CF 7-5A – Calculating NPV of a Cost-Reducing Project – Corporate Finance

Let's examine problem 75A. The first two parts of this problem are straightforward. We've done similar problems to them before, 74 is very similar, but part C is like a level of difficulty higher, significant step up in difficulty because it's an algebra puzzle to be solved. Let's do parts A and B and I think you'll find pretty similar if you got 74, no problem. Part C though, a little bit more difficult. Let's jump into it. Your company is considering purchasing a new piece of equipment that will significantly increase efficiency and reduce costs. The equipment costs $400,000 and it will be depreciated using the straight line method to zero over its four year life. So okay, $400,000 piece of equipment, it's going to zero and it's going to do that over four years. That is $100,000 per year in straight line depreciation, which of course is a number that's going to come in handy for us. The equipment will be sold for $20,000 at the end of four years and the reason for the purchase is because it will reduce other costs by $130,000. So it doesn't increase revenue, it just makes us run more efficiently, reduces costs. So similar to 74 per year and we'll reduce working capital needs by $40,000. That's a first. Most of the time we enter these projects and it increases working capital. This one decreased the working capital, which is good for cashflow in the short term. The company's tax rate is 20% and the required rate of return, the discount rate is 10%. Should we accept the project, we'll use NPV, net present value to figure out if we should accept the project or not. And to get to NPV, we got a bunch to do. We got to do a pro forma income statement is a great starting point. So how does this project affect sales? It does. No change in sales. How does it affect costs? Well, if successful, costs are actually reduced by $130,000. So a reduction in costs means earnings before interest, taxes, depreciation, and amortization EBITDA should go up by $130,000. That's a savings, right? What about our depreciation? Well, our depreciation is also going up by $100,000, meaning our EBIT leaves us $30,000 better off. Earnings before interest and taxes is up 30. Well, interest is not considered here, even though we may be borrowing money. Interest would be considered in a future calculation. For this though, we're looking at the project on its own terms. We're not considering interest because that's a financing option, right? That's not really intrinsic to the project. If you could finance it with your own cash, you don't pay any interest. Does it mean it's a better project? No. So, okay. EBIT is 30. So we ignore interest. If there was any, there's none mentioned here, but we have to deal with taxes. Taxes are 20%. 20% of 30,000 is 6,000, leaving us net income of 24K, $24,000. Okay. Well, remember, if we're doing net present value, we need the cash flows here. So, we go from our income statement to operating cash flows. And a way to calculate operating cash flows that we've been doing, standard way to do it, EBIT plus depreciation minus taxes equals OCF. So here's our little operating cash flow calculation. Our EBIT here, just taking it off that income statement, $30,000. Am I, is that visible? It's hard for me to tell. I think it is. Depreciation, we add 100,000. Take away taxes, take away 6,000 and 30 plus 100, 130 minus six is 124. Okay. Our operating cash flows are 124 and it's just savings, right? We're saving money. It says 130 grand, but in cash, we're actually saving 124 grand each year. So what does this look like? I always like to draw a bit of a timeline. So year, this is zero. What happens? Well, I've got to buy an asset, buy $400,000 worth of equipment. That's negative cash flow. But I also save on working capital. This reduces working capital, which is good for cash flow. It's a plus 40,000 to our cash flow. Year one, we just save our OCF, 124 grand, right? That's why we're doing the project. That's why I'm willing to put out 400 grand is because I save costs as I go. Year two, I save 124. Year three, save 124. And year four, I save $124,000. Okay. So those are my savings. Also in year four, other things happen. One, I have to get my working capital back to normal. The project's over. If the project was freeing up working capital, ending the project ties up that working capital again. So got to put, get the working capital reacquired. And the other thing is I get to sell the equipment. You couldn't sold for 20,000. You know, and I know I don't get all 20,000. If I depreciated the thing down to zero, I'm going to experience a $20,000 accounting gain. It's taxable at 20%. So there's $4,000 in additional taxes. So I don't get all 20. I only get 20 minus four. I get 16 of the 20 in terms of the equipment sale. Okay. That's it for my cash flows. So 400 minus 40 is 360 negative. Negative 400 plus 40 is negative 360. 