Conceptual Framework

First, assume that a loan of amount A is made at interest rate i; it is disbursed in full immediately and amortized over N years, to be repaid in equal annual installments of principal plus interest. If P is defined as the value of this annuity, then

If

then

The variable D_N is defined as the discount factor that gives the present value of one unit (if the loan above is for A dollars, then it is one dollar) payable yearly for N years. Alternatively, 1/D_N is the annual payment necessary to pay off a loan of one dollar over N years.

Analogously, if the same loan were made at market rate i*, then under market conditions the equation would be

Since the presumption is that i* > i, the subsidy element each year is

If both loans, however, were made in terms fixed until maturity, then 5, which gives the annual subsidy, grossly understates the present value of the total subsidy involved in making the loan at the lower rate because this would accrue every year, for N years. Thus, it is necessary to calculate the capitalized value of this stream of annual differences in payments, where the discount rate used is based on the market (or “true” opportunity) cost of capital:

where Δ = the grant or subsidy value in the loan.

The proper interpretation of Δ is that the two situations are identical in the following sense: it is precisely equivalent either to grant the N-year loan at rate i when the opportunity cost rate is i* or to grant the N-year loan at rate i* and to provide a cash grant of Δ. Thus, provision of a “low-interest” loan is equivalent to providing a “pure loan” combined with a “pure subsidy.” Alternatively, one can think of the reduction of i below i* as inducing an excess expansion of credit in the amount of Δ, which is defined as the credit subsidy.

Another form of presentation is a table of cash flows (Table 1) designed to obtain the present value of the total debt payments.⁹ This is useful because it allows the structure of the loan to be changed easily, so that the new subsidy value can be computed. Such a technique is necessary when attempting to sum subsidies across a wide variety of lending programs, because lending terms frequently differ dramatically. Education or construction sector loans, for example, often involve grace periods, whereas housing loans rarely do.

Table 1.

Cash Flow of Simple Annuity Loan

Table 1.

Cash Flow of Simple Annuity Loan

Years	Receipts (1)	Annual Payments (2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+ A	1	A
1 to N	—	A/DN	-A/DN	$D_N^{*}$	$-A\cdot \frac{D_N^{*}}{D_N}$
Total (Δ)					$A\left(1-\frac{D_N^{*}}{D_N}\right)$

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Table 1.

Cash Flow of Simple Annuity Loan

Years	Receipts (1)	Annual Payments (2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+ A	1	A
1 to N	—	A/DN	-A/DN	$D_N^{*}$	$-A\cdot \frac{D_N^{*}}{D_N}$
Total (Δ)					$A\left(1-\frac{D_N^{*}}{D_N}\right)$

For example, if A = $10,000,000, i = 10 percent a year, N = 20 years, and i* = 12.5 percent a year, then Δ = $1,494,334, which is 14.9 percent of the loan’s face value.

Now, if the above loan were structured so that the payback period were preceded by a grace period (of n years) during which nothing were payable, then the cash flow table would appear as shown in Table 2.

Table 2.

Cash Flow of Loan with Full Grace Period

¹The subtraction of the discount rates may appear confusing in years n to n + N. In the former example, $D_N^{*}$ was used to obtain the present discounted value of the stream of net payments for the simple loan. The problem now is just that this stream occurs n years in the future, so it must be discounted again by 1/(1 + i)ⁿ:

$\begin{array}{cc}{D}_{N+n}^{*}=\frac{1-\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}{i^{*}}& \begin{array}{c}\left(1\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_n^{*}=\frac{{\displaystyle 1-\left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]}}{i}& \begin{array}{c}\left(2\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{{\displaystyle \left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]{\displaystyle -\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}}}{i^{*}}& \begin{array}{c}\left(3\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{1}{{\left(1+i^{*}\right)}^n}\left[\frac{{\displaystyle 1-\frac{1}{{\left(1+i^{*}\right)}^N}}}{i^{*}}\right]& \begin{array}{c}\left(4\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}={\displaystyle \frac{1}{{\left(1+i^{*}\right)}^n}{D}_N}& \begin{array}{c}\left(5\right)\end{array}\end{array}$

Table 2.

