Transcription
In this video, we're going to work on some word problems using the compound interest formula. There are two equations that you need to know: A = P * (1 + r/n)^(n*T).
Now, in this formula, P is basically the principal; that's how much you would deposit in a savings account or a checking account, something like that. A represents the future value of that amount in the account after it's been credited for interest over a period of time. So you can think of P as the present value—how much you put in in the present—and A is like how much it will be worth 10 or 20 years later. R is the annual interest rate. So let's say if the annual interest rate is 8%, you need to plug in 0.08 for R. You need to convert 8% into a decimal. To do that, you would divide by 100. N is how many times you receive interest in a given year. So let's say if it's compounded monthly, that's 12 months in a year, so n would be 12 for monthly. Now, what about for weekly? What is the value of n? There are 52 weeks in a year, so n would be 52. Daily, n is 365. Quarterly, n is 4. Semiannually, n is 2. And annually, n is 1. And T is basically the time in years.
Now, there's one more equation that you need, and it's this one: A = P * e^(RT). A and P are basically the same as the last equation; P is the principal, that's the amount that you put into an account. A is the future value after some time, maybe 10 or 20 years later. E is basically the inverse of the natural log function. So if you have your calculator to find e, you may have to type in shift natural log or second Ln, something like that. R is the annual interest rate as a decimal, and T is the time in years.
So when do we use this equation compared to the other equation? Now, if you hear the keyword compounded continuously, use this equation. If it's compounded times something else—let's say monthly, daily, weekly, quarterly—you would use the other equation. But the only time you would use this equation is if the problem says it's compounded continuously.
So let's work on some problems. Susan puts $20,000 in a savings account paying 8% annual interest compounded monthly. At this rate, how much money will be in the account after 40 years? So it's not compounding continuously; therefore, we need to use this equation. So P is the principal; she puts in $20,000 in the account. R is the annual interest rate, which is 8%; if we divide that by 100, that's 0.08. And she receives that total 8% annual interest in 12 months. So basically, that 8% is divided into 12, so her account is credited with interest every month. So we're going to divide this 0.08 by n, which is 12, and we want to find out how much money will be in the account after 40 years. T is 40. 8/12 + 1 is basically 1.006 repeating. 12 * 40 is 480. So if you type this in the calculator, you should get a value of around $485,467.79. So that's how much money will be in the account after 40 years. So as you can see, it pays to save early.
Here's another problem. You can pause the video and work on it. John wants to have $2 million for retirement in 45 years. He invests in a mutual fund paying an average of 9.5% each year compounded quarterly. How much should he deposit into his mutual fund? So we need to use this equation: A = P(1 + r/n)^(nT). So we know the future value; he wants to have $2 million in his account. So he needs to decide how much he should put in now to get to that level. So we're looking for P in the problem. R is the annual interest rate: 9.5 divided by 100 is 0.095. And it's compounded quarterly, that is four times a year. So four times a year, his account is credited with interest. So we're going to divide it by four, and then it's raised to the nT, or 4 * 45. T is the time in years. So you could type it in exactly the way you see it. Let's find out what this value is equal to. First, 0.095 / 4 is 0.02375, and let's add one to it, so that's 1.02375. And 4 * 45 is 180. So this is going to be P * 68.37652. So to solve for P, we need to divide both sides by this number. So P is $2,000,000 divided by 68.37652, so that's going to be about $29,249.96. So if he invests about $29,000—let's round it to 250—if he invests that much, and if he finds an account paying an annual interest rate of 9.5% compounding quarterly, then in 45 years he should have $2 million in his retirement. So if he starts investing, let's say in his 20s, by his mid or upper 60s he can have that much in savings. So as you can see, due to the effect of compound interest, it pays to save early.
Sarah wishes to turn her $10,000 investment into $100,000 in 20 years. How much interest does she need to receive compounded annually to goal? So in this problem, we need to solve for R. So let's use this equation again. So A is the value in 20 years; she wants $100,000. P is her initial deposit, the principal, which is $10,000. R is the annual interest rate, which we're looking for. And N is one since it's compounded annually, which means that she receives interest once per year. And T is 20. So the first thing we should do in order to solve for R is divide both sides by 10,000. $100,000 / $10,000 is 10. So basically, she wants to multiply her investment by a factor of 10. So now what can we do to solve for R? How can we get rid of this exponent? In order to open the parentheses, we need to turn the 20 into a one. To do that, raise both sides to the reciprocal of 20, or 1/20. 20 * 1/20 is 1. So what we have is 10^(1/20) = 1 + R. So to solve for R, we need to subtract both sides by one. So it's 10^(1/20) - 1. 10^(1/20) is about 1.122. And subtracted by 1, this is equal to 0.122. Now, to turn it into a percentage, multiply by 100%. So R is 12.2%. So if she wants to multiply her investment by a factor of 10, she needs an account that is paying 12.2% annual interest. If she can find that, then in 20 years she's going to multiply her investment by a factor of 10. So if she invests $100,000, in 20 years it's going to be $1 million. If she invests $200,000, in 20 years it's going to be $2 million.
