another issue with internal rate of return is scale problem but scale problem happens only if you have mutually exclusive projects so remember I told you that you Chile exclusive projects are the ones that cannot be taken simultaneously so you need to choose one or the other so in other words you implement so-called ranking criteria so let me explain the problem so suppose I give you an option I give you you give me one dollar and I give you two dollars back okay so let's say we have something like this you have option number one and then you have option two so option number one you invest $1 ok $1 and you get back two dollars a year from now in one year just to make it simple okay so you give me one dollar I give you two dollars back in a year option number two you give me a thousand dollars okay and I give you back one thousand and five hundred dollars in a year which option would you choose well I hope that your gut feeling would tell you that you have to choose option two because you're gonna make a lot more money but then you start thinking about internal rate of return if you're really big fan of IRR you would say wait a minute the internal rate of return here in option one is equal to hundred percent while the internal rate of return of option two is equal to only 50 percent so IRR 1 is bigger than IRR - so looks like on the basis of internal rate of return you should go with option one and you would be absolutely correct if indeed you are using internal rate of return rule but that's where you are in trouble because internal rate of return just simply looks at overall percentage rate of return without taking into account the overall scale of the product or of the of the project so again your your intuition should tell you that project two is actually way better option two is way better even though it has lower rate of return because you're operating on such largest scale even though again IRR is lower your overall net present value of the project is going to be much higher so for example if I assume that my required benchmark is equal to ten percent the NPV of option one would be minus one dollar plus two divided by 1.1 NPV of option two is gonna be minus one thousand plus fifteen hundred divided by 1.1 and we don't even need to do much calculation to see very clearly that NPV of option two is going to be larger than NPV of option one so as you can see net present value here is a conflict with internal rate of return according to an NPV rule you should go with option two according to IRR all you have to go with option one so which one would you choose and the answer is always go with what NPV tells you because net present value looks at not only if you beat the market but it also take into account the actual absolute amount of your economic profit so again think about it this way yeah you give me one dollar you get two dollars so percentage-wise you're making a lot but your other option even though it's less profitable in terms of rate of return it gives you so much more money and luckily net present value captures that so NPV rule allows you to actually take that scaling into account all right so now why is it important for mutual exam you Chile exclusive projects only well because if they were independent then you would go with both option one and option two you would take one dollar from me excuse me you would give me one dollar and get two dollars back and you would also give me a thousand and got 1500 back but if they are mutually exclusive if you cannot choose one and the other if you have to make a choice then IRR internal rate of return actually fails it here so what you need to do you need to actually focus on net present value so the question is under what circumstances the situation like that might actually arise in practice most of the time this particular problem applies to projects that have different scale of operation for example you could have small scale or large scale think about it this way do you wanna run small plant and operate on small scale basis or you wanna up size and make you plant a lot bigger and operate and on a larger scale so that's where that issue of IRR might arise so let me let me give you two projects here so project let's say a and then you have project B okay and then I'll just list the the cash flows so for example cash flow project a requires one thousand dollar investment and you're gonna get back $2,000 a year from now okay now let's say project B will require two thousand dollar investment okay twice as much but it's gonna give you three thousand and five hundred a year from now so that would be your actually you know what let me let me do it this way instead of calling it project a I'm gonna call it small project and instead of project B I'm going to call it large project so things like that I mean the the situation like that often happens when you run small business and you're thinking about upsizing you're wondering should you expand should you make your project larger should you operate at a larger scale so what I'm gonna do next is I'm gonna calculate npv okay actually I'm going to start with IRR and then I'm gonna do net present value so I'm going to assume that your benchmark R is equal to ten percent as always so let me start with internal rate of return well very clearly you don't even need to do calculation with small project your RR is equal to 100% okay now with large project looks like it's equal to 75% okay so if you go from 2,000 to 3,000 that's 50% and then another 25 percent so it's total 75 so looks like IRR small it's bigger than IRR large so so far it looks like small project is preferred but again you got to be very careful that's where this ranking might be very tricky you start wondering okay let's just do NPV let's confirm and make sure that net present value basically confirms our conclusion so I'm going to do NPV small that is equal to minus one thousand plus two thousand divided by 1.1 so let me quickly calculate I'm going to use calculator in this case so it's two thousand divided by 1.1 okay so it's eight hundred and nineteen dollars approximately so I'm going to put 819 here and my NPV large is gonna be minus two thousand plus three thousand five hundred divided by 1.1 and that would be equal to so it's one over one point one one thousand 182 okay so one 182 so now you have a conflict NPV large is greater than and P V small so what's happening here as I mentioned before just like in our previous example NPV takes into account your overall economic profit your