Cash Flow of Loan with Full Grace Period

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate1 (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to n	—	—	—	$D_N^{*}$	—
n to n + N	—	A/DN	−A/DN	$D_{n+N}^{*}-D_n^{*}$	$-A/D_N(D_{n+N}^{*}-D_n^{*})$
Total (Δ)					$A\left[1-\left(\frac{D_{n+N}^{*}-D_n^{*}}{D_N}\right)\right]$

¹The subtraction of the discount rates may appear confusing in years n to n + N. In the former example, $D_N^{*}$ was used to obtain the present discounted value of the stream of net payments for the simple loan. The problem now is just that this stream occurs n years in the future, so it must be discounted again by 1/(1 + i)ⁿ:

$\begin{array}{cc}{D}_{N+n}^{*}=\frac{1-\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}{i^{*}}& \begin{array}{c}\left(1\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_n^{*}=\frac{{\displaystyle 1-\left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]}}{i}& \begin{array}{c}\left(2\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{{\displaystyle \left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]{\displaystyle -\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}}}{i^{*}}& \begin{array}{c}\left(3\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{1}{{\left(1+i^{*}\right)}^n}\left[\frac{{\displaystyle 1-\frac{1}{{\left(1+i^{*}\right)}^N}}}{i^{*}}\right]& \begin{array}{c}\left(4\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}={\displaystyle \frac{1}{{\left(1+i^{*}\right)}^n}{D}_N}& \begin{array}{c}\left(5\right)\end{array}\end{array}$

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Table 2.

Cash Flow of Loan with Full Grace Period

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate1 (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to n	—	—	—	$D_N^{*}$	—
n to n + N	—	A/DN	−A/DN	$D_{n+N}^{*}-D_n^{*}$	$-A/D_N(D_{n+N}^{*}-D_n^{*})$
Total (Δ)					$A\left[1-\left(\frac{D_{n+N}^{*}-D_n^{*}}{D_N}\right)\right]$

¹The subtraction of the discount rates may appear confusing in years n to n + N. In the former example, $D_N^{*}$ was used to obtain the present discounted value of the stream of net payments for the simple loan. The problem now is just that this stream occurs n years in the future, so it must be discounted again by 1/(1 + i)ⁿ:

$\begin{array}{cc}{D}_{N+n}^{*}=\frac{1-\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}{i^{*}}& \begin{array}{c}\left(1\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_n^{*}=\frac{{\displaystyle 1-\left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]}}{i}& \begin{array}{c}\left(2\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{{\displaystyle \left[\frac{1}{{\left(1+i^{*}\right)}^n}\right]{\displaystyle -\left[\frac{1}{{\left(1+i^{*}\right)}^{n+N}}\right]}}}{i^{*}}& \begin{array}{c}\left(3\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}=\frac{1}{{\left(1+i^{*}\right)}^n}\left[\frac{{\displaystyle 1-\frac{1}{{\left(1+i^{*}\right)}^N}}}{i^{*}}\right]& \begin{array}{c}\left(4\right)\end{array}\end{array}$

$\begin{array}{cc}{D}_{N+n}^{*}-D_n^{*}={\displaystyle \frac{1}{{\left(1+i^{*}\right)}^n}{D}_N}& \begin{array}{c}\left(5\right)\end{array}\end{array}$

Continuing the previous example, if n = 5 years, the new subsidy value is $5,280,002, or 52.8 percent of the loan’s face value. Thus, the addition of a “pure” grace period can be seen to affect the subsidy calculation dramatically, raising the subsidy element by 38 percent of the loan’s face value.

Consider next a loan with a grace period of n years, but only with interest payable during the grace period at rate i_n. In this case, the cash flows would be as shown in Table 3.

