You are saving up $12,000 for a luxurious European vacation tw...

Question

You are saving up $12,000 for a luxurious European vacation two years from now. How much money would you need to invest today at Clutch Bank, earning their juicy 10% annual interest, to have enough for your vacation? How much would you need to invest today, if instead you could only earn 6% interest?

Accepted Answer

Alright, so let's see how you guys did with this one. You are saving up $12,000 for a luxurious European vacation 2 years from now. How much money would you need to invest today at Clutch Bank earning their juicy 10% annual interest to have enough for your vacation? Alright, so notice what they're asking us for in this problem. They're asking us for an amount today, right? They're saying, how much would you need to invest today? So what we're looking for today, we're looking for a present value, right? The question is asking us to find a present value. But what information did they give us? They told us $12,000 What is that $12,000 That's an amount of money that we want to have at a future point in time, right? 2 years from now, we wanna have $12,000 so we can go on this vacation. So this $12,000, well, that's the future value, right? We wanna have at some future point in time, $12,000. So how much do we need today? So we wanna have a future value of 12,000 and what other variables did they give us? They gave us 2 years, right? 2 years from now, well, that's going to be our n, right? N, the number of periods that we need, that need to pass between the present value and the future value. And they also gave us the interest rate. They said, there's a 10% annual interest rate. 10%, that's going to be our R, our interest rate R. Okay? So notice what they did. They gave us 3 out of the 4 variables, right? They're always gonna do that. They gave us the future value, they gave us the interest rate and they gave us the number of periods, so we can solve for present value, alright? So remember, in this equate in this problem, what we're looking for is the present value today. So let's go ahead and do real quick a timeline just to visualize this. So we've got right now, year 0, then 1 year from now, and 2 years from now. Alright? So 2 years from now, we wanna have $12,000 and we're earning 10% interest, right? So there's 10% interest every year. So what is that worth today? If we can earn 10% each year, we can bring this back in time 2 years using our formula and it'll tell us what this $12,000 is worth today and that's the amount that we're going to have to put in the bank to earn that interest for 2 years, alright? So let's go ahead and use our formula and that is the present value. Actually, I'll do it down here. Present value equals future value divided by 1 plus R to the end, right? So let's fill that in. We know the future value was 12,000 and our denominator, we've got 1+r, so that's one plus 0.10, right? 10% as a decimal is 0.10. So 1 +0.10 and it's 2 years. So our n is 2. So it's gonna be 12,000 divided by 1.1 squared. So the first thing we want to do is find out what that denominator is. What is 1.1 times 1.1? It's 1.21. So 12,000 divided by 1.21 that's gonna give us our answer here, and that comes out to I'm gonna round it here to $9,917. Right? If we were to invest $9,917 today, well, it would earn interest for 2 years and it would be worth 12,000 after 2 years and we can confirm that, right? Let's go ahead and see what happens with the $9,917 after 1 year. So if we take 9917 times 1.1 and remember, we're rounding here, right? So we're going to earn interest for 1 year, so 9917 and we're going to add the 10% interest, right? So 9917 times 1.1 and that tells us what it's worth after 1 year. 10,00908. And that's just basically using our compounding equation, right? Where now, instead of dividing by 1 plus r to the n, we're multiplying by 1 plus r to the n like we did previously when we were compounding. So now that if we take that 10,908 and we multiply that times 1.1, Well, there we go. We're back up to 12,000. Obviously, we're off by a few pennies here and there because we're rounding. But that's essentially what's happening here, right? We take that $12,000 in the future and we bring it back 2 years and it's worth $9,917 today. And that's exactly what we would have to do. If we deposit $9,917 in Clutch Bank today, after 2 years, we'll have the $12,000 that we need for our vacation. So the answer here is $9,917 is what we need to invest today to have $12,000 in 2 years. Cool? Let's pause and and let's see if you guys got the second one. Now that you've seen it go through one time, the next one we're just changing the interest rate. So let's go ahead and see what happens there.

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