沃顿商学院全套笔记-六-

沃顿商学院全套笔记（六）

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P104：2_有用的技巧.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we talked about compounding or the。

process of moving cash flows forward in time。 Today I want to present several。

useful shortcuts to compute the present value and future value of common streams。

of cash flows that we see often in practice。 Let's get started。 Hey everybody。

welcome to our third lecture on the time value of money。 So last time we talked。

about compounding or the process of moving cash flows forward in time， excuse me。

to find their future value whereas in our first lecture we moved cash flows。

back in time via discounting to find their present value。 So what I want to do。

today is I want to give you some useful shortcuts for computing the present value。

or the future value of some streams of cash flows that commonly arise in， practice。

So let's get started。 The first thing I want to talk about is an， annuity。

An annuity is a finite stream of cash flows of identical magnitude。

and equal spacing in time and I've highlighted key elements of this， definition。

So here's a timeline representing an annuity。 First key。

aspect or feature of an annuity is or are cash flows of identical magnitude。

These are all of these cash flows are the same number。 So identical， magnitude。

Secondly this is a finite stream of cash flows。 It ends at some。

point in time and that might seem like an unnecessary or obvious assumption but。

you'll see that we'll deal with infinite cash flow streams in a little bit。

And then the last assumption is that the spacing between the cash flows has to be， equal。

So it's always a year。 We always get the cash flow after one year， two， months。

whatever that spacing it is it has to be the same。 And it turns out that。

this cash flow stream arises in a number of situations in practice。 So。

insurance companies sell a product called an annuity representing its， cash flow stream。

Home mortgages are an example of an annuity stream。 Auto leases。

certain bond payments and amortizing loans are annuities。 So it's actually。

fairly common in practice。 Now if we wanted to find the present value of。

these cash flows we could we know how to do that。 Right we can brute force it。

we can take each cash flow and discount it back to today。 Right so imagine I had a。

second cash flow。 I had the second cash flow here。 I could take go CF divided by。

1 plus R squared that would bring it back to today。 I could take this cash flow CF。

over 1 plus R to the T minus 1 that would bring it and I do that for all the cash。

flows and then I could add them up here。 That would give me the present value。 But。

that's a bit burdensome especially when T is big。 So what I'd like to show you is。

a shortcut or a simple formula to compute the present value of this cash flow。

stream and here it is。 We take the cash flow CF。 We take divided by the discount。

rate and multiply it by this term in parentheses here。 Now if I move the R。

over here I can re-express the present value of the annuity formula as just the。

annuity cash flow times this term here which is called an annuity factor。 That。

will give me the present value of this cash flow stream。 One thing to keep in。

mind though is that in order for this formula to make sense not only do all the。

features defining an annuity stream have to be true but we're assuming that the。

first cash flow arrives one period from today。 So for example if my cash flow。

stream look like this this is an annuity stream of cash flows but this formula is。

not going to give me the present value of this cash flow stream。 Actually what I。

would have to do is I could just add CF because this is the present value。 It's， coming today。

Let's do an example。 How much do I have to save today to withdraw。

$100 at the end of each of the next 20 years if I can earn 5% per annum？ Well。

step one is draw a timeline。 I'm trying to figure out how much I have to save。

today in order to pull out $100 every year over the next 20 years。 Well we know。

how to do that by brute force。 We can simply discount all of the cash flows。

back into today's time units and add them up。 More elegant solution that we just。

learned of course is to apply our present value of annuity formula。 Right the cash， flow is 100 CF。

The discount rate R is 5% and the time of the cash flows is 20， years。

Plugging all those numbers into the formula and computing we get the。

present value of these cash flows is $1，246。22。 That's how much money I have to。

save today in order to withdraw the $100 every year。 Now let's turn to。

something called a growing annuity which is as the name suggests just like。

an annum and witty but for the fact that the cash flows are growing。 So it's a。

finite stream of cash flows。 Okay evenly spaced through time。 Okay but now the cash。

flows aren't constant they're growing at a constant rate G。 And this this type of。

cash flow stream pops up in a number of instances in practice。 Certain income。

streams for example your work right you might imagine that your work your。

salary grows at some constant rate G or approximately some constant rate G。

Certain saving strategies maybe you want to save a certain amount each year but。

you want that amount to grow with your growing income stream。 In corporate。

finance certain project revenue and expense streams will often grow at a， near constant growth rate。

So it's a really useful approximation to many cash， flow streams will come across in practice。

And like our annuity stream we can， represent the present value of this cash flow stream with a simple formula as。

follows we take the cash flow as of the first period divided by the discount。

rate less the growth rate times this factor here。 That will give us the present。

value of this cash flow stream right here。 Okay。 But remember a critical。

assumption is that the first cash flow arrives one period from today。 Okay。 Let's， do an example。

How much do we have to save today to withdraw $100 at the end of， this year？ $102。50 next year $105。

06 the year after and so on for the next 19， years if we can earn 5% per annum。

Well let's draw a timeline and what we see， is that our first first withdrawal of $100 occurs one year out then 102。

5 and， on and on and on。 What we can discern from this problem is that these cash。

flows are growing at a constant rate G equal to 2。5% per annum。 So this cash flow。

stream satisfies all of the requirements needed to use the present value of a。

growing annuity formula。 So our first cash flow of $100 our discount rate of 5%。

and here's our growth rate of 2。5%。 That's going to get us a present value of， $1，529。69。

That's how much we would need to withdraw $100 growing at 2。5%， every year。

Now let's talk about a perpetuity。 So perpetuity is just like an。

annuity except the cash flows go on forever。 We get the same amount of money。

equally spaced in time forever。 So where does this thing come up in practice？

Well oddly enough it actually does and something called perpetuities or console。

bonds which exist over in the UK and interestingly enough the formula for this。

cash flow stream is very simple。 It's just the cash flow divided by the discount， rate CF over R。

Let's do an example。 How much do you have to save today to。

withdraw $100 at the end of each year forever if you can earn 5% per annum？

Well the timeline looks as follows。 $100 every year forever and clearly the。

brute force method of discounting each cash flow one at a time is never going。

to work it's just impossible。 So we have to use our formula。 We take the $100。

divided by the discount rate of 5% and that gives us $2，000。 We need $2，000 to be。

able to withdraw $100 a year forever assuming that money can earn 5% per annum。

And intuitively what's going on is once we get out 100 years 200 years。

whatever the present value of that money is so small it's very close to zero which。

is why you don't need an infinite amount of money。 And the perpetuity's。

cousin a growing perpetuity is just that it's an infinite stream of cash flow that。

grows at a constant rate G and that are evenly spaced out through time。 So here's。

a visual representation here's our timeline of a growing perpetuity。 What's。

an example of a growing perpetuity in practice？ Well dividend streams are much。

like a growing perpetuity。 They're a useful approximation don't take it。

literally companies don't last forever but we can treat them as such because。

there is no finite end date to most companies。 Apps in some events such as a。

bankruptcy or an acquisition or a take over something like that。 So what's the。

formula for a growing perpetuity？ Well it's just the cash flow that we're going。

to receive in the first year divided by the discount rate minus the growth rate， of that cash flow。

Again we're going to have to assume that the first cash flow。

arrives one year from today to use this formula as well as having all the other。

requirements being satisfied。 The cash flow is being evenly spaced and then。

growing at a constant rate。 So let's do a little example now。 How much do you， have to save？

How much do you have to save today to withdraw $100 at the end of， this year？ 102。5 next year， 105。

06 the year after and so on forever if we can earn， 5% per ann。

Well let's draw our timeline so we're going to get $100 here in year， one。 102。

5 in year two that's growing at 2。5% as is the 105。06 if I had written it， here。 So our G is 2。5%。

The first cash flow comes one year from today。 There's， our first cash flow。

This goes on forever and the spacing is equal so this is a。

growing perpetuity to which we can apply our growing perpetuity present value， formula。

So we take the first cash flow $100 divided by the difference between。

the discount rate and the growth rate to get $4，000。 In other words if we have， $4。

000 today and it's earning 5% interest per annum every year thereafter we can。

pull out $100 next year and have that amount grow by 2。5% every year thereafter。

So let's summarize。 We learned a couple of useful shortcuts today。 We talked about。

an annuity and its present value formula。 We talked about a perpetuity and its。

present value formula。 We talked about their growing cousins the growing。

annuity and the growing perpetuity and the present value formula for those guys。

And while that might seem somewhat esoteric and bland or boring we also。

discussed some of the applications that you might see in practice these cash flow， streams arising。

And where these shortcuts are really useful more than just。

finding the present value or the future value is in finding the cash flow。

associated with the stream。 So being able to manipulate these formulas is very。

important and so in the problems you're going to spend some time in real life。

contacts or at least as close to real life as we can get manipulating these。

formulas to derive certain aspects of interest whether it's the cash flow or。

the amount of time or the discount rate or the growth rate。 So tackle the。

problem set it really brings the material to life but be very careful that applying。

these formulas takes care。 Don't blindly apply them to any setting because as we。

discussed certain characteristics of the cash flows have to be met in order to or。

certain requirements of the formula have to be met in order to use it。 So good luck。

with the problems。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P105：3_税收.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we talked about some useful shortcuts。

to compute the present value and future value of cash flow streams that we。

commonly come across in practice。 Streams like perpetuities and annuities and。

growing perpetuities and growing annuities。 This time I want to shift。

gears and talk about taxes and their impact on our dollar returns。 Let's get， started。 Hi everyone。

welcome back to Corporate Finance。 Last time we talked。

about several useful shortcuts for computing the present value of cash flow， stream。

We talked about annuities and growing annuities and。

perpetuities and growing perpetuities and we also discussed that these cash flow。

streams were representative of cash flow streams we might find in practice。 What。

I want to do now is I want to shift gears and talk about taxes。 In particular I。

want to talk about how taxes impact our discount rate or our cost of capital and。

ultimately how they impact our dollar returns。 So let's get started。 So I want。

to start with a little bit of motivation。 I know we don't have a lot of time but。

it's worth mentioning a few things。 This is a picture of the top statutory tax。

rate on different sources of income in particular dividends which is the blue， line。

capital gains which is the red line and interest income which is the green， line。

I don't want to make too much of this picture but I just want to emphasize a， couple of things。

