Hello I'm Professor Brian Bueche, welcome back. This is the final video in our trilogy on time value of money. Now maybe this trilogy hasn't been as entertaining as the Lord of the Rings trilogy. But it certainly been shorter, and we'll have much more beneficial impact on your career. So in this last video, we're going to talk about the concept of annuities, which is a concept we run into a lot in doing financial calculations. So let's get to it. Annuity is a constant stream of future cash flows, so it's the same payment every period. Now there's two different types of Annuities. There's an ordinary annuity, or a annuity in, a rears, where you get the payments at the end of each period. Or an annuity due or an annuity in advance where the payments come at the start of each period. So you almost never see in any of the applications we do annuity due. Everything we're going to do is going to be an ordinary annuity so the payments are going to come at the end of the period. So an example of how this would work. How do you figure out the present value of an ordinary annuity of $500 with three periods at an interest of 8%? Well, you're getting the present value of $500 one year from now at 8%. Which would be 500 times the present value table rate of.92593. Plus the present value of the $500 that you'll receive two years from now, discounted back 8%. Plus the present value of the $500 that you receive three years from now, discounted back at 8%. That's the same as saying it's $500 times 2.57710, which is the combination of those three discount factors. Gives you the present valued annuity of 1288.55. So the nice thing about annuities is, if it is the constant payment. We don't have to do three separate present value calculations for each of the three payments. Instead, we're going to have a table that'll give us one factor that will allow us to do one calculation.
To figure out what's the present value of getting that annuity of $500 at the end of each of the three, next three periods. >> Does it have to be the same payment each period to use the annuity formula? >> Yes unfortunately the annuity formula will only work if the payment is the same each period. If not then you have to take the individual present value of all the individual payments. But fortunately, many of the applications that we'll look at will have constant streams of the same payment over some number of periods. So if we go back to our time value of money calculation abbreviations or elements. In the prior videos we had present value future value interest rate or discount rate and number of periods. With annuities, we're adding one more element, and this will be the last element, which is PMT for payment. This is the periodic payment for an annuity, and unless specified, assume it's going to be received at the end of each period. So it will be one of these ordinary annuities. So let's try doing some calculations of present values of annuities. First thing you could do is use the formulas, which is present value equals payment divided by r times [SOUND]. You know, I think we're going to stop using formulas at this point. I'm not going to go through how this is derived. This would be very tedious to try to figure out on your calculator. So we're going to stop with the formulas, and instead rely on either the tables, so present value equals payment. And then we're going to pull a factor from table four for annuities. Now, I realize I skipped table three, we'll come back to table three later. Or, we're going to use that same Excel function PV instead of having payment zero. This time we're going to put in a number and we'll set future value to zero. And we'll find out that Excel can kick out the Present Value of the annuities, so I'll show you both ways to do it.
