makes the subsequent crisis less likely, because of the real depreciation assumed in (3.3) [13].
Therefore, by borrowing reserves in order to protect the country from a crisis in t, we make the crisis in t+1more likely, because the probability of the "cleansing" effect of the crisis in tdecreases. The amount of the reserves needed to counterbalance the effect is equal to:
To complete the analysis one should take into account higher debt service cost in subsequent periods caused by the above-mentioned additional borrowing. The marginal total cost MTCof reserves, therefore, is equal to:
Quite clearly, the discount ratermatters. Large rmeans that the effects of future budgetary costs, and the lack of cleansing effect of the quick crisis do not bother the policy maker much. One could argue that the optimalr, which should be close to the inter-temporary consumption dis-count rate (probably related to the average real interest rate), may be completely different to the policy makers' r.
For example, if the government is on its way to loose the elections, it could risk postponing the (almost) inevitable currency crash, by borrowing foreign exchange reserves at a large cost, and making the crisis virtually certain, but only after the elections. In such case (3.6) collapses to (3.5).
3.3.3. Optimisation Problem
Combining (3.3) and (3.6) we are able to complete the analysis. Optimising policy maker tries to minimise the fol-lowing loss function by targeting the liquidity level:
(3.7) First term of (3.7) is the total cost of reserves (the area below the cost curve in Figure 3-4), while the second is (minus) total benefit from reserves.
Unfortunately, the problem is not solvable analitically.
There are two possible equilibria. One is at internation-al liquidity equinternation-al to zero. Increasing liquidity costs more than it brings (fundamentals are too bad for a slight improvement of liquidity to change the probability of a cri-β
δπt
r
t
++ 1
π 1
[12] Clearly this cost can be negative in a special case, when marginal costs are also negative, i.e, when the benefits of cheaper financing outweigh the cost of the additional unit of reserves.
[13] Here, we explicitly assume lack of two effects which make a crisis in subsequent period more likely after a crisis in t.First is the feedback effect on the reserves (crisis in tresults in the outflow of reserves and higher probability of the crisis in t+1). Second effect is the loss of reputation, which is more common than major reforms after the crisis.
[ ]
β
θ µ
λ φ
γ 1 1 1
1
) 1
( − − −
−
+ +
+
∂
=
∂ t t t
t
t l REER G
l y P
( )
( )
( )
∑
∞=−
−
−
−
− +
+
−
+ + +
∂ = +
−∂ +
=
0 n
n
1 t 1 t 1
t 1
t t
t t
r 1
r 1
G REER l l
1 y P TC
β δπ
β
δ θ µ
λ φ γ π
π
*
* )
(
(3.6)
( ) ( )
1
0 1
1 0
1
1 1
−
−
−
− =l
∫
− t +l∫
−∂∂t t tt dl
l dl E
MTC l
L
t
t χ
Figure 3-4. Evaluating optimal international liquidity - marginal cost and benefit
1 2 3 4 5 Reserves/GDP
50 100 150 200 250 300 350 400
US$m
Cost
Benefit
sis much). The second equilibrium is in the point where downward sloping marginal benefit and marginal cost curves cross. The second equilibrium is a global minimum only if L(lt-1)<0. In the example above, it is clearly the optimal level of liquidity: the total gain (which is the surface below the marginal benefit curve) exceeds the total cost.
The model has several advantages:
1. It takes into account not only the obvious costs of liq-uidity, but also the dynamic effects – the costs of postponing the crisis.
2. It is relatively straightforward to estimate. The bene-fit function (logit analysis of the reserves' impact on the probability of the crisis) is deeply rooted in the existing lit-erature on leading crisis indicators. Similarly, data for the depth-of-crisis function, and interest rate premium relation-ship is widely available for a wide range of countries.
3. The model allows for an estimation of the reputation cost of the foreign exchange crisis (which is the only missing data in the whole structure).