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In this paper, we attempt to obtain the exact probability distribution of the debt-to-GDP ratio in T years, assuming that 1) the primary balance is zero and 2) the interest rate and the GDP growth rate are given as exogenous random variables. With this approach, researchers can play the “Deficit Gamble” without conducting a Monte Carlo simulation. Calculating the distribution of the debt-to-GDP ratio would be useful for policy planning.

Recently, many countries provided fiscal stimulus packages to cope with the economic recession arising from the financial crisis. As a result, these governments’ debt increased, and the fiscal crisis became a very important issue, especially in the EU countries. Therefore, it is important to predict the future path of government debt. This paper provides a useful formula to obtain the probability distribution of the debt-to-GDP ratio in T years, assuming that (1) the primary balance is zero and (2) the interest rate and the GDP growth rate are given as exogenous random variables.

Given assumptions (1) and (2), the debt-to-GDP ratio in T years is a random variable. Thus, the time path of the debt-to-GDP ratio exceeding a target level given an initial condition is purely random. Therefore, this “game” is referred to as the “Deficit Gamble”. Ball et al. [

The rest of the paper is organized as follows: Section 2 derives a distribution of the debt-to-GDP ratio, Section 3 states a useful statistic for the debt-to-GDP ratio, and finally, Section 4 concludes our paper.

The deficit gamble has been suggested in Ball et al. [

The deficit gamble approach does not require any economic theory and, therefore, one can mechanically obtain information for debt accumulation path. However, an analysis using a micro-founded economic model is also important for considering how debt accumulates through an economic structure. In line with this, Sakuragawa and Hosono [

Our approach would be quite useful for supporting an argument for the sustainability of government debt. Related to this issue, Bohn [

Suppose that government debt is accumulated as follows:

where denotes the nominal government debt at the end of term, denotes the nominal interest rate, and denotes the primary balance, which is the government spending minus tax. From Equation (1), we get

where, defining as GDP, , , and. Our goal is to obtain the probability distribution of the debt-to-GDP ratio in T years with zero primary balance. Assuming that the current period is zero and that for all, the debt-to-GDP ratio in T years is as follows:

Following Ball et al. [

where is a normally distributed random variable with mean zero and variance. Taking the logarithm of Equation (2), we get

Since can be expressed as the following MA representation

Equation (3) can be written as follows:1

.

The above equation can be simplified as follows:

Note that the distribution we want to obtain is conditional on the current period’s information. Equation (4) implies that the distribution of conditional on is normal with mean

and variance

.

Noting that

and defining z as

We have the following proposition Proposition. If and is a sequence of independently normal distributed random variables with mean zero and variance, then as conditional on follows the standard normal distribution.

In practice, and are not known a priori and they have to be estimated on the basis of the available information. Regardless, since the estimates of and are fixed for years as non-random variables, the proposition still holds.

In this section, we define a useful statistic for the deficit gamble. Suppose that the deficit gamble fails if, where is some target level of the debt-to-GDP ratio that is set manually by researchers. Now define, which we call the “Deficit Gamble Statistic”, as follows:

In this setting, the probability of the debt deficit gamble failing in T years is calculated as, where denotes the standard normal distribution function. Here is an example.

Example. Assume with, , , and T = 25. Setting the target level f = 1.5, we obtain g = 2.6173. In this case, the probability of the debt deficit gamble failing in 25 years is 0.005.

In this paper, we obtain the exact distribution of the debt-to-GDP ratio in T years and further develop the formula for calculating the probability of the deficit gamble, wherein the debt-to-GDP ratio exceeds some target level, without conducting a Monte Carlo simulation. Calculating the distribution of the debt-to-GDP ratio would prove useful for policy planning.