DCF - Terminal Value - Gordon Growth Method Intuition

okay hello and welcome to our video here that is going to explain the intuition behind the Gordon growth method that is used in a discounted cash flow analysis to explain the terminal value calculation or to calculate terminal value now the reason why we're going through this is because it's very very common to use this formula and a DCF analysis but a lot of people a lot of books a lot of professors actually skip over the real explanation behind it entirely and that leads to a ton of questions out of all the DCF related topics I've covered in taught over the years the formula for terminal value as calculated with the Gordon growth method or the perpetuity growth method the long term growth method whatever you want to call it has generated more questions than probably anything else out there it's interesting to cover because it's not immediately obvious why the formula works the way it does so we're going to break it down in this lesson and first in step 1 we're going to go over the formula and the intuition behind it and then in step 2 we're going to go over the mathematical proof for it so I'll warn you in advance that there is going to be some algebra here there is going to be some math that we have to do but if you are more analytical and you want a rigorous proof of how this works we're going to cover that in step 2 so let's start this way off with step number 1 right here first now the familiar formula for this for terminal value as calculated with the Gordon growth method is just the final year of free cash flow times 1 plus the free cash flow growth rate and then you divide that whole term by the discount rate minus the free cash flow growth rate at the bottom now just to show you an example of how this comes up I'm going to jump into excel and bring up a file that has this and makes use of this formula so that's an example of how you use it in excel and the obvious question here is what does this actually mean why do we subtract the free cash flow growth rate in the bottom why do we have that bottom term discount rate minus free cash flow growth rate and then what's going on with the top term why do we take the final year free cash flow in the period we're modeling and then multiply it by the growth rate what's the deal with that what's really going on there and the basic idea behind this is as follows how much can we afford to pay for companies cash flows indefinitely into the future if we want a certain yield on our investment and that yield of course is called the discount rate in this analysis so to explain the intuition behind this I'm going to go through two scenarios the first one the first one here we're going to have no growth so the free cash flows we get from the company are going to stay the same every single year so we're going to skip the growth part now for now because it's easier to explain this when we start out with no growth and show you what the discount rate and the net present value in the formula means and then we're going to add growth once you understand that part so you understand the entire formula so here's an example in Excel now from Ralcorp Holdings a consumer retail company when the companies that we cover in some of our lessons and bonus case studies on the site now I'm not going to go over the concept for a DCF here I'm assuming you already know that but just to show you a real world example of where this comes up in this formula for terminal value over here if you look at this second option L 27 times 1 plus R 16 divided by R 10 minus R 16 well that is an example of how you use the terminal value formula based on the corden growth method the L 27 just a fun on your free cash flow one plus R 16 R 16 is the terminal growth rate then R 10 is the discount rate up here so that's a real-world example where use it and again the whole idea with the DCF is of course to take the present value of a company's cash flows and then the present value of the terminal value add those two up to get the total value of the firm so we're focused here strictly on the Gordon growth method used to calculate the terminal value and then which you discount to get the present value of for use in this formula in this analysis now the top term in that formula the final your free cash flow times 1 plus the growth rate that really just means the first year's free cash flow in the terminal period so if you have a DCF that you're projecting over 5 years or 10 years well this top term is really just the free cash flow in your six or your 11 or whatever the case might be one year beyond the period that you're actually projecting the free cash flow of so it's just the first set of cash flow that you get from the company in that terminal period which goes on indefinitely so that part is pretty straightforward to explain you're just trying to go one beyond the last year in your model we're going to assume a zero percent growth rate for now so we get the same exact free cash flow every single year and my question to you is how much would you pay for this so if we have the scenario we have no growth and we get a certain amount of cash though the same every single year we're targeting a certain percent yield how much would you actually pay for a company in this situation and to answer that question we're going to jump into Excel now and I'm going to break it down and show you exactly how much you would really pay for this okay so here we are in Excel and