36153
Yuri Dikhanov
DECDG
World Bank
ydikhanov@worldbank.org
January 11, 2005
HISTORICAL PV OF DEBT IN DEVELOPING
COUNTRIES, 1980-2002
Long-term debt sustainability analyses are usually based on nominal debt
numbers. While nominal debt is an important debt indicator, the present value
of debt payments [PV] is in many respects considered superior. However,
there have been no PV estimates in the literature spanning long enough time
periods and using consistently concurrent discount rates. This paper is
intended to fill this gap: it develops a methodology for historical PV
computations and applies it to the actual debt data.
The computation of PV is based on inputs from the Debt Reporting System
(DRS) maintained by DECDG Department of the World Bank. The DRS is a
unique depository of historical debt data for developing countries that allows
data mining, projections and detailed analysis. However, it is extremely time-
consuming to analyze its in-depth loan-by-loan information. This paper
pursues a different approach: instead of loan-by-loan data, in PV estimates a
somewhat more aggregated dataset is used. It turns out to be possible to
utilize average terms for loan aggregates for the loans with similar terms
(average grace, maturity and interest rate) to achieve sufficiently precise PV
estimates for the aggregate. 1
1 The author would like to thank Ibrahim Levent and Vinoda Basnayake [DECDG] for valuable
inputs into this paper.
1. DRS Data Sources and Aggregation Methodology
The World Bank is the sole repository for statistics on the external debt of
developing countries on a loan-by-loan basis. The Debtor Reporting System
(DRS) was set up in 1951 to monitor these statistics. All DRS data that are used
in the PV calculations are reported directly by the debtors except for lending by
some multilateral agencies in which case data are taken from the creditors’
records. These creditors include the African Development Bank, the Asian
Development Bank, the International Monetary Fund (IMF), the Inter-American
Development Bank, the International Bank for Reconstruction and Development
(IBRD), and the International Development Bank (IDA), (The IBRD and IDA are
components of the World Bank).
The DRS data is highly detailed and comprehensive. The loan-by-loan data
includes private, public, foreign, short term, long term, privately-guaranteed, and
publicly-guaranteed debt. In order to facilitate data manipulation and
intermediate computations, a smaller subset of the DRS data was extracted for PV
calculations. The DRS loan-by-loan data was aggregated by debtor country, pay
currency, effective date [commitment date] and creditor type. In addition, the
scope of analysis was limited to long term public and publicly-guaranteed [PPG]
foreign debt. This specific dataset is then processed using terms for average grace
period, maturity, and interest rate. The PV calculations are performed by
applying average terms to the loan aggregates with similar terms utilizing the
process detailed in this paper.
2. Derivation of PV
We start with definitions: Present Value (PV) is the discounted value of a payment or
stream of payments to be received in the future, taking into consideration a specific interest or
discount rate. Present Value represents a series of future cash flows expressed in today's currency
units.
Or, put differently, with obvious notation:
tN
DSt
PV = ∑
t = t0 (1 + d )t −t0
(1.1)
The continuous version of the formula will be, respectively:
Y.DIK HANO V (W ORLD BAN K ) 3 HIST ORIC A L PV
tN
(1.2) PV = ∫ ds (t )e d (t0 −t ) dt
t0
again, with obvious notation, where DS stands for debt service, and d stands for
discount rate. The formula (1.1) can be extended to cover monthly and daily [and
any arbitrary period] discounting.
From this presentation, it follows that the PV has the following properties:
(1) linearity wrt. debt service [multiplying debt service by a factor will result
in the PV greater than the PV of the original debt service by the same
factor] ;
(2) monotonicity wrt. debt service [ greater debt service results in greater PV];
(3) additivity wrt. debt service if discount rates do not change [i.e. the sum of
PVs of individual debt flows is equal to the PV of the sum of individual
debt flows; or, put differently, in order to compute the PV of a debt
portfolio, it is enough to group debt service into currency groups /or other
groups with the same discount rate/ and compute PV for those groups
only]
The PV is equal to the nominal [face] value of a loan if and only if the discount
rate is equal to the interest rate. In all other cases there will be a discrepancy
between the nominal (face) value of a loan and its PV. The difference is usually
called the Grant Element of the loan.
