Expert: Josh - 2/11/2004

Question
I want to thank you for your reply.

I do not understand the reason for using r=R/4% instead of r=R/4. The 4 represnts quarters per year not percents.  It seems confusing me to determine the quarterly rate with r=5%/4%.
Perhaps you can clarify the use of r=R/4% instead of r=R/4 or r=5%/4.

Once again, I thank you for your answer and assistance!
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Followup To
Question -
Hello:

I want to thank you for your reply.

In your answer you indicated the following: "The interest rate is usually quoted at R% per annum. However, if compounded quarterly, then, r=R/4% is the applicable rate."

1. Should r=R/4% be r=R/4, without the % sign? Doesn't the 4 represent four quarters as in rate per year divided by 4 quarters per year = 0.0125 or 1.25% per quarter?

2. A rate of 20% per year divided by 4 quarters per year would equal a quarterly rate of 5%. This amount would be $121.55 on $100.00? The same amount as 5% on $100.00 for four years. (True or False)

I thank you for your answers and assistance.

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Followup To
Question -
Hello:

Are the amounts of interest on $100.00 at 5% compounded quarterly for one year or compounded annually, that is once a year for four years, the
same amounts?

Example:

$100.00 at 5%

$105.000000 amount for first quarter or amount for year 1
$110.250000 amount for second quarter or amount for year 2
$115.762000 amount for third quarter or amount for year 3
$121.550625 amount for fourth quarter or amount for year 4

Answer -
No, you cannot compare the two, since the time horizons are different.

The problem amounts to knowing what the effective compound interest rate is, for each interval.

The interest rate is usually quoted at R% per annum. However, if compounded quarterly, then, r=R/4% is the applicable rate.

To make a fair comparison, suppose that we put $1000 in the bank for four years.

Scenario 1: 5% p.a., compounded quarterly, duration 16 quarters.
You get P(1+r)^16 = 1000*(1+0.0125)^16 = $121.98

Scenario 2: 5% p.a., compounded annually, duration 4 yrs.
You get P(1+R)^4 = 1000*(1+0.05)^4 = $121.55

Recall that we never use percentage in calculations. Percentages are expressed in decimal form (divide by 100).
Answer -
1. Since R/4= 1.25 percent, the value of r is 0.0125.
So, r=R/4 % is right in the context of the sentence.

2. (Sure, if anyone is going to pay you that much interest:)

Answer
Hi Anonymous,

I think you are analyzing things too much and it is holding you back.

In small matters such as this, mingling numbers, units and abstract symbols only add to the confusion. I honestly don't think that this is the way to go. You are making things more complicated than they need to be.

"%" is just a short-hand substitution for the word "percentage" in the sentence.
It plays no role and has no formal association with any of the entities in the fraction.

Strictly speaking, r=R/400 should be used. This is the correct formula. But it matters very little.
I inserted "r=R/4" as an object in the sentence to carry the same notion, just to sketch the idea. The percentage describes this entity, it doesn't go with 4.

When we calculate compound interest, r should be 0.0125.

But it makes sense to translate 0.0125 into natural language, so r=R/4=1.25 % is used.