The estimation of the grant element of loans reconsidered.

Yassin, Ibrahim Hassan

This paper examines critically the formulae which are frequently used in the calculations of the grant element of loans. Given the formula derived by Beenhakker (1976), which has been expanded into a more general form, the grant element of foreign assistance received by the Sudan during the period 1958-1979 is calculated. The grant element, was found to be low, reflecting hard terms of borrowing.

1. INTRODUCTION

It has been widely accepted that when loans are made on concessionary terms. they contain an aid component, or a grant element, which can be estimated in cash terms and regarded as a cost (to donors) or a benefit (to recipients) associated with such loans. The grant element is thus defined as the difference between the nominal value of the loan and the present value of all future repayments (amortization and interest) discounted by a proper discount rate.

The grant element method has the advantage of expressing the nature of loans (whether soft or hard) across donor sources, or of a whole loan programme, in terms of a single parameter. Thus, it facilitates the ranking of donors by their aid programmes and helps in distinguishing the desirable form(s) of credits as well as the corresponding sources.

Given the terms of borrowing, the grant element or the aid component of loans can be estimated by applying any appropriate formula. The impetus of the most commonly used formula goes back to Ohlin (1966). However, the application of this formula is limited to certain types of loans and hence it cannot be generalized. Therefore, the purpose of this paper is to examine critically Ohlin's formula and determine an alternative formulation which lends itself to a wider range of applicability by focusing on the less limiting formula of Beenhakker (1976).

The next section of this paper discusses the factors which determine the grant element of loans, while Section 3 examines critically the formulae by which the grant element can be calculated, and it determines the formula which has been applied in Section 4 to the case of the Sudan during the period 1958-1979. The final section offers some concluding remarks.

2. THE DETERMINANTS OF THE GRANT ELEMENT

The factors which determine the value of the grant element are mainly three: the rate of interest attached to the loan which is the major one, the grace period which lies between the date of disbursement until the repayments start (usually during this period only the interest is paid), and the maturity period by the end of which the repayments obligations terminate. These factors can be incorporated into this formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

A = (F - PV/F) 100 (2)

where A is the grant element as a percentage of the face-value of the loan, F is the face-value of the loan, [P.sub.n] is the total payment of principal and interest in year n, N is the maturity period, i is the discount rate, and PV is the present value of future repayments on the loan.

The lower the rate of interest and the higher the grace and maturity periods, the higher will be the grant element. Given these factors, the grant element can be calculated for different combinations of them in order to determine, for instance, by how many years a one percent increase in the interest rate can be offset by a corresponding increase in the grace and/or maturity periods.

3. THE ESTIMATION OF THE GRANT ELEMENT

Given the terms of borrowing, several formulations have been suggested for accomplishing the calculations of the grant element. As stated earlier, the most commonly used formula is attributable to Ohlin (1966). Assuming a constant stream of debt servicing payments (according to which the debtor will surrender a constant annual payment of the principal and interest when the grace period elapses), this form applies for long-term loans: (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where r is the interest rate attached to the loan, G is the grace period, and i and N are as defined before.

Ohlin's formula, like most of the conventional formulae, can be criticized for being very simplistic, to the extent that the accuracy of the estimated values of the grant element becomes questionable. In addition, formula (3) operates only when the discount rate is different from the interest rate attached to the loan (i.e., when i [not equal to] r). If i = r, any loan, irrespective of its length of maturity and grace period, will yield a zero grant element. But it is clear from Equation (2) that the grant element will be positive if PV < F, equal to zero if PV = F, and negative if PV > F, i.e., the interest rate is a necessary but not a sufficient determinant of the grant element. The effect of the maturity and grace periods is also important as the length of these periods may counteract any increase in the interest rate attached to the loan and hence maintain the value of the grant element all the same. (2)

Furthermore, when the grace period is equivalent to the maturity period (i.e., a bullet loan), formula (3) reduces to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

This formula, which is arrived at by applying L'Hospital's Rule to formula (3) when G = N, unnecessarily overestimates the value of the grant element as will be shown in the coming discussion. (3)

Moreover, Ohlin maintains that when the rate of discount is too high, it is possible to use the following 'rules of thumb': "each concession of one percentage point in the interest rate gives rise to a grant element of 4 percent of the face-value for a l0-year loan; 7 percent for a 20-year loan; 9 percent for a 30-year loan, and 10 percent for a 40-year loan" [Ohlin (1966), p. 103].

