Compensating balances are deposits the borrower (firm) keeps with the lender (bank) in non-interest bearing accounts on the term loan. See Ross, Westerfield and Jaffe ([12]), Downes and Goodman ([4]), Davidson III ([{3]). Historically, compensating balances have been frequently used as covenants in bank lending arrangements, as in Hagaman ([7]), Gallager ([6]) and Hingston ((8]). Under these arrangements, the borrowing firm is required to maintain a cash balance at the lending bank equal to a certain percentage of the loan balance during the life of the loan. See Booth and Chua ([1]). By leaving these funds with the lending bank without receiving interest, the borrowing firm increases the effective (true) interest the firm pays to the bank on a loan. See Kolodny, Seeley and Polakoff ({10]), Sealey and Heinkel ([13]), Lam and Boudreaux ([{11]) and Burr ([2]).
It is well known that, when there is no compensating balance, the effective cost of debt remains the same in the two different, frequently-used payment methods, i.e., full amortization method and bullet loan (bond) method. When a compensating balance is imposed, however, the effective costs of debt will surely increase for both types of loans. An important question arises: will the effective costs for both types of 84 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 a loan still be the same at a higher rate? If not, which one will involve a higher (or lower) effective borrowing cost? Answers to these questions are not obvious (at least to the authors). Moreover, we have not found a rigorous analysis of this issue in either any textbook or academic journal.
After setting the stage for discussion with a simple numerical example (Section II), we conduct a mathematical analysis of the patterns of the respective effective costs of debt when a compensating balance is imposed (Section III). Concluding comments are provided in Section IV. Appendix contains detailed proofs which supplement our results and an open question about the relation between compensating balance and loan period.
Consider a simple numerical example of borrowing $10,000 for three years at an annual quoted interest rate of 10% per year under two alternative ways, Al and B1.
Under A1, the loan is fully amortized and the annual payment at the end of each year (ordinary annuity) is $4,021.15. The pattern of annual cash flows (for the borrowing firm) is +10,000, -4,021.15, -4,021.15, -4,021.15. Both the quoted interest rate and effective interest rate (internal rate of return) are 10%: per year. Alternative B1 is a bullet (interest only) loan. The pattern of annual cash flows is +10,000, - 1,000, -1,000, -11,000. Both the quoted interest rate and effective interest rate (internal rate of return or IRR) are also 10% per year. Now the lending bank imposes a compensating balance of, say, $2,000 on both Al and B1. The borrowing firm deposits $2,000 at Year 0 (the beginning of the loan period) and gets it back from the bank at Year 3 (the end of the loan period). Al’s pattern of cash flows has now become: +8,000, -4,021.15, -4,021.15, -2,021.15, and the effective interest rate increases to 13.88%. Let us call this new alternative A2. B1’s pattern of cash flows has now become +8,000, - 1,000, -1,000, -9,000 with the newly installed compensating balance, and the effective interest rate increases to 12.50%. Let us call this new alternative B2. Note that Under B2, the borrower is effectively borrowing $8,000 for three years at a quoted annual interest rate of 12.50%.
Why does the effective interest rate for A2 (13.88%) become greater than that for B2 (12.50%), even though an identical compensation balance is imposed on both alternatives? Is it because under A2 the borrower pays back periodically part of the loan principal whereas under B2 the borrower keeps the entire principal for three years? The answer to this question is no, because, for both baseline alternatives, Al and B1, the effective interest rate is 10\% per year. Under B2, the firm effectively borrows $8,000 at an annual interest rate of 12.50% for three years. Under A2, the firm borrows the same effective amount of $8,000 for three years, but the amortization schedule is constructed for the loan principal of $10,000 (instead of $8,000) at an annual interest rate of 10% (instead of 12.50%). Why does the additional $2,000 used for the loan amortization more than compensate an interest rate reduction of 2.50% per year? The purpose of this paper is to provide a mathematical analysis that compares the effective costs of debt between the two alternatives for all possible combinations of the compensating balance and the loan period available in practice. Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 85
Consider borrowing a principal $P$ at a nominal interest rate of $R$ and pay it back in $n$ years at the end of each year. Depending on the payment method, we have the following four cases for cash flow chart and effective cost of debt.
A. Full Amortization Method
When an equal payment of A, is made at the end of each time interval, we have the following cash flow chart.
