Conclusion

The practical problem of banking can thus be solved by using a rather simple technique of optimization used so often in Physics.
The result is perfect in the sense that no approximation is involved in its derivation.
Also, the result is simple enough to be implemented by any practical banker.
The outcome is thus interesting not only for its mathematical exactitude but also for its easy applicability.

The Final Prescription

If the demand loan component were set at 44 units, the overall burden of interest would have been (44 ´ 10% + 1 ´ 12% ´ 6/12 + 6 ´ 12% ´ 6/12) = 4.82 units.
On the other hand, if the demand loan component were set at 46 units, the overall interest burden would have been (46 ´ 10% + 4 ´ 12% ´ 6/12) = 4.84 units.
Obviously, the overall burden due to interest is minimized if the demand loan is set at 45 units.

The Final Prescription

The readymade prescription for minimizing I now becomes:
(i)                  Select x at such a level that n/12 = a/(a+b) or, if a/(a+b) is not available, the next lower available value.
(ii)                If a certain range of values of x satisfies (i), choose the lowest value of x from the range. Once we set the optimal of x i.e. x0 at 45 units, the interest burden for the borrowing company over the next one year comes to (45 ´ 10% + 5 ´ 12% ´ 6/12) = 4.8 units.

The Implementation

B. If the levels of borrowing be 45 units for 6 months and 50 units for the other 6 months, n/12 = 1 if the demand loan component is set below 45 units; n/12 = ½ if the demand loan component is set at 45 units or above, but below 50 units.
Let a = 10% p.a. and a+b = 12% p.a. Then n/12 = 10/12 as per our formula. But, n/12 can be equal to only 1, ½, or 0.
We shall choose the next lower permissible value for n/12, viz. ½. Further, as n/12 = ½ for all values of the demand loan from 45 units and above but less than 50 units, we shall select the lowest permissible value, viz. 45 units as the desired value of x0.

The Implementation

A. Let the levels of borrowing projected by a company for the next twelve months be 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 and 52 units (the levels of borrowing need not occur in this chronological order). Also, let a = 10% p.a. and b=2% p.a.
In such a situation, n/12 = 10/(10+2), or n=10. In other words, the demand loan component should be set at such a level that the level of borrowing would exceed the demand loan for 10 months. Thus, x0 = 42 units or more but less than 43 units.
The actual solution will be corresponding to the lowest value of x0, i.e. x0 = 42 units

The Solution

In other words, for I to be minimum, x is to be so chosen that for n months Wi>x, where n/12 = a/(a+b)

The Solution

The Solution
Or,  
(5) n/12 = a/(a+b) for = 0

The Solution

The Solution
Or, 12
(4) = a -[(a+b)/12] Σ[ θ(Wi – x) + (Wi – x)δ(Wi – x)] i=1
 
12
= a – [(a+b)/12] n - [(a+b)/12] Σ (Wi – x)δ(Wi – x)
i=1
 
= a – [(a+b)/12] n
where n = number of months for which Wi >x

The Solution

If a company works out its requirement of working capital finance for the next twelve months i.e. Wi for i=1,2, …., 12, then the total interest burden for the company during the next twelve months works out to:
12 12
(3) I(x) = Σ x.(a/12) + Σ (Wi – x). θ(Wi – x). [(a+b)/12].
i=1 i=1
 
12
= a.x +[(a+b)/12] Σ (Wi – x). θ(Wi – x),
i=1

Formulation Of The Problem

Then,
(1) I=∫a.x.dt + ∫[W(t) – x].θ [W(t) – x]. (a+b) dt,
where θ[W(t) – x] is the well-known step function
Or,
12 12
(2) I=Σ (a/12).x + Σ [(a+b)/12].[Wi – x]. θ[Wi – x].
i=1 i=1
where Wi=level of working capital finance for the ‘i’th month

Formulation Of The Problem

Let W(t) = working capital finance required by a borrowing company, expressed as a function of time, over the next one year,
x = level of demand loan or fixed component,
W(t) – x [where W(t) > x] = level of variable component,
I = Interest burden for the company for the next one year
a = Rate of interest for the fixed component
(a + b) = Rate of interest for the variable component

Formulation Of The Problem

Once a customer submits the pattern of the working capital requirement over the next one year, the question arises as to how a practical banker will bifurcate the working capital requirement into a fixed and a variable component to minimise the annual interest burden

Tandon Committee Norms – Style Of Credit

It would, therefore, be in the interest of the borrowing company to ensure an efficient financial planning and to project correct levels for its projected fund requirement
Once the monthly requirement of working capital is submitted by a borrowing company, the lending bank has to work out the optimum level for demand loan that will minimize the annual interest burden

Tandon Committee Norms – Style Of Credit

If a borrower projects the demand loan component at a “higher than necessary” level, he would end up paying interest on amounts not actually required by it.
If he projects the demand loan at a low level, much of its withdrawals will attract a higher rate of interest and the overall interest cost over the year would not be minimized.

Tandon Committee Norms – Style Of Credit

(i)                  A fixed or demand loan component, interest on which is to be charged at a certain fixed rate throughout the year
(ii)                A variable or cash credit component, interest on which is to be charged at a somewhat higher rate. This component would indicate by what amount the level of borrowing for a particular customer exceeds the demand loan component

Tandon Committee Norms – Style Of Credit

The first and the most substantial work in the field of working capital finance in India was done by the Tandon Committee
The Committee aimed at inculcating in the borrowers the habit of making an effective financial planning through a system of reward and penalty
It suggested that the working capital facility should be bifurcated into two components

Working Capital Loan And Cash Credit System

Any arbitrary break-up of working capital facility into fixed and variable components will thus not be in line with the spirit of the policy of the central bank.