PV is often considered the single most important indicator of indebtedness used in
debt analysis. Commercial banks use it in most of their operations. It is preferred
over debt outstanding and simple sum of debt repayments because these
categories do not account for differences in repayment terms.
Accordingly, most of the debt operations, including those of Paris Club, are tied
to the PV (or, as it is sometimes [incorrectly] called, the NPV).
3. Properties of the Present Value
The most important property of the PV concept is equality of PV and face value
of debt (initial DOD) whenever interest rate is constant and equal to discount rate.
We will provide a proof of this property both for discreet and continuous cases
below:
Notation:
Y.DIK HANO V (W ORLD BAN K ) 4 HIST ORIC A L PV
discreet case:
dod stands for debt outstanding and disbursed,
inp – interest payments
prp – principal payments
tds – total debt service
continuous case:
p – principal payment flow
y – debt outstanding and disbursed
ν - interest payment flow
i – interest rate
the first time period is 0, the last one is N.
DISCREET CASE:
dod t = dod t −1 − prpt
inpt = i ⋅ dodt −1
tdst = i ⋅ dod t −1 + prpt
tdst = i ⋅ dod t −1 − dodt + dod t −1
tdst = (i + 1) ⋅ dod t −1 − dodt
t
dod t = dod 0 − ∑ prpτ
1
N N
∑ tds = ∑ [(i + 1) ⋅ dod
1
t
1
t −1 − dodt ]
N
tdst N
(i + 1) ⋅ dod t −1 − dod t
PV = ∑ =∑ =
1 (1 + i ) t
1 (1 + i )t
N
dodt −1 dod t dod N
= ∑[ t −1
− ] = dod 0 −
1 (1 + i ) (1 + i ) t
(1 + i ) N
PV ≡ dod 0 , as dod N is equal to zero
CONTINUOUS CASE:
Y.DIK HANO V (W ORLD BAN K ) 5 HIST ORIC A L PV
− p = y (principal payments)
&
ν = i ⋅ y (interest payments)
hence
tN
∫ (− y + i ⋅ y ) e
− it
PV = & dt , or
t = t0
tN
∫ d(-ye
− it
PV = ) = - ye− it tN
t0 = y (t0 ),
t = t0
as y (t N ) is equal to zero.
Related to the PV is the concept of Grant Element (GE). The Grant Element of a
loan is the difference between its PV as computed using actual interest rates and
using some accepted discount factor (such as CIRR or another market discount
rate).
Grant Element reflects financial terms of commitments: interest rate, maturity and
grace period (interval to first repayment of capital). It measures the
concessionality of a loan, as the difference between present value at a particular
interest rate versus that at the market rate. For example, the market rate is
assumed to be 10 per cent in DAC statistics. Thus, the Grant Element is nil for a
loan carrying an interest rate of 10 per cent; it is 100 per cent for a grant; and it
lies anywhere in between between these two limits for a soft loan. If the face
value of a loan is multiplied by its Grant Element, the result is referred to as the
grant equivalent of that loan.
