Expert: Josh - 12/10/2010

Question
QUESTION: Hello:

I read about this method in an old business mathematics book and was curious about how it works to produce the correct answer.

"Percent-day method is based on 360 days‚ time and 1% interest. Any interest
rate multiplied by any number of days is equal to so many %-days."
Example: 82 days at 5% equals 410%-days.

"The rest of the calculation to find interest using his method involves multiplying the amount of interest at 1% by the ratio of %-days to 360."

Here is an example calculation: An investor invested $720 for 92 days at 7%.
What is his interest?

Solution:
$720 for 92 days at 7% = 7% X 92 days = 644%-days.
644/360 X $7.20(1% for 360 days) = $12.88

Can you explain the logic of this calculation? I can understand that days multiplied by percent equals %-days, but I do not understand how $720 becomes
$7.20 and why is (1% for 360 days) used?

I am mostly confused by the following:

Why does the method involve multiplying the amount of interest at 1% by the ratio of %-days to 360?

I personally would not use this calculation, but I am simply curious to learn more about it.

I thank you for your reply.



ANSWER: I agree with you. I personally would not use this calculation too. The interest may be calculated in a variety of ways, and this is probably not the most intuitive way.

In general, the interest is given by the formula:
I = P x r, ......[1]
where P and r represent the "principal" and "effective interest", respectively.

If the interest rate is given as R% per year, and the principal is invested for only a fraction of a year [let's say D days], then the effective interest is given by r = R * (D/365) /100......[2]
The division by 100 is necessary to convert a percentage into a decimal representation. After all, 1 corresponds to 100%, 0.01 corresponds to 1%...

From [1], the interest (I) is given by the product of P and r. That is, I = P x R *(D/365)/100 using [2]. This may be rearranged as I = ((P/100)/365) x (R*D).

So, in the final analysis, we can pre-compute the second term R*D and call this [percentage.day]. In your example, this amounts to R*D = 7x92 = 644 [percentage-day]

Remember that we still need to compute the first term ((P/100)/365). This is where (720/100)/365 = 7.2/365 comes from. Finally, I = 7.2/365 x 644 [dollar].

For convenience, your textbook considered that there are 360 days in a year. So, all the "365"s are replaced with "360" in the previous paragraphs.

---------- FOLLOW-UP ----------

QUESTION: Hello:

I am still confused. Perhaps you can reword some of the information.

Why is (1% for 360 days) used in the calculation below?
644/360 X $7.20(1% for 360 days) = $12.88

"The rest of the calculation to find interest using his method involves multiplying the amount of interest at 1% by the ratio of %-days to 360."

Does the ratio of  %-days to 360 represent 644%/360? I do not understand '...multiplying the amount of interest at 1%.."

If possible. try to use basic arithmetic so that I may better understand.

I thank you for your follow-up reply.  

Answer
Kenneth,

If you attach specific words to numbers, it will confuse you more than anything. There is no reason for associating "1%" with "360 days".

The bottom line is that all the operations involved in the conventional formula for interest, I = P x (R/100) x (D/360) are replicated by the so called "Percent-day interest method". It is just a matter of rearranging the terms and grouping them differently.

First of all, you do understand that I = P x r [Interest (I) equals principal (P) times the effective interest rate (r)]

All I have done, is simply take into account that the money is invested for only a fraction of a year. So, if R is given as 12% p.a., and the money is invested for 73 days, the effective interest rate is 12 x (73/365) = 2.4%. Now, 2.4% is divided by 100 because it needs to be expressed in decimal, viz., 2.4/100 = 0.024. In general, r = (R/100) x (D/365). This is what you will eventually come to accept after some thoughts. So, the overall interest calculation is

I = P x (R/100) x (D/365)  ...[#1]

where I=interest amount, P=principal, D is the period of investment in days, R is the interest rate given as a percentage per annum.

I reiterate once again, that the "percent-day" method is nothing new. All it does, is rearranging the terms in the standard equation [#1].

They reorder the terms and group them as:

I = (R x D)/365 x (P/100) ...[#2]

[Please, do not mix words into the equation. An equation is all numbers.]

Because multiplication is commutative, we can perform multiplication in any order we like. It becomes confusing only when you give particular meaning and attach units with numbers, for instance, with (P/100) there is no meaningful relationship between days and percentage per se, it must be viewed in the wider context of the equation. The standard formula and the "percent-day" approach are EQUIVALENT, but the former is more intuitive as the latter group together incongruent terms in a non-intuitive way.

Using P=720 [dollar], R=7 [percent], D=92 [day], we get
(P x R) = 7 x 92 = 644. So, (R x D)/365 would explain where the term 644/365 came from. The remaining component in the interest equation [#2] is (P/100), it arises out of necessity to reconcile with the standard formula. This is how (720/100)=7.2 was born.

This totally explains all terms in 644/360 X $7.20(1% for 360 days) = $12.88, if you would leave out (1% for 360 days) which is not part of the equation. It ("1% for 360 days") actually makes no sense.

Don't read too much into it. Don't try to look for reason why P=720 was multiplied by 0.01 (or 1 percent as you put it). This cannot be looked at in isolation. To understand how this comes about, we must look at how the terms are reordered and regrouped in the standard formula, as I have shown above in [#1] and [#2].