[MUSIC] Question number 1, this asks you to define an annuity due, which is a series of equal amounts and equal time periods that occurs at the beginning of each period. Now I'd like you to visualize this on a timeline and use arbitrary numbers for the following three periods. And I'd like you to note that for each amount, we need to locate that amount at the beginning of each period, which makes an ordinary annuity into an annuity due. So this is what it would look like. We would have a timeline, in this case let's say it is a three period annuity. So we mark the three periods, 1, 2, and 3. And for the amount, notice this is the beginning of a period, the end of the period. So if it's an annuity due, the amounts are commencing at the beginning of the period. And then for the second one, beginning of the period. And then for the third one, beginning of the period. Now for question 2, let's understand each statement. The first one asks whether we can calculate the present value of a perpetuity. In fact, we can, if we use the following simple, but very powerful formula. So the formula for a perpetuity is that you can calculate the present value of perpetuity by simply identifying the annuity. Those are the equal amounts in equal time intervals that occur forever, and divide that by the interest rate. So if we just put some numbers in here, we can say that if an annuity of let's say $1 divided by an interest rate of 10%, well that will give us a value of the perpetuity to be $10. Note how this formula gives us a very intuitive understanding of the present value, V, as it relates to the interest rate, right? So V increases if r decreases, or V decreases if r increases. There is an inverse relationship between value and interest rates. Let's look at Statement B. This one is easy to calculate because we can use the same formula. And here we're given the present value and the amount so we can look for the interest rate. Statement C cannot be calculated because it asks for the future value of a perpetuity. That's impossible because a perpetuity never ends so we cannot calculate its future value. Just remember that on a timeline, if you want to visualize this once again as we saw before, a perpetuity is an annuity that simply does not end. This one would just keep on going indefinitely. We cannot calculate the future value of this because it does not end. For statements D and E, we can easily compute these using the formulas we learned in the time value lectures. For statement D, we calculated the present value of annuity due by multiplying the annuity with the present value annuity factor. That is, we take the annuity, A, and multiply by the annuity factor. And that annuity factor can be computed by this formula, 1- 1 over 1 + r raised to the power t, the whole thing divided by r. This here is the present value annuity factor for r% and a given time period. Now, how do we adjust this for an annuity due? Very easily, we simply take this annuity factor, and we multiply this by 1 + r, to reflect the extra period of compounding, okay? So let's put this in an example so this drives home. If we take a three period annuity, [COUGH] 0, 1, 2, and 3, and remember we are now looking for an adjustment for annuity due, so that means the amounts are at the start of each period. Start of period 1, start of period 2, start of period 3, and then apply this to our formula. So if we want the present value of the series, we take the annuity which in this case is $10. We're going to take that $10 here, and then we're going to multiply it by the annuity factor. Let's assume an interest rate of 10%. So we have 1- 1 over 1 + interest rate, this case raised to the power of 3 because it's a three-period annuity. Divide the whole thing by the interest rate, 10%. So this our annuity factor, and then, because it's an annuity due, the only adjustment we make is multiply it by 1 + the interest rate. And that will take care of the annuity due. In statement E we can calculate the future value of an ordinary annuity by multiplying the annuity with the future value annuity factor. Okay, so what am I talking about here? Well, we simply take the in this case the annuity, and we multiply it by the future value annuity factor. What is that? The future value annuity factor can be computed using this formula, 1 + r raised to the power t- 1 over r. So this is also known as the future value annuity factor, r% t time periods. And if we want to visualize this, once again, we can put it on a timeline. [COUGH] Let's just take the same example of a three period annuity. This time it's an ordinary annuity so it occurs at the end of each period. So if the amounts are still $10 per period, 10 at the end of year 1, 10 at the end of year 2, 10 at the end of year 3. What we're trying to do this time, unlike the previous problem we want the future value, not the present value. Last time we did the present value, this time we're doing the future value. How do we compute the future value? Well, we have the formula in front of us. So if we plug the numbers in, again assuming an interest rate of 10%, let's plug it in, the annuity is $10. So $10, here we have it. Multiply this by the annuity factor, 1 + the interest rate raised to the power 3. I's a three period problem, -1 over r, which in this case is 10%, sorry about that. [COUGH] So there you have it, the annuity factor multiplied by the annuity and that would give us the correct answer. Okay, for question 3, why don't we put all this information we have on a timeline so we can clearly visualize what exactly we're trying to do? Okay, so this is a four period problem. So we draw a four period timeline, so this is 0, 1, 2, 3, and 4. And we've been given a present value amount, which is $5,979, right. [COUGH] We've been given future values, 1,000 for year one. We don't know the value for year two, this is what we're tying to compute. And we know the values for year three and four are 2,000 each. So, what we're trying to do is calculate this missing value in the series which has a present value. Present value of this, plus this, plus this, plus the fourth amount, are equal to 5,979. So how do we depict that in terms of a formula? Very easy, because we know the present value of this series, each one of them. So we can take 5,979. This equals to 1,000, back for one year, so we bring it back by dividing the future amount by 1 plus the interest rate, which is going to be 12%. So this is for one year. Then we do it again for the second year we don't know this amount call it x 1 plus the interest rate for 2 years. Same for the third amount which is 2,000. Bring it back for 3 years and then finally the fourth amount bring it back for 4 years. So we have an equation with one unknown. We solve for it. If you actually want the numbers, this works out to about 893, approximately. This is going to be x/1.2544. This is if you work this out. And this one works out to about 1,424. This one about 1,271. By the way if you're wondering, in most examples we do go to two decimal places, I'm just trying to