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

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 5000 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 5000 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 100 percent and then we add three percent. So, each year we’re trying to work out 103 percent of the amount that we had at the beginning of the year. And to do that, we multiply our balance by 103 over 100. Now, 103 over 100 is equal to 1.03. So, our multiplier is 1.03.

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 5000 times 1.03 to the power of 𝑦 for this problem. So, 5000 was our initial investment. 1.03 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 6149 dollars and 37 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 100 percent and we add three percent, which means we’re working out 103 percent of the number. And we said that that meant that the multiplier was 103 over 100, which when we work it out is 1.03.

Now, let 𝑟 be the interest rate, so we’re looking at the general case rather than the specific case of three percent. So, now, we’re starting off with 100 percent and we’re adding 𝑟 percent. So, we’re finding 100 plus 𝑟 percent. So, the multiplier is gonna be 100 plus 𝑟 over 100. And we could split that calculation into two separate fractions. So, 100 plus 𝑟 all over 100 is the same as 100 over 100 plus 𝑟 over 100. And of course, 100 over 100 is just one. So, we can rewrite that as one plus 𝑟 over 100.

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 100 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. Then, I’ll let 𝑛 be the number of compound periods per year. So, for example, if we were doing compounding weekly, there would be 52; so 𝑛 would be 52. If we were doing it monthly, that would be 12 months in a year; so 𝑛 will be 12, 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 100 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 6000 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 12. We’re gonna have 12 bits of compounding going on per year. She’s investing 6000 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 100 all over 𝑛 all to the power of 𝑦 times 𝑛.

So, let’s think about this. 𝑛 was equal to 12, there was- it was compounded monthly. So, that’s 12 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 6000 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 6000 times one plus five over 100 all divided by 12 all to the power of seven times 12. 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 8508 dollars and 22 cents. So, with monthly compounding, we will end up with 8505 dollars and 22 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 6000 dollars. And the multiplier one plus five over 100 is 1.05, 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 8442 dollars and 60 cents, rounding to two decimal places.

So, comparing this amount to the amount that she’d have got for the interest being compounded only once a year, she’s got 65 dollars and 62 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 100000 dollars to invest for five years. She needs to end up with 120000 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 then. We’ve got 100000 dollars to invest. We’re gonna do that for five years. And our target value is 120000 dollars. Now, we’ve gotta work out the interest rate that’s going to achieve that. But in fact, we’ve got to 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 100 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, 120000; 𝑝, the principal sum, 100000; 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 100000, on the left-hand side we’ve got 120000 divided by 100000, which is just 1.2. And on the right-hand side, we had 100000 times one plus 𝑟 over 100 all to the power of five divided by 100000. So, the 100000 cancelled out, just leaving us with one plus 𝑟 over 100 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 100 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 1.2. 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 100. Now, if I subtract one from each side of that equation, on the left-hand side, I’ve got the fifth root of 1.2 take away one. And on the right-hand side, one plus 𝑟 over 100 subtract one is just 𝑟 over 100.

Well, we’re nearly there. I just need to multiply both sides of my equation by 100 now, and I’ve got an expression for the value of 𝑟. So, 𝑟 is equal to 100 times the fifth root of 1.2 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 3.71. 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’ll need to get an interest rate of 3.71 percent or more in order to achieve her savings target of 120000 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 52, so 𝑛 is equal to 52. 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, 120000 dollars, is equal to the initial amount, 100000 dollars, times one plus 𝑟 — which we don’t know, we’re trying to work that out — over 100 divided by the number of periods in a year, it’s 52, all to the power of five years times 52 periods.

Well, again, I’m just gonna divide both sides of my equation by 100000 and gradually unpick this until we’ve got an expression for 𝑟. Well, the left-hand side divided by 100000 is just 1.2 again. And the 100000 would cancel from the right-hand side, which was of course, was the whole point of dividing both sides by 100000. And the power of the bracket there, five times 52 is 260. Now, the right-hand side is to the power of 260. So, I’m gonna take the 260th root 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 260th root of 1.2. And on the right-hand side, the 260th root of something to the power of 260 is just that thing. So, I’m just gonna get one plus 𝑟 over 100 over 52. Now, I can subtract one from each side, which means I’ve got the 260th root of 1.2 minus one now, on the left-hand side. On the right-hand side, I’ve just got my 𝑟 over 100 all over 52. So, now, I’m gonna multiply both sides by 52 to simplify the right-hand side. And we’re nearly there now. I just need to multiply both sides by 100 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 3.65 percent. So, if Agatha’s got an account that compounds interest annually, and she wants to end up with 120000 dollars, she needs to get an interest rate of 3.71 percent. But if she can find an account which compounds the interest weekly, it would only need to have an interest rate of 3.65 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 12. If it’s weekly, 𝑛 will be 52. If it’s daily, it will be 365. 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.

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