First you should see that the tax rates are moving all over， the place。

There's an enormous amount of variation in taxes over time number one。

and number two you can see that historically tax rates have gotten， really high。

Now admittedly it's not clear how to how many people those tax。

rates of 90 95% actually applied but taxes are substantial and they move， around a lot over time。

So it's important to understand how they impact our。

investment income。 So let's do illustrate this by example and I want to。

revisit an example we looked at in the past in particular in our first lecture， on discounting。

So the question is how much do you have to save today to。

withdraw $100 at the end of each of the next four years if you can earn 5%， per annum？

Well recall the first thing we do is we lay out a timeline and here are， cash flows right。

We're going to be withdrawing $100 a year over the next。

four years and the question is asking how much do we need today。 So what we did is。

we discounted each of each cash flow each of the $100 by the 5% discount rate。

to get the present value。 So for example we took this first 100， 100 divided by， what？ 1 plus 0。

05 and that got us $95。24 approximately。 And we took the second， cash flow 100 divided by 1 plus 0。

05 squared that's a 2 and that got us the， second cash flow and on and on and then we could add these cash flows because。

they're all in the same time zero units and that produced our answer of $3。54。60。

Okay then when we thought about what was going on in the bank account recall。

right we would insert $354。60 into the account it would it would then earn。

interest at 5% we'd add that to the previous balance and we'd pull out some。

money and that would reduce our balance and we would continue that for four。

years at the end of the four years we would be less left with nothing。 But now。

consider what happens when we have some taxes。 See now we put our $3。54。60 into the。

account that earns interest at 5% but now we have to pay taxes on that。

interest income and I'm gonna assume the tax rate we're facing is 35% which isn't。

too far up from the top statutory rate currently prevailing。 So that 35% of the。

1773 is 621 in taxes that's gonna reduce our pre withdrawal balance to， $3。66。

12 then we pull out $100 and then we're left with $2。66。12 at the end of the。

first year which by the way the $2。66。12 is less than the $2。72。32 because of the， taxes。

Now if we continue that process we'll get down to zero but only because in。

the last year we can only pull out $83。06 that's all we have left in that last， year。

So on net we're $16。94 short the taxes are reducing the funds available。

for withdrawal we're running out of money early。 So the lesson here is that。

taxes reduce the return on our investment the dollar return on our。

investment and one way to account for that is an after tax discount rate I'll。

call it Rt and that equals our usual discount rate times 1 minus the tax rate。

So for our example the discount rate R was 5% and the tax rate T was 35% for an。

after tax discount rate of 3。25%。 Now let's revisit the problem using this， after tax discount rate。

So now when I discount all of these cash flows back to， today I'm going to use Rt which equals 3。

25% instead of the R which equals 5%。 When we do that when we do the arithmetic and add all these numbers up we get。

369。50 a number that's bigger than the 354。60 we originally found with the 5%， discount rate。

Now let's run through the exercise of looking at what happens to， the savings account。

So now we put in 369。50 that's going to earn interest at 5%。

we're going to pay taxes on that interest at a rate of 35% which。

generates a pre-withdrawal balance in the first year of 38151。 We pull out the。

hundred dollars and then we're left with 281。51。 We continue that process for a。

few more years and at the end of four years we're left with nothing in the。

bank account and we've been able to pull out a hundred dollars each year。 So what's。

the implication？ Well we need to save more to withdraw a hundred dollars per。

year after taxes right in particular we need to save 369。50 as opposed to the， 354。

60 and it's also interesting to note that the difference in how much we had。

to save the difference between the 369。50 and the 354。60 which equals 14。90 that。

equals the present value of the taxes at 5%。 Check this discount these cash。

flows at 5% and see what the present value is it should be 14。90。 Okay let's。

summarize and summarize this and bring it back together。 Taxes reduce our dollar， return。

The after tax return is less than the pre-tax return and it's less by a。

factor of 1 minus the tax rate that you're facing。 So we can discount by this。

after tax return to see how much money we're going to have once we've swept out。

the effect of taxes and given tax rates in the US not to mention most of Europe。

and many other countries taxes are significant and can have a huge impact。

on our cash flow streams our savings behavior and ultimately our decision， making。

So looking ahead I want you to dive into the problems work on the。

problem set and then we're going to look at inflation and investigate how。

inflation affects our cash flow stream and our decision making。 So thank you and I。

I look forward to seeing you in the next class。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P106：4_通货膨胀.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we talked about taxes and their。

impact on our dollar returns。 This time I want to talk about inflation and its。

impact on our real returns or consumption。 Let's get started。 Hey。

everybody welcome back to Corporate Finance。 So last time we introduced taxes。

and we explored the impact of taxes on our cost of capital or discount rate。

and our dollar investment returns。 What I want to do today is I want to think。

about the impact of inflation on our returns and ultimately on our decision。

making。 So let's get started。 And I want to start with a picture as I did with。

taxes to illustrate the importance of inflation。 This picture shows over the。

last approximately 110 years inflation in the United States。 And what you can see。

is a couple things。 First of all over the recent period inflation has been， relatively low。

But if you go back beyond say the last 30 years you can see。

periods of really high inflation including deflation。 And so what I want。

to emphasize is that while inflation doesn't seem particularly important， these days it can be。

And if we move outside of the US and some Western。

European countries inflation becomes incredibly important。 So let's understand， it。

How does inflation impact our returns in particular？ All right so let's。

revisit the example we've done over the last few lectures。 This example just to。

refresh your memory was you know how much do we have to deposit or save。

today in an account earning 5% if we want to withdraw $100 every year over the， next four years。

And the answer to that question was $354。60。 We deposit $354。60。 That earns interest at 5%。

That increases our balance。 We withdraw the money。 That。

reduces our balance and we continue over the next three years。 Until we drive the。

account balance down to zero。 Exactly。 Now the first lesson is inflation is not。

going to affect the money we earn。 It doesn't affect the interest over here。

We're still going to earn 5% every year a nominal rate of return。 What。

inflation is going to do is it's going to affect what we can buy with the money， we're pulling out。

It's going to affect the value of that money。 And so we'd like a。

way to quantify and understand the impact of inflation on that value。 So I'm。

going to introduce the concept of a real discount rate。 I'm going to denote it by， R。R。

And one plus the real discount rate equals one plus the nominal， discount rate。 Plus， I'm sorry。

divided by one plus the expected rate of， inflation which I've denoted by pi。

And a commonly used approximation that you'll， see though I'll emphasize this is an approximation is that the real rate。

equals the nominal rate minus the expected rate of inflation。 So in our， example we might have。

our discount rate was 5% and if expected， inflation is 2。

5% which is approximately what it's been over the recent history， in the US。

We get a real rate of return on our investment of 2。44%， substantially lower。

Now that's important because even though the， inflation is not impacting our account balance。

how much money any dollars we， have， it is impact， it is impacting what we can do with that money。

what we can， buy with it。 And that's ultimately what we care about it。 So let's try discounting。

our cash flows now by the real rate of return R。R which we just showed was， equal to 2。44% right。

And the discounting proceeds mechanically in the exact same， way as it was before。

Only now I'm using the real rate instead of the nominal， rate。

And if we do a little bit of arithmetic， we get the present values。

of all of these future cash flows。 We add them up and we get a value of， 376。75。

That's the present value of the sum of all these cash flows。 Now。

taxes affect dollars but inflation does not affect dollars， it affects consumption。

So each nom or the we earn a nominal return but we can't buy as much with it。

Let me illustrate this。 So let's insert here into our savings account the， 376。

75 that we just computed using the real discount rate。 And see what happens。

when we're withdrawing $100 every year？ Well that money is going to earn interest， at 5% every year。

We pull out $100 and what's going to happen is we're going to be， left with this surplus。

But that makes sense right because inflation doesn't， affect the dollars。

it affects what we can do with these things that we pull out。

See we have extra money here so what we really want to do to address inflation is we want to。

increase how much we pull out every year。 I don't want to pull out just $100 each year because prices。

are going up so that $100 say here in year two can't buy as much food or housing or clothes or。

whatever we need to buy。 So let's think about what kind of cash flow stream we might want。

to address inflation。 And one way to do that is to simply solve for the cash flows that we want to withdraw each year。

given a nominal discount rate are of 5%。 What is CF？ Well we can solve this。

This is just elementary algebra right？ And we want to use the nominal rate here since that's reflecting the dollars that we're earning。

So solving this for CF or cash flow we get 106。25 which is greater than the 100。 That makes sense。

We're putting in more money at the beginning right？ Remember originally I think we were putting in。

354。60 if I remember correctly。 So that we put in more which means we can take out more than we。

used to take out。 And let's see what happens now。 So we put in the 376。75 now we're going to。

withdraw 106。25 each year and we see that we're going to drive the account balance exactly to zero。

with nothing left over。 But ideally we want our withdrawals to increase each year to accommodate。

inflation right？ I want these withdrawals to go up every year to account for the increase in prices。

of the goods that I'm going to purchase， goods and services that I'm going to purchase with that money。

So let's think about this。 Well if prices are going up at 2。5% per year that means what I could buy。

with a hundred dollars today I'm going to need a hundred and two dollars and fifty cents next year。

because prices went up by 2。5%。 They're going to go up by 2。5% again so I'll need a little bit more。

and a little bit more and a little bit more each year。 See what this sequence of withdrawals。

maintains our purchasing power。 We'll be able to buy the same amount of food， the same amount of。

housing， go on the same vacations assuming the prices are all going up by 2。5% the expected rate of。

inflation。 Now these are all nominal。 These are all nominal values corresponding to the real。

one hundred dollars of purchasing power。 And so if we take the present value of these nominal。

dollars at the nominal discount rate we get the 376。75。

We discount nominal cash flows by the nominal， rate。

Keep that in mind it's important to emphasize that。 The present value of nominal cash flows。

at the nominal discount rate that's going to equal the present value of the real cash flows。

at the real rate。 Remember that when we were withdrawing a hundred dollars each year but we。

discounted these at the real rate of return which I think was 2。44% we got 376。75。

And all that's going， on is that the inflation term and the numerator and denominator of the real computation is they're。

canceling one another。 Okay so let's go back to our savings account。 We insert the 376。75。

We're now going to withdraw money that's growing at a rate of 2。

5% per year to keep up with inflation。 But our money in the account is earning the nominal rate of return of 5%。