So, here's the first question we're going to look at if this course gets you an extra 5,000 per year in salary until retirement. So basically everything that you learn in this course, you will go and apply that on your job. And you'll get paid an extra $5,000 than you would've other wise. How much would you've been willing to pay for that? >> Dude I like do not understand the question. How much would you pay for it? Aren't we like receiving the extra salary? >> I agree the terminology's a little strange, but it's the way we tend to think of these problems so let me explain it a little bit. So the idea is we figure out the present value of that extra $5,000. You'd be willing to pay for this course any amount up to. That present value. If you did then it would be a positive MPB project. The initial cash outflow for paying for the course would be less than the present value of the future cash inflows, the $5,000 over time. If you paid more for the course than the present value of those cash flows, the future, salary increases, then it's a negative MPB project. You wouldn't want to do it. So the assumption is here that the fair price would be that we would charge you for this course, exactly what you would get in future benefits. And the future benefits are the present value of that $5,000 of extra salary every year until you retire. So getting back to the question. We're going to assume again 20 years to retirement, and inflation's expected to be 15%. So here the relevant rate that we use, need to use to compute the annuity is the rate of inflation. Because that's what's going to make a dollar 20 years from now worth less than a dollar today. Prices would be going up over the next 20 years by 15% per year. Here is table four, which is the present value of an annuity table. To do this calculation, we want to go to the row for 20, 20 periods, 20 years. Go across to the column for 15%. And the number that we'd
want to pull is 6.2593. So then going back to our problem we're going to have the present value equals 5,000. Which is the payment times the factor we just looked at, 6.2593. Which means the present value is $31,297. So in other words you'd be willing to pay $31,297 in today's dollars to receive $5,000 per year over the next 20 years if inflation is 15%. >> $5,000 for 20 years is. Is $100,000. Of course, we would be willing to pay only $31,297 to get $100,000. That is a fantastic deal. Seems too good to be true. >> Now it's actually a fair deal because remember. The present value represents the value in today's dollars of that $100,000 that you'll receive in future dollars. And remember with 15% inflation. Those future dollars are not going to be worth the same as a dollar is today. So think back to the gas prices. Remember, gas was $1.53 in 1980, $4.14 in 2011. Well at 15% inflation, price of gas is going to be $67 20 years from now. So that $5,000 of extra salary is not going to go that far when gas is $67 dollars a a gallon. And that's why we have to discount it back to today's dollars to get a sense of it's real purchasing power. The present value represents that purchasing power in today's dollars. Before we move on let me quickly show you how to do this in Excel. So we push the function button, and we look for PV present value. So our rate, again, is 15%, 20 years until retirement, we're getting 5,000 per year and there's no future value, there's no lump sum at the end. Type is left blank because it's an ordinary annuity. We hit OK. If you don't like to see the negative, we'll put the little negative sign there 31,297. So we get the same answer using Excel. So let's do some more practice here, and I'll give you a chance to try to do some of these calculations on your own. So what if the inflation rate was only 5%. So again we're getting an extra $5,000 per year for
20 years, but instead of 15% inflation is only going to be 5%. So why don't I bring up the table and pause sign and have you take a crack and answering this one. So here again we want to look at the 20 year row because we're going out 20 years. But now it's only going to be five percent inflation so we look at the row 20, column 5%. We see that the factor is 12.4622. So we have our formula present value equals 5,000 times that factor, which is 12.4622 which means the present value would be 62,311. So what's happened is the present value of the $5,000 now is higher than it was before. What's happened is the inflation rate is now lower. So that $5,000 that you're getting in the future is worth more than when inflation was 15%. Right, so inflation is lower, those future dollars are going to be worth more to you than if inflation is high. And if those future dollars are worth more to you, it's going to mean that your present value is going to go up. What if you plan to retire in 10 years? So now we're going to continue with $5,000 extra per year but you're only going to get it for 10 years. Instead of 20 years, the inflation rate will go back up to 15 percent, which is what we had in the original case. So I will bring up the present value table, and the pause sign, and have you take a crack at this one. Okay, to solve this one, we need to go down to the row for 10 years. And then across to the column for 15% and we see at the intersection of the row in column, he has 5.0188. So in our formula, present value equals 5,000 times that factor, which is 5.0188, so the present value would be $25,094. Now this present value is lower than the present value we had in the base case. What's happened is we get fewer years of the payment. So instead of getting 20 years of $5,000, we only get 10 years and a result, as a result the present value of that. Future stream of payments, then the annuity goes down
relative to the base case. Yeah, so that's see how this generalizes when you look at things like 30 years, or 25%, or even a higher payment. So instead of $5,000, you get $10,000 new extra salary. So here's a little sensitivity analysis I did in Excel. And what, yeah, want you to look at is, let's first focus on where the inflation rate is the same. So we can look at the 5% at the top. As the number periods goes up with the same payment, we see that the present value goes up. So what that's in effect is, you're getting the payment over more years, so the value of that annuity's going to go up. Now if you look at the same number of years, but changed the interest rate. So if you look at 30 years for 5%, 15% and 25%, what you'll notice is the present value goes down. So holding the payment constant and the number of years as the interest rate goes up. The present value goes down and of course vice versa. And then the last three wells show you see what, shows you what happened if we increase the payment. So if you compare 5% 30 years 5,000 to 5% 30 years 10,000 you see the present value goes from 76,862 to 153,725. What happens there is that as the payment goes up, you're getting a bigger amount each year. Which makes the value, the present value of that annuity go up. So what we find is the present value is inversely related to the discount rate or inflation rate. As inflation goes up, present value goes down. As inflation goes down, present value goes up. And that's because inflation affects very directly how much that payment is going to be worth to you in current dollars. The present value is positively related to the payment and the number of periods. And that's because with annuity, you're getting the same payment every year. So if you get a bigger payment or you get more payments, it's going to increase the present value.