Table 3.

Cash Flow of Loan with Interest-Only Grace Period

Table 3.

Cash Flow of Loan with Interest-Only Grace Period

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to n	—	in⋅A	−in⋅A	$D_n^{*}$	$-A\cdot i_n\cdot D_n^{*}$
n to n + N	—	A/DN	−A/DN	$D_{N+n}^{*}-D_n^{*}$	$-A/D_N(D_{n+N}^{*}-D_n^{*})$
Total (Δ)				$A[1-i_n\cdot D_n^{*}-\left(\frac{{\displaystyle D_{n+N}^{*}-D_n^{*}}}{{\displaystyle D_N}}\right)]$

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Table 3.

Cash Flow of Loan with Interest-Only Grace Period

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to n	—	in⋅A	−in⋅A	$D_n^{*}$	$-A\cdot i_n\cdot D_n^{*}$
n to n + N	—	A/DN	−A/DN	$D_{N+n}^{*}-D_n^{*}$	$-A/D_N(D_{n+N}^{*}-D_n^{*})$
Total (Δ)				$A[1-i_n\cdot D_n^{*}-\left(\frac{{\displaystyle D_{n+N}^{*}-D_n^{*}}}{{\displaystyle D_N}}\right)]$

To continue the example, if i_n = i_N = 10 percent a year, then the subsidy value would be $1,719,391, or 17.2 percent of the loan’s face value. Alternatively, if i_n < i_N, say i_n = 8 percent a year in this example, then A becomes $2,431,505, or 24.3 percent of the loan.

Note that the results in this extended example (Table 4) conform to an intuitive a priori impression about the relative “softness” of the loan terms.¹⁰

Table 4.

Comparative Subsidies in Different Loan Structures¹

¹Based on the extended example in the text.

Table 4.

Comparative Subsidies in Different Loan Structures¹

Loan Structure	Subsidy Value (Δ) (In U.S. dollars)	Subsidy Element (Δ) (As Percentage of Face Value)
Simple annuity	1,494,333	14.9
With grace, where in = iN	1,719,391	17.2
With grace, where in < iN	2,431,505	24.3
With grace, where no payments	5,279,959	52.8

¹Based on the extended example in the text.

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Table 4.

Comparative Subsidies in Different Loan Structures¹

Loan Structure	Subsidy Value (Δ) (In U.S. dollars)	Subsidy Element (Δ) (As Percentage of Face Value)
Simple annuity	1,494,333	14.9
With grace, where in = iN	1,719,391	17.2
With grace, where in < iN	2,431,505	24.3
With grace, where no payments	5,279,959	52.8

¹Based on the extended example in the text.

Finally, consider a case in which the loan is set up like a bond—that is, interest only is payable throughout the life of the loan and the principal is repaid in one lump sum at the maturity date (Table 5).

Table 5.

Cash Flow of Bond-Type Loan

Table 5.

Cash Flow of Bond-Type Loan

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to N	—	in⋅A	−in⋅A	$D_n^{*}$	$-i\cdot A\cdot D_n^{*}$
N	—	A	—A	(1 + i*)−N	−A ⋅ (1 + i*)−N
Total (Δ)				$A\left[1-{\left(1+i^{*}\right)}^{-N}\cdot i\cdot D_N^{*}\right]$

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Table 5.

Cash Flow of Bond-Type Loan

Years	Receipts (1)	Annual Payments(2)	Annual Net Receipts (3)	Discount Factor with Market Interest Rate (4)	Net Present Value (5) = (3) × (4)
0	A	—	+A	1	A
1 to N	—	in⋅A	−in⋅A	$D_n^{*}$	$-i\cdot A\cdot D_n^{*}$
N	—	A	—A	(1 + i*)−N	−A ⋅ (1 + i*)−N
Total (Δ)				$A\left[1-{\left(1+i^{*}\right)}^{-N}\cdot i\cdot D_N^{*}\right]$

Although a fairly simple loan structure has been retained for exposition, the principles involved remain the same, even if loan structures are different and more complex. Most generally stated, the subsidy value of a loan is the difference between the present value of its disbursements and the present value of its service payments, discounted at the market rate of interest. The grant element is defined as the value of the subsidy or grant as a percentage of the present value of the disbursements. According to these definitions, therefore, a loan made at the market rate of interest carries a subsidy value and grant element of zero, while a pure grant has a subsidy element of 100 percent. For a “soft” or “concessional” loan, the grant element lies somewhere between these extremes.