And what happens is we， exactly exhaust our funds at the end of four years。

And we've been able to do so by increase our， withdrawals each year to keep up with inflation。

So let's summarize this。 Inflation does not affect。

dollar returns。 It's not affecting the money in the bank account or the rate at which it's growing。

What it's affecting is the purchasing power of that money。

So when I pull it out and go buy something， I can buy less of that good or that service with that same dollar year after year after year when。

we face inflation。 So we introduce the idea of a real rate of return that takes into account the。

effects of inflation。 Next time we're going to turn to a new topic interest rates。

And we're going to， build on what we've already learned to understand how to discount and value cash flow streams that。

don't happen every year that are irregular in their timing。 And how to deal with different。

compounding periods as opposed to annual compounding which is what we've implicitly been doing all。

along thus far。 So I look forward to seeing you next time。 Thanks。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P107：5_APR 和 EAR.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Today we're going to turn to a new topic。

interest rates， but before doing so， I want to briefly review our last topic。

the time value of money。 As you'll recall， we started off with some intuition， then。

we introduced the tools associated with the time value of money， namely the。

discount factor and the timeline。 Then we applied those tools to move money。

back in time via discounting and forward in time via compounding。 We。

discussed some useful shortcuts to compute the present value in the future。

value of cash flow streams that we commonly come across in practice， streams。

like an annuity and a perpetuity。 And then we closed out the topic by talking。

about taxes and inflation and their impact on our dollar returns and our。

ability to consume goods and services。 In this topic， interest rates， I want to。

start off by talking about interest rate quoting conventions and I want to talk。

about how to compute the present value and future value of streams of cash。

flows when they arrive at irregular times， non-annual， and when the compounding is。

non-annual as well。 So let's get started。 Hi everyone， welcome back to Corporate， Finance。

Today we're going to be turning to a new topic， interest rates， but before， doing so。

I want to briefly review our first topic， time value of money。 So we。

started off that topic by introducing the concept with some intuition。 What we did。

is we showed that cash or money received or paid at different points in time， has。

a different time unit and as such can't be added， much like different currencies， can't be added。

Then we introduced the tools associated with the time value of， money。

namely a timeline which is just a visual representation of when money is。

coming or going and a discount factor which is our exchange rate for time。 It's。

what we used to move cash flows forward and backward in time。 And we use those。

tools to discount cash flows that has moved them back in time and compound cash。

flows or move them forward in time。 We introduced some useful shortcuts， namely。

the present value of an annuity formula and the present value of a perpetuity， formula。

as well as formulas for the present value of a growing annuity and a， growing perpetuity。

Then we closed off the topic with a discussion of taxes and。

inflation and investigated how those those two concepts would impact both。

our dollar return and the purchasing power of those dollars。 Now I want to turn。

to interest rates in this lecture and this isn't so much a new topic as much as it。

is really an extension of the time value of money to incorporate institutional。

details and make things a little bit more realistic。 So we're going to talk about。

interest rate quotes and we're going to learn how to deal with cash flows that。

don't arrive once a year but may maybe arrive monthly or semiannually。 And we're。

also going to discuss how to deal with compounding of interest when it's not。

just annual。 So let's get started。 So here's a snapshot from December of 2014 of。

rates on five-year Jumbo CDs where CD is just stands for certificate of deposit。

It's a savings vehicle that most banks offer and here's one， two， three， four， banks。

Now the Jumbo that just refers to I think a minimum deposit of a hundred， thousand dollars。

So these are big deposits。 And one of the things you notice when。

looking at these rates is when you're when you're looking at these savings。

vehicles is that each one has two different rates。 It has a rate and an APY， and these numbers 2。37。

2。4， they're different。 So that begs the question is， why are they different？ How are they related？

And most importantly which one， is going to tell me how much money I'm going to make when I invest in this。

product。 Well let's go through this starting with the rate。 The rate refers to。

the APR the annual percentage rate。 That measures the amount of simple interest， earned in a year。

Simple interest is just the interest earned without， compounding， ignoring compounding。

And if you're wondering what compounding is， we're going to talk about it but just as a preview notice underneath the rate。

we have compounded daily。 Simple interest ignores that compounding frequency。 Now。

many banks quote interest rates in terms of an APR。 The problem is the APR is。

typically not what we're going to earn or what we're going to pay。 For that we。

have to turn to the APY or the annual percentage yield which is really just。

another way of saying EAR or effective annual rate。 See the EAR that measures。

the actual amount of interest earned or paid in a year。 That's what we care， about。

The EAR that's the number we care about。 The rate or the rate or the APR。

that's just a quoting convention。 Now how are these different rates related？ Well。

before showing you the explicit mathematics which frankly are almost。

trivial let me emphasize the following。 The EAR is a discount rate。 The EAR is what。

matters for computing interest and discounting cash flows。 The APR is not a， discount rate。

It's a means to an end。 It's a quoting convention。 We're going to use。

APR in conjunction with compounding frequency information to get at an EAR。

or at a periodic discount rate which I'll introduce in just a moment。 So。

remember EAR equals discount rate。 APR equals quote。 Now how do we get from one， to the other？

How do we move from APR to EAR or vice versa？ Well here's that simple。

mathematical formula I referred to just a moment ago。 The EAR is related to the。

APR by this equality。 Well what's going on here？ K that's just the number of。

compounding periods per year。 So imagine we had monthly compounding。 That would， imply K of 12。

How about semi-annual？ That would imply K of 2。 And I'm going to。

introduce a little bit of notation here。 I is the periodic interest rate or the。

periodic discount rate and that equals APR over K。 Let's do an example。 So。

imagine we're investing $100 in a CD offering 5% APR with semi-annual， compounding。

How much money will we have in one year？ Well there's actually two。

ways to approach this problem and we're going to do each in term。 The first thing， we do。

first thing we always do is we draw a timeline and so today period zero。

we're going to invest $100 and the question is asking how much money are we。

going to have in one year？ Now I've left this as a question mark to emphasize the。

fact that there are two ways to go about this。 Let's go about this the first way。

Go about answering this problem。 The first is to work in periods。 With semi-annual。

compounding that means there are two periods per year since I'm interested in。

how much money I have after one year that's after two periods。 And these。

periods are every six months period one， period two。 So if I'm going to work in。

periods I better compute a periodic discount rate。 That's I which we know is。

APR over K in which in this case reduces to 2。5%。 In other words I'm going to earn。

two and a half percent over the first six months and I'm going to earn two and。

a half percent over the next six months。 So I take my initial $100。

investment I multiply it by my periodic discount rate and after one period I've， got $102。

50 then I repeat I take that $102。50 I multiply it by my periodic， discount rate to get $105。

00 and a little over 6 cents。 In other words the future。

value two periods hence of $100 in this setting is just 100 times 1 plus I， raised to the power 2。

We're working in periods so I is our discount rate and 2。

is the number of periods we're moving the money forward in time。 Now let's。

approach the problem from the perspective of years。 Now we're looking for how much。

money we have after one year but if we're going to work in years okay now we need。

to be consistent so after six months this isn't one period this is half a year and。

because we're working in years our discount rate isn't going to be the。

periodic interest rate I it's going to be the ER the equivalent or the。

effective annual rate excuse me which we know from earlier is just one plus the。

periodic rate raised to the number of compounding periods per year so one plus。

I to the K which in this setting comes out to 5。0625 percent。 In other words over。

this entire period I'm going to earn 5。0625 percent which turns out to equal， $105。

06 a little over 6 cents that is the future value one period one year。

excuse me one year from now of a hundred dollars is a hundred dollars times one。

plus the EAR raised to the power one we're working in years which gives me the。

exact same answer we got before the one hundred five dollars and six then a。

quarter cents so the lesson if you discount cash flows using the EAR then。

you better measure time in years if you discount cash flows using the periodic。

interest rate then you better measure time in periods and the equality between the。

two that we just showed with our simple example is much more general and given。

by this straightforward proof which I'm not going to discuss in detail but I'll。

show you there if you're interested in looking at it so just to summarize we can。

work in periods right one two with a periodic interest rate or we can work in。

years with our effective annual rate and notice I measure everything。

consistently periods versus years so let's go back to our original example。

excuse me we had an APR of 2。37 percent we have a compounding frequency daily。

which I'm going to assume 365 days though you should be aware it could be 360 days。

or it could be 252 business days it depends upon the convention used for that。

product by that institution so this these two pieces of information the APR and K。

allow me to compute the periodic rate I which is 0。006741 percent now that's a。

tiny number but we're computing interest every single day here which means the。

effective annual rate which equals 1 plus i to the K or which in this case is 1， plus 0。

006714 percent raised to 365th power gets me to just under 2。4 percent， so when we round the 2。

398 we get about 2。4 percent。

all right so let's summarize we learned that there are two discount rates depending。

on what units of time we want to work with if we want to think in terms of。

periods we want to use a periodic discount rate that's I if we want to。

work in years we want to use the ER the effective annual rate which relies on。

cash flows measured in years both of these are discount rates they'll both get。

us to the same goal or the same end result but we have to be consistent in。

terms of how we measure time with which discount rate we use APR that's a。

quoting convention that's a means to an end we use APR in conjunction with the。

compounding frequency to get our discount rate whether it's the ER or the。

periodic interest rate or the periodic discount rate high and we can move。

between the APR and I and the ER by a couple of very simple mathematical。

relations that we discussed now next are a bunch of great problems that I want you。

to dive into and then after you're done with those move on to the second part of。

interest rates in which we're going to investigate the term structure of。

interest rates and talk about the yield curve let you know what these things， mean thanks so much。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P108：6_期限结构.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we talked about interest rates。 In。

particular， we talked about how interest rates are quoted versus how。

interest rates are used to discount cash flows。 We also talked about how to。

deal with cash flow streams when the cash flows arrived more than once a year。

or less frequently than once a year and when the compounding frequency wasn't， annual。

This time I want to talk about how interest rates or discount rates。

can vary over time and how that relationship is captured via a term。

structure of interest rates and yield curve。 Let's get started。 Hi everybody。

welcome back to Corporate Finance and our second lecture on interest rates。 So。

last time we introduced the topic by talking about interest rate quotes versus。

discount rates and we talked about an APR annual percentage rate which was a。

means of quoting interest rates that financial institutions often use and。

that's often distinct from what we care about for discounting cash flows which is。

an EAR effective annual rate or a periodic discount rate and we talked about how。

to move between these concepts or these constructs and then we showed how to。

apply them to non-annual cash flows and in situations where we have non-annual， compounding。