N
tdst N
(d − ∆ + 1) ⋅ dod t −1 − dodt
PV = ∑ =∑ =
1 (1 + d ) t
1 (1 + d )t
N
(1 + d ) ⋅ dod t −1 − dodt N ∆ dod t −1
=∑ −∑ =
1 (1 + d )t 1 (1 + d )
t
N
dodt −1
= dod 0 − ∆ ∑ , where ∆ is the difference
1 (1 + d )
t
between interest and discount rates. And Grant Element is:
dod t −1
N
dod 0
GE = ∆ ∑
1 (1 + d )t
Interestingly, the GE is thus exactly proportional to the difference between
interest and discount rates. Operationally, it means that one can compute effect of
interest rate changes on PV quite simply. Also, once GE for any given interest
rate is known, one can always reestimate GE for any other interest rate value
using proportional change in interest rate (for any given discount rate):
Y.DIK HANO V (W ORLD BAN K ) 6 HIST ORIC A L PV
i1 − d
GE (i1 ) / GE (i2 ) =
i2 − d
Now, using a finite series formula we can express
PV and GE of a loan:
n −1
a − [a + (n − 1) r ] q n rq (1 − q n −1 )
∑ (a + kr ) q k =
0 1− q
+
(1 − q ) 2
dodt −1
N
dod 0
GE = ∆ ∑
1 (1 + d )t
dod 0
dod t = dod 0 − t , where m is principal repayment length
m
and q = 1/(1 + d )
then,
N −1 N −1
1− t / m
GE = ∆ q ∑ = ∆ q ∑ (1 − t / m) q t =
0 (1 + d )t 0
N −m m −1
= ∆ q ∑ q t + ∆ q N − m +1 ∑ (1 − t / m) q t =
0 0
N −m m −1
1− q
= ∆ q[ + q N − m ∑ (1 − t / m) q t ], or
1− q 0
N −m N
i−d 1 1
GE = 2 [dm − + ]
d m 1+ d 1+ d
and, respectively, PV can be expressed as follows:
PV = dod 0 (1- GE )
As we can see, the above formulas allow efficient calculation of the PV and GE of
a loan without going through tedious year-by-year computations.
All of the above terms can be obtained from the DRS database. Moreover, it is
also possible to obtain the GE estimates based on the DAC’s 10% discount rate,
which caouldbe used to correct our estimated values. So the overall formula used
in the estimations would be:
GE final = GE + ∆
where
N −m N
i−d 1 1
GE = [dm − + ] , and
1+ d 1+ d
2
d m
N −m N
i − 0 .1 1 1
∆ = GE estimated by DRS at10% − [dm − + ]
1 + 0 .1 1 + 0 .1
2
0 .1 m
Y.DIK HANO V (W ORLD BAN K ) 7 HIST ORIC A L PV
Limit cases of the GE function are interesting per se [see Annex for details].
Several cases that have operational significance are discussed, including ones
where i->0 (interest rate approaching 0), d->0 (discount rate approaching 0), n-
>M (length of payments approaching maturity, i.e. grace approaching zero), m->0
(length of payments approaching 0, i.e. bullet payment). These cases allow for
very simple presentations.
4. Some results of computations of PV using DRS data
The results are presented in this section. Summary results are shown in Tables 1,
1a and 1b as well as in Figures 1, 1a and 1b.
Figure 1 shows overall developments of the debt situation during 1980-2002.
One can see clearly from there that there has been uneven growth in PV and
DOD. To grasp these differences better, Figure 2 is intended to show growth rates
of PV and DOD. Here we can clearly see the interesting picture where PV may
grow but DOD falls and the other way around.
The differences in growth rates between DOD and PV are shown in Figure 1b.
Here we can see drastic differences: from +15% to -8%. Such differences would
amount to equally drastic consequences for modeling debt sustainability.
Numbers for individual regions are presented in tabular form in Tables 1, 1a and
1b. Table 1b showing the differences in growth rates between PV and DOD
indicates a rather uniform picture across regions. The differences are the function
of discount rates, currency composition, maturity and grace period. So, the
uniformity is explained by the high correlation of developments on international
financial markets.
As many economic models operate with differences, using PV instead of DOD
may drastically change our understanding of the debt outlook, especially for
severely indebted countries.
Even long-term nominal growth for PV and DOD shows substantial differences:
during 1980-2002 DOD grew at 6.8% annually, whereas PV increased at 8.1%
annually. Corrected for inflation, these numbers become 3.8% and 5.1%,
respectively, this is equivalent to providing a different answer to the basic
question: whether the debt burden relative to the GDP has increased or decreased
during 1980-2001.