A more general formula which deals with the different forms of loans, and also operates even when the rate of interest attached to the loan is equal to the discount rate (i.e., the effect of the grace and maturity periods is accounted for), has been derived by Beenhakker (1976). It is clear from Equation (2) that the grant element can be calculated by determining first the present value of all future repayments (PV). This can be done by using the "Zeta" transformation for discrete time-series analysis (see Appendix 1), and the resulting formula would be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where all the variables are as defined before.

However, formula (5), like Ohlin's formulation, does not also deal with the case of a bullet loan, i.e., when G = N. Therefore, L'Hospital's Rule is also applied to formula (5) and the resulting form [as derived in Appendix (2)] that can be used to determine the present value of debt servicing on loans when G = N is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

It can be noticed that formulae (4)and (6)are derived by applying L'Hospital's Rule to formulae (3)and (5), when G = N, respectively. As pointed out earlier, when G = N, Ohlin's formula [i.e., formula (4)] would overestimate the values of the grant element. In Table 1, the values of the grant element obtained through the application of formulae (4)and (6)are compared. It is clear that formula (4)overestimates the values of the grant element.

4. THE GRANT ELEMENT OF FOREIGN ASSISTANCE TO THE SUDAN 1958-1979

The grant element of loans to the Sudan is calculated from the recipient's point of view (in order to assess the embodied benefit or the concessional element in loans) by applying formula (5) to Sudanese data. The data are based on contracted official loans because exact figures on the terms of borrowing and conditions were not available on the basis of actual flow of funds.

The three different forms of official foreign assistance contracted by the Sudan during the period 1958-1979 are compiled in Table 2. The Sudanese currency is used as a unit of measurement in order to account for the effect of the various exchange rates, i.e., all the foreign loans were converted into domestic Sudanese pounds, according to the prevailing exchange rates when these loans were contracted. It can be seen from Table 2 that the share of bilateral assistance in the total flows was the highest, followed by the share of multilateral aid, and then borrowing from private sources. The amount of loans contracted with the Arab countries was the largest, and it constituted 35.9 percent of the total flows.

On the other hand, Table 3 reveals that the grant element of contracted Sudanese loans fluctuated greatly over time, and so did the terms of borrowing. (4) The overall averages of the grant element tend to be 31 percent and 39 percent at the discount rates of 8 percent and 10 percent, respectively. (5) Regarding the terms of borrowing, the average rate of interest, the length of maturity, and the grace period were 4 percent, 16 and 6 years, respectively.

5. CONCLUDING REMARKS

This paper has examined critically the conventional formulae which are frequently used in the estimates of the grant element of loans. Particular attention is given to the commonly used formula of Ohlin (1966) which is constrained by several limitations. Consequently, a more practical formula, which was originally developed by Beenhakker (1976) and expanded into a more general form, has been applied to the foreign capital inflows received by the Sudan during the period 1958-1979.

The results show that the grant element was low, averaging 31 percent and 39 percent at the discount rates of 8 percent and 10 percent respectively. On the other hand, the terms of borrowing averaged 4 percent, 16 and 6 years, regarding the interest rate attached to the loans, the length of maturity, and the grace period respectively.

Appendix 1

THE DERIVATION OF FORMULA (5) USING ZETA TRANSFORMATION FOR TRUNCATED FUNCTIONS

The zeta transformation of the discrete time-series function f(nT), which was used by Beenhakker (1976) to derive formula (5), can be defined as:

z {f (nT)} = [[epsilon] summation over (n = 0)] f(nT) / [(1 + zT).sup.n] ... ... (7)

where z is a variable, n is an integer, and T is a constant length of time. If z is replaced by the interest rate i and the constant rime interval T is equal to unity (i.e., the compounding period is a unit of rime), then Equation (7)can be written as

z {f (n)} = [[epsilon] summation over (n = 0)] f(n) / [(1 + i).sup.n] ... ... (8)

which describes the present value of cash flows over time in the same manner as Equation (1). Since in loan agreements the function f (n) may change over time, the notation t = h will denote the time when the function starts, and t = k - 1 the time after which the function terminates, i.e., f (n) will have non-zero values over the interval h [less than or equal to] n [less than or equal to] k - 1, and otherwise zero values. Therefore, the zeta transformation, or the present value of f(n) starting at n = h and ending after n = k - 1, can be used to determine the present value of cash loans, assuming a "constant" stream of debt-servicing payments. As such, a typical schedule of a loan repayment during a given period of time can be mapped out in the table below (where all the variables are as defined before):

Years Interest Debt Servicing

 1 rF --
 2 rF --
 . . .
 . . .
 . . .
 G rF --
 G+1 -- rF[(1+r).sup.N-G+1] /
 [[(1+r).sup.N-G+1] - (1+r)]
 G+2 -- rF[(1+r).sup.N-G+1] /
 [[(1+r).sup.N-G+1] - (1+r)]
 . . .
 . . .
 . . .
 N -- rF[(1+r).sup.N-G+1] /
 [[(l+r).sup.N-G+1] - (1+r)]