The Development Assistance Committee (DAC) of the Organization for Economic Cooperation and Development (OECD) and the External Debt Division of the World Bank regularly compute the degree of concessionality in their own and other foreign lending by computing the grant element (based on a fixed discount rate of 10 percent a year) and simply defining any loan for which this value is greater than 25 percent as a concessional loan.¹¹ Most lending covered is assumed to be structured around equal principal repayments.

The major difficulty with the DAC and World Bank procedure is that the unchanged discount rate of 10 percent a year is too arbitrary and rigid. Other rates would have been (and will be) more appropriate at various times, depending on the conditions prevailing in world capital markets. For example, in the early 1980s, the 10 percent rate clearly was much too low and presumably resulted in significant underestimates of concessional lending. Moreover, the massive volume of loans analyzed by these organizations requires that simplifying assumptions be made regarding the uniform structuring of loans, particularly that all loans are disbursed immediately. Calculations done by the author suggest, however, that grant elements—and particularly, grant equivalents (the subsidy values)—are quite sensitive to the way loans are structured. Therefore, elements of error (of unknown magnitude) are introduced when the analysis is not done on a disaggregated basis in order to retain a high degree of accuracy. Since the emphasis of these organizations is on the grant element, the problem is less critical than when the focus is on calculating subsidy values. Both of these difficulties, however, introduce an element of uncertainty into the published concessional lending figures and make unambiguous interpretation of them rather difficult.

Before turning to a discussion of the problems that arise in practice, mention should be made of the general relationship between the subsidy element of a loan and the interest spread between its quoted rate and the opportunity cost interest rate. The regularity and shape of this relationship have important implications for applications of the subsidy estimation technique and for interpretation of the results. Chart 1 shows the subsidy (or grant) element as a function of this spread for a bond, where A = 100, i = 15, and n = 20. This function crosses the zero axis when the opportunity cost rate of interest (i*) is set equal to the effective rate of interest (ERI) on the loan (see below), which in this case also equals the quoted rate. When i* is below the loan’s ERI, then profits can be made in the government’s borrowing-cum-lending operations. This is indicated in the figure by negative subsidy rates at these levels.¹² Alternatively, when i* rises above the ERI on lending, then positive subsidy rates are implied; the higher the i*, the greater is the grant element of the loan. Note, however, that the relationship is highly nonlinear. Even a relatively small interest spread (5 percent) gives a reasonably large (25 percent) subsidy element. Similarly, once a certain spread has been attained (say, 20 percent, i.e., when i* equals 40 percent), then increases in the spread do not give rise to much additional subsidy flows. In the limit, of course, when i* is infinitely high, the subsidy element approaches its maximum limit of 100 percent.

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Subsidy Element and Interest Spread

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Chart 2, which presents the same relationship for different types of loan structures, shows that this shape is not unique to the bond structure assumed. Specifically, this figure plots the examples given earlier in this section: for A = 10,000,000; i = .10; N = 20; and i* = .125; A1 plots the simple amortized loan, and A3 the bond. The same general terms are assumed in plotting A4 but a grace period of five years is added and the equal payments structure is used. Amortized loans with a grace period are shown in A2.A, A2.B, and A2.C; the same general terms are assumed, but the five-year grace period added is characterized differently: it is assumed in A that full interest (0.10 percent a year) is paid during the grace period; in B that reduced interest (0.08 percent a year) is paid; and in C that nothing is paid.