I'm going to show you a quick demonstration to show you more of the intuition behind this and answer that question of how much you would pay for a company that generates a fixed amount of free cash flows each year if you're targeting a certain percentage yield now here as you can see our discount rate so the yield were targeting is going to be 10% we're assuming no growth and the first year free cash flow so really that term of final year free cash flow times 1 plus the growth rate because it's the first part of our terminal period it's going to be a hundred right here so all these years your one over two year 100 this is our terminal period what we're not modeling and the DCF analysis what goes beyond the end of our model there so for the free cash flow let's just plot this out and see what happens so we have 100 and remember we're saying no growth so we're just going to take our previous term times 1 plus the growth rate which of course is just zero but we're going to have that anyway and let's take this all the way over to year 100 and as you expect it's a hundred each year so now the question is how much would you pay for this and you can see the answer down here the free cash flow each year divided by the discount rate and why is that well very simply because if you invest a thousand you get a 10% yield on it that gives you a hundred dollars per year in cash flow and if you don't believe me I'm going to show you how Excel calculates it and then I'll show you our numbers up here and show you how very close if not the same they are so the net present value free cash flows here is just the initial free cash flow divided by the discount rate so it's just how much we get each year divided by the yield we're targeting that tells us how much we can pay for now technically I should factor in the growth rate here but we're just ignoring it here just to show you this basic concept first so it gives us thousand and then in Excel the way you do this is you use the NPV function NPV stands for net present value which I define down here how much week for it to pay today for a series of cash flows that go into the future and definitely if we're targeting a specific yield to get those cash flows from our initial investment and if we use the NPV function look at this I'm going to say 10% and then for the values let's take this whole series of cash flows over here and look at this excel gives us almost the same number nine nine nine point nine three so we had a few more years or a couple hundred more years or something like that it would actually come out to a thousand we made it go into infinity but you can see right here the concept that in Excel it matches up because Excel knows what's going on here it knows what we're doing and the way we calculate this with this formula is pretty much the same the only difference is that we only have a hundred years in Excel but we have an infinite number of years with this calculation above so these match up and it tells us clearly that the NPV of a series of cash flows at a certain discount rate is really just equal to if we have no growth the initial free cash flow that we get each year which is the same in all years here divided by that discount rate and that is the intuition behind this part okay so that explains how much you'd pay for a company with the same cash flows each year and the yield that you're targeting so we assume ten percent in that case now what we're going to do now is go into scenario two which is what happens when we have growth so when the company's free cash flows you're actually growing each year so now let's say the company is actually growing it's free cash flow it's still initially $100 but now it's growing that free cash flow by three percent each year just to give us a low relatively straightforward number to work with here now you're still targeting that same 10% yield on your investment so we're still using the same numbers here and my question to you is if the company's free cash flows are growing could you pay more or less or the same as that one thousand dollars and still get that same ten percent yield so you still want that same 10% yield on your investment whatever your investment is the company starts out at $100 our free cash flow but it grows that by three percent every single year indefinitely so would you pay more or less than that thousand dollars to get that same ten percent yield and the answer of course is more you could pay more the that now because there is growth so that $100 per year that you start out at initially well that's going to grow to a lot more than $100 per year eventually if you're growing at 3% per year indefinitely and that's the point of subtracting the growth term in the denominator here when you subtract that growth term so you have a 10% discount rate while you subtract 3% there you get 7% in the denominator now when you divide something by 7% it's going to be a lot bigger than when you divide it by 10% and you're saying effectively we can now pay a whole lot more because we have this growth involved and we're getting more for our money here so I'm going to go into Excel now and show you how this works there okay now this next scenario down here scenario number 2 now we're going to say as I say here that the company's cash flows are actually growing each year and the question that I just posed to you is could you pay more or less to get this same 10% yield that we're targeting and of course the answer is more