Direct proportions are very easy to comprehend. This is when a change in one quantity produces a proportional variation in the other. For example, if one tomato costs Sh5, then two tomatoes will be Sh10. The total cost is directly proportional to the number of fruits.
Unfortunately, most things in life are not related that way. Consider the time it takes an object to fall to the ground. It is obvious that the higher the dropping height, the greater the amount of time it takes. However, the time taken is not directly proportional to the height. It changes as the square root of the height.
Thus, while an object falling from, say, five metres takes one second to reach the ground, it does not take two seconds to fall from 10m: It takes 1.4 seconds (square-root of two). The challenge with this kind of a relationship is that the ordinary calculator doesn’t have a square-root function. As such, many people work with direct proportions.
I think this is the reason there have been many complaints from bank customers that the law capping the interest rate is not being implemented fairly. They claim that the new instalments demanded by the banks do not “reflect” the changed rates.
The problem arises from the assumption that the monthly payment is directly proportional to the interest rate. Well, it is not! And it is easy to understand why. The instalment has two components: the interest and the principal. Obviously, the latter does NOT depend on the interest rate.
Suppose you take a Sh1 million loan for five years at 20 per cent per annum; the monthly instalment would be Sh26,494. If the bank suddenly became “sufficiently philanthropic” and waived the interest completely, would you expect to pay nothing? Of course not! You would still have to pay Sh16,667 every month for the five years to clear the principal sum.
By the same logic, it is unreasonable to expect that, when the interest rate drops from 20 to 14.5 per cent, the instalment should decrease by the same proportion (from Sh26,494 to Sh19,208). The correct new monthly instalment is Sh23,528 – just Sh2,966 lower.
Still, things get more complicated: the new instalment also depends on how long you’ve had the loan. That is, if you took the Sh1 million a year ago had been paying the Sh26,494 over the last 12 months, the principal balance is Sh870,586 by the time the rates change. Applying 14.5 per cent to this for the remaining four years brings the instalment to Sh24,009. This is almost Sh500 higher than that for a person taking a new five-year loan.
Clearly, the formula for calculating the balances is not a straightforward, direct proportion equation. I have designed an Excel workbook that does the calculations to answer the most common questions that I get from readers. You can download it for free from my website, www.figures.co.ke.
– This article is by Mungai Kihanya. It was first published in the Sunday Nation.