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Video: Compound Interest (with Interest Compounded More Than Once a Year)

Tim Burnham

We will learn how to solve compound interest problems in which the annual rate of interest is compounded more than once a year, so we will use the formula V = [initial amount] * [1 + (r/100)/n]^(y*n), where n is the number of years and r the annual rate of interest.

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Video Transcript

Before watching this video, you should already be familiar with exponential growth and problems in which interest is compounded just once a year. We’re gonna look at situations where interest is compounded multiple times in a year.

First, remember that we can calculate the value of an investment, which is compounded just once a year, by applying our knowledge of exponential growth. For example, I invest five thousand dollars in an investment account which gives three percent interest, compounded annually. If I leave the investment for seven years, what will be its value?

So I’m starting off with five thousand dollars. I’m adding three percent interest each year and I’m doing that for seven years. Now to add three percent to a number, we start with a hundred percent and then we add three percent. So each year we’re trying to work out a hundred and three percent of the amount that we had at the beginning of the year. And to do that, we multiply our balance by a hundred and three over a hundred. Now a hundred and three over a hundred is equal to one point o three. So our multiplier is one point o three. Now let’s just define a couple of variables. If we let 𝑣 be the value of the investment and 𝑦 be the number of years that we’ve invested it for, then, in general, 𝑣 is equal to five thousand times one point o three to the power of 𝑦 for this problem. So five thousand was our initial investment, one point o three is the multiplier which is gonna add the three percent every year, and 𝑦 is the number of years. And 𝑣 is the value of the investment. Now we’re looking for the particular case where we invest it for seven years. So 𝑦 is gonna be equal to seven. And popping that into our calculator and then rounding the answer to two decimal places because it’s money, gives us an answer of six thousand one hundred and forty-nine dollars and thirty-seven cents.

Now let’s just take a slightly closer look at how we calculated that multiplier when we were adding three percent. So the process of adding three percent, we start off with a hundred percent and we add three percent, which means we’re working out a hundred and three percent of the number. And we said that that meant that the multiplier was a hundred and three over a hundred, which when we work it out is one point o three. Now let 𝑟 be the interest rate, so looking at the general case rather than the specific case of three percent. So now we’re starting over the hundred percent and we’re adding 𝑟 percent. So we’re finding a hundred plus 𝑟 percent. So the multiplier is gonna be a hundred plus 𝑟 over a hundred. Now we could split that calculation into two separate fractions. So a hundred plus 𝑟 all over a hundred is the same as a hundred over a hundred plus 𝑟 over a hundred. And of course, a hundred over a hundred is just one. So we can rewrite that as one plus 𝑟 over a hundred. So now we’ve got a general formula for the multiplier, if we’ve got an annual interest rate of 𝑟 percent and we’re doing compound interest once a year.

So now we’ve got a general formula: the value of an investment is equal to the initial amount invested times one plus the interest rate over a hundred all to the power of 𝑦.

Well, that’s fabulous. But in the world of finance, companies often offer investments which are compounded more frequently than once a year, for example quarterly. So that’s every three months, or four times a year, or monthly, or even daily. In fact, some even offer interest rates which are compounded continually, so infinitely many times a year. Well, we’ll talk about that one in another video. But for now, let’s look at situations where you’re given an annual rate of interest but it’s compounded 𝑛 times a year.

Now let’s define what letters we’re gonna use to represent different things. So we’re gonna have 𝑝, which is the principal amount invested. So that’s our initial investment. Some people call it the principal sum, or the principal amount invested. 𝑟 is gonna be the annual rate of interest that we’re giving and that’s gonna be presented as a percentage. We’ll let 𝑛 be the number of compound periods per year. So for example if we were doing compounding weekly, that would be fifty-two; so 𝑛 would be fifty-two. If we were doing it monthly, that would be twelve months in a year; so 𝑛 will be twelve and so on. 𝑦 is gonna be the number of years that we’re investing that money for. And we’re gonna let 𝑣 be the value of the investment.

Now we’ve got a general formula for the value of our investment which is compounded 𝑛 times per year. It’s 𝑣 is equal to 𝑝 times one plus 𝑟 over a hundred all over 𝑛 all to the power of 𝑦 times 𝑛. Now let’s see that in action.