Mary invests $50,000 into an index annuity that's averaging 8.4% per year compounded semiannually. At this rate, how many years will it take for her account to reach $1 million? So let's write the equation: A = P * (1 + r/n)^(nT). So her goal is to reach $1 million; that's the—that's the A value, so to speak. Her investment, the principal, is $50,000. The interest rate is 8.4%, which is 0.084. And it's compounded semiannually, which means she receives interest twice a year, so n is 2. So what we need to do is solve for T. First, let's divide both sides by $50,000. So what's $1,000,000 / $50,000? That's equal to 20. So she wants to multiply her investment by 20. 0.084 / 2 + 1 is 1.042. In order to solve for T, we need to use logarithms. So let's take the log of both sides. So on the left, we're going to have log 20, and on the right, we're going to have log 1.042^(2T). A property of logs allows us to take the exponent and move it to the front. So therefore, what we now have is log 20 is equal to 2T * log 1.042. Now, to get T by itself, let's divide by 2 log 1.042 both sides. So therefore, T is equal to—I'm going to take it one step at a time—log 20 is about 1.30103. Log 1.042 * 2 is 0.0357354. If you divide these two numbers, you should get 36.4 years. If I typed it in correctly—mistakes do happen—but this is how long it's going to take her to multiply her investment by a factor of 20. So in 36.4 years, if she can find an account that is averaging 8.4% per year in interest, she could turn this $50,000 investment to a million.
Juliet invests $100,000 in an account paying 7.2% interest compounded continuously. How much money will be in her account after 30 years? Now, anytime you see this key expression, compounding continuously, this is the equation that you need. So we're looking for the future value of her account 30 years from now. So we're solving for A. We have our principal investment; it's $100,000. And the interest that she's receiving is 7.2%, or 0.072 as a decimal. And her account will be active for 30 years. 0.072 * 30 is basically 2.16. And e, which is the inverse of the natural log function, e raised to 2.16 is about 8.6711376 something times $100,000. So her investment is going to be worth $867,113.77.
Mark wants to have $1.5 million in 50 years. How much should he invest now in an account paying 12% interest compounding continuously? So here is our key expression, which means we need to use this equation again. So we have the future value, the value in 50 years, so that's A. In this problem, we're looking for P. We need to know how much he should deposit into his account in order to reach this goal. R is 12%, or 0.12, and the time is 50 years. So first, let's multiply 0.12 * 50, and that's equal to 6. Now, to get P by itself, let's divide both sides by e^6. So P is going to be $1,500,000 / e^6, and basically, this is equal to $3,718.13, which seems very, very small. But the reason why this small amount turns into this large amount is because of the time. 50 years is a long time. That's one. And two, the interest rate is much higher than the interest of the other problem, which were like 7, 8%. A 12% interest rate compounded continuously will greatly increase his account value over a long period of time. As you can see, a small investment was greatly multiplied over 50 years.
John invests $5,000 in an account paying 11% interest compounded continuously. How long will it take for his investment to turn into $2 million? So let's try this problem. So we have the same formula: A = Pe^(rt). And we have the future value of $2 million, and his deposit of $5,000. R is 0.11. But this problem, we're looking for T. So let's begin by dividing both sides by $5,000. So if you wish to do this in your head, you can get rid of three zeros. So you have 2,000 / 5. 2,000 is basically 20 * 100, and 20 / 5 is 4. 4 * 100 is 400. So if you take $2 million and divide it by $5,000, it will give you 400. So we have 400 = e^(0.11T). Now, instead of using log, we're going to use natural log. The reason being is the natural log of e is equal to 1. So natural log 400 is equal to the natural log e^(0.11t). So whenever you have a variable in the exponent, you can use the log function or the natural log function. But when you're dealing with e, it's easier to deal with or use the natural log function. So what should we do now? Once you get to this part, take the exponent and move it to the front. So we have the natural log of 400, and that's equal to 0.11T * the natural log of e. Now, the natural log of e, as you mentioned, is 1, so that's just going to disappear. So our last step is simply to divide both sides by 0.11. Natural log of 400 is about 5.99146. If we divide that by 0.11, this is going to be 54.47. So that's how long he needs to invest if he wants to have $2 million. Now, $5,000 is a small investment, but if he invests early, that's the key; he can take the advantage of the effect of compound or compound interest. His money will grow to $2 million if he does it in—if he invests early, let's say 54 years early. So as you can see, whenever you invest early and you can use time to help multiply your investment. Now, granted, this effect will be greatly increased if you can find an account, a savings account or a checking account that's paying a very high interest rate. Typically, mutual funds and index annuities are probably the best place where you can get such high interest rates. But that is it for this video. Thanks for watching, and have a great day.