round numbers off. If you work this out what you will get is the value of X is equal to about $3,000. For question number four let's visualize once again the information on a timeline. And in this case you're making 36 equal payments in the future whose present value is $1,000. Okay, so lets do that. So we've got a timeline that goes from time zero until 36 payments in the future. So lets just depict that, say we've got three periods that continue to go on until the future. Which ends with the 36th payment and so we're looking for those amounts for each of those payments right until the 36th period. And the present value of all of this of course given to us which is $5,000. So we'll just put that amount here, time zero, we know it's 5,000. So, how do we find those particular payments? Well, this looks like an ordinary annuity, we want the present value of that annuity. So we're going to equate this present value with the formula for the annuity and we know that all too well by now. So we take the annuity which is what we're looking for, in this case it's x. And we're going to multiply this by the annuity factor which is 1 minus 1 over 1 plus r raised to the power t the whole thing over r. Remember this is the present value annuity factor for r % t periods. Right, all we have to do now is plug the numbers in and if we do that what do we get. Well 5,000 equals to x. We're still looking for that, multiply that by the factor which is 1 minus 1 over 1 point. In this example we have to be careful and use the correct monthly rate. We know the annual rate. The annual rate has been given to us, which is in fact 9%. So, the annual rate is 9%. We want to convert this to a monthly rate and the frequency of compounding of course is 12, 12 months in a year. So that we adjust r by m and that would be 9% divided by 12, which gives us the monthly rate of 0.0075. That's the rate we want to plug in here. So 1- 1 / 1 + the rate 0.007, missed a zero there, 0.0075 raised to the power 36. And the whole thing of course divided by R which is .0075, right? We solve for this, we know this equals to $5,000 and your payments are going to work out to be $158.99 or about 159. Question number five. Again, we can visualize the information on a timeline. You're making equal biweekly deposits for 35 years in the future. And this time, you're given a future value which is $1 million. All right, but if this is bi-weekly lets start with something we know which is that there are 52 weeks in the year. And since you make deposits every two weeks you will make 26 deposits in a year for a total of 26 deposits times 35 years which is 910 deposits, right? So let's put this on a timeline. We've got from zero, right, until 910 deposits, right? And if you're wondering how I did the math quickly for 910, remember that we have 52 weeks in a year. And we're biweekly deposits that means every two weeks. So if we divide that by two we get 26 deposits, and 26 deposits is occurring over 35 years. And that gives the 910 periods. So now that we have figured the number of periods out, we also need to use the appropriate interest rate. Now we know that the interest rate given in the problem is equal to 9%, we need to a convert 9% annual rate into a biweekly rate. So, if this is annual, we simply, again, use M, which in this example is 26 times in a year. And so we'll just divide 9% by 26. And we have our biweekly rate of 0.0046. So we're looking for these particular deposits for each of the next 910 periods. And these deposits must accumulate to an amount which is $1 million. That's what we want these deposits to add up to. So we're going into the future. We must use a future value formula and if we use the future value formula, remember that this is an annuity and ordinary annuity. So, in terms of formulas we're talking about computing a future value that we already have. Future value equals to the annuity multiplied by the annuity factor 1 + r raised to the power t minus 1 over r. That's our future value, annuity factor at r% for t periods. All we have to do now is plug the numbers in, and this becomes a pretty easy problem. So we know the future value It's a million, let's put that in. And equate that with our annuity, which is what we're looking for. Multiplied by the appropriate factor, which in this case is 1, plus the interest rate we just computed, .0046. Remember the time periods, we also computed on 910 and subtract 1 and divided by the interest rate. And so now we have an equation that we can solve. And of course, if you do that, you're going to see your deposits in fact work out to $70 $70.96, or about $71. If you deposit $71 every two weeks for the next 35 years, you will be a millionaire. It's as simple as that. Question number six. So here we have the Classic Credit Card Problem we have a debt of 1250 we know we have the debt today so that is the present value. And then we have 12 payments that are made in to the future and so we can put this information on a timeline very easily. So we've got a 12 period problem. Here we go. 0, 1, 2, 3, all the way up to 12 periods. We know each of those payments that we're making is $110. 110, 110, 110 and so on right up to here. 110. Now, we know that the present value of all of these payments is $1250. That's the present value that's given to us, and these are the future Payments that we're making. What we're looking for, of course, in this problem is the interest rate. What is the interest rate implicit in this particular transaction? So what we could do is set up the equation for, again, present value for an annuity. This is an ordinary annuity, equal amounts. It's equal time intervals. And solve for the interest rate. So, let's remind ourselves again of that formula. We know that the present value is equal to the annuity times one minus one over one plus r raised to the power t over r. Hopefully this formula is getting familiar to you. And simply plug the numbers in from our timeline to the equation and then go for the solution. So let's plug the numbers in. We know the present value is 1250, and we know. The annuity is 110. We multiply this by 1 minus 1 over 1 plus R, which is what we're looking for. We have 12 periods in the problem, divide that by R. Of course, mathematically this is not easy to solve but with a financial calculator, this is a piece of cake. If you plug the numbers in the calculator and that whole procedure is shown to you using the tutorial. What you will get is a monthly interest rate. Remember this is a monthly interest rate. It works out to 0.00. We must convert this monthly interest rate into an annual rate, so how do we do that? That's very easy, you'll recall we need another formula for that. The formula for an effective annual rate is equal to 1 plus the nominal rate divided by m is to the power m -1. Right? Plug the numbers in. We have the monthly nominal rate of .0081, so this will be 1.0081 raised to the power twelve minus one, and this will work out to 10.18 percent. That is the interest rate you're in fact paying on this particular transaction.