overall abnormal profit by how much you beat the market in dollar terms so in this case you're actually better off going large even though on paper it looks like your internal rate of return for small project is higher but the reason why it's so high is because you operate on a very small scale sometimes it's beneficial to become bigger and perhaps sacrifice a little bit of your IRR but in the process getting just bigger overall economic profit okay so again if you have a trade-off like that always go with excuse me always go with what NPV tells you so what you go ahead and do you choose so you go with large project okay so that's that's what happens if you have a scaling problem so what would be your first clue because you might not even realize that there might be an issue here if you simply use IRR you could just jump to the wrong conclusion so your first clue if you look at your initial investment and you see a substantial difference and again it's not clear what substantial means you know fifty percent difference twenty percent one hundred percent in this case it's obvious the large project is twice as large so you should be already suspicious so again there is no clear threshold in terms of what is considered substantial difference so if you want to be very conservative just always check with net present value so even though you know you might have size per size problem or scale problem or you might not have it you you're okay you're welcome to do IRR but always confirm with NPV and make sure that the result that you get in using internal rate of return is consistent with NPV if it's not it might be signal it might be an indication that something's going on that potentially you have the Desai's is you okay or scale issue all right so now to deal with scale problem alternatively you could calculate something that is called incremental IRR okay you don't have to do it honestly I prefer to just use net present value but if you still want to use some kind of notion of incremental IRR so what you're going to do you will just create a fake project which is called large - small its incremental so in other words intuitively you are asking the question what it will take what additional investment would be required to go from small to large and what additional income I'm going to get as a result so you will see that going from small to large you would need to invest additional thousand dollars but you're going to get additional fifteen hundred dollars of actual cash inflow and you could easily see that internal rate of return on that differential project is gonna be I'm gonna call it IRR incremental and that's gonna be equal to fifty percent so it's greater than your benchmark so you go ahead and go with large projects so you decide to up side up sighs so you decide to increase the scale of your operation because incrementally that additional investment is clearly justified okay so you put additional thousand dollars but you gain you know additional fifteen hundred which is fifty percent rate of return okay but again you don't have to make that loss last step you could just do NPV and you would come to appropriate conclusion all right finally the last thing we're gonna have here is the timing problem okay so that's probably the most complicated one so again it concerns mutually exclusive projects if they're independent it's not an issue as long as IRR beats the benchmark you should be okay but if they are mutually exclusive so that you need to compare their IRS and rank them the timing problem might create some issues for you so let's look at at these two projects project a and project B so if you look carefully you notice a very drastically different pattern so even though initial investments are the same project a gives you a lot in the beginning but then less in the further two periods project B on the other hand gives you very little in the beginning but a lot in the end okay and everything else equal you would prefer project a because getting money earlier is always better than getting money later but the catch is project B gives you more money later so even though you know it's distributed you know more to the to the end of the project the absolute amount is bigger so what is better to get ten and then one one or getting one one but then twelve so what's going on here you have drastically different distribution of cash flows over time okay now why is it a problem let me try to demonstrate it to you using some kind of graphical analysis so okay so here's what normally happens if you have so here is your net present value profile so that's your discount rate and this is your NPV okay so normally when when cash flows are distributed relatively similarly over time you will have project a you will have project B you know something like this and then that net where net present value profiles are not going to be inter and not going to intersect so for instance that's your project a that's your project B so what it also means is that if this is your IRA and this is your IRR be knowing that IRR a is greater than IRR be automatically implies that NPV a is always greater than NPV be okay so in other words just one point the intersection point IRR internal rate of return uniquely identifies the position of net present value profile relative position for project a versus project be a regardless of discount rate which means that if discount rate is low project a is still preferred to project B if discount rate is a little bit higher still project a is better than project B and so on and so forth so for any R so for any our project a is preferred over project B okay so that's what happens normally okay but now imagine that we have our situation where the cash flows are extremely different if distributed very very differently so here's your discount rate here's your NPV okay and now let me check quickly just to be sure okay so green is okay so green is project B okay I just want to be consistent with what you guys have so now what happens here is your project B and here is your project a okay so here's your a and here's your B so as you can see the net present value profiles are intersecting and the reason for it is because they have drastically different slopes so why does that happen look carefully at the the actual cash flows project a because its first cashflow is so big it's NPV is not going to be very sensitive to changes in our because this sooner you get your money right the more close the closer your initial cash flow is to to now the less volatile