[MUSIC] The final segment of Time Value will help you to understand a very special type of annuity that in fact never ends, and it's commonly known as a perpetuity. This means equal amounts made in equal time intervals forever. There are two common examples of perpetuities in reality that include securities. One is a special type of bond, another is a special kind of stock. The special type of bond is known as a console. That was a name given to a government issued bond that was first distributed in 1751 by the Bank of England. The bank promised to pay a fixed rate of 3.52% each year forever. So while the original coupon amount, which is the rate multiplied by the denomination of the bond, has changed over time, the government has bought back some of the remaining consoles and completely redeemed them in 2015. The question is, how does one go about determining the market value of this particular security? The same question can be asked about the second type of perpetuity, commonly known as a preferred stock, which you will see in the balance sheet of many, many companies. A preferred stock is issued by companies to investors who can expect a fixed dividend each period, forever. For example, the National Bank of Greece preferred series A has a fixed dividend of 2.25 euros, which based on the current share price of 10 euros yields a pretty handsome rate of return, that's 22.5%. Now before you run off to buy what many regard as a super risky investment, let's look at how perpetuities are valued. The formula on discounting a fixed series of numbers that continue on forever,is deceptively simple. The formula actually can be written out like this. The value of a perpetuity is simply equal to the annuity divided by the discount rate. Notice the golden relationship here, in fact, the inverse relationship between interest rates and value. Interest rates go up, values goes down, interest rates go down, values go up. So for our example, first the console and the preferred stock that we talked about, let's draw some timelines to visualize the information that I presented. For the console, we're going to assume a $1,000 denomination for each of those bonds that the Bank of England issued. So on a timeline it might look something like this. So we have time periods here. We can continue these, remember this is a perpetuity where we have equal time periods that go on indefinitely, forever. So the denomination of the console is as we mentioned 1,000, and the interest rate we said that the Bank of England was paying is 3.52%, which multiplied by 1,000 gives us the 35.2 pounds that each of these bonds was paying. So this is the annuity, 35.2, 35.2, 35.2, and so on and so forth, forever and ever, that the bondholder would receive from the British government if they were to purchase the bond. So what is the value of this bond? Well, we can use our deceptively simple formula here. The value of the bond is going to be equal to the annuity which we've defined as 35.2 pounds. And the interest rate is the key question. That of course is going to vary, it varies every day sometimes every hour. Let's assume the interest rate is 5%. If it's 5%, the value is going to work out to 704 pounds, okay? Now you can see as I was mentioning this fantastic relationship that we can observe right away. If the interest rates decrease, if the interest rates decrease down to let's say dramatically to 3%. If the interest rates go down to 3%, the value will jump up a higher percentage, in fact it works out to 1,733 pounds. Let's use the same formula now for the preferred stock example. In that particular example, again we had some information that we can depict on a timeline that goes on forever so we have time period 0, 1, 2, 3, and so on and so forth. And for each period the Bank of Greece was going to pay 2.25 euros, [COUGH] 2.25, 2.25 for each period forever. Given this annuity of 2.25 euros each year forever that the Bank of Greece is promising, what is the value of it today? Well since this is a real example, we can observe the value of this preferred share series is in fact euros 10. It's about 10 euros. And it's paying 2.25 per year. So you can see what the implied interest rate would be. The implied interest rate, of course, works out to 22.5%. Now that's a huge rate to earn. Well that's because this rate has to do with remember there are three aspects to the rate. There is a real aspect, there is an inflation aspect, and then there is a risk aspect. This is something we explore later on when we do interest rates. Finally, there is a special perpetuity that a lot of finance people work with. This is known as a growing perpetuity, which simply means adding an assumption that the series of numbers will grow at a constant rate. Let's do one example. Suppose we were to calculate the value of receiving $1 each year forever at 20% interest rate. Now you know how to do this problem because it is a familiar one. This is the definition of an ordinary perpetuity. So we have $1 each year that goes on forever. We're given an interest rate of 20%. We want to know the value today. And the value today, we can compute easily by simply taking the annuity, in this case, $1, the perpetuity excuse me, and dividing it by the interest rate, which gives us the value of 5 today. What if however, that this $1 would grow each year? So we have a growth assumption of 10%. How do we build the growth assumption into this series? Well, that's what a growing perpetuity is. Now, if I was to do this just for a few years, you could see that