Say monthly， semi-annual， whatever you might have。 Today I want to。

talk about the term structure of interest rates and the yield curve and this。

lecture is a little bit different in that it's not geared towards solving。

problems per se at least not directly because that's going to take us into a。

fixed income valuation which is beyond the scope of this course。 Rather what it's。

going to do is it's going to be important to understand what these concepts are。

because they're going to be used for corporate decision making later on。 So， let's get started。

All right， thus far we've assumed discount rates are constant。

through time they just don't change and what do I mean by that？ Well if I look at。

my present value formula right I take my cash flows and I discount them by one。

plus the discount rate but notice that's the same discount rate it's the same。

number for every cash flow regardless of when the cash flow arrives。 Now in reality。

it seems as if interest rates vary with the term of the investment。 Let me give you， some examples。

Here's a screenshot of home mortgage refinancing rates that I。

took not too long ago and you can see that as the term of the mortgage。

refinancing varies so too does the rate or APR and consequently so does the EAR。 The discount rate。

Likewise when I looked at fixed term CD rates where I remember。

CDs or just certificates of deposits their rates tend to vary with the term of。

the investment as well and there's a lot of numbers here so let me focus your， attention here。

Here we have the term of the investment and here we have the APR。

and the EAR and you can see that as the term increases or changes so too do the， interest rate。

Now what's the point here？ Well as the term of the investment。

changes quite often but not always quite often the interest rate will change and。

the term structure is nothing more than the relation between the investment。

term and the interest rate。 The yield curve is just a graph of that relation so。

let me show you a treasury yield curve from July 24th 2014 on the horizontal。

axis we have the maturity of the treasury security so the here here's a one month。

T-bill right here's a 30 year T bond here's a five year T note so these are just。

different treasury securities that vary by maturity across the horizontal axis and。

on the y-axis is the yield which for the time being and in this context you can。

think of loosely as the discount rate R and the point is is that as the maturity。

of the security varies so too does the discount rate right in other words when。

the when the government borrows for 30 years it's getting and or paying an。

interest rate just above 3% whereas when it's borrowing over a。

short horizon say 30 days it's paying basically zero now let me come back to。

this notion of a yield and what a yield is a yield y is the one discount rate。

that when applied to the promised cash flows of the security recovered the。

price of the security and that is a mouthful so let me actually show you a。

little formula that you're familiar with to hopefully clarify this see I take。

the price of the security on the last slide the price of the T-bill the price。

of the T bond and I lay out all of the promised cash flows to the security I。

discount them back and I ask what is the discount rate such that when I discount。

these cash flows I get the price that is the yield or yield to maturity so to。

build the yield curve that's just a matter of simply computing the yield for。

securities of different maturities and so without getting into the institutional。

details and the semi-annual compounding and quoting conventions associated with。

treasuries let me just let me just talk conceptually so if I want the one year。

the one year yield I would just take this say cash flow at year one divide by one。

plus y one set it equal to the current price and solve for y if I want the。

second healed I would take the second which the second security with maybe。

annual cash flows of one and you know CF one and CF two and I would solve for the。

one discount rate y two let's call it such that when I discount these cash flows。

CF one and CF two I get the price P2 and we do that for all different。

maturities and the why is that come out Y1 Y2 Y3 dot dot dot dot all the way down。

to Yt that represents the yield curve those are the points on the yield curve。

but that's the same as computing the discount rate for securities with。

different maturities hence the link between discount rates and yields now。

there is a difference between discount rates and yields having to do with。

promise versus expected cash flows but that's a little bit outside the scope of。

the course so let's leave that aside for now and in the context of treasuries yields。

and expected returns are relatively close now one thing I want to emphasize is。

that yield curves in other words interest rates they move around a lot or at least。

they can and so to illustrate that fact I've plotted three yield curves here the。

purple one is from 2012 the blue one is from 2000 and the red one is from 1981。

and you can see that the rate at which the government was borrowing has very。

dramatically over time right today for short-term loans they're not paying any。

interest the interest rates basically zero back in 1981 the interest rate was。

around 15 percent the other thing I want to point out is that the the。

relationship between the short end of the yield curve short-term interest rate and。

long end of the yield curve long-term interest rates that can vary over time。

as well here at least in 2012 we see that the curve is upward sloping so that。

interest rates short-term loans to the government are less costly than long-term。

loans whereas back in 1981 we see that long-term interest rates were actually。

below shorter-term interest rates at least over a stretch here so it was it was。

cheaper for the government to borrow at longer rates or over longer terms at least。

at a lower interest rate now this raises the question well what does this mean。

what does the what does the upward sloping curve in 2012 mean as opposed to the。

downward sloping curve in 1981 well there's a lot of academic debate about。

that but one popular opinion is that the slope reflects expectations of future。

interest rates so when the yield curve is upward sloping as it is in 2012 this。

suggests that future interest rates are likely to be higher whereas back in 1981。

this downward sloping portion also actually in 2000 it looks slightly。

downward sloping as well the downward sloping curve suggests that future。

interest rates are gonna be lower treasury yield curves what they're doing is。

they're graphing the relationship between interest rates on on risk-free loans and。

loan maturity so that they have a very special meaning when we talk about the。

risk-free rate we're really talking about the interest rate or the yield on。

treasuries but we can plot yield curves for a host of different securities and。

that's what I've done on this next slide is I've got three different yield curves。

the blue curve is the yield curve for high quality corporate debt so you can。

see that for what do I mean by high quality that's investment grade think。

triple B or higher so when high quality or highly credit rated firms were。

borrowing for say 29 years it's costing them about 5% per annum at least as of。

July 2014 whereas on the when they're borrowing short term say two years it's。

a little over a percent the green curve is the yield curve from。

municipal bonds triple a rated municipal bonds as of July 2014 and then the red。

curve is the treasury curve and what's sort of interesting to note is that the。

yields or the cost of borrowing was higher appears to be higher for the。

federal government over at least certain periods than it was for municipalities。

which is strange if you think that our federal government is is a much safer。

bet than a municipality even a triple a rated municipality but what's going on。

there is mostly a tax differential as well as some liquidity issues all right so。

what's the lesson here well look yields vary by maturity and risk so to。

illustrate that point I'm gonna look at corporate bond the yields on corporate。

bonds and you can see as the ratings credit rating improves from triple B to。

triple A which is the best rating the yield goes down and I know drawing the。

up arrow isn't helping but but the numbers are getting smaller and they're。

getting smaller within each maturity bucket one to two two to five five to ten。

ten plus and that's a result of decreasing or credit risk right these are。

these are safer bond these are safe these are less safe but still relatively safe。

the other thing to notice is that within a credit rating say triple A the。

yields or the interest rates are different now in this case they're。

increasing but as we showed a few slides ago right they don't always have to be。

increasing with maturity they could be decreasing the point is that they're。

different across the maturity spectrum now all of these interest rates that we've。

discussed as far referred to as spot rates what are spot rates they're the。

interest rate for a loan that's made today now typically there's a different。

spot rate for loans of different maturities and different risks that's what。

we showed right here these are spot rates they vary by credit risk and they vary。

by maturity now what's one of the big punchlines or takeaways from all of this。

aside from just general knowledge and information understanding interest。

rates well this present value formula that we've been working with is in many。

ways just an approximation because we know interest rates vary over time so。

it's really an approximation for this now it's not such a bad approximation when。

the yield curve is flat so when we have a yield curve here's maturity here's。

yield when the yield curve is flat r1 r2 r3 are the same so these two things。

would be equal the problem let's see if I can erase some stuff here the problem。

is when the yield curve is say upward sloping severely or even downward。

sloping now there's going to be a pretty big difference from using some sort of。

average discount rate r as a proxy for all of the different spot rates okay so。

let's wrap up here what did we talk about today we talked about the term。

structure of interest rate which captures the relation between interest。

rates and the investment term right loans or savings of different。

maturities different terms will often have different interest rates and that。

relation is captured by the term structure the yield curve simply graphs。

the term structure right it plots on the horizontal axis the maturity on the。

vertical axis the yield or the interest rate and that shows us the relationship。

between investment term and interest rate we also learned that in the interest。

rates will vary by the risk of the investment though we discussed that。

earlier on back in our I think our first lecture and we talked about spot rates。

which are the interest rates for a loan that's made today so what's up next we。

turn to a new topic discounted cash flow I look forward to seeing you then thanks， a lot。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P109：7_折现现金流决策制定.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 This time we're going to turn to a new topic。

discounted cash flow analysis。 But before doing so， I want to briefly recap our， last topic。

interest rates。 We started off discussing interest rate quotes and。

how they differed from the interest rates we use to discount cash flows。 We。

then discussed how to discount cash flow streams when the cash flows。

arrived at frequencies other than annual and when the compounding of。

interest was something other than annual as well。 We closed off the topic by。

discussing the term structure of interest rates and the yield curve both of。

which summarized the relationship between interest rates and the term of， the investment。

Today we turn to discounted cash flow analysis and I want。

to start the topic off by discussing how firms or people more generally should be， making decisions。

Let's get started。 Hey everybody， welcome back to Corporate， Finance。

Today we're going to be shifting gears to a new topic， discounted cash， flow analysis。

But before doing so， I want to briefly recap our last topic， interest rates。

So in that topic we started off by introducing the notion of。

APR and EAR and a periodic discount rate。 The difference being as follows。

APR is our annual percentage rate or a means to quote interest rates。 Whereas， EAR。

effective annual rate and the periodic interest rate are actually， discount rates。

They're economically relevant。 They're what we care about。

they're what we use to discount cash flows and make decisions。 Then we use。

those discount rates to value and assess cash flow streams that were no longer。

arriving on an annual basis and to deal with the compounding of interest on a， non-annual basis。

Then we closed off the topic with a discussion of the term。

structure of interest rates which is the relation between the interest rate and。

the term of the investment or the duration of the investment。 And we discussed， yield curves。

loosely how to construct them， what the yield means， where it comes。

from and how to interpret these graphs。 Now we're going to switch over to。

discounted cash flow analysis and I want to start by motivating the topic and。

providing a little bit of background by discussing decision making。 So let's get， started。

