Debt Development index

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below is also feasible, any debt level below the frontier may reflect internal credit rationing, where-as the frontier itself reflects external credit rationing; iv) implies that if a given debt level is fewhere-asible at a given input level, it is also feasible at any input level at or above; v) closedness is a mathemati-cal requirement with no important financial implications; vi) implies that unlimited debt levels are not possible with a given set of inputs; vii) implies that any weighted average of two levels of debt capacity is feasible. This requirement implicitly means that all outputs and inputs are continuously divisible.

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Figure 2.1: Decomposition of the Debt Development index.

The interpretation of equation (5) can be illustrated by the following example: Suppose you are a farmer at time @1, your total liabilities are given by A^WA, and amount to € 1,000,000. Your Z\@1 frontier debt capacity is given by A^WA^{^} and is € 2,700,000. The Z\@2 frontier debt capacity is giv-en by A^WA^{^} and is € 4,300,000 which is the frontier debt capacity at time @2, if you have exactly the same collateral as you had at @1. At time @2 your total liabilities are given by B^WB. Now let’s sup-pose they have risen to € 2,000,000 and your collateral has also risen which is reflected in the hori-zontal movement from A^W to B^W. The Z\@1 frontier debt capacity is given by B^WB^{^} and is

€4,000,000, the Z\@2 frontier debt capacity is given by B^WB^{^} and is € 6,300,000.

Numerically, the Debt Development index can now be calculated as follows:

--MO (P_N^{5 5}, ⁵, P⁵, ⁵) = {[(2 4) ⁄ ⁄(1 2.7)⁄ ] × [ (2 6.3)⁄ ⁄(1 4.3)⁄ ]}^½ = 1.36 (10) Debt Development indices which are above one indicate relatively increased indebtedness, while indices which are below one indicate relatively decreased indebtedness. By relatively we here mean the change in the debt of the farm relative to the frontier known as the debt capacity. Hence, the

57

DDi is the development in the relative indebtedness from @1 to @2. In the example above, the farm at point B in Figure 2.1 is 36% more indebted than the farm at point A.

Following Färe et al. (1994) and Wheelock and Wilson (1999), the estimate of the DDi can be de-composed. This decomposition enables us to express the debt development with respect to a debt frontier with varying returns to scale (VRS). By VRS it is meant that the relation between debt and collateral on the frontier does not have to be constant. This is a weaker and more appropriate as-sumption for the shape of the debt possibility frontier. The data is more tightly enveloped assuming VRS which yields a smaller and more conservative estimate of the debt possibility set K(x⁵) than an estimate based on the assumption of CRS. The Wheelock and Wilson (1999) decomposition goes further and is followed in the DDi reinterpretation of the Malmquist index:

--MO = ∆ -‚@ ",@P F@,ƒ,ƒ,@,„… _N⁵

× ∆-‚@ ",@P F@,ƒ,ƒ,@,„… †ƒ@ @„ Eƒ

× ∆-‚@ ",@P × ∆ -‚@ ",@P †ƒ@ @„ Eƒ

(11)

Where:

∆ -‚@ ",@P F@,ƒ,@,„… = Q-R

ˆ5(_S⁵, P_S⁵) -R

ˆ5(_S⁵, P_S⁵)T (12)

∆ -‚@ ". F@,ƒ,@,„… †ƒ@ @„ Eƒ = Q-R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵) -R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵)T (13)

∆ -‚@ ",@P = Q-R

ˆ5(_S⁵, P_S⁵) -R

ˆ5(_S⁵, P_S⁵)×-R

ˆ5(_S⁵, P_S⁵) -R

ˆ5(_S⁵, P_S⁵)T

(14)

∆ -‚@ ",@P †ƒ@ @„ Eƒ

= Q-R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵) -R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵) ×-R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵) -R

&5(_S⁵, P_S⁵)/-R

ˆ5(_S⁵, P_S⁵)T

(15)

The decomposition is illustrated in Figure 2.1. The components related to scale capture the fraction of the change which is related to scale change. These parts are not recognized as pure change in

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debt capacity or pure change in the debt capacity utilization and do not have an important economic interpretation, but they are important controls in the shift from CRS to VRS. The ∆ -‚@ ",@P is the shift in the VRS debt frontier measured as the geometric mean of the shift between the fron-tiers measured at the position of the farm at the two different points in time. In the two dimensional illustration of Figure 2.1 it is the geometric mean given by:

∆ -‚@ ",@P = {[(X^WX X⁄ ^WX^∗⁄(X^WX X⁄ ^WX^∗∗)] × [ (V^WV V⁄ ^WV^∗)⁄(V^WV V⁄ ^WV^∗∗)]}^½ (16)

The change in debt capacity is the change in the frontier, that is the relative movement of the fron-tier or the change in how much the most indebted farmers can borrow.

The ∆ -‚@ ",@P F@,ƒ,@,„… is the change in the debt capacity utilization relative to the VRS debt frontier measured as the debt capacity utilization at the second period over the debt capacity utilization in the first period. In Figure 2.1, this is illustrated by:

∆ -‚@ ",@P F@,ƒ,@,„… = (V^WV V⁄ ^WV^∗∗)⁄(X^WX X⁄ ^WX^∗) (17) The change in debt capacity utilization is the relative change in the actual to maximum debt ratio.

This is somewhat similar to a change in a debt-to-asset ratio. However, here assets are substituted by the estimate of maximum debt.

One important weakness in the Wheelock and Wilson (1999) decomposition is that points like C in Figure 2.1 will have no frontier projection for the Œ\⁵frontier, and ∆ -‚@ ",@P will be ill-defined in this case. It is a methodological trade-off between a reliance on the CRS frontier or the VRS frontier, with the possibility of ill-defined change in debt capacity scores. We consider the latter alternative to be the best and we acknowledge the discussion in the Malmquist index litera-ture. We omit farms with ill-defined change in debt capacity scores.

The output-oriented efficiency score in the production economic application can be reinterpreted as the debt to debt capacity ratio, or more loosely defined as the management adjusted debt to collat-eral value ratio. We assume that collatcollat-eral and earnings (EBIT) are the fundamental factors, which determine the debt capacity or loan approval on the individual level. The non-utilized amount of

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debt capacity can be interpreted as credit reserves and in Figure 2.1 the credit reserves at time @1 for the farmer represented by the point A are illustrated by the distance between A and A*. This is a useful proxy for in the Gabriel and Baker (1980) model shown in (14).

( + − ≤ ) ≤

[(̅ + − ) − ] ≤ (18)

where:

α = probability that some critical cash demand cannot be met (default risk) cx = net cash flow

cx = expected net cash flow μ = liquidity reserves

I = fixed debt servicing obligations z = minimum liquidity requierment

σ = the subjective variance of net cash flow

Gabriel and Baker (1980) used land price change as a proxy for change in μ in a preliminary empir-ical test of their risk balancing hypothesis. We provide a more refined and micro-oriented measure of change in credit availability.