An investment account offers an annual rate of interest of five percent, compounded monthly. Amera invests six thousand dollars in this account. How much will she have in the account after seven years? Compare this to the amount she would have if the interest were only compounded once a year.

Okay, so let’s pick out the important information. The annual interest rate is five percent and its compounded monthly, so that means that 𝑛 is twelve. We’re gonna have twelve bits of compounding going on per year. She’s investing six thousand dollars in the account and she’s gonna have it for seven years. Well the formula for the final value is 𝑣 is equal to 𝑝 times one plus 𝑟 over hundred all over 𝑛 all to the power of 𝑦 times 𝑛. So let’s think about this. 𝑛 was equal to twelve, there was- it was compounded monthly, so that’s twelve times a year. 𝑟 is five; the rate of interest is five percent per year. 𝑦 equals seven because we’re talking about seven years. And the principal, the initial, amount invested was six thousand dollars. Well, replacing all those letters in the formula with the numbers that we’ve just put there, we’ve got 𝑣, the final value, is equal to six thousand times one plus five over hundred all divided by twelve all to the power of seven times twelve.

So popping all that into the calculator and then rounding our answer to two decimal places, because we’re talking about money, that gives us eight thousand five hundred and eight dollars and twenty-two cents. So with monthly compounding, we will end up with eight thousand five hundred and eight dollars and twenty-two cents in the account. Now we’ve got to compare that with annual compounding.

So the formula for annual compounding — we still got our initial principal sum of six thousand dollars and the multiplier one plus five over a hundred is one point o five, so we’re adding five percent every year. And we’re just doing that at the end of the year. And we’re doing that seven times here, so the power there is seven. If we put that into our calculator, we only get eight thousand four hundred and forty-two dollars and sixty cents, rounding to two decimal places.

So comparing this amount, the amount that she’d have got for the interest being compounded only once a year, she’s got sixty-five dollars and sixty-two cents more than if the interest were only compounded once a year. So that’s the difference between those two amounts. So it’s worth pointing out then, that looking at how often these things are compounded, can make quite a big difference to the amount of money that you get back on your savings. So it’s definitely worth checking out.

Okay then, let’s look at one final quite tough question. Agatha has got one hundred thousand dollars to invest for five years. She needs to end up with a hundred and twenty thousand dollars. If the account she invests in only compounds interest annually, what is the minimum rate of interest she would need in order to achieve her savings target? Also, if the interest rate were compounded weekly, instead of annually, what would be the minimum interest rate required then? Give your answers to two decimal places.

Right. Let’s pick out some relevant information there. We’ve got a hundred thousand dollars to invest. We’re gonna do that for five years. And our target value is a hundred and twenty thousand dollars. Now we’ve gotta work out the interest rate that’s going to achieve that. But in fact, we’ve gotta do that twice. Firstly, we’ve gotta do it once when we compound the interest just once a year. And then secondly, we’re gonna do it again when the interest is compounded weekly. So let’s call these parts 𝑎 and 𝑏; 𝑎 is when we’re doing the interest annually and 𝑏 is when we’re compounding it weekly.

So the formula we’re gonna use in the first case is, the final value is equal to the principal sum that we invest times the multiplier which, remember, is the interest rate divided by a hundred. And then we add one to that. And that’s to the power of 𝑦, the number of years that we’re investing for. Now we know the value of 𝑣, and 𝑝, and 𝑦, and we’re hoping to work out what the value of 𝑟 is. So we’re substituting 𝑣, the final amount, a hundred and twenty thousand. 𝑝, the principal sum, a hundred thousand. And 𝑦, the number of years, five. And then rearrange that to work out the value of 𝑟.

