0:02

Hello I'm Professor Brian Bueche, welcome back.

Â This is the final video in our trilogy on time value of money.

Â Now maybe this trilogy hasn't been as entertaining as the Lord of

Â the Rings trilogy.

Â But it certainly been shorter, and

Â we'll have much more beneficial impact on your career.

Â So in this last video, we're going to talk about the concept of annuities,

Â which is a concept we run into a lot in doing financial calculations.

Â So let's get to it.

Â 0:32

Annuity is a constant stream of future cash flows, so

Â it's the same payment every period.

Â Now there's two different types of Annuities.

Â There's an ordinary annuity, or

Â a annuity in, a rears, where you get the payments at the end of each period.

Â Or an annuity due or

Â an annuity in advance where the payments come at the start of each period.

Â So you almost never see in any of the applications we do annuity due.

Â Everything we're going to do is going to be an ordinary annuity so

Â the payments are going to come at the end of the period.

Â 1:02

So an example of how this would work.

Â How do you figure out the present value of an ordinary annuity of $500

Â with three periods at an interest of 8%?

Â Well, you're getting the present value of $500 one year from now at 8%.

Â Which would be 500 times the present value table rate of .92593.

Â Plus the present value of the $500 that you'll receive two years from now,

Â discounted back 8%.

Â Plus the present value of the $500 that you receive three years from now,

Â discounted back at 8%.

Â That's the same as saying it's $500 times 2.57710,

Â which is the combination of those three discount factors.

Â Gives you the present valued annuity of 1288.55.

Â So the nice thing about annuities is, if it is the constant payment.

Â We don't have to do three separate present value calculations for

Â each of the three payments.

Â Instead, we're going to have a table that'll give us one factor that will

Â allow us to do one calculation.

Â To figure out what's the present value of getting that annuity of $500 at

Â the end of each of the three, next three periods.

Â 2:19

>> Yes unfortunately the annuity formula will only work if

Â the payment is the same each period.

Â If not then you have to take the individual present value of

Â all the individual payments.

Â But fortunately, many of the applications that we'll look at

Â will have constant streams of the same payment over some number of periods.

Â 2:42

So if we go back to our time value of money calculation

Â abbreviations or elements.

Â In the prior videos we had present value future value interest rate or

Â discount rate and number of periods.

Â With annuities, we're adding one more element, and

Â this will be the last element, which is PMT for payment.

Â This is the periodic payment for an annuity, and

Â unless specified, assume it's going to be received at the end of each period.

Â So it will be one of these ordinary annuities.

Â So let's try doing some calculations of present values of annuities.

Â 3:14

First thing you could do is use the formulas,

Â which is present value equals payment divided by r times [SOUND].

Â You know, I think we're going to stop using formulas at this point.

Â I'm not going to go through how this is derived.

Â This would be very tedious to try to figure out on your calculator.

Â So we're going to stop with the formulas, and

Â instead rely on either the tables, so present value equals payment.

Â And then we're going to pull a factor from table four for annuities.

Â Now, I realize I skipped table three, we'll come back to table three later.

Â 3:44

Or, we're going to use that same Excel function PV instead of

Â having payment zero.

Â This time we're going to put in a number and we'll set future value to zero.

Â And we'll find out that Excel can kick out the Present Value of the annuities, so

Â I'll show you both ways to do it.

Â So, here's the first question we're going to look at if this course gets you

Â an extra 5,000 per year in salary until retirement.

Â So basically everything that you learn in this course, you will go and

Â apply that on your job.

Â And you'll get paid an extra $5,000 than you would've other wise.

Â How much would you've been willing to pay for that?

Â 4:27

>> I agree the terminology's a little strange, but it's the way we tend to

Â think of these problems so let me explain it a little bit.

Â So the idea is we figure out the present value of that extra $5,000.

Â You'd be willing to pay for this course any amount up to.

Â That present value.

Â If you did then it would be a positive MPB project.

Â The initial cash outflow for paying for the course would be

Â less than the present value of the future cash inflows, the $5,000 over time.

Â If you paid more for the course than the present value of those cash flows,

Â the future, salary increases, then it's a negative MPB project.

Â You wouldn't want to do it.

Â So the assumption is here that the fair price would be that we

Â would charge you for this course, exactly what you would get in future benefits.

Â And the future benefits are the present value of that $5,000 of

Â extra salary every year until you retire.

Â 5:21

So getting back to the question.

Â We're going to assume again 20 years to retirement, and

Â inflation's expected to be 15%.

Â So here the relevant rate that we use,

Â need to use to compute the annuity is the rate of inflation.