Last thing to look at would be the future value of annuities. So I'm not even going to show you the formula. There's two ways to do this. The future value is the payment times table three factor. So table three will have the annuity factors for future values. Or we can use that same formula as, in Excel as we did a couple videos ago. Except we replace zero with the amount of the payment and then we drop the present value term and I'll, I'll show you how to do that. So here the question would be if this course gets you an extra 5,000 per year in salary until re, retirement, how much is this worth when you retire. So instead of figuring out the present value of that annuity, let's figure out the future value of the annuity. How much dollars will you have in the future, when this is all said and done. So we're going to assume 20 years to retirement, inflation is expected to be 15% per year, so here's table 3, which is our future value of annuity table. So we go down to the row for 20 years. And across to the column for 15% and we see that the intersection, the factor is 102.4436. So using our formula, that's future value equals 5000, the annuity payment. Times this factor, which is 102.4436, which is $512,218. So again, let me quickly jump in and show you how to do it in Excel. So we hit the function button. We look for future value. We've got 15 percent rate of inflation, 20 years to retirement, $5,000 extra salary. There's no present value lump sum. There's the type is left blank because it's ordinary. We hit okay. Don't like to see the negative so put the negative in front. $512,218, so we get the same answer in Excel as we did with the future value into each table. Now if you recall, a few videos ago, we did how much would you have if you invested $10,000 in the stock market with 15% rate for 20 years. And we came up with $163,635. So getting the extra 5,000 year in salary is worth much more to you in the future. Then would be just investing $10,000 into
the stock market and letting it grow. >> Dude, are you like saying that taking this course is like better than investing in the stock market? >> Actually, the best thing to do is use your knowledge from this course to both get a higher salary at work. And be a better investor in the stock market, and then you get higher future values on both fronts. So far we've been working with annual compounding. So we have annual interest rates. Interest is compounded annually. But oftentimes we have to do problems where the interest is compounded more frequently than that. For example, with bonds, which we're going to see a lot, it's semi-annual compounding, so every six months, the interest gets applied to the bond. So in that case what we need to do is take the interest rate, which is always given as an annual rate. And divide it by 2, and take the number of years for the bond and multiply it by 2 to translate it from an annual stream to semiannual stream. Now let's do a couple examples. So what's the present value of a $1,000 five-year, 12% savings bond with. Annual compounding. So this is what we saw before, where the present value's going to be $1,000, which is the future value. Then we would go and look up the factor from table 2 for 5 years, 12%. No I'm not going to do that, but I'll leave it for you to double check. And we come up with 567. Now if this was semi annual compounding which is more typical for the bonds we're going to see. It's the same formula present value equals the future value of a 1,000 times now we want to do 6% for ten periods. So, we have instead of five years we have ten semi annual periods ten half years. And we don't get 12% for each half year, we only get 6%. And we get a different present value, 558. You could do this with monthly compounding, where, now, you would divide the interest rate by 12%. So, instead of 12%, it would be one percent. You'd multiply the number of periods by 12.