because there's growth as a result of that growth the denominator in our equation is going to be smaller and since the Dominator is smaller we're dividing by a smaller number the entire term is going to be a bigger number we can pay more for the company up front now I'm going to go into Excel and show you and prove to you why this works the way it does and why the NPV formula is really equivalent to the formula that you use in the terminal value when you're calculating it with the Gordan growth method so free cash flow let's take a look at this we're going to take this initial number and then remember this time we have 3 percent growth each year so we have to factor this in and let's take our previous term multiplied by 1 plus the growth rate over here and make sure we anchor this and then copy this all the way across the Year 100 so that we can see it goes all the way up to close to $2,000 by year 100 right here but of course the relevant question is what is the net present value because of course money in your 100 is going to be worth far less than money today because we could have invested it and earned something on that in the meantime the question here if we're targeting the same 10% yield and the cash flows are growing how much would you pay and the answer is the initial free cash flow in this period so the year 1 free cash flow 100 divided by the discount rate minus the free cash flow growth rate which comes out to 1429 here and so let's take a look at this formula and see how this works and then I'm going to prove to you that if you do this in Excel by using the NPV function it comes out to the same thing so we take this 100 we divide by the discount rate minus the free cash flow growth rate 14:29 and then let's take a look at this in Excel NPV we have the 10% discount rate and we have these free cash flows through year 100 and comes out to almost the same number it's a little bit different but very nearly the same number and let's just change the decimal places there so 14 27 verses 14 29 so not exactly the same but you can see it's clearly going to converge on the same exact number if we had a few hundred more years it would do that but you get the idea from this and the reason of course is that the net present value is higher because we can pay more to get that same equivalent yield every single year now because there is growth involved if we invest 14 29 we get 100 the first year no that's not a 10% yield you can look at it yourself 100 divided by 14 29 it's actually run a 7% yield but the relevant question is what is that yield in each year going forward and the answer is in your to take and divide by this well we get to 7% and I'm actually going to increase the decimals on this so you can see it a little bit better and take a look at this the yield here is going up every single year because we're getting more free cash flow from this initial investment and that is why we can afford to pay more than a thousand because the free cash flow is going up each year the yield is going up each year and no due to the time value of money that money in your 10 or your 50 or 100 is worth less than what is shown in Excel but when all is said and done we could still pay substantially more and get that same 10 percent yield equivalently with these changing cash flow numbers now so that is how this works now the mathematical proof and the intuition for this which is the next part I'm going to go into I'm going to start out very simply here and what I have here above is just our free cash flow numbers which are exactly the same as what I linked to above and then I have the present value free cash flow to get that present value we just take our discount rate one plus the discount rate here and raise it to the power of the year we're in so up here I have a hidden row with the year numbers over here so we're just raising it to the power each time and what I want to do is show you what the cumulative sum of the present value these free cash flows comes out to each year and show you how it changes so let's do this and let's say some I'm going to anchor this first part so this doesn't shift around at all have this and let's just copy this all the way across through your 100 if that and so what I want to draw your attention to is the following the present value free cash flows and the cumulative sum look at how these change now over time present value free cash flow starts off at a decently high number but it definitely declines a lot all the way to your 10 and then meanwhile the cumulative sum keeps changing by a smaller amount and if you keep scrolling over take a look at this by the time we get all the way out to your 80 or so your 81 the present value of free cash flow is almost zero and the cumulative sum is barely even changing so what's going on here is that this cumulative sum is converging on a specific number and the present value is getting closer and closer to zero so we have an infinite series where the present value the term in the series is getting closer and closer to zero as time approaches infinity and the sum of the series converges on a specific number so it's a geometric series converging on a specific sum over time and you can see my explanation down here but what is happening here is that that number two converging is the net present value of free cash flows for this cash flow profile at this discount rate and so what we can do now is use some algebra to actually calculate what this is and to derive