between :
_ Cov(ω*,R*MVt*)
Pω*,R*MV * —
σω* cr*mv *
where σx denotes the standard deviation of x.
Another way of capturing the ability of debt management to insulate debt levels from dehcit
fluctuations is to examine the relative volatility of the total deficit to the primary deficit. The
more successful debt management is at exploiting a negative covariance between interest∕coupon
payments and the primary deficit then the less volatile the total deficit (including interest pay-
ments but excluding capital gains - ωφ) is relative to the primary deficit. Further the better the
government is at exploiting a negative covariance between holding period returns and the primary
deficit then the less volatile will be changes in the end of period market value of debt relative to
the primary deficit. This suggests the following unconditional variances as additional measures of
fiscal insurance:
σω*τ ∕σω*
σAMV* ∕σω*
Our final measure focuses on the dynamic response of debt to shocks to the primary fiscal
deficit. Using the period by period budget constraint (1) recursively we arrive at :
j ∙γ j j 1 γ
MV*+j — MV* ∏flt+fc —t- + ɪʃ . ∣[ Rt+1-1 v ' ) (2)
k=1 Yt+k n=1 l=n Yt+1-1
— MVt0Qt.+j + Φu+j (L)ωt+j
where φot+j — ∏k=ι Rt+kYt∕Yt+k and φιt+j(L)ω*+j = £П=1 ~Γn(∏j n Rt+i-iYt∕Yt+i-φ. τhis
equation defines the t + j level of debt∕GDP as equal to i) the level of debt at time t multiplied by
the relevant compound rate of return ii) plus the sum of all intervening primary deficits grossed up
by the relevant compound interest rate. We therefore have :
dMVt+j _ mv*dφQt+j ∂φιt+ι(L) /W+
я ω* — MVt ω ω* + ω ω* ωt+j + φιt+jlL) a ω*
(3)
d⅛+ι ∂φf,+ι ∂φf,+ι j j ∂φf,+ι
In other words, the impact of deficit shocks (≡ω+1) on the level of debt can be decomposed into :
(a) a term reflecting how interest rate charges on the original level of debt alter in response to such