Actuarial Outpost Compound interest and Simple Interest
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#1
10-05-2019, 05:04 AM
 alvarezlr SOA Join Date: Oct 2019 Posts: 1
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?
#2
10-05-2019, 06:19 AM
 Underpaid Member SOA Join Date: Oct 2016 Posts: 891

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)
#3
10-05-2019, 10:24 AM
 Steve Grondin Member SOA AAA Join Date: Nov 2001 Posts: 7,038

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).
#4
10-05-2019, 10:38 AM
 Patience Member SOA AAA Join Date: Sep 2001 Location: a kinder, gentler place Favorite beer: Scotch Posts: 50,022

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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#5
10-06-2019, 09:31 AM
 Vorian Atreides Wiki/Note Contributor CAS Join Date: Apr 2005 Location: As far as 3 cups of sugar will take you Studying for ACAS College: Hard Knocks Favorite beer: Most German dark lagers Posts: 66,910

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:

$\frac{i}{n}\;\times\; t\ <\ i\;\times\; t \text{ for 0 < t < 1 and n > 1}$
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#6
10-06-2019, 10:12 PM
 Caffeine Junkie Member SOA AAA Join Date: Oct 2001 Location: P.O. Box 4947, Austin, TX 78765, USA Posts: 370

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.
#7
10-07-2019, 12:05 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 9,145

Quote:
 Originally Posted by Caffeine Junkie 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).
#8
10-08-2019, 10:38 AM
 manaknight14 Member CAS Join Date: Nov 2012 Location: 0100111001001010 Studying for Exam 9 College: tDEC survivor Favorite beer: Capt'n Eli's root Posts: 1,311

Quote:
 Originally Posted by Patience 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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#9
10-08-2019, 11:57 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 9,145

Quote:
 Originally Posted by manaknight14 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.
#10
10-08-2019, 03:59 PM
 Steve Grondin Member SOA AAA Join Date: Nov 2001 Posts: 7,038

Quote:
 Originally Posted by Academic Actuary 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.

 Tags compound, interest, simple

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