Â Because that's what's going to make a dollar 20 years from

Â now worth less than a dollar today.

Â Prices would be going up over the next 20 years by 15% per year.

Â Here is table four, which is the present value of an annuity table.

Â To do this calculation, we want to go to the row for 20, 20 periods, 20 years.

Â Go across to the column for 15%.

Â And the number that we'd want to pull is 6.2593.

Â So then going back to our problem we're going to have the present value

Â equals 5,000.

Â Which is the payment times the factor we just looked at, 6.2593.

Â Which means the present value is $31,297.

Â So in other words you'd be willing to pay $31,297 in today's

Â dollars to receive $5,000 per year over the next 20 years if inflation is 15%.

Â >> $5,000 for 20 years is.

Â Is $100,000.

Â Of course, we would be willing to pay only $31,297 to get $100,000.

Â That is a fantastic deal.

Â Seems too good to be true.

Â 7:00

>> Now it's actually a fair deal because remember.

Â The present value represents the value in

Â today's dollars of that $100,000 that you'll receive in future dollars.

Â And remember with 15% inflation.

Â Those future dollars are not going to be worth the same as a dollar is today.

Â So think back to the gas prices.

Â Remember, gas was $1.53 in 1980, $4.14 in 2011.

Â Well at 15% inflation, price of gas is going to be $67 20 years from now.

Â So that $5,000 of extra salary is not going to go that far when gas is

Â $67 dollars a a gallon.

Â And that's why we have to discount it back to today's dollars to get a sense of

Â it's real purchasing power.

Â The present value represents that purchasing power in today's dollars.

Â 7:52

Before we move on let me quickly show you how to do this in Excel.

Â So we push the function button, and we look for PV present value.

Â So our rate, again, is 15%, 20 years until retirement,

Â we're getting 5,000 per year and

Â there's no future value, there's no lump sum at the end.

Â Type is left blank because it's an ordinary annuity.

Â We hit OK.

Â If you don't like to see the negative,

Â we'll put the little negative sign there 31,297.

Â So we get the same answer using Excel.

Â So let's do some more practice here,

Â and I'll give you a chance to try to do some of these calculations on your own.

Â So what if the inflation rate was only 5%.

Â So again we're getting an extra $5,000 per year for

Â 20 years, but instead of 15% inflation is only going to be 5%.

Â So why don't I bring up the table and pause sign and

Â have you take a crack and answering this one.

Â 9:04

So here again we want to look at the 20 year row because we're going out 20 years.

Â But now it's only going to be five percent inflation so

Â we look at the row 20, column 5%.

Â We see that the factor is 12.4622.

Â 9:34

So what's happened is the present value of the $5,000 now is higher than it

Â was before.

Â What's happened is the inflation rate is now lower.

Â So that $5,000 that you're getting in the future is worth more than when

Â inflation was 15%.

Â Right, so inflation is lower,

Â those future dollars are going to be worth more to you than if inflation is high.

Â And if those future dollars are worth more to you,

Â it's going to mean that your present value is going to go up.

Â 10:04

What if you plan to retire in 10 years?

Â So now we're going to continue with $5,000 extra per year but

Â you're only going to get it for 10 years.

Â Instead of 20 years, the inflation rate will go back up to 15 percent,

Â which is what we had in the original case.

Â So I will bring up the present value table, and

Â the pause sign, and have you take a crack at this one.

Â 10:29

Okay, to solve this one, we need to go down to the row for 10 years.

Â And then across to the column for 15% and

Â we see at the intersection of the row in column, he has 5.0188.

Â So in our formula, present value equals 5,000 times that factor,

Â which is 5.0188, so the present value would be $25,094.

Â 10:55

Now this present value is lower than the present value we had in the base case.

Â What's happened is we get fewer years of the payment.

Â So instead of getting 20 years of $5,000,

Â we only get 10 years and a result, as a result the present value of that.

Â Future stream of payments,

Â then the annuity goes down relative to the base case.

Â 11:20

Yeah, so that's see how this generalizes when you look at things like 30 years, or

Â 25%, or even a higher payment.

Â So instead of $5,000, you get $10,000 new extra salary.

Â So here's a little sensitivity analysis I did in Excel.

Â And what, yeah, want you to look at is,

Â let's first focus on where the inflation rate is the same.

Â So we can look at the 5% at the top.

Â 11:43

As the number periods goes up with the same payment,

Â we see that the present value goes up.

Â So what that's in effect is, you're getting the payment over more years, so

Â the value of that annuity's going to go up.