Out An An An An tA Time 0 1 2 hace eee eee eee n-1 n In P
Net P -An PAN .............. -An -An
The net present value of this loan for the borrower is 0, hence epal TR and this again implies
1 1 1 1 1 () paa, { + tu. Nea, {i- ) 1+R_ (1+R)? (1+R) R (1+R)" This amount a! - oe ) , called the present value interest factor for an ordinary +R annuity plays an important role in this paper, so we give it a formal definition of f (R). Using this terminology, the end of year payment A =P/f (R). The effective cost of debt r, is the interest rate which makes the present value of cash-inflow and cash- outflow equal, and r,=R by our choice of A...
B. Bullet Loan Method This is a typical payment schedule of bonds. An equal payment of $PR$ (coupon) is made at the end of each time interval, and at the end of the loan period, the borrower pays back the principal $P$. The cash flow chart is the following. 86 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98
Out PR PR PR PR+P A A— Time 0 1 2 aaa. ee aaa. n-1 n In P
Net P -PR PR a... -PR -PR-P
Fig.2: Bullet method without compensting balance
The geometric series makes the net present value zero, hence
1 1 1 P o=p-pr { + —_ +..+ —— 7-5 ) (2) +R (1+R? (1+R)° 7: 1+4R)Y
Again, by the construction of this payment method, the effective cost of debt r, is the same as the nominal rate R.
Now, assume the lender (bank) requires a compensating balance X which will be returned to the borrower at the end of the loan period. No interest will be compounded and the borrower will receive only the original amount X. Theoretically, the compensating balance can be any number between 0 and the principal P, but in practice it is usually less than 30% of the principal. Throughout this paper, we assume the compensating balance is less than half of the principal, X < P/2. This choice of P/2 guarantees the uniqueness of the effective cost of debt (Appendix B). Our goal is to prove that there is only one effective cost of debt, r, and r,, respectively, for each payment method considered, and compare their sizes. Despite our assumption throughout this paper that all the payments are made at the end of each year, in practice, such payments are often made monthly. This means some payment methods can have positive net cash flows at the end of the loan period (recall that the cash flow is calculated in the borrower’s position), causing more than one sign changes in the net cash flows. According to Descartes’ Rule of Signs, this can cause multiple effective costs of debt, and our restriction 0 < X < P/2 assures that this will not happen.
C. Full Amortization Method with Compensating Balance
Assuming a compensating balance X is imposed in case (a). Youngna Choi et al. / The international Journal of Banking and Finance 2 (2004) 83-98 87
Out xX An An An An A Time 0 1 2 .............. n-1 n
In P Xx
Net P-X -An Al ............. -An X-An
The end of year payment A, remains the same and r, is decided by
1 1 1 Xx 0=(P-X)-Aj — + t.t pt +r, (1+r,)? (ltt, ’ l+r,)° or equivalently xX P P-X)+ =Af(r,) where A, = — Po) + oe AH = ER) @)
D. Bullet Loan Method with Compensating Balance
Finally, we have a bullet loan method with a compensating balance X imposed.
Out x PR PR PR PR+P a Time 0 1 2 eee ee n-1 n In P Xx
Net P-X PR PR ............. -PR PR-P+X
88 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98
1, is decided by zero net present value of the cash flow,
1 1 1 P Xx 0=@-x)-PR{ + t.t ). + l+r, (141, Ar) Ar)! (1H)
or equivalently P xX
(P - X)- + = (Ar)! (At)
PE (r,) (4)
The function representing the present value interest factor for an ordinary annuity
1 1 . : . 4 . f=()= 7 “a 3 is a decreasing function of x (Appendix A) and plays an important x +x role throughout the paper.
Now we prove that the effective cost of debt r, under full amortization method is always greater than the nominal interest rate R and r,. We,also show that under the bullet payment method, the effective cost of debtr, depends gnly on the compensating balance X and is independent of the loan period n.
PR
Theorem 1.r, >Rand r,= —— P-R)
Proof. Rewriting equation (3) using A - aS ,
1 ) P P-xy{1- =—f ( if any, in 4 6)
1 . (any lies between 0 and 1 for any positive interest rate , so the left hand side of A
(5) is less than P, hence f,(r,) < f,(R).