124 for year one, 124 for year two, 124 for year three and 124 plus 16 is 140 minus 40 is 100 even, okay. So now these are all my cash flows. I just present value it and decide if it's a good idea or not. So to present value, you just divide by one plus the discount rate to the power of however many years you're discounting. So our initial cash flow is 1.1 to the power of zero, which is one. So it's negative 360 divided by one. It'll be negative 360. This is 124 divided by 1.1 to the power of one, 1.1 to the power of two, divide by 1.1 to the power of three, then divide by 1.1 to the power of four. We could do this by hand. It would take a minute. I'm going to do it in my fancy financial calculator here. So I just enter cash flow, second function clear. So cash flow at time zero 360 negative enter cash flow one, 124 enter. How many times did that one occur? One, two, three. So I'm going to enter three as the frequency there. My next cash flow to be entered. That's not cash flow at time two. That's just the next cash flow after the 124s is a hundred thousand and it happened once and there we have it. So we've, we've done all of our cash flow. I'm going to say, give me the present value of that at a discount rate of 10%. Enter hit the down arrow, compute NPV. I get 16, 6, 7, 1, 16, 6, 7, 1. That is my net present value. It's positive. And so should the project be accepted? Yes, except the project. I'll just note that number here. 16, 6, 7, 1. Positive that looks maybe like a minus there. So I put a plus beside it. Uh, yes. Okay. Um, so this was an annual savings of one 30. If the annual savings were 80, which is 50 grand a year worse, would you do it? And you can already just see the answer is going to be no way. It's going to be very negative net present value, but let's crunch the numbers to determine just how negative the net present value, uh, should be. So, um, here's how I'm going to do this. I'm going to just erase. I'm going to leave everything up kind of all the number, like all the, um, for example, sales costs, EBITDA, all the titles up, and I'm just going to erase the numbers and we're going to put new numbers in, but it's, it's the same question, just with slightly changed numbers. So no point in erasing, uh, all the guides here. So I'll leave this all up. Uh, we have the number 16, six, seven, one. And again, you can just rewind the video. If you want to go back to some number that was in there, I'm just going to erase all this. I'll leave the year one, two, three, four, and we can, uh, go again. So everything's the same, except for this is not one 30. This is eight, right? That number changes. Okay. So start from the start and we just do it again, do it again at 80 is the numbers actually come out a little weird here because we're going to end up negative numbers as you will soon see, but it's fine. So our sales are unaffected. Our costs are reduced by 80,000. So our EBITDA is up by 80,000. Unfortunately for us, our depreciation is a hundred. So our EBIT is negative 20. This is our first time with a loss before, uh, interest in tax, uh, having the loss, having negative EBIT means we have negative taxes, our taxes are 20%. So it's 20% of 20,000 is 4,000, but they're negative 4,000 in taxes. We get, we get a refund. It reduces our tax bill from, you know, the rest of our business. If we're losing money, uh, on this one, uh, at least accounting wise. So we end up at negative 16,000, negative 20 minus negative four is negative 16. It's, it is weird operating in negative numbers, but that's what's happening. So in terms of OCF, our EBIT is negative 20. Our depreciation is positive a hundred and our taxes are negative four. So it's minus negative four, right? We would subtract our taxes or subtracting negative four. Awkward construction here, but, uh, negative 20 plus a hundred is plus 80 minus negative four is plus four. So it's, it's plus 84 overall. Again, minusing