its value is going to be when interest rate excuse me when discount rate changes project B on the other hand is going to have much steeper slope because the more distant in the future the cash flow is the more sensitive its value its present value is to changes in interest rates or discount rates okay so what's going to happen is your project a has more flat or flatter slope project B has much steeper slope so what happens is even though IRR a so here's your IRA is bigger than IRR B okay so now we have IRR a bigger than IRR be alright just like here same thing IRA was bigger than IRB but now we cannot say that NPV a is always bigger than NPV be we cannot say that anymore and the reason for it is because just these two points you know they do not uniquely identify the position of curve a relative to curve B in fact what happens is at low R's somewhere here you actually prefer project B okay so at low are you prefer project B but at higher ARS somewhere here you actually prefer project a okay so why is it the problem because now if you don't see those graphs if you don't have any information so suppose I just turn this page and you don't know anything all you know is that irra is bigger than IRR B on the basis of that information only you cannot make a conclusion anymore you cannot fall into that trap because you might be inclined to simply say that you know choose a and you will be absolutely wrong because choosing a versus B will now depend on where you are in terms of your discount rate it will depend on where you are in terms of your benchmark okay so simply knowing the relationship between IRS does not tell you which project is preferred okay and that is why it's a problem because you need that ranking of projects on the basis of IRR when you're dealing with mutually exclusive projects so let's say again I identify those two projects the project a has higher internal rate of return should I jump to conclusion and choose it over B the answer is no you need to check with net present value because you have a so called cross over problem and cross over problem is again that knowing I are ours does not tell you which project is really better okay and again it happens only if you have drastically different slopes of net present value profile and as a result they intersect so just knowing their IRS is not enough to make conclusion about your preferences in terms of project a over project B okay so here is the chart which is very similar to what I did just you know in colors nicer so your IRA is 16:04 your IRB is 1294 but it doesn't mean anything okay if discount rate if your benchmark is low you know somewhere here you're actually going to prefer project B okay the green line is above the blue line but if you are is higher you start preferring project a so you go to the blue line okay so what do you do then always go with net present value as you can see for all the problems that we have with our internal rate of return the answer is always the same check with net present value it's not going to fail you it will always give you a right the right solution so whatever conclusion you come up with using that using internal rate of return make sure that it's backed up by calculation of NPV and every time you see that net present value is inconsistent with internal rate of return you should always favor NPV okay that's the rule so always go with net present value all right so if you want to you could calculate that crossover rate it turns out to be somewhere here it's equal to 10 point 55 so what is crossover rate it just tells you remember I told you that when R is low you go with project B when R is higher you go with project a but how low and how high well crossover rate actually you know answer that question so what it means it means that if benchmark R is less than 10 55 you go with project B if R is greater than 10 55 you go with a and of course if it's exactly equal to 1055 you are indifferent between these two projects so do you need to calculate crossover rate not necessarily it's not very useful because if you are going to check with NPV anyway all you have to do is just to estimate your actual discount rate R and then check and see which project has higher NPV at that rate okay but if you do want to calculate crossover rate here is how to do it it's very similar to incremental IRR so you have project a you have project B now you create a fake project that actually looks at differential cash flows either a minus B or B minus a it doesn't really matter so whether you do a minus B or B minus a then you apply internal rate of return rule to those incremental cash flows and you should be able to obtain that cross over 8 that is equal to 10 40 at 10:55 okay but again I'm giving it to you for completely for completeness honestly you don't need that part you don't need that calculation that's more advanced but it's not very necessary as long as you check with net present value you should be okay so let's summarize now looks like internal rate of return is extremely appealing it's very nice way to summarize your project because it gives you that nice sharp measure it tells your percentage rate of return on investment that could be directly compared to your benchmark and as a result it gives you very intuitive information about your project and luckily in most cases NPV and IRR will give you same decisions however under certain circumstances you have to be extra cautious with applying internal rate of return those circumstance are when you have sign changes okay with your cash flows in which case you might have multiple internal rates of return which means that none of them are interpretable if that's the case you just go with net present value or alternatively you could try to do modified IRR when you combine those cash flows but that's again that's not very desirable another problem with internal rate of return concerns mutually exclusive projects in which case you may have scale problem okay so we call it scale problem so with scale problem you might be tempted to go with smaller projects simply because they have higher internal rate of return but according to net present value you might actually be better off going with larger project even though it has lower IRR it will give you bigger overall value measured by a net present value and finally you have timing problem also called crossover problem that happens when you observe drastically different distribution of cash flows over time so one project is really you know skewed towards early cash flows