Video: Finding the Minimum Rate of Interest Using Compound Interest

Agatha has $100000 to invest for 5 years. She needs to end up with $120000. 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? If the interest were compounded weekly, what would the minimum interest rate required? (Give answers to 2 dp).

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Video Transcript

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.

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