Â Now if you look at the same number of years, but changed the interest rate.

Â So if you look at 30 years for 5%, 15% and

Â 25%, what you'll notice is the present value goes down.

Â So holding the payment constant and

Â the number of years as the interest rate goes up.

Â The present value goes down and of course vice versa.

Â And then the last three wells show you see what,

Â shows you what happened if we increase the payment.

Â So if you compare 5% 30 years 5,000 to 5% 30 years

Â 10,000 you see the present value goes from 76,862 to 153,725.

Â What happens there is that as the payment goes up,

Â you're getting a bigger amount each year.

Â Which makes the value, the present value of that annuity go up.

Â So what we find is the present value is inversely related to the discount rate or

Â inflation rate.

Â As inflation goes up, present value goes down.

Â As inflation goes down, present value goes up.

Â And that's because inflation affects very directly how much that payment is

Â going to be worth to you in current dollars.

Â The present value is positively related to the payment and the number of periods.

Â And that's because with annuity, you're getting the same payment every year.

Â So if you get a bigger payment or

Â you get more payments, it's going to increase the present value.

Â 13:21

Last thing to look at would be the future value of annuities.

Â So I'm not even going to show you the formula.

Â There's two ways to do this.

Â The future value is the payment times table three factor.

Â So table three will have the annuity factors for future values.

Â Or we can use that same formula as, in Excel as we did a couple videos ago.

Â Except we replace zero with the amount of the payment and

Â then we drop the present value term and I'll, I'll show you how to do that.

Â So here the question would be if this course gets you an extra 5,000 per year in

Â salary until re, retirement, how much is this worth when you retire.

Â So instead of figuring out the present value of that annuity,

Â let's figure out the future value of the annuity.

Â How much dollars will you have in the future, when this is all said and done.

Â So we're going to assume 20 years to retirement, inflation is expected to be

Â 15% per year, so here's table 3, which is our future value of annuity table.

Â So we go down to the row for 20 years.

Â And across to the column for 15% and

Â we see that the intersection, the factor is 102.4436.

Â So using our formula, that's future value equals 5000, the annuity payment.

Â Times this factor, which is 102.4436, which is $512,218.

Â So again, let me quickly jump in and show you how to do it in Excel.

Â So we hit the function button.

Â 14:55

We look for future value.

Â We've got 15 percent rate of inflation, 20 years to retirement, $5,000 extra salary.

Â There's no present value lump sum.

Â There's the type is left blank because it's ordinary.

Â We hit okay.

Â Don't like to see the negative so put the negative in front.

Â $512,218, so

Â we get the same answer in Excel as we did with the future value into each table.

Â 15:27

Now if you recall, a few videos ago, we did how much would you have

Â if you invested $10,000 in the stock market with 15% rate for 20 years.

Â And we came up with $163,635.

Â So getting the extra 5,000 year in salary is worth much more to you in the future.

Â Then would be just investing $10,000 into the stock market and letting it grow.

Â 16:01

>> Actually, the best thing to do is use your knowledge from this course to

Â both get a higher salary at work.

Â And be a better investor in the stock market, and

Â then you get higher future values on both fronts.

Â 16:15

So far we've been working with annual compounding.

Â So we have annual interest rates.

Â Interest is compounded annually.

Â But oftentimes we have to do problems where the interest is

Â compounded more frequently than that.

Â For example, with bonds, which we're going to see a lot, it's semi-annual

Â compounding, so every six months, the interest gets applied to the bond.

Â So in that case what we need to do is take the interest rate,

Â which is always given as an annual rate.

Â And divide it by 2, and take the number of years for the bond and

Â multiply it by 2 to translate it from an annual stream to semiannual stream.

Â Now let's do a couple examples.

Â So what's the present value of a $1,000 five-year, 12% savings bond with.

Â Annual compounding.

Â So this is what we saw before, where the present value's going to be $1,000,

Â which is the future value.

Â Then we would go and look up the factor from table 2 for 5 years, 12%.

Â No I'm not going to do that, but I'll leave it for you to double check.

Â And we come up with 567.

Â 17:20

Now if this was semi annual compounding which is more typical for

Â the bonds we're going to see.

Â It's the same formula present value equals the future value of a 1,000 times now we

Â want to do 6% for ten periods.

Â So, we have instead of five years we have ten semi annual periods ten half years.

Â And we don't get 12% for each half year, we only get 6%.

Â And we get a different present value, 558.

Â You could do this with monthly compounding, where, now,

Â you would divide the interest rate by 12%.

Â So, instead of 12%, it would be one percent.