negatives, never a comfortable thing to do, but there you have, we've done it. So what does this look like in terms of cash flows? Well, suspiciously similar to the last one. The only difference is this OCF. So the year zero cash flows still negative 400 K, um, plus the 40, we save reducing our working capital needs, uh, year one plus 84 year two plus a four year three plus 84 and year four plus 84, but minus, uh, the 40 of working capital that we have to reacquire. So again, we reduced our capital at the start projects over the project that caused us to reduce our working capital. We're going to need to reacquire that working capital. Maybe we're carrying more inventory now or something like that. Um, we also, uh, do exactly the same calculation for selling the asset. And it means we got 20 grand for it, but four grand went out in tax. So we get 16,000 from here is very much the same exercise as before. Minus 360, uh, plus 84 plus 84 plus 84. And then 84 plus 16 is a hundred minus 40 is 60. So this is plus 60 and same discounting as we did before. 1.1 to the zero, 1.1 to the one, 1.1 squared, 1.1 cubed and 1.1 to the fourth power, I won't do it in my calculator or I won't do it by hand, but I'll do it by my calculator and we'll get the right answer here. Hopefully let me clear what was in there. Second function clear. Uh, three 60 negative is my cashflow at time zero. My cashflow. Number one is 84,000 positive. The frequency on that is three. There's three payments of that amount or three cash flows that amount. And my second cashflow is 60,000. My second unique cashflow is 60,000. I only have one of those. Let's compute our net present value. Our discount rate is 10. Hit the down arrow, hit compute, and we get 1, 10, 1, 2, 4, and it's negative. So this is a money loser and we suspected it would be. It was negative 1, 1, 0, 1, 2, 4. Uh, would you accept the project? No. Okay. So let's just a bit of a step back at, uh, $130,000 per year in savings. Lost my mouse at $130,000 a year in savings. This was a great project. We made some money at $80,000 a year in savings. This was a bad project. The third question by far the hardest. So if you're struggling so far, buckle up, this is harder step. Uh, it says at what level of savings would you be indifferent as to whether or not the project should be accepted? At what level of savings would you be indifferent as to whether or not to accept the project? Okay. I'm going to erase something. I'm actually going to leave a few things up here, but I'm going to erase this. Again, you can just rewind if you need some of those numbers back up. I'm going to erase that. Uh, I'm going to erase these. Actually, why not just erase the whole thing? And the only things I wanted to leave up were the ones that didn't change. So the negative 400 and negative or 40 and, and what's down here doesn't change between the two. Um, so what is this asking? It's saying at what level of savings would we be indifferent as to whether the project would be accepted? Well, it's the net present value where we are indifferent. What's the net present value where we go? Oh, I could take or leave the project. That would be when net present value equals zero, we are indifferent. So we're looking for NPV to be zero. So we just did a present value on this information, but what we're missing is the savings, the operating cashflow, which is driven largely by the savings. So if I could figure out what is the operating cashflow and remember those were the same amounts in the first part, they were like one 24, one 24, one 24. In the last problem or the last version, part B was 84, 84, 84, 84. Um, so they're all the same amount. So I'm going to just say there's one variable to be solved for. Uh, let's figure out what, uh, what that X needs to be to make net present value zero. Now, how do I do it? Well, I know that I got to put out $360,000, just adding those two together, right? Negative 400 plus 40 is negative 360. And I can take the present value of these two as well. Negative 40 plus 16 is a negative 24,000. Let's present it, value it up to there. So you just divide by one plus the discount rate to the power of the number of years you want to discount it. 1.1 to the four. So 24,000 divided by 1.1 to the four to get that present value. And the idea is we just want to get the X's alone if we could. Right. And that's sort of, again, I mentioned, this is a challenging algebra question. It's challenging for me. You might be seeing that in my explanation. Um, but anyway, well, let's get the X's. 1.1 to the power of four, 1.461. We hit one over X. We multiply by 24,000 negative. 