Because f(x) is a decreasing function, r, is greater than equal to R. The equality holds when X=0.
1 1 (P -X)-(P-X) ea — Ar)" At)"
so equation (2) becomes Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 89
1 1 1 ea yer {i ) @ Ar) Ig (At) PR. PR therefore P - X= — implies Ts — : I, P-X
Note that this theorem proves the uniqueness of r, but not that of r,. It only proves that r, is greater than the original interest rate R. Next theorem is the main result of this paper that the effective cost of debt under full amortization method is also unique and always higher than that under bullet payment method.
To prove this theorem, we introduce a function which is derived from the net present value of an investment project. Consider a full amortization payment method set throughout the paper, then the net present value of the borrower is a function of the loan period $n$, the compensating balance X and the discount
P where A = At) *
When we fix n and X, NPV(n,X,r) becomes a function of r which we denote as NPV _ ,(r). Likewise, when we fix X andr, NPV(n,X,r) becomes a function of n, rate r. Define NPV(n,X,r) = (P-X) - A.f.(r) + and we denote this function as NPV_,(n). In terms of NPV(n,X,r), r, is a zero of the polynomial equation NPV(n,X,r) = (P-X) - Af,@ + =0 . As
(1 +4,)" mentioned earlier in Section II, we assume that the compensating balance is less
P than half of the principal, 0 < X < “7 and this guarantees that NPV, ,(r) is increasing in r (Appendix B), hence the effective cost of debt r, is unique.
Theorem 2.1, <r,. The equality holds when r ‘a=1,=R, that is, when the compensating balance X=0.
Proof. Consider NPV(n,X,r) = (P-X) - Af@+
. This is an increasing (i +4,)" function of r which meets the vertical NPV, ,, -axis at
1 1 1 Xx NPV, ,(0) = (P-X)- A,¢ —— + t.t ) - (7) . 1+0 (1+0) (1+0)¥Y (1+0)
=p-X-An+X=P- nA, and the horizontal r-axis at T, (by the definition of r,).
NPY, x(t)
—X+ x (1+R)" P-nA,
Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98
NPV, ,(r) is increasing in r implies r, > r,, if and only if NPV, ,(r,) < 0. By the definition of r,,
Then
P-X (P-X) - dap - PRf (1) -0
+1) or equivalently, P (P-X) + - PRE) + —— Ar) At) xX P NPV, ,(r,) = (P-X) - Af (ty) + = PRE (r,) + “Af (tn) (1+1,), At),
P PR(1+R)* = (PR-A,)f (ty) + = ( PR- a ft)
-R 1 (i+r,)-1 01 P -R (1+R,)*-1 (he fh (1+R)"-1 1, (1+r,)" (It) 1+4r,)° r, (1+R,)™-1 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 91
(+R)? P (1+R)-1-1 (14R)" P - rr - ) f(t,)+ —— =PR ——_____ f(r) + __ (1+R)- A) (1+R)"1 Ar)
Hence NPV, ,(r,) £ O if it can be shown that
< or equivalently —_< ———— (1) r (14R)'-1 Rr, (1+R)1
Replace r, by Be , then equation (11) becomes
P PR PX ((*pxyA, ) R (4+R)"-1 which is equivalent to PR_,, : 13 (P-X)(1+ En )* - P (P-X) > P(1+R)" - P. (13)
PR Let G(R) = (P-X)(1+ Px: (P-X) - P(1+R)"+P. Then
PR P G(R) = (P-X) n (1+ py yet - Px Pn(1+R)"?
zeae 14 =Pn{(l+ poy -(1+R)"} >0 because (14) _,,PR_«, P - lox =l+ Px R>+R.
Moreover,
G’(R) > 0 for all R > 0 and G(0) > 0 implies G(R) > 0 for all R > 0, hence equation (13) has been proved. Because F(R) > 0 for all R> 0, hence NPV, ,(r,) < 0. NPV, ,(r) is an increasing function and NPV, ,(t,) <0 =NPV_,(r,), therefore r, <r,.
PR... . . . PR isanincreasing function of X andr, is always greater than r,, but these two facts do not guarantee that r, is an increasing
It has been already shown that r,= 92 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 function of X. Figure 6 shows an example which satisfies our result so far, yet r is not an increasing function of X. The following corollary proves that r ',is an increasing function of the compensating balance X, hence such case as Figure 6 never takes place.