Â You'd multiply the number of periods by 12.

Â So, instead of five years.

Â It would be 60 months.

Â We would get a present value of 550.

Â You can even do daily compounding.

Â So take the 12% interest rate and

Â divide it by the number of days in a year, which I think is about 365.

Â Take the five year period and figure out the number of days, 1825.

Â And basically figure out what's the present value,

Â if you had daily compounding of interest.

Â Which would be at the daily interest rate for the number of days.

Â And you end up with 549.

Â 18:46

>> So continuous compounding literally means that interest is

Â always being compounded.

Â The way it was calculated is, I,

Â I don't know if you know in mathematics the number e.

Â But if you take e raised to the interest rate,

Â that gives you what the interest would be if it was compounded all the time.

Â But as you can see on the slide, as we did more frequent compounding periods,

Â it had less and less of effect on present value.

Â So the reason banks can promise this,

Â is it wasn't requiring them to pay that much more interest.

Â Than say, daily compounding or even weekly compounding.

Â So you would've been better off with the toaster.

Â 19:23

So let's wrap up this video.

Â And this unit on time valued money by looking at the example of

Â pricing a money a bond.

Â And, and don't worry you're going to get a lot more practice with this down the road.

Â So how much would you pay to buy a newly issued three year bond.

Â That pays coupon payments of $250 every six months and then $10,000 at maturity.

Â Current market interest rate is 5%.

Â So notice that there's really two different payment streams here that we

Â need to take the present value of.

Â First, we have an annuity.

Â We're getting $250 at the end of every six months.

Â So we can use our present value of an annuity calculation.

Â Then we're getting $10,000 in a lump sum of maturity,

Â there were going to look at the present value of a future value.

Â 20:09

But what complicates this is that it's semi-annual.

Â So we need to double the number of periods.

Â So, instead of a three year bond, we have to view this as six semi-annual periods.

Â And then divide that annual interest rate by 2.

Â So instead of the market rate being 5% per year,

Â we're really getting 2.5% compounding every six months.

Â 20:33

Now we can figure out the present value of the payment at maturity.

Â So the $10,000 lump.

Â So we're looking for the present value, we know the future value is $10,000.

Â Interest rate is 2.5%.

Â There are six semi-annual periods, and we set the payment equal to zero now.

Â And you could use the formula with your calculator, or

Â the present value table or Excel.

Â I'm going to pop to Excel to solve this.

Â 21:04

So for the present value of the $10,000 value principle.

Â We'll bring up our function.

Â Present value.

Â We've got a rate of two and a half percent, which we enter as 0.025.

Â Six semi-annual periods.

Â No payment at this point.

Â And a future value of 10,000.

Â That's what we're getting at maturity.

Â 21:25

And then we'll slip in the little minus here, so

Â that we don't have to look at a negative number, 8,623.

Â And so as we see the present value of the payment at maturity is $8,623.

Â Then we have to

Â 21:40

figure out the present value of that annuity of semi annual payments.

Â So here, we're looking for present value.

Â The future value set to zero.

Â Interest rate again, is two and a half percent.

Â Six semi-annual periods, and

Â we're getting $250 at the end of every semi-annual period.

Â So let's pop out to Excel and solve this one.

Â 22:13

We have six semi-annual periods.

Â We're getting a payment of 250.

Â We're going to leave future value out.

Â And leave type alone.

Â I'll go ahead and put the minus in.

Â And so we end up with 1,377.

Â And so carrying that over, the present value of that annuity.

Â Is $1,377.

Â So the price of the bond, the amount that you're willing to pay,

Â would be the present value of the payment at maturity, 8,623.

Â Plus the present value of that annuity, 1,377, which would equal 10,000.

Â Which just so happens is the same value of the payment of maturity.

Â When we talk about bonds, we'll explain why that's the case.

Â But before we do that, let me pop out to Excel to show you one more thing.

Â 23:15

Now going back to the formula,

Â the reason why the formula provides both payment and future value is.

Â You could actually price a bond in one step by just filing that in.

Â So it'll present value both streams.

Â So if we put in the rate as 0.025, two and a half percent.

Â Six semi-annual periods.

Â $250 payment.

Â And a $10,000 future value.

Â We end up with, low and behold, $10,000.

Â So you can actually price a bond, which is a combination of the annuity and

Â the lump sum, just using the one pass through the Excel formula.

Â 23:58

And that wraps up our look at the basic concepts of time, value money.

Â Still need more practice?

Â No problem.

Â We still have plenty of more applications to look at when we

Â talk about accounting for bonds and for leases.

Â See you then.

Â