16, three 92. And so that goes up negative 16, three 92. So we've moved those out of this section. Uh, so the payments that we're aware of, or the present value of, of all the stuff that we already know is negative three 76, three 92, that's just negative 400 positive 40 and now 16, three 92. The present value of the other year for stuff. Okay. So if these are negative three 76, three 92, and I want net present value to be zero, it means the present value of these X's has got to be positive. Three 76, three 92. And what we have is a pretty straightforward annuity, right? Year one, this positive cash flows plus year two's cash flow plus year three plus year four has got to have a present value of three 76, three 92. We can solve for an annuity. I could do this by hand, right? There is present value of an annuity formula and we're solving for the payment amount, the regular payment amount, but I've got the financial calculator here. So I'll use it and I'm just going to fill in information. So what is my N? My N is four, four years. What is my I Y well, that's the discount, right? It's 10%. What is my PV? Well, you just said the PV has got to be three 76, three 92, because the present value of the other stuff is negative three 76, three 92. We want it to work out to zero. So three 76, three 92 positive PMT. We don't know. That's what we're solving for. And our FV is zero. Okay. So let's figure out what that payment amount is. Uh, four goes in as our N 10 goes in as our I Y PV three 76, three 92. And, uh, zero is our FV. Now we compute the PMT. I get one 18, seven, four, one, one, one eight, seven, four, one. That is my operating cashflow. That is the OCF one, one, eight, seven, four, one. Now remember that's not our savings. Our savings is like up here. We got to go all the way back. So we're just stepping backwards through the problem. Uh, unfortunately for us, we can't recalculate our taxes. We have our depreciation. Our depreciation is a hundred thousand dollars, but we can't recalculate the taxes. There's too many variables out there, uh, for us to figure out the taxes because we don't know our costs. Our cost savings and our cost savings creates our taxes. So there's, it gets awkward. The best way to do this is to actually use the other formula. We learned this chapter for calculating OCF. I told you this was challenging. So if you look back to problem seven, one, a, you'll see, we calculated OCF both the standard way. That's what we're doing here. And we also did it, the tax shield approach to calculating OCF. And let me just remind you how the tax shield approach works. So tax shield approach, tax shield approach. I can already feel I'm 20 minutes deep in the video. And we're like going back to an old problem is it's a stretch. This is a challenging problem. No doubt. So I hope you're up for it. If you're hanging in there, thank you so much for hanging in there. Uh, okay. So the tax shield approach says. Take our sales minus our costs times one minus the tax rate. And we add in our tax shield. And it's also called the depreciation tax shield because depreciation saves us some tax dollars. So first let's calculate our depreciation tax shield, which we can do. It's a hundred thousand dollars a year in depreciation taxes are 20% depreciation saves us $20,000 in taxes. So the tax shield, $20,000, uh, sales minus costs. That's an interesting one. Sales. We know are unaffected. We're, we're going after costs here. That's what we want to know. So our sales are unaffected. Our costs. Well, we're, that's what we're solving for. Uh, times one minus the tax rate. So, uh, one minus tax rate, tax rates, 20%, one minus the tax rate is 0.8. Plus 20,000 equals, and this is our OCF. We just calculated our OCF. We said our OCF is 1 1 8 7 4 1. It's what's in the calculator. 