where V ar(.) and Cov(.) denote variances and covariances conditional on the information
available at time t and Pr is the probability of a debt crisis as perceived by the government.
The optimal debt shares depends on both risk and cost considerations. Risk is minimized
if a debt instrument provides insurance against variations in the primary budget and the
debt ratio due to output and inflation uncertainty and if the conditional variance of its
returns is relatively low. This is captured by the first two terms in equations (19)-(21).
Equation (19) shows that floating-rate debt is optimal for risk minimization when the
Selic rate and thus the interest payments are positively correlated with unanticipated output
and inflation. This allows the government to pay less interests when output and inflation
and thus the primary surplus are unexpectedly low. More importantly, since lower output
growth tends to increase the debt ratio, instruments with returns correlated to nominal
output growth help to stabilize the debt ratio, thus reducing the risk of a debt crisis.
However, the case for indexation weakens as the conditional variance of the Selic rate
increases, thus producing unnecessary fluctuations in interest payments.
Equation (20) shows that the optimal share of dollar denominated debt increases as the
exchange rate co-varies positively with output and inflation. If the exchange rate appre-
ciated at times of unexpectedly low output —-an unlikely event—, cyclical variations in
the government budget could be hedged by dollar denominated debt. To the extent that
exchange rate depreciation is associated with inflation, foreign currency debt helps to sta-
bilize the debt ratio. Clearly, exposure to exchange-rate risk becomes less attractive as the
volatility of the exchange rate increases.
Equation (21) shows that the optimal share of price- indexed debt increases with the co-
variance between output and inflation. If this covariance is positive, lower interest payments
on price-indexed debt provides an insurance against the cyclical deficit due to unexpected
slowdowns in economic activity. However, inflation-indexed debt is optimal even if the co-
variance between output and inflation were zero. The reason is that price-indexed debt
provides the perfect hedge against an increase in the debt ratio due to lower than expected
nominal output growth.
Risk minimization also depends on the conditional covariances between the returns on
the various debt instruments. For instance, a positive covariance between the returns on
two types of debt makes the two instruments substitutes in the government portfolio. This
is captured by the third and fourth terms in equations (19)-(21).
Leaving aside cost considerations, the government should choose the debt composition
which offers the best insurance against the risk of deflation and low growth. But insurance
is costly; higher expected returns are generally required on hedging instruments, and this
leads on average to greater debt accumulation. Debt stabilization thus implies a trade off
between cost and risk minimization. The effect of expected return differentials (or risk
premia) on the optimal debt composition is captured by the last term in in the right-end-
side of equations (19)-(21). This term increases with the risk premia, TPt, FPt and IPt,
more precisely, with the excess return (as perceived by the government) of fixed-rate bonds
relative to the instrument considered. As shown in equations (15)-(17), the impact of the
excess return on the optimal share depends on the marginal increase in the probability of
a debt crisis. The latter has been written as a function of the expected debt reduction
Et(At+1 - ∆BtT+1) and the probability of a debt crisis, Pr, as perceived by the government.
(It is worth noting, that the probability Pr also depends on the expected debt reduction, so