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## Finance and capital markets

### Course: Finance and capital markets > Unit 1

Lesson 4: Continuous compound interest and e# Formula for continuously compounding interest

Learn how to calculate interest when interest is compounded continually. We compare the effects of compounding more than annually, building up to interest compounding continually. Created by Sal Khan.

## Want to join the conversation?

- Try as I might, I cannot understand why this formula is correct

P(1 - r/n)^t*n

And this one isn't

P(1-r/n*t)^t*n

What is the significance of the n vs. the t? Why aren't they used as a single variable?(20 votes)- Good answer....but more simply it's because (1+r/n) represents a single period (ex. one MONTH); (1+r/n)^n represents doing it for a full cycle ('n' times , ex. one YEAR) ; (1+r/n)^tn represents doing it for several cycles (ex. Several YEARS)(13 votes)

- At2:27, Sal explains pretty well that the formula would be 50 (1+0.10/4) raised to the power of 12 (or 4 times 3 to be precise). However, if I compute that in my calculator, I get 67.2444 as a result, and not 67.49 as he indicates at the very end...Any idea why?(9 votes)
- Because at2:27Sal is taking n=4 but in the end is taking n=infinity (continuously compounding interest)(14 votes)

- I want to know why the rate is divided by time (r/n)? If somebody could explain how that is derived?(5 votes)
- Using the video's example, the rate is divided by 4 because it's a yearly rate spread over 4 periods within the year, 3 months each period.

The interest is compounding every period, and once it's finished doing that for a year you will have your annual interest, i.e. 10%. In the example you can see this more-or-less works out:

(1 + 0.10/4)^4

In which 0.10 is your 10% rate, and /4 divides it across the 4 three-month periods. It's then raised to the 4th power because it compounds every period.

If you do the above math you'll find (1+0.10/4)^4 = 1.1038, which we could round to 1.10, which ends up at your 10% rate.

So the example's fancy compounding rate every 3 months effectively amounts to the same thing as a 10% rate for a year's loan.

It's only if somebody borrowed for a longer time period that it would make more of a difference. For example, borrowing at this rate for three years would not mean just paying 3 * 10% on your original amount or something like that. In fact in 3 years the interest would've compounded 12 times, since there's 4 periods every year, and in the end you'd actually be paying 34% on your original amount.

Anyway hopefully that gives some idea of where r/n came from in this case.(13 votes)

- I don't understand how "n" just disappeared from the last formula and still the result was approximately the same. I understood it like "t" in the last formula was n*t in the first and that the "t" represents the period in which the interest is coming. Sal said that it was years but in the first case the period is 3 months not 1 year. Sorry if my English is bad i hope you understood my question :)(7 votes)
- You are right, in that the n "disappeared." It disappeared at2:28when the limit was taken as n goes to infinity. At this point, we are now dealing with a different formula than the original: we are not compounding over n=4 periods, but compounding over n=infinitely many periods. If this seems strange, it's because it is. Check out the previous two videos, if you haven't already; they explain the derivation of e. By taking this limit to compound continuously, you then yield a slightly different answer than if you had just plugged the numbers from the formula at2:18into your calculator. This seems like a small difference, and it can be seen as such given the small percentage difference between the two answers. But if one is dealing with much larger principal and much longer time, the difference will then be exaggerated likewise.(6 votes)

- At, 2 minutes it says that the fraction inside the () is 0.10 / n but it is over 3 years so would't it be n * 3 (years)(4 votes)
- No,
`n`

is the number of*compounds per period*, and`r`

is the*interest per period*. And`t`

is the number of*periods*.

So`1 + r/n`

is the*interest per compound*(note that "per period" divided out). And`n * t`

is the total number of compounds.(5 votes)

- Is there a practical use of continuously compounding interest in real life? Banks wouldn't want customers to get that kind of interest. Where do we use this in real life?(3 votes)
- Banks actually do use this for demand deposits. They also use it for many loans which they give out, most notably credit card loans. Of course, loans that have a fixed payment schedule, like mortgages, normally won't compound continuously, but instead every payment period (month normally).(6 votes)

- why continuously compounding interest is useful?(4 votes)
- If you are the lender, it's very useful because you earn more interest!

When interest compounds, you are paying interest on interest. The more frequently that happens, the more you end up paying.(3 votes)

- Will I survive without understanding this?(4 votes)
- Picture in your head a rectangle. Imagine slicing up a rectangle into tiny rectangles. Imagine money flowing out of each of those tiny rectangles. That is your answer.(0 votes)

- What is the definition of 'e'?(2 votes)
- This is the best explanation of "e" I have found to date: https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/(2 votes)

- Do you really use this in your live?(2 votes)

