[MUSIC] Final quiz, question number 1. The rates of return are expressed in annual terms. That is, percent means per annum, unless the problem states otherwise. The formula to calculate the rate of return is also known as the return on investment. And that's calculating the price change divided by the initial investment. Return on investment. In this problem, the price changes by $1. From $10 it goes up to $11, which is divided by the original investment of $10, and that gives us the 10% rate of return. So the formula for rate of return or return on investment, we can then summarize, is the change in price, price 1 minus price in time 0, over the price in time 0. In our problem, what was the data? Well, we invested $10 and that $10 became $11. So we see the price change is 1/10 gives us the answer of 10%. Now, you think the problem ended here, the solution ended here, but in fact, this happened for half a year. So it took 6 months for this to occur. But because we're expressing this as a rate annually, and there are, as we know, two six month time intervals for one year. If this represents 10%, we must multiply this by two to get the annual rate of 20%, which is the correct answer, number E. For question number 2, the statement in this question is followed by several choices. So this is best answered by visualizing the amounts on a timeline and the statement is asking us to consider future cash flows, and decide how their present value can either increase or decrease, based on one of the four statements that is provided. So let's look at each of those statements, visualize this scenario, draw a timeline, and we can come up with arbitrary numbers to represent future cash flows, to see what will have to their present values. Let's start with the first statement, which states, if the discount rate decreases. So, what is the impact of a change in the discount rate, for a future amount in terms of its present value? Why don't we visualize that? We can take our trusted timeline. In this case, we can say, what if we had a one period problem? Here is the future amount, say $10, and if the discount rate decreases, what would that do to its present value? Well, what we can do at this point, is just recall our present value formula. Present value is equal to future value, divided by one plus r, raised to the power t. Right? So, we know, the future amount of $10. We can choose an interest rate if we like, and see what would happen if we use our formula. So let's assume the interest rate is, in fact, equal to 10%. Now, if we look at this problem the present value, what would happen if we just apply the numbers in here. We have a future amount of 10, the interest rate is 10%, so 1 plus the interest rate, and the time period is 1. And if we do this math, of course, it works out $9.10. Now what would happen if the interest rate became lower. If the interest rate became, let's say 5%. Let's just change this number to 5%. So we would have a denominator, now, of 1.05 raised to the power 1. Clearly, when the interest rate goes down, the present value should go up. And in fact, if you do the math for this problem, 10 over 1.05 works out to $9.50. So you can see the increase compared to the previous problem here, interest rate goes down, present value goes up. Now, let's look at the second statement, which reads ,the amounts occur closer to times zero. Again, we can visualize that on our timeline. So we have a timeline here, and an amount in the future. What if it occurred closer to today? Well, if we simply take this amount and say, instead of receiving it one year from now, we receive it right now. Well clearly, the present value of $10 today is $10, compared to this $10 one year from now, which we knew worked out to $9.10. So, if we have amounts that are closer to today, the present value, of course, goes up. Therefore, any future amounts that are going to be closer to today, the present value will increase, and that's why this statement is correct as well. Question number three asks if you invest $1,000 at 4% compounded annually how much interest was earned in year five? So, in order to answer this question, let's visualize the information on a timeline, as we usually do, and here is the timeline. So we have a five period problem. One, two, three, four, and five. And if interest, you know if we just take a step back and say to ourselves, interest is just not being compounded, this would be such an easy problem, because we're simply taking $1,000 today, and the interest rate is given to be, I believe it's 4%. So 4% of 1000 is going to $40. And that's the 40 you earn each year for the next five years. And we would have our answer. But this is not the case because interest is being compounded each year. So in order to find out what interest is being earned in the last period here, what we could do, is identify what's the value of that $1000 in year five? So if we compound $1000 for five years, what is that value? Future value in year five, and let's compound that $1000, in year four. What's the value in the future at the end of the fourth year? Of course the difference between these two values will give us the amount of interest that's earned in that year. So let's go ahead and do that. We know, for example, the formula for future value, what is that formula? Future value equals present value into 1 + r raised to the power t. In this case, we just have to plug the numbers in and we can get the results very quickly. For example, future value for year five is going to be the present value, 1000, multiplied by one plus the interest rate. For year five. [COUGH] If you did that calculation, this would work out to $1,216.65. Let's do that again now for year four. At the end of year four, plug in numbers for the same equation, 4%, but this time it's for year four. Do the math and it works out to $1,169.86. Take the difference, we have $46.79 and voila, we have the answer, which is answer number B. Question number four. So we're still working with the mechanics of time value. Hang in there, you're going to get really good at this by the time we're through. What are we doing in this problem? We're depositing $50 each week for the next 45 years. Okay, why would you be doing that? Well, of course you do that if you wanted to put aside some money for your retirement, and putting aside $50 is not a bad idea. So this problem going to tell us if we did that, what kind of money would be available after we, let's say, stop working in 45 years, what would we have accumulated? Well, for this problem to be visualized once again, we draw a timeline. And in this timeline let's make sure we identify the correct number of periods, okay? The problem did state that you're going to be [COUGH] making deposits, Every week, okay. It's every week, okay. >> Let's pick it up with the problem did state. >> The problem did state that the frequency of compounding is every week. Now we noted earlier that the frequency of compounding can be depicted by the letter m. So in this example, m=52 because there are 52 weeks in a year. We know that the time period is for the next 45 years. So if we have 45 years and each year has 52 weeks. Then our total is going to be 45 times 52, and that is going to give us the total number of periods which is 2,340. That's the number of periods on the timeline. So what we have to do here is go from 0 to, well, I'm