this $1, if it was growing each year at 10%, that series of numbers would look something like this. The $1 would become 10% more, so it would be $1.10. And that $1.10 would again grow for another year at 10% would become a $1.21. And that 1.21 would grow for another 10% to $1.33, and this would continue to happen forever and ever. How do I find the present values of this series that is growing at 10%? That is the problem. The answer however is deceptively simple because we can use the formula, the value of a growing perpetuity is equal to the annuity multiplied by 1 + this constant growth rate, divided by r- g. So let's plug the numbers into our example here. We have [COUGH] the annuity of a dollar, it's growing at 1 plus the interest rate. The interest rate remember is 20%- the growth rate of 10%. And this works out to $11, which of course is worth more than the $5 that we saw earlier on. That's because this series of numbers was not growing, and this series of numbers is growing. As the series of time values comes to a close, I'd like you to focus on the toolbox of formulas that we have been filling up with several concepts behind these to really take stock and see the big picture that we don't really have that many formulas. In fact, we have six. And I'd like to summarize those formulas for you so that they can become a handy reference to use again and again as this specialization progresses. So let's begin with the first one. The first formula that we started off with was future value formula. Future value equals present value into 1 + r raised to the power t. Of course, a lot of the calculations that we do in finance have to do with today or the present value, so we just work with the same one. And simply bring future value on this side and divide that by 1+r raised to the power of t. So far so good, we really worked with one formula so far. From this we extracted something very critical in this formula that has to do with compounding and discounting, and that was the role of the interest rate. We needed to convert these annual percentage rates and express them as effective annual rates. So we had this formula, 1 + the annual, this is the annual percentage rate, which is the advertised rate. We divide that by the frequency of compounding within a year, raise it to the power of M and subtract 1. So we can express annual rates with effective annual rates. Then we looked at the whole notion of annuities and perpetuities. And again we worked with future values and present value. If you wanted to compute a series of numbers that were equal in equal time intervals in the future, you would use the future value annuity formula. In the future value annuity formula, we're simply taking the annuity, the equal amount and equal time intervals, multiplying it by the future value annuity factor, which is 1 + r raised to the power t- 1 over r. Again, we can reverse that and ask ourselves, well if we have a series of numbers that we expect, we could be a business. We could be an individual, we could be any entity expecting a series of numbers in the future. What is it worth today? And that could be representing a whole number of decisions that we are going to [COUGH] apply as we move forward in this course. So the present value of an annuity is again the series of equal amounts in equal time intervals multiplied by 1- 1 over 1+r, raised to the power t over r. Both of these assume ordinary annuities. We made a little distinction and said, multiply this by 1+r if it is an annuity due, if the amounts are occurring in the beginning of each period. So this adjustment only if we have an annuity due. [COUGH] And that takes us to the last set of formulas with a perpetuity. We had two types of perpetuities. We had an ordinary perpetuity, and we had a growing perpetuity. For the ordinary perpetuity, the value of this perpetuity is the annuity divided by the interest rate, whereas the value of a growing perpetuity is the annuity times (1+r) divided by r-g. [COUGH] In all of these formulas it's important to remember that this frequency is what troubles a lot of folks when they start to do the calculations. The golden rule with the frequency of compounding is really, that when you have m, always think of m in terms of adjusting your interest rate. You always divide the interest rate by m. And adjusting your time period, you always multiply the time period by m. [COUGH] And then fit them into your formula and off you go to the races. Now, the formulas themselves, you don't have to be a math wizard, all of these are programmed into your calculator,. I wanted to show you what's behind the engine of the calculator, so you know what it's actually doing. So that you can make adjustments and understand how the calculations interact with the information that you work with. This seems like a lot, but it really isn't, because all of finance, the complex word of finance is built on something as simple as the formulas that you see here. So it's really important to start with a good sound understanding of this space. Practicing problems on the time value of money is the best way to increase your confidence. Use the handful of these formulas to help the mechanics. And this is going to help us to see this bigger picture, what we talked about earlier in terms of opportunity cost. And what Mr. Maclaurin's warning that we should pay heed to, we must be shaping the tools instead of the tools shaping us. Let's look forward to this bigger picture in making financial decisions that's going to make a difference in your