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Comparative Patterns of Subsidy Element and Interest Spread

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Complications in Applications

The preceding section glosses over a number of difficulties that need to be addressed when the method is applied. It is necessary to know a great deal about specific program borrowers to properly estimate the income transfers, the impacts on credit allocation, and the total interest subsidies involved. As Break has noted, official lending operations conducted with a fixed supply of total credit result in income transfers to inframarginal borrowers of the difference between the interest expense they would have incurred on private credit and that paid to the government, while the amount transferred to submarginal borrowers is whatever income is yielded by their projects after repayment of their government loans. Moreover, loans to this latter group create a real transfer of credit, while lending to the former involves no reallocation of credit at all, unless these borrowers incur more debt than they otherwise would have in private markets. Therefore, precise quantification of the economic role of official lending operations requires reliable information about the credit status of the borrowers, the alternative interest rates they would have paid in private markets, the interest rate elasticity of their credit demands, and the profitability of their projects.¹³

Of course, most of these data do not exist, and their estimation over a wide spectrum of groups at the microlevel is not practical. Furthermore, even if available, their usefulness would be limited by several factors that will be discussed in the following section (e.g., different stages and lag patterns among programs, etc.). Thus, it seems inevitable that if any useful descriptive figures are to be generated at the aggregate level, while minimizing the data requirements and assuring some simplicity in the analysis, a less ambitious goal than estimating the total subsidies must be accepted. In these circumstances, it makes sense to focus on the explicit subsidy transfer alone and forgo the desirable aim of measuring the larger flows and real impacts.

This focus solves some, but not all, of the difficult operational problems. If the method is applied directly to government lending, taking the variables A, i, N, and n, from observable official loans and using the government’s marginal borrowing rate for i*, then the major assumption required is that the borrowers are inframarginal relative to the government’s borrowing rate. This seems to be true for most capital-scarce countries.

Chart 3 helps to set the context of this discussion, depicting the explicit subsidy measure as well as the inframarginality assumption. If dD and SS are the market demand and supply schedules, then G would be the free market solution without government intervention; OC of credit would be extended at rate i_FM. If the government decides to make D’D of credit available and selects recipients at random from among all those who demand credit at the government’s subsidized lending rate ($i_L^G$), then the fraction D′D/$i_L^G$D of all those demanding credit at any rate of not less than $i_L^G$ receive official subsidized loans, with the remaining demand satisfied in the unassisted private market along D’d. However, D’d determines only the residual quantity of credit supplied in the market. To fix the rate that unassisted borrowers must pay to private lenders, the total demand for credit must be obtained by adding back in the quantity channeled to the subsidized borrower, D’D. The intersection of this combined schedule (d’D) with the original supply schedule determines the rate for unsubsidized borrowers ($i_{\mathit{FM}}^{\prime }$) who demand OA of credit, out of a total of OE, where AE equals D’D. In most applications, $i_{\mathit{FM}}^{\prime }$ would not be directly observable, but it would be some weighted average of rates in various markets—for example, in many developing countries on the unorganized money market and in the banking system. The value of the total interest subsidy (explicit plus implicit) is represented by the area IGDD’.

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The Explicit Subsidy Measure

Source: Adapted from von Furstenberg (1976), Figure 1.

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If $i_B^G$ represents the government’s marginal borrowing rate, then YBDD’ represents the value of explicit credit subsidies received by government loan recipients. In most applications, the present value of this area would be approximated by the present value of the slightly different area ACDD’ (equals TAD’X) on the assumption that all borrowers are inframarginal to the government’s borrowing rate $i_B^G$ (i.e., all lie on demand schedule dD above B).