not going to write out 2,340, but you get the picture. This goes right up to 2,340. That's how many deposits we're making. Well, how much money are we putting in? We already said we're saving $50 every week. $50 every week goes on, and we keep depositing this. And of course, what we want to know is what happens in the 45th year? What's our bank balance looking like in the 45th year? What's the future value right here, in the 45th year, or after 2,340 periods? Well, all we need now is a formula and an interest rate, and off we go. The formula is going to be using the interest rate of 10% that has been given to us. Here the interest rate is 10%, but remember that it's per year. What we need to do is convert that interest rate for every week. And the way to do that is you always take the interest rate and divide it by the frequency of compounding. In this example, it's 10% divided by 52, and that gives us a weekly rate of 0.0019. Now the formula. What is the formula that converts a series of numbers, equal numbers in equal amounts in the future? Well, that is the future value of an annuity. The future value of an annuity simply takes the annuity and multiplies it by the future value annuity factor. Which in this case is (1+r) raised to the power t minus 1 over r. There you have it, there is the annuity factor. We can plug that in very quickly with our information in this problem. We know the annuity is the series of numbers $50. And we can multiply it by 1 plus the interest rate, which is right here, for each week, 0.0019, raise this to the power of the total number of periods, 2,340, subtract 1 and divide this thing by this interest rate 0.0019, and then [COUGH] see what the result is. And behold the result! Ladies and gentlemen, believe it or not, you have saved $2,304,353. Not so bad if you decide to put aside $50 each week for the rest of your working time period if that coincides with 45 years. Good news at the end. All right, so in question number 5 we've got two options, and we want to see which one of these two options is going to have a higher present value. So in both cases, why don't we set up the timeline. In the first case, we're getting this cash prize of $150 annually forever, right, so let's do that. We've got a timeline that goes from 0 annually of $150 forever, indefinitely, we're going to keep getting this 150 forever, right. And we've been given a interest rate, and in this example the interest rate is 6%, but it's compounded semi-annually, which means m=2. Now if we're going to use an annual rate compounded twice a year, we have to revert back to our effective annual rate formula and convert this annual percentage rate into an effective annual rate. You'll recall we can do that conversion EAR, effective annual rate, is equal to 1 plus the nominal rate, in this case 6%, divided by m, which is 2, raised to the power m, again 2, minus 1. And if we work this out it works out to 6.9%. Sorry, 6.09%, important difference. So now I have a perpetuity, in fact, that I need to value back in today's dollars and I have my interest rate and the formula for the perpetuity is the most beautiful formula ever. It is so simple, so elegant, and there you see it, it's the annuity divided by the discount rate. So what is the annuity in this example? We know the annuity is 150 and the discount rate is 6.09%. We do the math and we come up with $2,463.05. That's the result for alternative number one. We want to compare this alternative With the alternative number two which is to receive $600 every six months for the next 25 years. So every six months we're going to be getting $100 for 25 years. You can see in this second variation, what we have is a time period of 25 years and says, each here has two six months intervals. Again, M is equal to two so in total 25 years times two gives us 50 periods. So when we draw this timeline once again, we want to go from 0, 1, 2, 3 all the way to 50 periods. And what are we receiving in each of those 50 periods? Well we're receiving $100 in each of those 50 periods. So, unlike the first part, where we have a perpetuity, this time we have a finite time period, and this is known as an annuity. So, we pull out from our tool kit the annuity formula. So, I'm going to find some space on this board and write that formula out for you. The present value for an annuity is the annuity multiplied by [1-(1/(1+r)^t))/r], right? Working with this formula, we have the annuity, we have the time period of 25 years which is 50 periods. All we need is an interest rate, the interest rate in this problem is 6% compounded semi annually. So if it's 6% per year, every six month it's going to be half of that. Remember, whenever we deal with a frequency of semiannual, in this case compounding, if the interest rate is 6%, then we must divide that by m, which in this case is 2 and that gives us the 3%. So why don't we plug this information into our formula here and off we go to the races. So what will it be? Well the annuity we see is 100 multiply that by 1-1 over 1.03. Remember we've got 50 periods, that's what we plug in here divided by the interest rate 0.03. Work this out and you get $2,572.97, there we go. So what would you rather have $2,572 or $2,463? The answer is clear. You know what choice to make. Alright, we've got some space to complete the quiz with question number 6. Now here you've got a fairly common situation for lots of people who need money for what ever reason and they go to somebody that will front them some money. In this case you're borrowing a $100 and you promise to pay back a $105 one month later. Now you might say that's quiet reasonable, what's $5 for something I needed right now and i could use for the whole month. We'll look at the mechanics behind that and what we've learned so far, specially with this particular formula for the effective annual rate of interest, which I am going to rewrite for you here. Effective annual rate is equal to (1+ r/m)^m-1. So what happens here? Well, we borrowed, $100 today. So a time period of zero, we've borrowed 100. We have 100 in our pocket and after one month, we're going to pay back minus 100, right? What am I doing. Okay so let's put this information on a timeline. Today, which is time zero, we have borrowed $100 and that's what we get in our pocket today. But after one month, we're going to pay back, $105. And we can see that represents 5 over 100 which is 5%. You might be quite satisfied with this, but that's just a month. Now if I was to tell you, that clearly, there are 12 months in a year and what you're really paying is 60% per year. You might say, that's a very high interest rate to pay, 60%, nobody charges that. In fact, it is illegal to charge an interest rate depending on your jurisdiction or you're not allowed to have these exorbitant rates. And let's not forget our formula here, what the formula is going to show us is that we're paying even more than 60%, right? So, this 60%, actually it's even higher than 60%, and that's because of our formula here. Our formula is suggesting that we're, in fact, compounding this 5% each month for the next 12 months. So, what is the annual rate? Well, if we simply take, this is representing our 5 percent. One plus the 5%, raised to the power of 12, minus one actually works out to 79.59%, almost 80%. That's what you're paying for this one month $5 interest loan.