These estimates provide a significantly different picture of the debt burden
dynamics in the developing countries. Using DOD instead of PV in debt analyses
not only underestimates the severity of the debt problem, but often assumes
Y.DIK HANO V (W ORLD BAN K ) 8 HIST ORIC A L PV
wrong dynamics for many crucial periods, which, in turn, will undermine the
analyses’ outcomes in those countries.
ANNEX
Limit properties of the GE Function
Some of the limit properties of the GE function are interesting per se. Presented
below are cases, where i->0 (interest rate approaching 0), d->0 (discount rate
approaching 0), n->M (length of payments approaching maturity, i.e. grace
approaching zero), m->0 (length of payments approaching 0, i.e. bullet payment).
Derivatives can be presented as follows:
Y.DIK HANO V (W ORLD BAN K ) 10 HIST ORIC A L PV
Figure 1. Developing World DOD & PV, 1980-2002
1,600 120
1,400
100
1,200
80
1,000
800 60
World PV
600
World DOD 40
400 PV/DOD
20
200
0 0
1970 1975 1980 1985 1990 1995 2000
50% Figure1a. Growth rates of World DOD & PV, 1970-2002
40%
World DOD
World PV
30%
20%
10%
0%
1970 1975 1980 1985 1990 1995 2000
-10%
Y.DIK HANO V (W ORLD BAN K ) 11 HIST ORIC A L PV
Figure 1b. Difference between World DOD & PV
15% growth rates, 1970-2002
10% PV/DOD
5%
0%
1970 1975 1980 1985 1990 1995 2000
-5%
-10%
Table 1. Public and Publicly Guaranteed Debt,1980-2001
US$ millions
Year
REGION 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Afric a DOD 42,113 47,721 55,590 62,176 64,027 77,370 96,032 122,209 124,061 129,607 144,436 149,823 146,307 153,906 166,074 174,402 170,023 163,785 170,427 156,274 160,123 151,765
Afric a NPV 31,664 34,115 43,306 54,161 54,999 67,495 88,628 115,308 113,822 110,264 115,106 119,176 118,843 135,238 141,280 151,164 150,975 146,410 157,438 145,030 140,122 138,571
Asia DOD 36,330 41,936 50,042 59,698 64,222 79,881 102,554 129,075 132,046 157,077 173,200 184,286 196,260 216,292 242,042 257,475 257,998 268,707 278,647 305,726 286,652 276,228
Asia NPV 30,282 35,461 46,122 59,638 63,879 79,431 105,197 134,444 135,423 146,363 151,659 160,652 179,104 210,027 230,900 252,947 257,254 275,129 284,976 315,771 290,238 292,944
L.America DOD 130,523 150,856 177,841 222,906 249,297 274,213 306,713 339,666 336,146 330,410 328,013 330,829 328,983 333,723 352,829 376,242 377,152 362,079 391,935 399,317 402,752 397,856