So, instead of five years. It would be 60 months. We would get a present value of 550. You can even do daily compounding. So take the 12% interest rate and divide it by the number of days in a year, which I think is about 365. Take the five year period and figure out the number of days, 1825. And basically figure out what's the present value, if you had daily compounding of interest. Which would be at the daily interest rate for the number of days. And you end up with 549. >> Back in the 1970's, I remember that my bank started offering continuous compounding on its savings accounts. Instead of a free toaster, what does continuous compounding mean? >> So continuous compounding literally means that interest is always being compounded. The way it was calculated is, I, I don't know if you know in mathematics the number e. But if you take e raised to the interest rate, that gives you what the interest would be if it was compounded all the time. But as you can see on the slide, as we did more frequent compounding periods, it had less and less of effect on present value. So the reason banks can promise this, is it wasn't requiring them to pay that much more interest. Than say, daily compounding or even weekly compounding. So you would've been better off with the toaster. So let's wrap up this video. And this unit on time valued money by looking at the example of pricing a money a bond. And, and don't worry you're going to get a lot more practice with this down the road. So how much would you pay to buy a newly issued three year bond. That pays coupon payments of $250 every six months and then $10,000 at maturity. Current market interest rate is 5%. So notice that there's really two different payment streams here that we need to take the present value of. First, we have an annuity. We're getting $250 at the end of every six months. So we can use our present value
of an annuity calculation. Then we're getting $10,000 in a lump sum of maturity, there were going to look at the present value of a future value. But what complicates this is that it's semi-annual. So we need to double the number of periods. So, instead of a three year bond, we have to view this as six semi-annual periods. And then divide that annual interest rate by 2. So instead of the market rate being 5% per year, we're really getting 2.5% compounding every six months. Now we can figure out the present value of the payment at maturity. So the $10,000 lump. So we're looking for the present value, we know the future value is $10,000. Interest rate is 2.5%. There are six semi-annual periods, and we set the payment equal to zero now. And you could use the formula with your calculator, or the present value table or Excel. I'm going to pop to Excel to solve this. So for the present value of the $10,000 value principle. We'll bring up our function. Present value. We've got a rate of two and a half percent, which we enter as 0.025. Six semi-annual periods. No payment at this point. And a future value of 10,000. That's what we're getting at maturity. And then we'll slip in the little minus here, so that we don't have to look at a negative number, 8,623. And so as we see the present value of the payment at maturity is $8,623. Then we have to figure out the present value of that annuity of semi annual payments. So here, we're looking for present value. The future value set to zero. Interest rate again, is two and a half percent. Six semi-annual periods, and we're getting $250 at the end of every semi-annual period. So let's pop out to Excel and solve this one. So let's look at the present value of that $250 semi-annual payment. We bring up the function button.
Go over to present value. We've got our two and a half percent interest rate again. We have six semi-annual periods. We're getting a payment of 250. We're going to leave future value out. And leave type alone. I'll go ahead and put the minus in. And so we end up with 1,377. And so carrying that over, the present value of that annuity. Is $1,377. So the price of the bond, the amount that you're willing to pay, would be the present value of the payment at maturity, 8,623. Plus the present value of that annuity, 1,377, which would equal 10,000. Which just so happens is the same value of the payment of maturity. When we talk about bonds, we'll explain why that's the case. But before we do that, let me pop out to Excel to show you one more thing. So here's what I want to show you, so you're going back to these two components, if we add them up, we get $10,000 is the price of the bond. Now going back to the formula, the reason why the formula provides both payment and future value is. You could actually price a bond in one step by just filing that in. So it'll present value both streams. So if we put in the rate as 0.025, two and a half percent. Six semi-annual periods. $250 payment. And a $10,000 future value. We end up with, low and behold, $10,000. So you can actually price a bond, which is a combination of the annuity and the lump sum, just using the one pass through the Excel formula. And that wraps up our look at the basic concepts of time, value money. Still need more practice? No problem. We still have plenty of more applications to look at when we talk about accounting for bonds and for leases. See you then. >> See you next video.