it is now necessary to isolate the constant terms, dividing the former expression as:
Zt+1 = [(1 — b ) ( γYβ — H ) + (1 - q ) κH ] rB+1 — ( γYβ — L ) eL+1 + κ ( rRq — rD ), (38)
C = (βγγ — 1) [z — b ] — κu.
(39)
We are assuming that λ1 < 1andλ2 > 1. The right-hand side can be solved forward, applying
the algorithm developed by Sargent.16 The expression can be rewritten, using the properties that
1 „ _ ~(λ2L) 1 _ _ a ατlя 1 l,V _ - -(λ2L) 1 V _ _L ŋ∞ 1 1 ʌ V _ ,vl,,u.,4 n
(1 -λ 2 L ) a 1 - ( λ 2 L )-1 a 1 -λ 2 ,and (1 -λ 2 L ) bXt b 1 - ( λ 2 L )-1 Xt b ^i =1 (4λ 2) Xt+i +1, where a
and b are arbitrary constant terms. Applying the transversality condition of the problem (discussed
in the Appendix), Equation (31) can be solved as:
Ft+j+1 = ɪ Ft+j + βγY — 1 NW +
βγY γYβ
1 — (1 — q) κ — Yy C
(1 - Yy ) Yy βα
1 - (1 - q)κ ∑∞ ( 1 )iE [Zt+j+i+2]
βγγ = γ-) Et+i α J
∞ . 1 . λ∙ _ r∕'
(1 - q)(Yyβ - 1)
Yy β
∞i
∑ (w) Xt+j+i+1. (40)
YY
+TT- Σ( ɪ >+i [ Zt+1+1+1- ]
βYY i=1 YY α
In order to simplify the interpretation of the results, it can be assumed that interest rates follow
a random walk process, so that the deterministic component is assumed to remain constant. This
assumption is reasonable for market interest rates on bonds, but it is not acceptable for the rates
on deposits, since the equilibrium rate of deposits is obtained from the model. Just to simplify
the understanding of the results, we will disregard the relevance of the deterministic components,
assuming for simplicity that future rates on deposits are expected to behave like a random walk.
If we assume that interest rates are random walks, assuming that the correlation between
interest rates and default costs is time-invariant, we can rewrite the expression treating all the
terms as constants:
Ft+j +1 = ɪ Ft+j + 1 - ((1 q)K YY H {[(1 — d ) (Yy в — H) + (1 — q) κH] •
βYY (1 - YY )YY βα b
[rB+ j+1 + COV(rB,1 )] + κ(qrR+ j- rD+ j)} + (1 -YY)((1γ-βYY)1) (g3rD+ j — g4rB+ j) +
— [ γYβ — 1][1 — (1 — q )κ - γY] COV ( LD, 1 ) + 1 - (1 - q )κ - γY C + βγγ-l Nwi
(41)
βYY (1 - YY)α α (1 - YY)YYβα YYβ
To understand the results we have to remind our assumption that YY > 1and
We can observe that interest rates on bonds at time t + 1 have a positive sign, so they always
increase the quantity of bonds held in the portfolio at time t + 1. Rates on bonds at time t have
a negative sign, in the first term, showing a positive effect on time t holdings of bonds. We can
observe though that under our basic assumption,17 higher rates produce an increase in the holdings
of bonds. The other terms in the interest rate at time t have a positive sign, and this means that
there is even a positive dependence on the lagged value of the interest rate. We can conclude that
both current and lagged values of interest rates increase the quantity of bonds held in the portfolio
by the bank.
Industrial costs and interest rates on deposits have a negative sign, since the profit margin
becomes tighter as they increase. On the contrary higher industrial costs on loans have a positive
sign, since they increase the relative appeal of bonds.
The covariance between the rate of interest and the reciprocal of the cost function has a positive
sign. This implies that a positive correlation between the rate of interest and the default cost has
a negative effect on the purchase of bonds. the sign of the other covariance is the opposite. This
highlights the fact that when the demand for loans is correlated with default costs, the bank issues
a proportionally lower quantity of loans, buying more bonds instead.
16See Sargent [24] p.176.
17That the nominal rate of growth is positive and larger than the discount rate.
11