Let's start with a question。 How should we make financial decisions？

Well a reasonable answer might be to undertake those actions that create value。

value for those affected by the decision， value for the owners of a firm for example。

But which actions create value？ And well that's a complicated question， a sort of。

general answer that seems reasonable would be to consider actions in which the。

benefits exceed the costs。 But here's a little wrinkle。 What if the costs and。

benefits arrive at different times？ Well actually we're well equipped to deal。

with that wrinkle because we can compare the present value of the benefits to the。

present value of the costs。 See because by computing the present value we know。

that the discount rate R will adjust both for the timing and the risk of the， cash flows。

So that brings us to our first lesson which is that the NPV。

decision rule that is NPV which stands for net present value NPV。 The NPV。

decision rule says that we should accept all projects with a positive NPV and。

reject all projects with a negative NPV where NPV is nothing more than the。

difference between the present value of the benefits and the present value of， the costs。

In other words if the present value of the benefits is big is bigger。

than the present value of the costs that's a good thing that creates value and we。

want to undertake that project or make that decision that leads to positive NPV。

And while this formula right here looks somewhat vague and ambiguous it's。

actually masking something that we're already very familiar with and that's。

just discounted cash flows。 NPV is nothing more than discounted cash flows。

except now instead of cash flow CF I have this F in front of all the cash flows。 So， it's F CF。

The F simply stands for free。 So these are free cash flows but they're。

still just cash flows nonetheless。 There's nothing special there。 I'm going to。

formally define and discuss how to compute free cash flows in our next。

lecture but for the time being just recognize we're doing nothing new here。

We're simply taking a stream of cash flows FCF not FCF one through FCF T and。

we're discounting them back to today by a discount rate R to get a present value。

a net present value。 That's it。 And so the decision rule just says if the NPV is。

greater than zero except the project if the NPV is less than zero reject the。

project sorry reject the project and while that seems fairly easy and straightforward。

actually implementing it is quite a bit more subtle and we're going to talk about。

those subtleties as we move along throughout this topic。 So before moving on。

to some of the mechanics of a DCF analysis I want to talk about decision。

making in practice briefly because it's useful to motivate it and to understand。

what people in different areas of both finance and non-financial sectors are。

doing when they're making decisions。 So the first thing I want to look at is this。

or show you is this survey evidence from a survey by John Graham and Cam Harvey。

former colleagues at Duke in which they surveyed CFOs from the Fortune 500 and。

other domestic US domestic firms and they asked them how frequently do you use。

capital budgeting different capital budgeting techniques and what you can see。

is that there actually a variety of responses I've listed six here but the。

majority or the predominant number of responses point to net present value and。

internal rate of return as the most popular decision criteria that CFOs use。

that's followed somewhat closely by the payback rule and to a lesser extent the。

discounted payback rule and in our in our discussion of DCF well NPV is going to。

be center stage we're actually going to spend a fair amount of time discussing。

all of these decision rules because each one actually has important。

information for the decision-making process though each one also has certain。

limitations that we need to understand now that's in the non-financial corporate。

sector if we look at what in vain investment bankers do when they are。

value in companies in fairness opinions you can see that the large majority of。

investment bankers are also relying on discounted cash flow analysis you know。

something akin to NPV but they're also using a host of comparables methods or。

relying on multiples and while this is going to take us a little bit outside of。

the scope of our course I might mention if we have some time how we use multiples。

in in in valuation exercises if not I might post some additional material and。

finally there's a recent survey by Paul Gomper Steve Kaplan and Vladimir。

Mukaraliya Mav I hope I'm not butchering that name too poorly but they have a。

really interesting survey of private equity firms and in which they investigate。

among among many things which criterion PE investors use to evaluate an。

investment and what's interesting here is is the vast majority rely on internal。

rate of return by a wide margin over any other criterion so let's。

summarize this so we're gonna move through this topic of DCF discounted。

cash flow emphasizing to a certain extent the NPV rule because that is the。

optimal rule in terms of always leading you to making the decision that creates。

value that said I'm gonna take a much more practical perspective to corporate。

decision-making or decision-making more financial decision-making more broadly。

and recognize that other rules are still informative they're still useful and so。

it's not surprising that we see their use by practitioners whether it's PE。

private equity investors or investment bankers or CFOs they're still。

informative but the important thing to keep in mind is that these other rules。

have certain weaknesses that we need we need to understand and we need to。

recognize the limitations of these rules so that we can use multiple rules in。

conjunction to come to the best decision so what's next we're gonna dive into。

how to compute free cash flows are the first stage in understanding how to。

execute a DCF so I look forward to seeing you in the next lecture thanks so much。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P11：10_为直销喝彩.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

So while the Harris and Tesco stories are terrific， and we'll provide a point just to。

some books that summarize each of those stories quite well， I want to emphasize that they're。

not the only ones who have built a business around a deep understanding of their customers。

and by no means are they the first。 In fact， the first companies that actually built a business in this manner around their。

customers has happened many， many years ago， and it emerges from the sector of direct marketing。

When I say direct marketing， most people don't have a real positive association with it。

They think about low-end products， they think about infomercials and other not great marketing。

activities。 It's not the kind of industry that you aspire to be associated with or learn from。

But when you strip away what most customers see from direct marketers and look at the。

actual business practices below the surface， you realize that it's actually quite impressive。

If you look at what direct marketing is really all about， it is really building the business。

around the customer， but not just the customer in some generic sense， but around each and。

every customer。 It's about understanding the relationship with each different customer who has bought。

what from us， for how much， what kinds of products have they inquired about， what kinds。

of products have they returned， what interactions have they had with customer service。

That's what direct marketing is all about。 It's having that much richer relationship between the company and the customer。

What's interesting about it is that direct marketing is not a new concept。

It's been around since 1967 when Lester Wonderman looked at these emerging set of data-driven。

business practices and said， "You know what？ There's actually a lot we can do。

We can actually formalize some of these business practices and come up with some best practices。

associated with them。"， But even if you don't spend a lot of time thinking about direct marketing。

a lot of the， words and the concepts have already filtered their way into today's everyday marketing。

conversation。 So a lot of these segmentation concepts that Barbara discussed are often associated with。

direct marketing， but even other expressions like customer lifetime value， something that。

you've heard about before and that we're going to spend more time talking about， that comes。

directly from the direct marketers。

So the direct marketers were the first ones who said， "You know what？

We can collect all this data about our customers， about each and every one of them。

And we can actually build a business by understanding who the valuable customers are， who the less。

valuable ones are， which messages we should be sending to which customers at which time。

and importantly， what kinds of products we can develop and deliver in order to create。

more value for our most valuable customers and to try to attract more customers like， them。"。

So the Harris and Tesco stories are wonderful， but they're not unique。

And so I want to spend a lot of time celebrating some direct marketing practices。

And I want to emphasize that a lot of firms out there today might not aspire to be direct。

marketers， but they don't realize it， but they are。 Any company that's operating on the internet。

any company that has the capability to track， a particular customer over time has the capability to learn from direct marketing。

And I encourage all of you to read books on direct marketing， even if you don't think。

about yourself that way。 There's just so many concepts that you can learn and leverage。

especially as we enter， this world of big data。 [MUSIC]。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P110：8_自由现金流.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we introduced the new topic discounted。

cash flow analysis by discussing how corporations or people more generally。

should be making decisions。 We discussed several different decision criteria。

focusing in particular on three the NPV rule， the IRR rule and the payback， period。

We also discussed how in practice we should be taking a practical。

approach and using the relevant information from all three rules。 Today。

I want to talk about free cash flows， a critical element in implementing any of。

those decision rules。 Let's get started。 Hey everybody welcome back to Corporate， Finance。

Today we're going to start in on lecture two of discounted cash flow， analysis。

Up before doing so let's do a brief recap of our last lecture。 So we。

introduced the topic discounted cash flow analysis by discussing decision， making。

And we talked about several different decision rules used in。

practice beginning with the NPV rule which is nothing more than the difference。

between the present value of costs and the present value of benefits。 And while。

most academics including me will argue that this is the best rule that you， should always use。

I take a practical view and recognize that other rules such as。

internal rate of return and payback period contain relevant information for。

decision making that if used wisely and in recognition of their limitations can。

make can lead to better decisions。 And so that's sort of the overarching。

approach we're going to take to DCF is that there are several different decision。

rules will rely on NPV but use what we can from other decision rules。 Now in this。

lecture I want to talk about sort of the bedrock or the found one of the key。

components of any DCF and any sort of decision making and that's free cash。

flow specifically what is it。 So let's get started。 So remember there are two， components to NPV。

There are free cash flows and there's a discount rate。

because NPV as you recall from last time is little more than just a discounted。

stream of cash flows in this case free cash flows。 So recall NPV was formally。

defined as follows and we recognize that this is really nothing new。 This is。

what we've been doing all along。 What I'm going to focus here is on the。

numerator the free cash flows。 How do we get at these free cash flows in a。

corporate setting when whether it's capital budgeting or valuation more。

broadly how do we compute these FCFs。 That's what this lecture is really about。

So let's let's take a look。 Free cash flow begins with revenue or sales。 We。

subtract off costs and then we subtract off depreciation actually。

depreciation and amortization but mostly depreciation by depreciation or， amortization both。

Now you might wonder well first what is depreciation。

Depreciation is just a countenance way of recognizing the loss in value the。

deteriorate deterioration and value of physical assets like plants and。

equipment but it's sort of an accounting notion。 It doesn't represent a true。

cash flow right when a plant depreciates it's not as though money is leaving the， company。

So you might wonder why we're even considering it。 Well the reason we're。

considering it is because we're going to take this term in parentheses that I've。

now bracketed and multiply it times 1 minus TC。 TC is the marginal tax rate。

See even though depreciation doesn't represent any dollars flowing out of。

the company or away from a project it's not a literal cost in terms of， dollars。

What it does is it does reduce our taxable income it provides a tax。

shield and so we have to take that into account when computing free cash flows。

because taxes are actual dollars leaving the company or the project。 Now this。

quantity here goes by several names they're all synonyms。 Unlevered net income。

net operating profit after taxes or notepad or earnings before interest after， taxes EBA。