In terms of practical applications, therefore, selection of the government’s marginal borrowing terms remains as the final major operational issue, and this issue is far from trivial because the results are very sensitive to variation in the i*’s.¹⁴ The major alternatives available are the terms on official external and internal borrowing. In general, the internal rate, for example, a government bond yield, should be preferred because it entails the same currency in which the lending occurs, thus minimizing errors that might be introduced through incorrect estimates of exchange risk.¹⁵

In applications, however, economists may have to use an external borrowing rate because a domestic counterpart is unavailable. But even though governments still issue a large amount of fixed term foreign bonds,¹⁶ this market may be effectively closed to some countries. They may have access only to floating rates, or to adjustable rate markets (e.g., the Eurodollar market). In this case there are three alternatives available, none of which is entirely satisfactory. First, the observed rate could be used, assuming that it would not change, at least on average, over the period of analysis. Second, forecasts could be made of expected changes in the rate, perhaps by using different values to determine a range of subsidy values. Finally, if there seemed to be too much uncertainty in either of the former strategies, it might be preferable to forgo estimation of the capitalized value of the subsidy stream and simply to do the analysis on an annual basis. Computationally, this is much easier as it involves only the calculation of the service payments on outstanding loan balances at the mean lending rate and again at the mean borrowing rate, and then the subtraction of one from the other. The difference is the subsidy value for that particular period.¹⁷

It is well known that unanticipated inflation can result in a shift in real income from lenders to borrowers as the real value of debt contracted in fixed nominal terms is eroded. However, if the government is both borrowing and lending at fixed rates over similar periods then, as a first approximation, it can be assumed that there are no net inflationary effects on the subsidy calculations because what is lost on the lending side is gained on the borrowing side. Similarly, if it is assumed that nominal interest rates are adjusted to incorporate fully actual inflationary developments under flexible rate loans, then more rapid rates of amortization will be implied than would have occurred under comparable fixed rate loans.¹⁸ But again, if the government is both borrowing and lending in this way, then the net effect on the subsidy transfer should largely net out. Only if there were some mixed combination of structures would there seem to be a significant impact on the real value of the subsidy transferred through the government. The more common configuration probably would be the case of foreign borrowing at variable rates, but domestic lending on fixed terms. In this case, an increase would be expected in the real value of the subsidy transferred through the government due to inflation, and calculations based solely on nominal values would, therefore, underestimate the real transfer. Since the capitalized value of the subsidy stream is being computed, attempts to correct for these effects would have to include inflation forecasts over the relevant time horizon.

Effective Rate of Interest

The most important operational issue in the foregoing analysis is selection of the correct opportunity cost interest rate, i*. In some cases, the correct choice of the discount rate may be so difficult that it cannot be done with any confidence. Or, it might be that all rates are floating and an annual type of analysis must be used. In these instances, an alternative approach that does not involve a discount rate, that is, the ERI, might be preferred. The ERI bears the same relationship to the former analysis as the calculation of the internal rate of return does to the net present value in project analysis.¹⁹ The analogy results because a loan is just like a “backward project” in the sense that the benefits of the “project” (loan disbursements) come initially, while the costs (debt service payments) are spread over the lifetime of the loan.

The ERI is defined as that rate of interest which, if used as a discount rate, would reduce the net present value of the loan to zero. Thus, the ERI is the same as the rate of interest on any loan on which interest must be paid on outstanding balances at all times (and on which there are no other costs payable).²⁰ Therefore, there may be no need to calculate the ERI because it is the same as the quoted rate on the loan. However, when loans include special arrangements, such as grace periods, during which payments are reduced below interest at the quoted rate, then the ERI diverges from this quoted rate and thus must be calculated directly.

Calculation of the ERI involves solving the following equation for r:

where

LD	=	loan disbursements;
DSP	=	debt service payments;
t	=	time period; and
n	=	maturity

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LD	=	loan disbursements;
DSP	=	debt service payments;
t	=	time period; and
n	=	maturity

Estimation of credit subsidies using ERIs is done in the same way as was described above for annual subsidies. The ERIs in both government borrowing and lending are calculated, then, the cost of servicing the outstanding balance of loans is calculated for each. The difference between the borrowing and lending cost is the subsidy.