## Video transcript

Let's say that we're
looking to borrow $50. We can say that our principal is $50. We're going to borrow it for 3 years. Our time, let's say T in years is 3. Let's say, we're not going
to just compound per year. We're going to compound 4 times a year, or every 3 months. Let's say that our interest rate ... if we were to only compound once per year, it would be 10%. Since we're going to
compound 4 times a year, we're going to see ... We're going to divide this by 4 to see how much we compound each period. 10% is the same thing as 0.10. Let's write an expression. I encourage you actually
to pause this video and try to write an expression for the amount that you
would have to pay back if you were to do this. If you were to borrow $50 over 3 years, compounding 4 times a year, each period you would be compounding 10% divided 4%. How much would you have
to pay back in 3 years? Let's write it out. $50, that's your principal. You're going to multiply that, so you could compound it. Each time, each period, each of these 3 x 4 periods. You have 3 years, each of them divide into 4 sections, so you're going to have 12 periods. Each of them you're going to
compound by 1 plus this R. I'll write that as a decimal. 0.10 divided by the number of times you're compounding per year to the ... Well, you would be raising
it to the nth power, if this was only over a year. There's 4 periods and you would raise it to the 4th power if it was only a year, but this is 3 years. You're going to be doing this 3 x 4. You're going to have 4 periods, 3 times. Let me write this. It's going to be 4 ... Actually, instead of N right over here let me write the 4, so you
can see all the numbers. You're going to do this 4
x 3, to the 4 x 3 power. I encourage you, if you want, you could pause the video and you can use your calculator to actually calculate what that is. The whole point of this is
just to use real numbers to see why this actually makes sense. This is your principal. Each time you're going
to be multiplying that times 1.025. You're going to be growing it by 2 1/2% and you're going to do this 12 times, because there's 12 periods. 4 periods per year times 3 years. This is going to be how
much you have to pay back. If we wanted to write this in a little bit more abstract terms, we could write this as P(1 +). I'll do this a close parentheses,
since it's the same color. R over N to the N x T power. You could pick your P,
your Ts, your Ns and your R and you could put it here and that's essentially how much you're going to have to pay back. An interesting thing, and you saw that we had this up here from a previous video, where we took a limit as
N approaches infinity. Let's do the same thing here. Let's think about what that would mean. If we took the limit as
N approaches infinity, if we took the limit of this
as N approaches infinity, what is this conceptually? We're dividing our year into more and more and more chunks, an infinite number of chunks. You could really say, "This would be the case where we're doing continuous compound interest. Which is a fascinating concept to me. You're dividing your time period in an infinite number of chunks and then compounding just an infinitely small extra amount every one of those periods. You can actually come up
with an expression for that. As we see, that this actually doesn't just go unbounded and
give us crazy things, that we can actually use this to come up with a formula for continuously compounding interest. Which is used heavily
in finance and banking and, as you can imagine,
a bunch of things, actually many things outside
of finance and banking, exponential growth, etc., etc. Let's see if we can
actually try to evaluate this thing right over here. The one thing I am going to do to simplify this, is to do a substitution. I'm going to define a variable. The whole goal is so that
I can get it into a form that looks something like this. I'm going to define a variable X. I'm going to say that X is
the reciprocal of R over N, so that I can get a 1
over X right over here. I'll write that as N over R. X is equal to N over R, or we could write this as N is equal to X x R. If we make that substitution the limit is N approaches infinite. If we make the limit as
X approaches infinite, then N is going to go to infinite as well. If N goes to infinite, then X is going to go to infinite as well. R, right over here, is just a constant. We're just assuming that that's a given, that N is what we're
really seeing what happens as we change it. We could rewrite this
thing right over here. I'm doing it. I'm not being as super rigorous, but it's really to give you an intuition for where the formula we're
about to see comes from. Let's rewrite this as the
limit is X approaches infinite. The limit of constant
times some expression. We could take the constant out. We could say that's going to be P times the limit as X
approaches infinite of 1 plus. R over N is 1 over X. 1+1 over X to the ... N is X x R. N is X x R, so let me write that, to the X x R, R x T power. All of this business is
the exact same thing. Let me rewrite this. Let me copy and paste
this part right over here. Copy. This is the same thing. This is equal to P times (let me put some parenthesis here) times (maybe that's too
big) times the limit. This limit right over here. If I raise something to
the product of these, I'm taking X x R x T, that's the same thing as doing this whole thing to the X and then raising that to the RT power. This comes from exponent properties, that you might have learned before. These 2 things are equivalent. I'm doing a couple of
steps in the process here, but hopefully this seems
reasonably intuitive for you. I'm really just using the property. The limit as, let's say,
X approaches C of F of X to the, let's call it, to the XRT power. This is the same thing as the limit as X approaches C of F of X to the X and then all of that
raised to the RT power. What is this stuff right over here? What is all of this business
that's inside the parentheses? We've seen that before. All of this, all of that is equal to E. We can write this. This is exciting. This is formula for continuous
compounding interest. If we continuously compound, we're going to have to pay
back our principal times E, to the RT power. Let's do a concrete example here. If you were to borrow $50,
over 3 years, 10% interest, but you're not compounding
just 4 times a year, you're going to compound
an infinite times per year. You're going to be continuous compounding. We can see how much you would
actually have to pay back. It is going to be 50 x E to the ... Our rate is .1. Just let me put some parentheses here. 0.1 x time, so times 3 years. T as in years. We assumed it was in years. We get ... You would have to pay back $67. If we're to round ...
$67.49 if you were to round.