The complex mathematics of loans
The Sunday Nation
Nairobi,
07 October 2012
If mathematics was easy and straightforward, Charles Nyamai would be a very happy man. I say that because of the question he sent in: “Kindly advise me if there is a constant conversion factor whereby you can change a fixed interest rate to reducing balance or vice versa?”
The straightforward answer is that there is no fixed conversion factor. And the reason is not that mathematicians like to complicate things; no! It is that life itself is complicated and mathematics is an attempt to map out the complex relationships between different parameters.
Suppose you borrowed Sh100,000 for five years at 10 per cent interest per annum calculated on the reducing balance method. You would have to pay Sh2,125 every month in order to clear.
What if the lender used the “fixed interest” method? Well,10 per cent of Sh100,000 is Sh10,000 per year. In five years it comes to Sh50,000. This is the simple interest payable. Adding this to the borrowed amount, we get Sh150,000.
The duration of the loan is five years, or 60 months, thus the monthly instalment is Sh150,000 divided by 60. That is, Sh2,500. Obviously, this is a higher figure than the Sh2,125 for the reducing balance method.
The conversion factor that Charles is looking for in this case is Sh2,500 divided by Sh2,125. It comes to 1.176. But the question he asking is whether this factor remains constant when the starting figures are changed.
To find out, let’s first change the loan duration from five to ten years. In that case, the reducing balance repayment comes down to Sh1,321 while that of the fixed interest method also reduces to Sh1,667. The conversion factor now is 1,667 divided by 1,321, which comes to 1.262. This is significantly different from the 1.176 of the five-year loan.
Now let’s change interest rate from 10 to 15 per cent but keep the term of the loan at five years. The monthly instalment for the reducing balance method now comes to Sh1,613 while that for the fixed interest method is Sh2,917. The conversion factor this time is 1.808. This is again significantly different from both the first and second figures.
Clearly each time we change either the term of the loan or the interest rate, the conversion factor changes. The only time it remains unchanged is when the amount borrowed is changed – all other parameters being constant.
The reason for this complexity is that while the formula for calculating instalments under the fixed rate method is straightforward, the one for the reducing balance is not. In the latter, the interest rate is raised to a power equal to the number of months.
But despite the apparent erratic changes in the conversion factor, it is quite easy to evaluate the formula for determining it for any interest rate and any loan period. By that I mean that the process of getting the relationship is simple; however, the result is not a straightforward equation.
Raise your hand if you want the formula…
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