L.America NPV 118,525 137,276 180,064 253,234 281,394 303,163 350,176 380,090 360,518 333,938 312,561 314,617 330,458 352,678 352,506 391,659 403,440 397,258 439,325 448,817 439,465 464,349
M.East&N.Africa DOD 53,568 57,988 63,837 70,049 71,426 83,067 95,914 112,345 117,451 119,494 115,764 116,959 114,709 118,297 135,873 141,578 136,181 123,275 127,325 120,881 114,126 113,611
M.East&N.Africa NPV 41,816 42,566 51,143 63,310 64,554 77,406 94,222 112,984 115,524 109,314 100,873 101,478 102,039 112,077 126,763 134,450 129,633 119,086 125,265 119,541 109,308 113,821
S. Asia DOD 31,189 32,611 36,574 39,582 42,062 50,381 60,531 71,835 78,999 95,059 105,800 110,834 116,677 125,922 134,106 128,983 126,018 126,851 136,749 142,154 136,472 134,261
S. Asia NPV 18,054 17,914 22,032 26,695 28,358 35,597 46,731 58,124 63,974 71,056 75,468 81,081 90,874 107,351 109,971 108,152 108,510 113,011 126,203 131,680 120,751 124,303
East&Central Europe DOD 44,631 50,837 57,340 60,315 77,455 104,127 119,219 144,305 142,438 150,063 172,129 187,105 195,095 246,238 260,965 269,959 266,857 267,798 293,278 292,182 283,351 263,213
East&Central Europe NPV 39,808 43,680 53,813 62,371 80,525 107,484 128,222 157,314 152,582 151,239 162,403 177,459 186,008 248,837 254,541 268,118 272,168 276,113 314,290 314,240 298,818 291,632
World DOD 338,354 381,949 441,224 514,726 568,489 669,039 780,965 919,435 931,141 981,709 1,039,340 1,079,835 1,098,030 1,194,378 1,291,889 1,348,639 1,334,230 1,312,495 1,398,362 1,416,534 1,383,476 1,336,935
World NPV 280,149 311,013 396,481 519,409 573,708 670,575 813,176 958,265 941,842 922,175 918,071 954,464 1,007,326 1,166,207 1,215,962 1,306,490 1,321,981 1,327,008 1,447,497 1,475,077 1,398,703 1,425,620
Table 1a. Growth of Public and Publicly Guaranteed Debt,1980-2001
Per Cent
Year
REGION 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Afric a DOD 13.3% 16.5% 11.8% 3.0% 20.8% 24.1% 27.3% 1.5% 4.5% 11.4% 3.7% -2.3% 5.2% 7.9% 5.0% -2.5% -3.7% 4.1% -8.3% 2.5% -5.2%
Afric a NPV 7.7% 26.9% 25.1% 1.5% 22.7% 31.3% 30.1% -1.3% -3.1% 4.4% 3.5% -0.3% 13.8% 4.5% 7.0% -0.1% -3.0% 7.5% -7.9% -3.4% -1.1%
Asia DOD 15.4% 19.3% 19.3% 7.6% 24.4% 28.4% 25.9% 2.3% 19.0% 10.3% 6.4% 6.5% 10.2% 11.9% 6.4% 0.2% 4.2% 3.7% 9.7% -6.2% -3.6%
Asia NPV 17.1% 30.1% 29.3% 7.1% 24.3% 32.4% 27.8% 0.7% 8.1% 3.6% 5.9% 11.5% 17.3% 9.9% 9.5% 1.7% 6.9% 3.6% 10.8% -8.1% 0.9%
L.America DOD 15.6% 17.9% 25.3% 11.8% 10.0% 11.9% 10.7% -1.0% -1.7% -0.7% 0.9% -0.6% 1.4% 5.7% 6.6% 0.2% -4.0% 8.2% 1.9% 0.9% -1.2%