our formula I'm going to go back to PowerPoint and show you how to do that now what I want to conclude this lesson by doing is by going through some of the algebra behind it and going through more rigorous mathematical proof for this now it's not exactly a proof it's more of a derivation of the formula but I want to go through it because again it's just something that a lot of other training courses a lot of other companies books professors and so on just skip over entirely now a 1 year in advance that if you do not like math you don't like algebra that much you should probably just skip this part because we've already been over the intuition and that's really most of what you need to know but if you are more analytical you want something more rigorous we're going to go through that right now so the idea here is that really we have the sum of a geometric series and I showed you that before in Excel and basically what's going on is that over time the sum of the present value of free cash flows converges to a specific number and the reason that happens is because the present value of free cash flows gets smaller every single year so it keeps getting smaller eventually it's going to hit zero at some point far into the future and so the sum of all those present values of free cash flows approaches a specific number we have a geometric series now if you want more about this you can go look on Wikipedia I don't really have time to explain it here but you'll see what it is and it's basically just a series where the sum converges to a specific number over time and the sum of this you can calculate as a divided by 1 minus R what is a mean what is 1 minus R mean well a in this case is the present value of free cash flows one year into the terminal period of the model so this is basically the first term of this summation which of course is just a final year free cash flow times one plus the growth rate divided by one plus the discount rate why are we dividing by one plus the discount rate well normally we raise that to the year number the power of the year number but we're in year one so we just raised to the power of one and it's just one plus the discount right here and then R is the common ratio this basically means what you multiply each term in the series by to get to the next term so R here what is it in this context well in the first year it just means what do we multiply this term up here the final year free cash flow times 1 plus the free cash flow growth rate divided by 1 plus the discount rate what do we multiply that by to get to the present value of free cash flow in year 2 so how do we get to this what is actually mean here well in this case we know that the free cash flow grows by 1 plus the growth rate each year so that seems pretty straightforward and we know the present value of free cash flow is just equal to the free cash flow divided by 1 plus the discount rate raised to the power of the year that we're in so if we're in year 1 it's just 1 if we're in your queue we raise it to the power of 2 we square it your 3 we raise it to the power of 3 so the common ratio is simply 1 plus the free cash flow growth rate divided by 1 plus the discount rate what that means is that every single year to find the present value of free cash flow in the next year we multiply this year's term by this number 1 plus the growth rate divided by 1 plus the discount rate so over time our free cash flows are going to keep growing but the present value of those free cash flows each year is going to be shrinking because of the discount rate because money today is worth more than money tomorrow so this is as I say here how much the present value free cash flow changes over the previous years number each year so let's put it all together now and think about how to write this formula then we'll use some algebra to combine terms simplify and get down ultimately to a derivation of the real formula for terminal value as calculated by the Gordon growth method so a is just equal to the final year free cash flow times 1 plus the growth rate divided by 1 plus the discount rate straightforward R is equal to 1 plus the free cash flow growth rate divided by 1 plus the discount rate so a divided by the term of 1 minus R is this fraction that you see here on screen I'm not even going to attempted to read this off it's just easier to look at yourself and see it all visually so the question is how we go from this ugly-looking fraction to our actual formula how can we go from one to the other here and to do that we're going to have to use some algebra and rearrange and combine some of these terms and basically what we have to do is in the bottom here in this part we need to make this a lot easier and we need to get to the 1 plus discount rate that's at the bottom of this fraction at the top so I'm going to show you how to do that by using some math right now so the first thing we're going to do is rewrite the one term in the bottom the one minus that one part in the beginning as 1 plus the discount rate divided by one plus the discount rate so it's the same exact thing but we're doing that so we can get a common denominator at the bottom so this whole term that just becomes what you see on screen 1 plus the discount rate over 1 plus the discount rate minus 1 plus the growth rate over 1 plus the discount rate and of course you can combine those two terms and make it equal to discount rate minus free cash flow growth rate divided by one plus the