So this is the equation with those numbers put in there. And now if we divide both sides of the equation by a hundred thousand, on the left-hand side we’ve got a hundred and twenty thousand divided by a hundred thousand, which is just one point two. And on the right-hand side, we had a hundred thousand times one plus 𝑟 over a hundred all to the power of five divided by a hundred thousand. So the hundred thousands cancelled out, just leaving us with one plus 𝑟 over a hundred to the power of five. Now if I take the fifth root of both sides, then I’m going to be able to just have one plus 𝑟 over a hundred on the right-hand side, which will enable me to move forward with rearranging this and making 𝑟 the subject. So that’s what I’m gonna do. So on the left-hand side of the equation, I’ve got the fifth root of one point two. I’m not gonna evaluate that just yet. I’ll save that to the end and put it on my calculator. And on the right-hand side, I’ve got the fifth root of something to the power of five. So that’s just gonna be that something. That would just be one plus 𝑟 over a hundred. Now if I subtract one from each side of that equation, on the left-hand side I’ve got the fifth root of one point two take away one, and on the right-hand side, one plus 𝑟 over a hundred subtract one is just 𝑟 over a hundred. Well we’re nearly there. I just need to multiply both sides of my equation by a hundred now, and I’ve got an expression for the value of 𝑟. So 𝑟 is equal to a hundred times the fifth root of one point two minus one.

And when I put that into my calculator and round it to two decimal places, like it said to in the question, I get 𝑟 is equal to three point seven one. Now remember 𝑟 was the rate of interest in percent. So the answer to part 𝑎 then is, if she’s got an account which compounds interest annually, she will need to get an interest rate of three point seven one percent or more in order to achieve her savings target of a hundred and twenty thousand dollars.

Now we need to do our calculation again but with a different formula, with compounding weekly. So the number of compoundings that happen per year is fifty-two, so 𝑛 is equal to fifty-two. And we’ve still got the same initial investment, we’ve got the same target value for the end of this thing, and we’ve got the same number of years, five. So putting those values into the equation, we’ve got the target value a hundred and twenty thousand dollars is equal to the initial amount a hundred thousand dollars times one plus 𝑟, which we don’t know which one will work that out, over a hundred divided by the number of periods in a year, is fifty-two, all to the power of five years times fifty-two periods.

Well again, I’m just gonna divide both sides of my equation by a hundred thousand and gradually unpick this until we’ve got an expression for 𝑟. Well the left-hand side divided by a hundred thousand is just one point two again. And the hundred thousand would cancel from the right-hand side, which was of course, was the whole point of dividing both sides by a hundred thousand. And the power of the bracket there, five times fifty-two is two hundred and sixty. Now the right-hand side is to the power of two hundred and sixty. So I’m gonna take the two hundred and sixty as route of each side of the equation. Again, I’m not gonna do that on my calculator just yet. I’m gonna leave it in that-that format and then I’ll do the calculation at the very end. So on the left-hand side, I’m gonna get the two hundred and sixty as root of one point two and on the right-hand side, the two hundred and sixty as root of something to the power of two hundred and sixty, is just that thing. So I’m just gonna get one plus 𝑟 over a hundred over fifty-two. Now I can subtract one from each side, which means I’ve got the two hundred and sixty as root of one point two minus one now, on the left-hand side. On the right-hand side, I’ve just got my 𝑟 over a hundred all over fifty-two. So now I’m gonna multiply both sides by fifty-two to simplify the right-hand side. And we’re nearly there now. I just need to multiply both sides by a hundred in order to just give me an expression for 𝑟. What we’ve gotta do now is type that little line into my calculator and round it to two decimal places, which gives me three point six five percent.

So if Agatha’s got an account that compounds interest annually and she wants to end up with a hundred and twenty thousand dollars, she needs to get an interest rate of three point seven one percent. But if she can find an account which compounds the interest weekly, it would only need to have an interest rate of three point six five percent. So that’s a bit less.

So the advice here is, just looking at the headline interest rate isn’t necessarily enough to tell you how that account works. You need to know are they compounding weekly, daily, hourly, monthly, or annually. It will make a difference to how much interest you get on your money.

So in summary then, we’ve been using this formula to work out the value of an investment when interest is compounded more than once a year. 𝑛 is the number of times per year that the interest is compounded. So if it’s compounded monthly, 𝑛 will be twelve. If it’s weekly, 𝑛 will be fifty-two. If it’s daily, it would be three hundred and sixty-five. And we also saw how sometimes you have to rearrange this equation to find the missing unknown. And in our last example we had to work out the annual rate of interest, and that required quite a lot of rearranging in order to work out the value of 𝑟.