Once we have this we're gonna have to add back in depreciation and again。

that's just to net out the subtraction of depreciation here。 It doesn't represent。

a true cash flow what it does represent is a tax shield so we add back in the。

depreciation then we're going to subtract off you get rid of that capital。

expenditures or any investments we have to make and then finally we're gonna。

subtract off the change in networking capital and that sometimes throws people。

for a little bit what do you mean subtract off the change in networking， capital。

Well first what is networking capital？ Well networking capital NWC is。

equal to current yeah that that's supposed to be a U。 Current assets minus。

current liabilities and current assets that's loosely cash plus accounts。

receivable plus inventory and current liabilities we're gonna focus on， accounts payable。

Now some of you might say what about short-term debt or long-term。

debt that's coming due that's financing leave that aside that's a separate issue。

we'll talk about that a little bit in a little bit okay so what goes into the。

free cash flow calculation is not networking capital the change is important。

because these are all what economists call stock variables and we're trying to。

compute flows so we want to look at the change the period over period change in。

networking capital and subtract that from free cash flow。 So what what is free cash。

flow intuitively well it's the residual cash flow that's left over after all。

of the projects requirements have been satisfied and the implications accounted。

for it's the cash flow that can be distributed to the financial claimants of。

the company debt and equity that's another way to view it it's not the same as a。

counting cash flow from the statement of cash flows but we can derive free cash。

flow from the statement of cash flows actually with just a few steps now I want。

to be precise here the free cash flow we we're gonna compute here or is defined。

right here this is unlevered free cash flow and I say unlevered to distinguish it。

from free cash flow to equity or free or or levered free cash flow which we'll。

discuss on the next slide so free cash flow to equity starts with free cash。

flow the definition of free cash flow that's just this quantity right here and。

then we've added well we've appended two terms we're gonna subtract off the。

after-tax interest cost and add back in any net borrowing that is borrowing。

above and beyond any repayment of debt so another way to compactly write the。

free cash flow to equity is that it starts with unlevered free cash flow the。

free cash flow available to debt and equity holders and we subtract off the。

after-tax interest expense and add in the net borrowing so what is free cash。

flow to equity it's the residual cash flow leftover after all the projects。

requirements have been satisfied implications accounted for and all debt。

financing has been satisfied that's that's critical we take care of the debt。

holders for us because they're senior claimants free cash flow to equity is。

the cash flow that can be distributed to the shareholders equity holders of the。

project or the company and free cash flow to equity this is look at it this。

should be there we go is more precisely levered free cash flow because free cash。

flow to equity is affected by the the company's choice of financial structure。

it's a leverage decision how much debt it plans to take on so let me show you。

this what we've just discussed schematically in terms of a hierarchy that。

that's kind of getting us a little bit close to an income statement right。

remember free cash flow starts with up at the top with revenue we're gonna pull。

out the depreciation well yeah the revenue comes in we're gonna pull out the。

depreciation and costs we're gonna pull out our taxes that's gonna give us our。

unlevered net income this is our no-pat or our EBA then we're gonna add back in。

the depreciation because it's a non-cash expense and again that could be。

depreciation and amortization we're gonna subtract off our capital。

expenditures our investment requirements we're gonna subtract off our。

investment in networking capital the change in networking capital and that's。

gonna give us our unlevered free cash flows these are the cash flows that can。

go to debt holders and equity holders then we're gonna pull subtract out our。

after-tax interest expense and I say after-tax because remember interest is。

tax deductible right it provides us a tax shield and then we're gonna add back。

in and in that borrowing so if the company borrows some company borrows some。

company borrows some debt in excess of what it repays that's a cash inflow。

that's available for use by the equity holders and what we're left with down here。

at the bottom is the levered free cash flow or the free cash flow to the。

equity holders so strategic decisions a useful way to think about strategic。

versus financial decisions is that strategic decisions are gonna affect the。

inputs to free cash flow well and free cash flow to equity but they're。

impacting here right strategic decisions are gonna affect our。

market share and our revenue they're gonna affect our cost our。

investment decisions right investment decisions are our operations you know。

inventory and the like all of these strategic decisions are gonna impact our。

unlevered free cash flow and ultimately our levered free cash flow。

financing decisions are down here they're gonna impact our leverage choice the。

the after-tax interest expense the net borrowing okay and hence that that's why。

we refer to these cash flows here as unlevered they're unaffected by leverage。

choices now strictly speaking that's not always true in more advanced value。

valuation courses you might see financing decisions feeding back on some of the。

inputs into unlevered free cash flow such as the ability of a firm to invest or。

or its revenue growth but a useful benchmark and a common benchmark is。

just is just to recognize that financing financing decisions do not affect the。

unlevered free cash flow and that's a great way by the way to check your。

computations in any sort of valuation or DCF exercise at least one way is by。

recognizing that financial policy is not going to hit the unlevered free cash。

flows all right so let's let's summarize this now。

you know NPV is a decision rule that's going to quantify the value implications。

of decisions that that's what we learned last time and it there's two key。

components to it right there's free cash flows and discount rates this lecture。

was all about how do we put some meat behind that FCF term how do we actually。

compute it how do we derive it for a project or a firm more more more broadly。

speaking and what we saw is that there's a relatively simple formula for computing。

free cash flows and don't be misled by its simplicity it applies broadly yeah we。

can always use this definition the trick in practice is actually estimating the。

components and figuring out what hits unlevered free cash flow versus levered。

free cash flow what's relevant for cash flow what's not relevant for cash flow。

and the remainder of this topic is going to be geared towards answering those。

questions in the context of an actual example so coming up next we're going to。

start in on forecast drivers are trying to understand how we're going to forecast。

each of those components of free cash flow out into the future thanks so much。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P111：9_预测驱动因素.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。

Last time we introduced Free Cash Flow。 We talked conceptually about how to compute Free Cash Flow。

Today what I want to do is I want to introduce a capital budgeting application。

to illustrate discounted cash flow analysis， and in particular what I want to do is emphasize the role of forecast drivers。

or the assumptions we need to forecast free cash flows out into the future。 Let's get started。

Hey everybody， welcome back to Corporate Finance， and our third lecture on the DCF topic。

"Forecast Drivers"。 Let me start with a brief recap of what we talked about in our last lecture。

which was Free Cash Flow。 In that lecture we talked about what Free Cash Flow is， how to compute it。

We formalized it， we actually gave it a specific formula。

We also talked about Free Cash Flow to equity， the levered counterpart to Free Cash Flow。

or more precisely unlevered Free Cash Flow。 What I want to do in this lecture is talk about forecast drivers。

or the assumptions required to forecast each component of the Free Cash Flow formula。

out into the future。 We're going to do it by way of a specific example。 Let's get started。

Imagine we're a company in say 2008， and we're considering， building and selling a tablet。

Back in 2008 this was really just the start of the tablet market。

I've put our Free Cash Flow formula up here to highlight the components that we're going to need。

Remember， the question is， should we enter this market？ Should we produce and sell this tablet？

Well to make that decision， what we're going to want to do is a DCF。

and we're going to want to forecast each component of this Free Cash Flow formula into the future。

Let's do that。 We're going to start with revenue。 So revenue is market size。

how big the market is in terms of units。 Times market share。

what proportion of the market we're going to capture。

times our price per unit。 So let's look at our forecast here。 So I'll start with the market size。

Now， I'm going to assume that in the first year the market size is going to be a million units。

That's going to be the early adopters， the techies。

the people who are really excited about new technology， and the introduction of new technology。

So that's going to be one million units in year one。

but that's then going to grow quickly in subsequent years。 And you might say。

twenty five hundred percent， that's ridiculous。 And it's an absolutely enormous growth rate for sure。

But remember， this is the beginning of a new market that we're anticipating will take off in a big way。

So twenty five hundred percent in the second year followed by one twenty eight。

nine point four and three point five percent。 Now you might wonder。

where in the world am I getting these？ Well， I'm getting them from a variety of different sources。

I'm getting them from my marketing people and my strategy group within the company。

I'm getting them from industry analysts as well。 So there's a host of different sources of input that go into these numbers。

but at the end of the day they are just assumptions and I'm going to emphasize that fact later on。

So I can use these two assumptions about the market to compute a market size in terms of millions of unit。

just one million in the first year。 That's going to grow by twenty five hundred percent。

one hundred and twenty eight， nine point four， three point five。

So that's my forecast for the size of the tablet market。 Now what's interesting。

what I'm going to show you in this next slide is here is the actual market size for tablets。

And what you can see， even with a twenty five hundred percent growth rate。

I was conservative in my estimate。 Twenty six million as opposed to sixteen。

sixty million in sales in the second year followed by one sixteen， one ninety five， two twenty nine。

So that sales， sorry， not sales， the actual size of the market。

So you can see this market took off incredibly rapidly。 Next I want to look at my market share。

I'm going to assume an initial penetration of twenty five percent。

followed by annual growth of five percent each year thereafter。

That's going to allow me to forecast my market share， which I bring down to twenty five percent。

That's going to grow at five percent， five percent， five percent and five percent。

So this is just to be clear， twenty five percent times one plus five percent。

That'll get you to twenty six point three percent and likewise for the subsequent forecasts。

And finally we have to come up with a pricing strategy。

I'm going to price our tablet initially in year one at two hundred dollars per unit。

which I just bring down here。 I'm not going to experience any price change in the second year。

but then I'm going to increase prices in the third year by just under fifty dollars。

Now you might wonder what am I thinking consumer electronics prices go down and go down fairly rapidly。

But what I'm going to do is I'm going to build in some element of versioning here。

So just like we had the iPad， iPad 2， iPad Air， I plan on doing some R&D， which I'll discuss later。

that's going to develop new better versions， lighter， more powerful， better screens， etc。

So that's going to justify an increasing unit price over time。

Let's move on to costs。 So I'm going to start with cogs or costs of goods sold。

which I express as a margin here or a percentage of sales。

And I'm going to assume that's going to run at about eighty point six， six percent。

A number based on our company's experience working in the computer industry for some time based on estimates from our operations people and insights from other analysts and consultants。

My SG&A sales general and administrative expenses are going to run at about one percent of my current SG&A or sixty nine point five nine million dollars。

I'm just making that number up there with me。 So it's going to be sixty nine point five nine million in the first year。