Fig. 6: r, is greater than r,, but not anincreasing function of X
Corollary r, is an increasing function of X.
P Proof Recall that (P-X) =A f,(r,) - =. where A,= FRY Because f(x) is decreasing, NPV(n,X,r) = (P-X) - Af,(t,) + = is an increasing function of r A for fixed n and X.
It suffices to show X, < X, implies r, <1, where r, and r, satisfy NPV(n,X,, r,) = NPV(a,X,, r,) = 0.
Xx NPV(n.X.t,) PX) An) ye (16)
and Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 93
NPV(0,X,,1,) = (P-X,) - A,f(r,) + -y, RA
Ar)" an Therefore NPV(n,X,,r,) - NPV(n,X,,,)(P-X,)(P-X,) + roa, ye XX) x Sara ya -X,) (18)
1 1 “(aay - i} XX) -(i yaa
Both arr ay) and (X,-X,) are positive implies NPV(n,X_,r,) > NPV(n.X,,1,)
implies r, > r, because NPV(n,X,r) is an increasing function of r for fixed n and X, proving the corollary.
When a compensating balance is not imposed, the effective costs of debt for the two different payment methods, full amortization method and bullet loan method are the same as the nominal interest rate. When a compensating balance is imposed, their respective effective costs of debt become different. Both effective costs increase as the newly imposed compensating balance increases. This paper provided a mathematical proof that the true cost of a fully amortized loan is always greater than that of a bullet loan. Our findings lead to the following conclusion: whenever a bank imposes a compensating balance, the borrowing firm should prefer a bullet loan to a fully-amortized one to avoid an ambush posed by such a balance.
A. The Present Value Interest Factor for an Ordinary Annuity
1 . D. ; For 0<x<1, fn(x)= + {1- ——. } is decreasing in x andg,(x) = 1 - - is increasing in x. x (1+x) AR df (x) 1 n Proof YA rooi a xz {1 Gay oa } dd Al, x (1txy x (1 (A.1)
= -(1+x)"!+ (n+1) x41 (+x) 94 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98
Let h(x) = -(1+x)"! + (n+1)x+1, (O<x<1), then h’(x) = -(n+1)(1+x)" + (n+1) +0 = (n+1){1-(1+x)n} < 0 for O<x<1 and (A.2)
h(x) is a decreasing function on (0,1) and h(0)=0, hence for 0 <x <1, hence h(x)<0. F(x) h(x) dx
Therefore ——— * 2 +x <
Qand f (x) is a decreasing function.
d; nel ig (X) 04 2+x)
n dx AT = (axl? 0 for all 0 <x <1, hence g(x) is increasing.
x B. The IRR 7, = (P-X) - ADO +H 7
For a fixed n and X, NPV, ,(r) is an increasing function of r.
Proof. Write X = (1+a)A, for -1< a< -1_ . This range of decides the lower bound and upper bound of X so that 0 < x. < =. . Then d “ap NPV, x) =- ALO - xi (B.1) and this implies d -(tr)™! + (n+1)r+1 n ae NPM =A, a TFA, Taye -(1+r)™!- (n+1)r-1 \ sal Sint AH Gep™ a if (B.2) sa (Gen (n+1)r-1- (1+0)nr? \ ran /
Hence NPV, ’(r) > 0 if and only if Ana Get (1to)nr ) - 0, which again (140) implies (1+r)"*! - (n+1)r-1-(1+0)nr? >0 .
n! Using the binomial formula At Papa Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 95 rewrite (145)! = 14(ntL)r + aa r+F(r), then
(14r)! - (nt 1)r-1-(1+0:)nr? = 1+ (n+ Dr + ve r+ F(r)
~(n-+1)r-1-(1-+00)nr f,(R) is a decreasing function (Part A of this Appendix) and lim f(t) = tim G4RE1 Lyi 0+ R)*1-0 rao" ro RUI+R)" +o 1(+R)"+Rn(+R)™ n(1+0)"! a 1(1+0)8+0n0+0)"! = yA (B.4)
Hence (B.3) is greater than Zinta) sen >0 because F(r) is a polynomial in r with positive coefficients. This proves NPV’ ,(r) > 0, therefore NPVn,X(r) is increasing.
