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  #1  
Old 10-05-2019, 05:04 AM
alvarezlr alvarezlr is offline
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Default Compound interest and Simple Interest

Hello!

I know this might be a really silly question, but I can't find the answer even thought i've been done my searching.

So, we're all familiar with the following graph that shows both simple and compound interest curves.



I want to focus now on the period that goes from 0 to 1 of the X-axis . As It can be seen, simple interest accumulation is larger than compound interest accumulation for values of t between 0 and 1.

Now, I understand the mathematics behind this. I understand, mathematically speaking, why simple interst is higher than compound interest.

What i'm trying to find is the finantial reason of why this happens. Could it explained in words? How would you explain it to someone who is not familiar with Derivatives, nor exponentials or so?
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  #2  
Old 10-05-2019, 06:19 AM
Underpaid Underpaid is offline
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By describing the rate as r per n time units, (in your example n=1) this feels very artificial, because there is nothing special about 1.

For them to have the same return at time 1, and compound interest to build on itself, it must accumulate more slowly at first, otherwise, it would pass the simple interest before 1.

It feels like an intermediate value theorem question, where you could define a function f(t) = p_simple(t) - p_compound(t)
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Old 10-05-2019, 10:24 AM
Steve Grondin Steve Grondin is offline
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There is nothing magical about simple vs compound. They are just a way of defining intermediate (between periods) values. Why compound is lower than simple between 0 and 1 for the same rate? The way I think about it is compound means interest is earned on interest, while interest earns nothing (intraperiod) in simple. That means the growth in compound is increasing over the period, while the growth in simple remains the same. So if the value is growing faster and will be at the same level at 1, it needs to be lower before 1. (Okay, that's a little like calculus, but more wordy).
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Old 10-05-2019, 10:38 AM
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Isn't it as simple as if the Annual Effective Rates are the same after 1 year - the Nominal Compound Rate has to be lower than the Nominal Simple Rate to get to the same place at t =1.

If interest accumulates monthly and I = 6%. Compound is approximately .487% monthly and Simple .500%

of course after that the Compound works off a larger capital.

And if the nominal rates were the same, you would not see the same effect
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Old 10-06-2019, 09:31 AM
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What is the period for compounding?

Recall that under simple interest, you're effectively getting the interest at the end of the investment period (and it's calculated based on the starting value).

Under compound interest, you get your interest at a predetermined time interval; and subsequent interest calculation is based on this new principal amount (hence the term "compound interest").

And the more frequently the compounding occurs before time 1, the smaller that initial interest paid will be:

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Old 10-06-2019, 10:12 PM
Caffeine Junkie Caffeine Junkie is offline
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For the same reason that if you race a bicycle against a Boeing 747, the bicycle will be ahead of the jet for a very short distance.

The example works if you assume the bicycle instantly achieves its top speed (i.e. zero acceleration thereafter), and assume the jet has constant acceleration during the race. A graph of distance travelled vs time will look similar to the above graph. At some point early in the race they will be exactly tied - before that point the bicycle is ahead, and after that point the airliner is ahead.

Not a perfect example, but it works in my mind.
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Old 10-07-2019, 12:05 AM
Academic Actuary Academic Actuary is offline
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Quote:
Originally Posted by Caffeine Junkie View Post
For the same reason that if you race a bicycle against a Boeing 747, the bicycle will be ahead of the jet for a very short distance.

The example works if you assume the bicycle instantly achieves its top speed (i.e. zero acceleration thereafter), and assume the jet has constant acceleration during the race. A graph of distance travelled vs time will look similar to the above graph. At some point early in the race they will be exactly tied - before that point the bicycle is ahead, and after that point the airliner is ahead.

Not a perfect example, but it works in my mind.
Very good example if you specify that the bicycle and plane will be tied at a specified time (like a year in the interest case).
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Old 10-08-2019, 10:38 AM
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Quote:
Originally Posted by Patience View Post
Isn't it as simple as if the Annual Effective Rates are the same after 1 year - the Nominal Compound Rate has to be lower than the Nominal Simple Rate to get to the same place at t =1.

If interest accumulates monthly and I = 6%. Compound is approximately .487% monthly and Simple .500%

of course after that the Compound works off a larger capital.

And if the nominal rates were the same, you would not see the same effect
This. The graph in the OP assumes it's the effective rates that are the same. If the nominal rates were the same, compound interest would always outpace the simple interest.
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Old 10-08-2019, 11:57 AM
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Quote:
Originally Posted by manaknight14 View Post
This. The graph in the OP assumes it's the effective rates that are the same. If the nominal rates were the same, compound interest would always outpace the simple interest.
As a technical point, it would require more than one compounding per year. If you had equal nominal rates and one compounding every two years, the simple interest would be ahead for two years.
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Old 10-08-2019, 03:59 PM
Steve Grondin Steve Grondin is offline
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Quote:
Originally Posted by Academic Actuary View Post
As a technical point, it would require more than one compounding per year. If you had equal nominal rates and one compounding every two years, the simple interest would be ahead for two years.
Yes. The OP's graph implies continuous compounding. Although it is a bad graph (likely hand drawn) because the minimum value is not actually at 0.
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