1 1 8 7 4 1. I know it's in there as the negative number. It's not negative operating cash flow. Just funny things happen as you use your financial calculator is positive 1 1 8 7 4 1. So zero minus costs times 0.8. So zero times 0.8 is zero. I'm just going to get rid of the zero. Negative cost times 0.8 is 0.8 negative 0.8 costs. Just multiplying it in plus 20,000 equals 1 1 8 7 4 1. Uh, let's take away 20,000 1 1 8 7 4 1 minus 20,000. It's 98 7 4 1 equals 0.8 costs and negative 0.8 costs. So let's divide both sides by negative 0.8 divided by 0.8 negative. And I get negative 1, 2, 3, 4, 2, 6. So costs equals negative 1, 2, 3, 4, 2, 6. Okay. That's our answer. I know it took a long journey to get there, but the question said at what level of savings would be, we be indifferent as to whether or not the project would be accepted. We would be indifferent. Net present value would be equal. If our cost savings was 1 23, 4 26. If that said 1 23, 4 26, our net present value would be zero. This is a big problem and a challenging one. If you got through it, good for you. Thank you so much for hanging in there with me. By the way, I'm going to end the video now. I'll give you permission to leave now. I just want to prove it. I want to like, I'm going to redo the problem one last time. Again, you don't have to stick around for this, but I'm going to redo it. As if this number had been 1 23, 4 26, and we're going to see that net present value is zero or close to, it'll be off by a couple of dollars. Cause we're, you know, it's 1 23, 4 26.25. And I've, I've missed a lot of decimals along the way. We're going to be close to zero as our net present value. So let's try this one last time with 1 23, 4 26 as our savings. We we've solved the question. The what level of savings? 1, 2, 3, 4, 2, 6. Okay. Goodbye. You know, this is just for my own fun. I know the things I do for fun, 25 minutes into a video, but five more minutes isn't going to kill me. I'd like to sort of see this one through. Okay. So if the question had said, we say 1, 2, 3, 4, 2, 6, our sales unaffected. Our costs are reduced by 1, 2, 3, 4, 2, 6. Our EBIT goes up by 1, 2, 3, 4, 2, 6. Our depreciation remains a hundred thousand. Our, or sorry EBITDA is 1, 2, 3, 4, 2, 6. Our EBIT is 23, 4, 2, 6. And our taxes here, 23, 4, 2, 6. Times a 20%. So times 0.2. 4, 6, 8, 5. Our net income then 4, 6, 8, 5 or a 23, 4, 2, 6 minus 4, 6, 8, 5. But if you can do this, you know, these things inside out, right? Like it's a challenging for me and I've done a lot of these. So as a rookie, you should be pleased with yourself. If you're getting through this, no problem. A very good chance. You're smarter than me, but lots of my viewers are definitely smarter than me. Uh, okay. 23, 4, 2, 6 is our EBIT. A hundred thousand dollars a year is our depreciation. Our taxes 4, 6, 8, 5. So I'll subtract that and add that. So 23, 4, 2, 6 plus a hundred minus 4, 6, 8, 5 equals 1, 1, 8, 7, 4, 1. Oh, that's a good number. That's a number I remember from when we did this before. Oh, look at this. I think we're going to get it. I have a great feeling about this. Okay. So our cash flows still were 400 negative and 40 positive there. But our savings is 1, 1, 8, 7, 4, 1 or our OCF rather 1, 1, 8, 7, 4, 1. 1, 1, 8, 7, 4, 1 and 1, 1, 8, 7, 4, 1. Oh, this is great. I'm pleased to see it's working. Uh, negative 360 is our total here. Uh, 1, 1, 8, 7, 4, 1 here, 1, 1, 8, 7, 4, 1, 1, 1, 8, 7, 4, 1. And the last one, 1, 1, 8, 7, 4, 1 minus 40 plus 16 is 94, 7, 4, 1. Okay. So again, we could present value this all by hand, divide by 1 plus the discount rate to the power of the number of years. We're not going to do that. I'm going to use my fancy calculator. CF clear what's in there. CF 360 negative enter down 1 cashflow 1, 1, 1, 8, 7, 4, 1, 1, 1, 8, 7, 4, 1. Uh, enter down. Now we had three of them. 1, 2, 3. So I'll put three in as my frequency. Cashflow at time 2, 94, 7, 4, 1 hit enter. And there's only one of those. And now I'm going to compute net present value. So I go NPV. It asks for the discount rate. It's 10 hit the down arrow. Please work, please work before I, before I receive it's works. Just hit one of those buttons for me. It's good luck. It's going to be good luck. We're hoping for net present value to be zero or close to it. Like, you know, a dollar, $2 under $10. I'm going to, I'm going to take a W here. I'm going to take a victory lap. Uh, I got to hit compute. Here we go. Compute 0.67. Yes, it worked. Everybody. It worked. Okay. But with that, the whole last five minutes was just for fun. If you had fun, thanks so much for watching. If you stuck around, uh, a challenging problem, I think part C part A and B totally reasonable part C heavy lifting. But if you made it to the end and you lifted the heavy stuff with me. Thanks for being here. Thanks for hanging in there. Have a great day. Bye bye.