C. Open Question and Conjecture
While working on the relation between the compensating balance and effective costs of debt, the authors became interested in another side of the effective costs of debt: when the compensating balance remains the same, are the effective costs of debt affected by the length of the loan period? Since we already know that the effective cost of debt under the bullet loan method depends only on the principal, the nominal interest rate and the compensating balance, our question boils down to the case of a full amortization payment method. From numerical experiments, the authors found the same pattern in effective cost of debt for other principal and loan period under full amortization method, which are summarized in the following conjecture.
Conjecture. There exists a loan period N, for which the effective cost of debt r ', for full amortization method is the maximum. r ', increases as the loan period n increases up to N,, then decreases as n increases past N, and approaches PR/(P-X) as n grows to co,
The authors have not been able to present a complete proof of this conjecture but can provide some facts about the NPV function when the compensating balance is fixed. 96 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98
-(P-X)- -_* aPyy_ Recall that NPV(n,X,r)=(P-X); ALG) Taye where A, R {1 aR
1 1 and f,()= t1- (a9 is an increasing function of r and the effective cost of debt r, is the value of r such that NPV(r,,X,n)=0. For a fixed compensating balance X, NPV(,X,n) is a function ofr and n and we write this as NPV,(r,n). Furthermore, for a fixed parameter p > R, NPV,(,n) is a function of n only. We can extend the domain of NPV,(p,n) to the set of all nonnegative real numbers so that NPV,(p,n) is a continuous function of n. For each n, there will be an r, such that NPV (r,.n) = 0, hence NPV,(p, n) > 0 if p >r, and NPV,(p,n) < 0 if <r,. Then we can prove the following Lemma.
Lemma. For NPV,(p,n), there exists n, (not necessarily an integer) such that . d dn NPV,(p,n) < 0 for n <n, and dp (pp) > 0 forn>n,.
As long as it can be proved that the above n, is independent of the parameter , one can use the following duality of r, and NPV,: it suffices to show that r, = r,(n) is increasing(decreasing, respectively) when NPV,(,n) is decreasing (increasing, respectively). Then r,(n) has its maximum value at n = 4, Where n, is the value decided by the previous lemma such that NPV,(p,n,) < NPV,(p,n) for all positive real number n.
The result is illustrated in Figure 7: the upper figure shows the result of Corollary and the lower picture is a gist of Lemma.
We prove by contradiction. Suppose there is a pair n, < n, < n, such that NPV,(p,n,) > NPV,(p,n,) andr,(n,)>r,(n,). By the previous Lemma, NPV,p(,n) is decreasing regardless of the choice of p for n <n,, so choose p such that r,(n,) > p > r,(n,). Then
NPV, (r,(n,),n,) = 0 > NPV,(p,n,) and NPV, (r,(n,),n,) = 0 > NPV,(p,n,) hence
NPV ,(p,n,) < 0 < NPV,(p,n,) contradicting our assumption.
' A term loan extends 1 year and up to 15 years, with the most common maturities falling between 1 and 5 years. This is the most common form of intermediate bank loan to commercial enterprises. It is appropriate to finance inventory, permanent working capital needs, or plant and equipment. The loan repayment schedule may require monthly, quarterly, semiannual, or annual payment. Johnson ([9]).
? For a three-year loan with $8,000 principal at a stated interest rate of 12.50% per year, the effective cost of debt is 12.50%.
3 Descartes’ Rule of Signs: when a polynomial f(x)=a, the coefficients, the equation can have up to k zeros.
+a, X+...a x" has k sign changes of ol Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98 97
Therefore for any n, <n, <n, and n, <n, <n,, r,(n,) < (n,) and NPV,(p,n,) > NPV,(p,n,) (C.3) and for the same reason r,(n,) <1,(n,) and NPV,(p,n,) > NPV,(p,n,) (C.4)
Then PR 1-(1+r,)" x
NPV, r,(n),n) = (P-X)- Car + Gr) = 0, hence lim NPV, (r,(n),n) = (P-X)- FR +X.0 = Oand this implies - A lim r,- Ba 2 . (C.5) Ty p PR Ken nn nn nn R no | n
NPV | p-py n(i+9)
In(I+R) , : : J
| Go
98 Youngna Choi et al. / The International Journal of Banking and Finance 2 (2004) 83-98