discount rate because the one term at on the top that just cancels out entirely because you have a positive 1 and then you have a negative 1 so we're still working with this ugly looking term that I have here on the top of the screen and now what you can do is rewrite this term because we just went through some algebra to rewrite the bottom part and you can rewrite this term as follows finally your free cash flow times 1 plus the growth rate divided by 1 plus the discount rate and then you can rewrite that whole bottom part as another fraction discount rate minus the free cash flow growth rate over 1 plus the discount rate so now you have a fraction that is comprised of multiple other fractions you have one fraction on the top and then you have another fraction on the bottom of course this is very confusing so we prefer to rewrite this as something else entirely and to simplify it even further fortunately that's very easy here because what you can do is just multiply these two fractions out you can flip the denominator and numerator in the second fraction and then just multiply the first fraction times the second fraction when you do that look what happens here the first fraction final your free cash though times 1 plus the growth rate over 1 plus the discount rate multiplied by 1 plus the discount rate over the discount rate minus the free cash flow growth rate and look at this remember from your algebra and how fractions work the denominator and numerator here cancel each other out because they're the same thing so you have a denominator and numerator in two separate fractions you're multiplying that are the same while they cancel each other out and look what happens we get to funnel your free cash flow times 1 plus the growth rate divided by the discount rate minus the growth rate at the bottom and that is our familiar formula for terminal value as calculated with the Gordon growth method and we're done so I know there was a lot of math it may not have been entirely obvious on screen how everything works but I just wanted to go through the derivation because I know I'm going to get some questions about this about a mathematical proof behind it or the derivation so for the record I just wanted to go through it and show you exactly how it works to be clear this is not at all required for interviews it's really just for your own information if you really want to understand the details of how this works if this came up in an interview the interviewer would not even know how to derive this 99% of people working in finance would not even know how to derive this at their fingertips so this is really more for your own information than anything else the intuition behind the formula is important but the derivation this is really just more fYI than anything else so I know we went through a lot in this lesson to sum up everything and to go through everything we've been through before so Gordon growth method why is it important it's very common in DCF as a way to calculate terminal value and you could just use the multiples method as a simpler method but it's good to know this as well and the thing is when you use a formula like this you don't want to just use it you want to understand why it works maybe you don't know the full mathematical derivation but at least you want to be able to understand the intuition behind it that's important whenever you have a formula like this that's not immediately obvious and you're using it in Excel the basic idea is that if we're targeting a certain percentage yield the discount rate and we know the cash flows we're going to get we can pay the cash flow each year assuming no growth divided by the discount rate and that's how much we're willing to pay to get those cash flows at the yield that we're targeting and then if there's growth we can pay more than that we can pay the first year free cash flow in that terminal period divided by the discount rate minus the free cash flow growth rate and we can pay that extra amount the denominator becomes smaller and so now we're paying more because we have growth we're still getting the same yield but because of that growth that's involved we can pay more upfront to get that series of close that is now growing so higher growth means smaller denominator and we can pay more and vice-versa if we have lower growth it means we have a bigger denominator and we can pay less so that is how the formula works that's the intuition behind it as well as the mathematical proof I hope this clarifies this point and better explains to you exactly how it works even if you didn't get all the mathematical derivation and the algebra behind it at least you should have a better idea of the intuition behind this formula and how to use it and why it works the way it does in a discounted cash flow analysis

Channel: Mergers & Inquisitions / Breaking Into Wall Street

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Keywords: terminal growth method, dcf intuition, perpetuity growth method, long-term growth method, discount rate, terminal value, gordon growth method, dcf concept, gordon growth derivation, gordon growth model, dcf, discounted cash flow valuation, discounted cash flow, cash flow, free cash flow, valuation, intrinsic valuation, Breaking Into Wall Street, Mergers And Inquisitions, Wall Street Prep, Excel Model Tutorial, WSTSS, TheStreet, WallStreetOasis, Investment, EduPristine

Id: hCGn1ejYs1I

Length: 24min 35sec (1475 seconds)

Published: Thu Jan 09 2014