And I'm going to assume that's actually going to grow by twenty five percent per year。

Because hopefully a sales in the size of this market grows， I'm going to need more。

I'm going to have more overhead to keep up with that growth。 Whereas I'm tying cogs。

which is loosely viewed as a variable cost， I'm tying that forecast to sales。 Now R&D。

well I'm going to have to spend two hundred million dollars up front in R&D to get this thing up and running。

And then I'm going to have an additional twenty， so that's this up front R&D。

then I'm going to have an additional twenty five million every year thereafter to deal with versioning and improvements。

Now let's turn to our investment needs。 I'm going to assume an up front investment of two hundred twenty seven point seven million dollars to build a plant。

get any necessary equipment for production。 My first year investment is going to be ten percent of that initial investment。

and then subsequent years are going to be annual growth of five percent， and then one one and one。

And you'll see how this rolls out in the next lecture when we actually forecast those free cash flows。

But these are just the assumptions， we're laying the groundwork here。

Now at the end of this project， which is five years， right， so I'm going to assume。

I mean assuming that this is a five year project。 At the end of this project I'm going to have all of this plant property and equipment left over。

Well let's focus on the plant and equipment。 It's going to be there。

it's not going to just disappear at the end of the project。

So I'm going to assume I can do something with it， I could sell it， I could rent it out。

I could redeploy it for another product。 But I'm going to take a step back from precisely what I do with it。

and just assume a liquidation value or a valuation for it in which I get fifty cents on the dollar。

It's just an assumption。 Now for depreciation purposes。

I'm going to assume that all capital depreciates on a straight line over five years。 That is。

it's useful economic life is five years， and so each year one-fifth of the capital theoretically disappears。

Again that's a non-cash expense， it's not costing me anything but it is going to provide a tax shield here that we have to recognize。

So we need to understand what our depreciation is going to be。

Now on to networking capital， we call networking capital is cash plus inventory plus accounts receivable minus accounts payable。

accounts receivable， accounts payable。 Let's go after each component here。

I'm going to assume this project is going to require a certain amount of cash which might seem a little bit odd。

but think of it as cash that I need to pay employees and ongoing bills such as utility costs。

IT expenses， whatever it may be。 And those requirements are going to be defined by 50% of my SG&A and 100% of my anticipated R&D expenditures。

Let's clear that， there we go。 My inventory forecast is based on inventory days or turnover。

How long does it take to get inventory in and then out？ I'm going to assume that that's 7。

58 days which by the way is very fast， you have to look to a company like Dell for that kind of turnover。

And finally I want to recognize that at the end of this I'm going to have some inventory left over which I assume I can get 25 cents on the dollar。

At that point it's going to be fairly obsolete， I might want to scrap it。

I might want to sell it on some secondary market。 Sorry about that， some secondary market。

So I'm going to assume 25 cents on the dollar for that remaining inventory at the end of this project。

My accounts receivable are based on days receivable or how many days it's going to take until I actually get some money in following a sale because remember a lot of sales are based on credit。

And likewise my accounts payable are based on days payable which is going to tell me roughly how long it takes for me to pay my suppliers。

I might be paying them using trade credit。

So that's our working capital but remember what we want is the change in networking capital。

So while these forecasts here will get me my networking capital what I'm going to want to look at in free cash flow is the year on year change from year t。

sorry from your t minus 1 to your t。

And then the last piece of this puzzle is taxes。 I want the marginal tax rate。

I want the tax rate on additional dollar of earnings， I'm going to assume that's 25。5%。

A lot of people will estimate this with an effective tax rate that is looking at the income statement。

tax expense over pre-tax income。 That's not strictly speaking what we want。

To get at the marginal tax rate you probably need access to the tax records of the company which are difficult if not impossible to get。

So a lot of people will also assume a top statutory corporate tax rate of around 35， 36%。 Alright。

so what we've done is we've gone through， let me just wrap up a little bit here。

We've gone through and we've produced or made assumptions about every component in our free cash flow formula。

And all of those assumptions are going to allow us to forecast dollar values into the future。

But let's take a step back。 This is nonsense。 That's a common look I get from students or practitioners when I discuss this framework because the argument is。

look， this is impossible to make accurate forecasts in the future。

It's hard enough to figure out what our company is going to do and how the economy is going to change and how that's going to impact our company。

A quarter out， much less five years。 And I agree， but I think that's not the point。

Let me be clear on what I agree。 It's not that this is nonsense。

but I do agree that it's very difficult， if not impossible， to make accurate forecasts。

But that's not the point。 That is just not the point of a DCF。

The point of a DCF is really to focus the discussion and analysis on the relevant issues。

To get away from decision making based on gut feelings， what the stars look like。

whether it's a full moon， other ad hoc rules of thumb。

just referring to nebulous experience and I know best， or I've been at the company for 20 years。

this is the way it is。 DCF provides a rigorous framework within which we can discuss what really matters for value creation。

And that's the whole point。 That's what I want to emphasize。

The goal of a DCF isn't to get one number that we argue is correct。

The goal of a DCF is actually to provide a host of numbers and a host of sensible and financially correct information from which we can make better decisions。

And I'm going to emphasize that throughout。 Another lesson I want to emphasize here is that no successful devaluation or DCF can rely solely upon input from financial personnel。

It's critical that we get input and insights from all aspects of the company or across division lines。

I mean， think about what we're forecasting。 We're forecasting revenues。

What's going on with the market？ So we want to talk to our marketing and strategy people。

We're going to be talking about investments and operations。

We need to understand what our inventory is going to be from our operations people。

from line managers， even from lower rank and file employees。

That's how you build a successful valuation。 That's how you build a successful DCF model with really good inputs from knowledgeable people。

Alright， let's wrap this thing up。 So today we discussed forecast drivers in the context of a specific example that being a tablet。

And these are nothing more than the assumptions that we're going to use to forecast the dollar values of each component of our free cash flow formula。

And again， the goal isn't to get the right answer。

It's to provide that framework within which we can discuss sensibly and logically the relevant issues for making decisions。

So in the next lecture， what I want to do is I actually want to use these forecast drivers to forecast the free cash flows。

the dollar values。 I look forward to seeing you then。

沃顿商学院《商务基础》｜Business Foundations Specialization｜（中英字幕） - P112：10_预测自由现金流.zh_en - GPT中英字幕课程资源 - BV1R34y1c74c

Welcome back to Corporate Finance。 Last time we talked about forecast drivers or the assumptions required to forecast free cash flows into the future。

Today， I want to apply the forecast drivers we discussed last time in the context of our tablet case and apply them to the free cash flow formula to forecast free cash flows into the future for the tablet project。

Let's get started。 Hey everyone， welcome back to Corporate Finance。

Today we're going to be talking about forecasting free cash flows， but before doing so。

let's start with our recap of the last lecture。 Last time we laid out our forecast drivers and those were the assumptions that are necessary to generate our free cash flow forecasts。

In particular， they're the assumptions about each of the components going into our free cash flow formula。

Today I want to apply those forecast drivers to actually forecast dollar cash flows。

dollar free cash flows over the five year horizon for our tablet project。 So let's get started。

So here's our free cash flow formula。 And again， the goal today is to translate the forecast drivers into dollar forecasts。

So as a brief reminder， here are our revenue forecast drivers。

We're going to start by forecasting the dollar revenues。

And we know that revenue equals market size times market share times price。 So for year one。

our revenue forecast is just going to be the year one initial market size， one million units。

times our share。 We're going to get 25% of that market。

And then we're going to multiply it by our per unit price of $200 for a revenue forecast in year one of $50 million。

If we repeat that process for years two through five。

we'll get our revenue forecasts for all five years。 And there they are。

Now let's move on to costs。 And we'll start with cogs， cost of goods sold。

You can see that our forecast drivers here are expressed as a percentage and in particular a percentage of sales。

So I'm going to repeat the sales forecast from the last slide here because we're going to need them to get at cogs forecast。

So if we look at year one， we can see that 80。66% of the sales are assumed to be cost of goods sold。

So we take our $50 million revenue forecast， multiply it times that 80。

66% and out pops an estimate or projection of cost of goods sold in the first year， $40。33 million。

If we repeat that process through years two through five。

we're going to get a full blown cogs forecast over the entire projection period。

Now let's move on to SG&A， still in the costs category。

Here we've assumed a first year SG&A expense of $69。59 million。

Now it's in dollars already so there's not really anything to do。

It was just an assumption I made based on 1% of the 2008 company wide SG&A expense。

a number I essentially made up just for illustrative purposes。

But going forward in years two through five， we're going to assume that SG&A expense grows at 25% per annum。

So to get year two， for example， we start with our $69。

59 million expense in year one and we compounded up by 25% to get an estimate of $87 million for our second year SG&A expense。

And if we continue with that 25% annual growth， we're going to get an actual SG&A series here。

Finally， moving on to R&D， there's really not much to do here because I've assumed $4。

00 both for the upfront R&D necessary to get the project off the ground。

as well as the subsequent R&D required for any versioning。 So here are our R&D dollar forecasts。

I'm going to take a step back and put this together into a somewhat familiar format。

and we're going to start with top line sales forecasts right here。

I'm going to subtract off my cogs to get at gross profit。

I'm then going to subtract off my SG&A expenses and my R&D expenses to get estimates of EBITDA。

And EBITDA is a mouthful once you say it。 It's earnings before interest， taxes， depreciation。

I'll abbreviate that and amortization。 Okay。 Now let's move on to capital expenditures。

So our upfront investment is going to be $227 million， well， a little over $227 million。

for the plants and the equipment。 Then we're going to invest in the first year of the project 10% of that amount。

followed by annual growth of 5%， 1%， 1%， and 1%。 So to get our dollar forecast。

there's nothing to do for year zero。 That's just the $227 million。 For year one。

we're going to take 10% of that $227 million to get $22。77 million。 And then for year。

let me clean that up a little bit。 For year two， we're going to grow that at an assumed 5%。

so we compound out up the $22。77 million at 5% to get an estimate or forecast of capital expenditures in year two of $23。

9 million。 And continuing that growth process for years three， four， and five at 1%。

we're ultimately going to arrive at our projected capital expenditure series。 There it is。 Now。

we're going to assume we're going to straight line depreciate this capital expenditures over five years。

