LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A sum today is not the same claim as the same sum on a later date, because the earlier sum can earn interest in the meantime. With a 4 percent annual rate and one compounding per year, 100 grows to 104 in one year. Compounding twice in that year grows it to 104.04. Letting the number of compoundings grow without bound does not send the result to infinity. The lecture calls that finite limit the continuously compounded case and leaves the proof for later.
Why it matters
Put today at one end of a line and a future date at the other. If a riskless rate of 4 percent a year is available, 100 in hand can be the 104 one year out. Offered a choice between 100 now and 100 in a year, the earlier 100 is the better claim under that rate, because it can become 104. Offered 102 or 103 in a year against 100 now, the earlier 100 still wins, because 102 and 103 are both less than 104. The comparison is not “more dollars on the page.” It is dollars after the interest the calendar allows.
The lecture treats the 4 percent as an assumed riskless rate for the arithmetic. It does not estimate a rate from a market, and it does not say every borrower actually receives 4 percent.
The idea in plain words
Write P for the starting principal, r for the annual interest rate, m for the number of compoundings in a year, and t for the horizon in years. Multiply P by a growth factor, and raise that factor to m times t. The factor itself is one, added to the rate after the rate has been split into m pieces. The exponent counts every compounding across the horizon.
Set P at 100, r at 0.04, m at 1, and t at 1. You get 100 times 1.04, which is 104. That matches the one-year story. Leave those inputs and change only m to 2. The lecture evaluates the new amount as 104.04. Interest is credited halfway through the year, and the second credit applies to a principal that already includes the first credit. That second credit is interest earned on interest inside a single year.
Hold r and t fixed and increase m. The ending amount rises. Twice a year beat once a year in the example, 104.04 against 104. The lecture states the pattern in general, not only for these inputs: more compoundings, higher ending amount, if the rate and the horizon stay put.
The interesting bound is what happens as m goes to infinity inside one horizon. The expression becomes a limit of that same principal, growth factor, and exponent as m grows without end. An unbounded number of compoundings sounds like an unbounded amount of money. The lecture says the limit is finite. The name it gives that finite result is the continuously compounded rate of interest. It does not expand the limit into a closed expression in this session. It marks the proof as the next piece of work.
How to use it
Use the formula when the contract tells you P, r, m, and t. Convert a percent to a decimal before you divide by m. Check the m = 1 case by hand, because it must equal P times (1 plus r) to the power t. Then increase m and confirm the amount moved up, not down. For the lecture’s inputs, the checks are 104 and 104.04.
What usually goes wrong
Comparing 100 now with 103 later as if the 103 were automatically larger ignores the 104 the earlier money can reach at 4 percent. The calendar is part of the price. Dropping it makes a worse claim look better.
Questions people ask
Why is 100 today worth more than 103 in a year in this example?
Because 100 invested at the assumed 4 percent becomes 104 in one year with annual compounding, and 104 is larger than 103. The same 100 also beats 102 and beats 100 received later.
What does compounding twice a year change?
With P = 100, r = 0.04, and t = 1, two compoundings produce 104.04 instead of 104. The extra 0.04 is interest earned on the interest credited at the first half-year.
Does infinitely frequent compounding explode?
No. The lecture says the limit as the compounding count goes to infinity is finite, and it names that limit the continuously compounded case. The proof is not in this session.