An international reserves variation threshold to increase loan funding

Abstract

In this article we propose a methodology for the calculation of an international reserves variation threshold that defines the lending decisions of international creditors to developing countries. If the change in net international reserves of the borrowing variation is above that of the threshold, the creditors are willing to lend more. Otherwise, if that change is below the threshold, there will be capital flight. Such a threshold depends on the stocks of debt and international reserves, as well as on the perception of the default risk of the country to honour its debts. Using that threshold, we perform a counterfactual exercise to calculate the time series of international reserve levels that minimize the total cost of reserves holding, namely, the cost of possible illiquidity of the country plus the opportunity cost of holding international reserves. We illustrate the methodology by applying it to five Latin American countries: Argentina, Brazil, Chile, Mexico and Peru.

Appendices

Appendix A: Theoretical model for the threshold

In this appendix we provide a theoretical justification for the existence of a strategy function for loans funding (strategy function for loans and the threshold of IR variation).

A representative international investor with expected utility u(W) = W^1−γ, (γ ≥ 0 the relative risk aversion coefficient) and initial wealth L > 0 (representing the international liquidity for loans) is deciding the rate of increase of loans to a country. The current debt is D = 1 and the perceived variation of IR is $\theta \in \mathbb {R}$. The rate of return of investments in the country in r and the recovery rate in case of default of the country is (1 − τ). Thus, the investor must decide the optimal level of new total investments in the country $D^{\prime }$.

The investor assigns a probability π(𝜃) ∈ [0, 1] to the event “the country defaults”. We will suppose that $\pi ^{\prime }(\theta )<0$. The final wealth of the investor in case of default is:

$$ W_{1}=L-(D^{\prime}-1)+(1-\tau) D^{\prime}=L+1-\tau D^{\prime} $$

(A.1)

and in the case of non-default:

$$ W_{2}=L-(D^{\prime}-1)+(1+r) D^{\prime}=L+1+r D^{\prime} $$

(A.2)

then, the investor problem is:

$$ \max_{D^{\prime}\geq 0} \pi(\theta) u(W_{1}) +(1-\pi(\theta)) u(W_{2}) $$

Using the first order condition for interior solutions and substituting W₁ and W₂ from Eqs. A.1 and A.2 we obtain:

$$ \Big(\frac{W_{2}}{W_{1}}\Big)^{-\gamma} = \frac{r^{-1}\tau \pi(\theta)}{1-\pi(\theta)} \Rightarrow \frac{L+1+rD^{\prime}}{L+1-\tau D^{\prime}}=\Big[\frac{r^{-1}\tau \pi(\theta)}{1-\pi(\theta)}\Big]^{-1/\gamma} \equiv K(\theta) $$

(A.3)

$$ \Rightarrow D^{\prime}= \frac{(K(\theta)-1)(L+1)}{(r+\tau K(\theta))}. $$

(A.4)

Notice that the function K(⋅) contains parameters like risk aversion, interest rate and recovery rate in case of default. In addition to those parameters, the optimal value $D^{\prime }$ includes the market liquidity L. The existence of f(⋅) and ξ of definition (2.2) is a consequence of the following two propositions.

Proposition A.1

If π(𝜃) is a strictly decreasing function, $\pi (-\infty )=1$ and $\pi (+\infty )=0$, then K(𝜃) defined in Eq. A.3 is a strictly increasing function and satisfies $K(-\infty )=0$ and $K(+\infty )=+\infty$.

Proof

The function r^− 1τπ(𝜃)/(1 − π(𝜃)) is strictly decreasing, since π(𝜃) is strictly decreasing; thus, K(𝜃) defined in Eq. A.3 is a strictly increasing function. The other two equalities result from the definition of K(𝜃) and using the values of π(⋅) in $-\infty$ and in $+\infty$. □

Proposition A.2

With the hypotheses of Proposition A.1, there exists ξ > 0 such that $[\theta > \xi \Leftrightarrow D^{\prime }>1]$.

Proof

Using (A.4) we have that:

$$ D^{\prime}= \frac{(K(\theta)-1)(L+1)}{(r+\tau K(\theta))} > 1 \Leftrightarrow K(\theta)>\frac{L+1+r}{L+1-\tau}\equiv \rho(L) $$

Since K(𝜃) is a strictly increasing function satisfying $K(-\infty )=0$ and $K(+\infty )=+\infty$., the inequality above implies:

$$ \theta>K^{-1}(\rho(L))\equiv \xi. $$

(A.5)

□

Since $x = f(\theta ) = D^{\prime }-1$, we can conclude that ξ and f(𝜃) given in Eqs. A.5 and A.4 respectively depend on the risk aversion, interest rate of loans, rate of recovery in case of default and the market liquidity, just as mentioned in Section 2.

A numerical example illustrates the results of Proposition A.2. Notice that from Eq. A.4, the rate of increase of credit concessions to the country is defined by:

$$ x = D^{\prime}-1 = \frac{(K(\theta)-1)(L+1)}{(r+\tau K(\theta))}-1\equiv f(\theta). $$

If we consider π(𝜃) = (1 + 𝜃)^− 1, γ = 0.1, L = 5, r = 0.01 and τ = 0.01, the Fig. 3 shows the graphic of x = f(𝜃). The value of the threshold is ξ ≈ 10.0185

Fig. 3

Investment strategy function and the threshold ξ > 0.

We can also provide a numerical example illustrating the existence of a negative threshold. Consider negative values for 𝜃 and suppose that π(𝜃) = 1 + 1.2(𝜃 − 0.55)^− 1, γ = 0.4, L = 5, r = 0.01 and τ = 0.01. The Fig. 4 shows the investment strategy function and the threshold level is ξ ≈− 0.7685.

Fig. 4

Investment strategy function and the threshold ξ < 0.

Appendix B: Stationarity of {𝜃 _t}

In this section we test for stationarity of the time series {𝜃_t}. We use the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski-Phillips-Schmidt-Shin(KPSS) stationarity tests and they indicate that the time series {𝜃_t} is stationary with significance of 5%, according to the Table 4.

Table 4 Stationarity test of the time series {𝜃_t}

In Table 4, the critical values with significance level of 5% of the ADF and PP tests for Argentina, Brazil, Chile and Peru are -2.89, -3.46 and 1.94 for models with intercept, trend and intercept, and none respectively. For the KPSS the critical values test are 0.46 and 0.15 for models with intercept, and trend and intercept respectively. For Mexico, due to the lower number of observations, the critical values of the ADF and PP tests are -2.93, -3.51 and 1.95 for models with intercept, trend and intercept, and none respectively. For the KPSS test the critical values are 0.46 and 0.15 for models with intercept, and trend and intercept respectively.

Once we have checked the stationarity of the time series, we select the more appropriate model using the Akaike, Schwarz and Hanna-Quinn information criteria (AIC, SIC and HQ respectively). Table 5 shows the results of that selection.

Table 5 Selection criteria for the appropriate dynamic model of {𝜃_t}

As we can see, the three criteria points out the AR(1) model as the more appropriate for the time series {𝜃_t} in each country except for Chile. For that country, the AIC would select the ARMA(1,1); however, we will use the AR(1), since the SIC and HQ criteria select it, that dynamic model is more parsimonious and the AIC statistics of the ARMA(1,1) is not far from that of the AR(1) model. This will allow us to maintain the uniformity of the dynamics in our analysis.

Appendix C: Optimal versus observed IR levels

In this appendix we show the optimal versus the observed path of IR in each country of our group. Figure 5 exhibits the evolution of those time series.

Fig. 5

Optimal vs. actual levels of international reserves stocks (US Millions)