you have a lot in the beginning and then very little in the end another project is the opposite you have very little in the beginning and then a lot in the end so when that happens the net present value profile curves intersect and as a result simply knowing the relationship between IRR of two projects does not tell you uniquely which project is better so in this case you still need to use NPV so let me just summarize if not sure about IRR always check with NPV simple as that okay so net present value is never gonna fail you alright okay finally let's consider the last the last tool which is called profitability index pio profitability index is not new measure it's just another presentation of net present value because P I profitability index has exactly the same ingredients as net present value the only difference is instead of looking at the difference between present value and initial cash flow you take the ratio of the two okay so again let me let me show it here so your NPV is equal to minus c0 plus present value of future cash flows and you compare it to zero now what I'm going to do instead of taking the difference between them I'll take this one and I'm going to divide by this one without minus sign of course because I don't need to take differences I'm just looking at ratio so P I your profitability index would be present value of future cash flows and you divide by CF 0 your initial investment so it's exactly the same thing except that instead of taking difference you look at the ratio and of course now you compare to one okay so if P I is greater than one you invest if bi is less than one you reject okay so that's your acceptance criteria so profitability index has to be greater than one and of course your ranking is such that you pick the project that has the highest profitability index so what's the appeal well the main advantage is that again it's easy to understand and communicate it's it's it's kind of um a scaled version of NPV so what you're doing instead of saying that well my net present value is equal to 1 billion dollars it's not clear if it's a lot or is it a little you say that my profitability index is equal to one point one which means that my present value of future cash flows exceeds my initial investment by 10% now be careful here it has nothing to do with rate of return okay it's not your internal rate of return it's just overall ratio of present value over initial investment so yes it has to be greater than zero but again interpretation is not that it's rate of return it's just simply the scaled version of NPV still it could be easier to understand and communicate than just your regular net present value also profitability and the index might be helpful when you have investments with limited funds so I could show you example but I don't think we need it really but one possibility is when you have some limited resources and you want to rank your projects on the basis of profitability index and then you pick the combination of projects that they'd give you the highest combined profitability index okay so excuse me not combined but you go with the highest rated the next highest rated and so on and so forth but you could achieve exactly the same thing if you simply find the combination of projects that give you the highest total net present value so I would say that the biggest advantage of Pi or profitability index is just interpretation that it's just a scaled version of NPV and it gives you that idea by how much you your your present value exceeds your initial investment not in dollars in dollar terms but as a ratio that's all now the disadvantage is it has all the problems with mutually exclusive investments that are very similar to our are in particular scale problem okay it's not going to have timing problem okay you wouldn't be you wouldn't worry about crossover problem because this thing already takes and take our present value in the right way but it will have problem with mutually exclusive projects of a very different size because again if you make $2 on $1 investment even though your NPV might be small the present value divided by initial investment will be very large okay because what's going to happen your profitability index would be artificially very high because very small scale so exactly the same issue as you had with IRR so from that perspective NPV is still a better measure so again profitability index is good but NPV is better because it's universal it works for everything it works for differential timing it works for differential scale it works when IRR does not exist at all so it beats IRR it beats profitability index it obviously beats the payback period which is absolutely inappropriate other than just an additional piece of information so the summary is that for most companies net present value is the preferred method but they also use IRR a lot okay so the most widely used is NPV and IRR now the reason why internal rate of return is so popular again is because it's very intuitive and it gives a very nice measure of overall rate of return so but again even those companies that do use internal rate of return they always try to make sure that it's not at conflict with net present value and by the way here you have reference to so-called accounting rate of return don't we never discussed it because it doesn't make any sense okay so some terms still use payback period but again only as an additional piece of information but you know the industry is dominated by IRR and NPV profitability index is okay but it's not as popular as IRR and and PD okay so sorry sorry about that so now let's just look at the at the summary I'm not going to go over it I just put it here for your reference so it's a good way for you to quickly look at all those tools and kind of identify the main characteristics when they're good when they're bad and what they do so you have an TV you have internal rate of return you have profitability index you have payback period and then you have discounted payback period so all of those rules that we talked about okay and in fact they're kind of ranked here in terms of their preferences and their reliability so number one choice you know that's your number one net present value internal rate of return is second best it has its issues but it's very intuitive and very popular profitability index goes next and there are the other two payback period and discounted payback period are sort of residual you know they're not very popular they used but only as additional piece of information