What that means is one-fifth of the capital stock is going to depreciate each year。

So what I'm going to do to make things a little bit easier is I'm going to create a row of accumulated capital expenditures。

So that's nothing more than the current plus all previous capital expenditures。 So this $250。

5 is the $227 plus the $22。8。 The $274。4 is the $250。5 plus the $23。9 and on and on and on。

And the reason I'm doing that is to avoid keeping track of different vintages of capital stocks。

since they all face the same depreciation schedule， at least by assumption。

And what I can do to compute the depreciation is simply divide last year's accumulated depreciations。

sorry， accumulated capital expenditures by five。 So the 45。

5 in year one comes from $227 divided by five。 Year two fifty point one comes from the two fifty point five divided by five。

And again， what we're doing is what's really going on is this capital stock is depreciating by another five。

another twenty percent。 And then this depreciates by twenty percent。

So we add those two to get the fifty point one。 And we continue that on and on and we're left with we get our depreciation series。

Now， here's a question。 What happens to all of this physical capital at the end of the project。

at the end of the five years？ It certainly doesn't evaporate， right？ It doesn't disappear。

We can sell it or we can redeploy it for another purpose。 So we need to recognize that。 Otherwise。

we're going to underestimate cash flows。 So what I'm going to do is I'm going to take all of that capital expenditure。

that accumulated capital expenditure。 And I'm going to subtract from it the accumulated depreciation。

That is， I'm just going to sum up all of the depreciation to get this two seventy four point eight。

The difference between the accumulated cap X and the accumulated depreciation is the book value of the assets are seventy two point eight million dollars。

Now， I'm going to assume that I can't sell that on a dollar for dollar basis rather than have to sell it at a discount。

Fifty cents on the dollar。 And so the liquidation value， what I can actually。

the money I can actually get for it， which is different from the book value。

is assumed to be thirty six point four million dollars。 But remember we have to deal with taxes。

And so what we're really interested in are the after tax proceeds from selling this。

So we're going to get thirty six point four million dollars。

But we're going to experience a book loss because the value for which we can sell the assets by assumption。

I might add。 This isn't always the case。 Now， is less than the book value of the assets。

I multiply that times the tax rate so we get a little bit of a tax shield here from our loss。

And the result are after tax proceeds that are actually greater than the liquidation value。

We're going to get forty five point six million dollars。

And so the end result is that we have our projected capital expenditures here。

But once I factor in the after tax proceeds from the sale of these assets。

we're actually going to have a negative capital expenditure in that last year， that fifth year。

which is going to reflect an inflow of money。

So now let's turn to networking capital more precisely the change in networking capital。

but we'll start with networking capital， which is。

you remember is cash plus inventory plus accounts receivable minus accounts payable。

And here were all of our assumptions， all of our forecast drivers for the components of networking capital。

Let's start with the top， the cash requirements。 We're going to require 50% of SG&A in cash and 100% of our R&D expenditures。

So we're going to need our SG&A forecast and our R&D forecast to back out the cash requirements。

Notably for year one， we're going to need 50% of our SG&A expenditures or 34。

8 million dollars held in cash。 But we're also going to need 100% of our R&D expenditures 100% of the 25 million or 25 million dollars。

We're also going to need to hold that in cash。 So our cash requirements are the sum of these two and are given by this row。

And again， we're just going to repeat that， sorry， year by year for two， three and four。

Turning to inventory， our forecast drivers lay out the days， the inventory days。

but the inventory days are based on COG。 So I'm also going to need my COG series。

which I put here for convenience。 So to compute the first year inventory requirements。

I'm going to take my days in inventory 7。5A。 I'm going to multiply that by my COG's expenses。

And I'm going to divide that by 365 to get an estimate of first year COGs at $837，000。

If I keep doing that year after year， we'll get our inventory requirements。

Turning to accounts receivable， we've got days receivable assumptions。 Those are based on sales。

So here's my sales forecasted sales。 When I apply the days receivable to the sales。

say for year one， for example， I've got 38。49 days receivable times the $50 million sales forecast divided by 365 days gets me a forecasted or projected accounts receivable of $5。

2 million， a little over $5。2 million at the end of year one。

We continue that process for years two through five。

and there's our accounts receivables projections。 And finally， we turn to accounts payable。

I've got days payable assumptions of 61。54 days。 This is based on COG。

So here's my COG series for convenience。 I apply my days payable forecast to our forecasted COG series and out pops the accounts payable's forecast。

Repeating that process again， years two through five， we get our accounts payable， there's a typo。

accounts。 We get our projected accounts payable series。 We can put this all together。

Our cash inventory accounts receivable and accounts payable forecasts to compute networking capital for each year。

Remember， it's going to be cash plus inventory plus accounts receivable minus accounts payable。

That gets us our networking capital。 But here's a question。

What happens to all this working capital here at the end of the project？ Specifically。

what happens to all the cash that's sitting there or the inventory or the accounts receivable。

the customers from whom we're waiting to receive payment？ Or the accounts payable。

the suppliers who are awaiting our payment？ What happens to all that networking capital？ Well。

like physical assets， it doesn't just disappear。 In fact， most of it is going to be recovered。

Not all of it， and I'll explain why in just a second。 In particular， right。

we're going to be able to recover the cash。 We're going to be able to recover some money from the inventory。

but by assumption， back in our forecast drivers， we're assuming we're only getting 25 cents on the dollar for this inventory。

That is probably obsolete and either not worth much in terms of scrap or on some sort of secondary market。

We're going to have to pay， well， excuse me， well。

we're going to have to pay all of our accounts receivable， but we're also going to have to collect。

Sorry， we're going to have to get rid of that。 We're going to have to pay our accounts payable and we're going to have to collect our accounts receivable。

The upshot of this is we're going to recover $51。375 million in networking capital at the end of the project。

So our change in networking capital， remember， we're interested in the year on your change。

is given by this row。 Let's just briefly discuss it。 Get rid of that guy。 So the first change is 59。

1 minus zero， which is 59。1。 The second change is the 49。7 minus the 59。1 gets me minus 9。4。

And on and on and on。 And in the last year， we would have 25。6 minus the 28。2。

but then we have to add back in the 51。3 to get networking capital of 48。6。

which reflects recovering all that working capital at the end of the fifth year。 Now。

there is a bit of an assumption。 Well， there's a lot of assumptions。

but one of the assumptions here is that we're recovering all the working capital right at the end of period five。

In reality， you're probably going to collect that within or during year six。

So I could have extended this to a sixth year， but it's not going to make too big of a difference。

The discounting effect is going to be small。 So I just wrap it all up in this fifth year here。

We have all the pieces now。 And what I want to do first is organize them into a useful and familiar format。

What I'll call a quasi income statement。 And it's a quasi income statement because it's missing an important item and it's applying taxes slightly differently。

I'll explain that when I get there。 So we start top line with sales， subtract off cogs。

That gets us gross profit。 We're going to subtract off our SG&A， subtract off R&D。

That gets us our EBITDA。 And sorry， ignore these numbers here。 What they are。

their references to rows in the spreadsheet。 They're just a useful way to keep track of things。

We're going to pull off our depreciation。 That's going to get us our EBITDA。

We're going to apply our tax rate to our EBITDA to get our tax expense。

which we subtract from EBITDA to get our no-pad。 Now。

I call this a quasi income statement because what's missing here is interest。 No interest。

That's financing。 We're after unlevered cash flow。 So that's not relevant here。

It's also quasi because the taxes are applied to EBITDA as opposed to pre-tax income。

which comes after interest。 So this is our no-pad series。

Let's carry that no-pad forward up to here。 Add back in depreciation。

subtract off capital expenditures， and subtract off the change in networking capital to get what we're really after。

or ultimately after， which are free cash flows。 These are the free cash flows to the project。

Now， some other things to keep in mind when you're going after free cash flows。

First of all are opportunity costs。 Alternative uses of resources。 They're not free。 Second。

project externalities。 And this is a biggie。 In our context。

you can imagine that the tablet might cannibalize or take away sales from other product lines。

maybe desktops， for example。 But you also want to be cognizant of spillovers。

You could imagine that in the process of developing and selling this product， this tablet。

we might learn something about our production process that applies to other product lines。

Or it might lead us to the development of an entirely new product line。

We need to recognize these externalities that are associated with this project。 So。

what do we mean by "sunk cost"？ Well， imagine we commissioned a marketing study back in 2006。

two years before we were valuing this project or deciding on this project。

The costs associated with that marketing project as of 2008 are completely irrelevant。 They're sunk。

They're irrelevant for a decision-making。 All that matters are the costs going forward and benefits。

Other non-cash items， such as amortization， we have to take that into account。

Particularly because that generates a tax shield。 But also other non-cash items like stock-based compensation。

stock grants， restricted stock awards， options， employee stock options and the like。

Salveage values。 We touched on this in this lecture。

Assets don't just disappear and either do liabilities。

So we have to take into account any salveage values or liquidation values associated with our assets。

Execution risk。 Look， this is a risky project。 But the way we're going to capture execution risk that is risk unique to this project is not through the discount rate。

And I'll come back to talk about that more later on in the class when we get to estimating the discount rate。

But this is specific to the project。 So it's got to affect the free cash flows。

And the way it's going to affect that is we're going to look at expected free cash flows。

That's the way to think about the numerator and the NPV calculation。

And we can also capture this through sensitivity analysis by looking at different scenarios or different forecasts for free cash flows。

That's going to come up in a little bit when we talk about sensitivity analysis。

And one last point is， you know， I've assumed an annual frequency here。

but there's nothing special about that。 We could have done this on a quarterly frequency。

We could have done it on a monthly frequency， semi-annual。 Take your pick。

What determines the cash flow frequency is really the situation。

It's project dependent and it's sector dependent。 So don't think there's anything special about annual frequency。

We know how to deal with non-annual frequency cash flows and discounting them。

So it's not a problem at all。

All right， let's summarize this。 What we did in this lecture is we actually forecasted the dollar values of the free cash flows。

And the way we did it is we built it up from our forecast drivers of the components within free cash flows。

This is one of the two basic inputs into a DCF， the other being the discount rate。

And the question now is， what do we do with these free cash flows？ Well。

that's the topic of our next lecture where we take a look at decision criteria。

So thanks for listening。 And I look forward to seeing you in the next lecture。