N Years rFG (NG) rF[(l+r).sup.N-G+1] /
 [[(1+r).sup.N-G+1] - (1+r)]

The table presents the cost of interest and a constant stream of debt-servicing payments. The total amount that the borrower has to repay during a period of N years is:

[rFG + (N - G) rF (1 +r).sup.N - G + 1] / [[absolute value of (1 + r).sup.N - G + 1] - (1 + r)]

It is clear that these future repayments do not reflect the true value of money at the "present" time and, therefore, these repayments must be converted into their present equivalent value and summed up. The conversion can be done by applying the zeta transformation of the function f (n) = c (as shown in the above table) to:

(1) The annual amounts of rF during years 1 through G (with c = rF, h = 1 and k - 1 = G).

(2) The annual amounts of rF [(1 + r).sup.N - G + 1] / [[absolute value of (1 + r).sup.N - G + 1] - (1 + r)] during years G + 1 through N (with c = rF [(1 + r).sup.N - G + 1] / [[absolute value of (1 + r).sup.N - G + 1] - (1 + r)], h = G + 1 and k - 1 = N).

Thus, the present value related to the arrangements of this kind of repayments is:

PV : rF [[absolute value of 1 -(1 + i).sup.-G] + [(1 + r).sup.N - G + 1] x [((1 + i).sup.-G] - [(1 + i).sup.-N]) / [((1 + r).sup.N - G +1] - (1 + r))] / i

Appendix 2

THE DERIVATION OF FORMULA (6)

Suppose we have two functions f (x) and g (x) which are zero when x = a. Although the ratio f(a)]g (a) is an undefined quantity [o/o], the limit of f(x)/g (x) as x [right arrow] a may exist nevertheless.

Consider the ratio of f (x) and g (x) and let both functions be expressed at the point x = a by using Taylor's Theorem.

Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

By assumption, f(a) = g (a) = 0, ... ... (10) and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Hence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Provided that g' (a) is non-zero, Equation (12) shows that the limit of the ratio of the two functions as x [right arrow] a, where both functions are equal to zero when x = a, is given by the ratio of the derivatives of the two functions, each being evaluated at x=a.

If, however, f' (a) = g' (a) = o, then the same procedure must be applied. Provided that the limit exists, it is usually possible to find a value of n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

This method of evaluating limits is normally expressed by rewriting Equation (12) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

which is known as "L'Hospital's Rule".

By applying this rule to formula (5) when G = N, the following set of equations can be obtained:

let f(x) = [(1 +i).sup.-x] - [(1 + i).sup.-n] ... ... (15)

since [a.sup.b] = [e.sup.b log a] ... ... (16)

then: f(x) = [[e.sup.xlog(1+i) - (1 + i].sup.-n] ... (17)

and g(x) = [(1+r).sup.n-x+1] - (1 + r) ... ... (18)

then: g(x) = [[e.sup.(n-x+l) log (1 + r) - [1 + r)] ... (19)

since f(n) = g(n) = 0 ... ... (20)

then: f (x) = -[e.sup.-xlog (1 + i)] . log (1 + i) ... (21)

[(1+i).sup.-x] log (1 + i) ... ... (22)

and g'(x) = --[e.sub(n-x+1) log (1 + r)] . log (1 + r) ... (23)

= -[(1 + r).sup.n-x+1] .log (1 + r) ... (24)

Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Hence, when G = N, the required formula would be:

PV = rF [1 - [(1 + i).sup.-G] + (1 + r) X ([(1 + i).sup.-G] log(1 + i)) / ((1 + r) log (1 + r))] /i

Author's Note: This paper is based on my Ph.D. thesis completed at the University of Kent at Canterbury. For useful comments, I would like to thank--but in no way implicate--Richard Disney and Allen Carruth. I am also indebted to anonymous referees for helpful suggestions.

REFERENCES

Beenhakker, H. L. (1976) Handbook for the Analysis of Capital Investment. Westport, Conn." Greenwood Press.

Ohlin, G. (1966) Foreign Aid Policies Reconsidered. Paris: OECD, Development Centre Studies.

Yassin, I. H. (1983) Foreign Capital Inflows and Economic Development." The Experience of the Sudan 1958-1979. Unpublished PhD. Thesis, University of Kent at Canterbury.

(1) Debt-servicing payments can also be made at an increasing rate (i.e., in each successive period the payable instalment increases) or at a decreasing rate over time.