L.America NPV 15.8% 31.2% 40.6% 11.1% 7.7% 15.5% 8.5% -5.1% -7.4% -6.4% 0.7% 5.0% 6.7% 0.0% 11.1% 3.0% -1.5% 10.6% 2.2% -2.1% 5.7%
M.East&N.Africa DOD 8.3% 10.1% 9.7% 2.0% 16.3% 15.5% 17.1% 4.5% 1.7% -3.1% 1.0% -1.9% 3.1% 14.9% 4.2% -3.8% -9.5% 3.3% -5.1% -5.6% -0.5%
M.East&N.Africa NPV 1.8% 20.1% 23.8% 2.0% 19.9% 21.7% 19.9% 2.2% -5.4% -7.7% 0.6% 0.6% 9.8% 13.1% 6.1% -3.6% -8.1% 5.2% -4.6% -8.6% 4.1%
S. Asia DOD 4.6% 12.1% 8.2% 6.3% 19.8% 20.1% 18.7% 10.0% 20.3% 11.3% 4.8% 5.3% 7.9% 6.5% -3.8% -2.3% 0.7% 7.8% 4.0% -4.0% -1.6%
S. Asia NPV -0.8% 23.0% 21.2% 6.2% 25.5% 31.3% 24.4% 10.1% 11.1% 6.2% 7.4% 12.1% 18.1% 2.4% -1.7% 0.3% 4.1% 11.7% 4.3% -8.3% 2.9%
East&Central Europe DOD 13.9% 12.8% 5.2% 28.4% 34.4% 14.5% 21.0% -1.3% 5.4% 14.7% 8.7% 4.3% 26.2% 6.0% 3.4% -1.1% 0.4% 9.5% -0.4% -3.0% -7.1%
East&Central Europe NPV 9.7% 23.2% 15.9% 29.1% 33.5% 19.3% 22.7% -3.0% -0.9% 7.4% 9.3% 4.8% 33.8% 2.3% 5.3% 1.5% 1.4% 13.8% 0.0% -4.9% -2.4%
World DOD 12.9% 15.5% 16.7% 10.4% 17.7% 16.7% 17.7% 1.3% 5.4% 5.9% 3.9% 1.7% 8.8% 8.2% 4.4% -1.1% -1.6% 6.5% 1.3% -2.3% -3.4%
World NPV 11.0% 27.5% 31.0% 10.5% 16.9% 21.3% 17.8% -1.7% -2.1% -0.4% 4.0% 5.5% 15.8% 4.3% 7.4% 1.2% 0.4% 9.1% 1.9% -5.2% 1.9%
Table 1a. Difference of DOD and PV Growth of Public and Publicly Guaranteed Debt,1980-2001
Per Cent
Year
REGION 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Afric a DOD 5.6% -10.5% -13.2% 1.4% -1.9% -7.2% -2.8% 2.8% 7.6% 7.1% 0.2% -2.1% -8.6% 3.4% -2.0% -2.4% -0.6% -3.5% -0.4% 5.8% -4.1%
Asia DOD -1.7% -10.7% -10.0% 0.5% 0.0% -4.1% -1.9% 1.6% 10.9% 6.6% 0.5% -5.0% -7.1% 2.0% -3.2% -1.5% -2.8% 0.1% -1.1% 1.8% -4.6%
L.America DOD -0.2% -13.3% -15.3% 0.7% 2.3% -3.7% 2.2% 4.1% 5.7% 5.7% 0.2% -5.6% -5.3% 5.8% -4.5% -2.8% -2.5% -2.3% -0.3% 2.9% -6.9%
M.East&N.Africa DOD 6.5% -10.1% -14.1% 0.0% -3.6% -6.3% -2.8% 2.3% 7.1% 4.6% 0.4% -2.5% -6.7% 1.8% -1.9% -0.2% -1.3% -1.9% -0.5% 3.0% -4.6%
S. Asia DOD 5.3% -10.8% -12.9% 0.0% -5.8% -11.1% -5.7% -0.1% 9.3% 5.1% -2.7% -6.8% -10.2% 4.1% -2.2% -2.6% -3.5% -3.9% -0.4% 4.3% -4.6%
East&Central Europe DOD 4.2% -10.4% -10.7% -0.7% 1.0% -4.8% -1.6% 1.7% 6.2% 7.3% -0.6% -0.5% -7.6% 3.7% -1.9% -2.7% -1.1% -4.3% -0.4% 1.9% -4.7%
World DOD 1.9% -12.0% -14.3% 0.0% 0.8% -4.5% -0.1% 3.0% 7.5% 6.3% -0.1% -3.9% -7.0% 3.9% -3.1% -2.3% -2.0% -2.5% -0.6% 2.8% -5.3%
Y.DIK HANO V (W ORLD BAN K ) 13 HIST ORIC A L PV