(2) It is implicit that Ohlin's formulation distinguishes between short-term and long-term loans. Hence, for short-term loans, this approximation is suggested:

A = (i - r) N/2

where all the terms are as defined before. In fact, this approximation is valid when iN < 1 and G = 0, which is a specific definition of short-term loans.

(3) Those who are interested in the derivation of formula (4) can pursue the exercise themselves. However, it should be mentioned that a similar application of L'Hospital's Rule is contained in Appendix (2).

(4) The grant element is calculated by using annual weighted shares of individual loans in the total inflows, i.e., by multiplying the grant element of each loan (A) received in a specific year by the nominal value of the loan (F) and dividing the product by the total annual amount of loans received during that year. This relationship can be expressed as:

(Am X Fm)/ [M.summation over (m = 1)] Fm

where m = 1.... M, is the number of loans received in a given year.

(5) The 8 percent discount rate is assumed to represent the world market rate of interest being proxied by the average annual rate of the UK money-markets and the Euro Dollar market during the period 1958-1979. This rate acts as the rate of interest at which the Sudan might have had to borrow in the absence of aid, and it has been calculated from the IMF International Financial Statistics; whilst the 10 percent is the standard discount rate of the Development Assistance Committee (DAC) of the Organization for Economic Cooperation and Development (OECD). For more detail on the grant element of loans to the Sudan, see Yassin (1983).

IBRAHIM HASSAN YASSIN, The author is Lecturer in Economics at the University of Gezira, Sudan.

Table 1

A Comparison of the Grant Element when the Maturity and the Grace
Periods are Equal

 Terms of Borrowing Grant Element Using

 Rate of Maturity Grace
 Interest Period Period Formula (4) Formula (6)

(Percent) Years At 10 Percent Discount Rate

 0.75 15 15 71.9 59.3
 1.0 10 10 56.9 40.6
 3.0 8 8 38.5 31.5
 3.5 5 5 25.6 17.7
 5.0 4 4 16.5 13.2

Source: Own estimates based on data compiled from various issues of
the Annual Report of the Bank of Sudan and the records of the Sudanese
Ministries of National Planning, and Finance and National Economy.

Table 2

Sources and Size of Official Loans Contracted by the Sudan during the
Period 1958-1979

 Percent
 Share in
 Total Amount Total
Source of Loans (L.S. Million) Borrowing

(1) Bilateral Agreements 802.76620 66.7
 (a) USA and West European Countries 245.73118 20.4
 (b) Arab Countries 432.42502 35.9
 (c) East European Countries 124.61000 10.4

(2) Multilateral Agreements 223.71700 18.6

(3) Borrowing from Private Sources 176.53000 14.7

Total 1203.01320 100.0

Source: Based on data from the same source as Table 1.

Table 3

Values of the Grant Element of Official Loans and the Terms of
Borrowing: A Weighted Annual Average (1958-1979)

 Total Amount Average
 of Contracted Average Rate Repayment
Years Loans of Interest Period
 (L.S. Million) (Percent) (Years)

1958 13.6 5.5 17
1959 7.8 3 8
1960 9.3 5.6 15
1961 31.8 4.2 17
1962 9.7 4.2 11
1963 7.4 2.7 13
1964 .27 5.75 40
1965 17.025 4.8 17
1966 6.62 4.9 9
1967 27.52 3.7 12
1968 24.9 3.9 16
1969 18.0905 4.7 8
1970 24.225 1.9 10
1971 46.1 2.6 6
1972 64.82668 3 16
1973 119.332 3.2 17
1974 216.72 4.9 15
1975 116.52 3.2 22
1976 108.865 3 22
1977 85.229 3 18
1978 204.91 3.9 16
1979 42.26 2.4 32

 Over-all 4 16
 Average

 Average Grant Element at a
Years Average Discount Rate, Based on
 Grace
 Period
 (Years) 8 Percent 10 Percent

1958 3 25.5 35.9
1959 8 23.7 31.5
1960 2 21.3 31.1
1961 6 34.1 42.8
1962 4 21.8 29.6
1963 6 32.6 40.8
1964 10 59.6 71.5
1965 4 28.8 38.6
1966 3 14.2 21.4
1967 3 25.0 31.9
1968 4 29.0 37.5
1969 2 16.0 22.0
1970 10 31.3 40.5
1971 8 18.5 24.8
1972 5 32.2 38.9
1973 7 34.5 41.7
1974 5 26.6 33.5
1975 6 40.9 48.4
1976 7 42.9 51.0
1977 5 37.6 46.6
1978 6 33.0 41.1
1979 7 54.1 62.4

 6 31 39

Source: Own estimates based on data from the same source as Table.