FIN 20 Present Value of an Annuity

Let's take a look at problem three, eight, a we're taking the present value of an annuity. An annuity is just a stretch of recurring cash flows. And this is such a common problem in corporate finance. I promise if you're doing finance, you will see this type of problem in your class where you've got a string of payments. So let's look at the situation and kind of contemplate how we're going to tackle it. You run a taxi company and are thinking of investing in all electric vehicles to change your fleet from gas powered to electric would cost a million dollars today, but would produce annual savings of $150,000 over the 10 year life of the cars. Company uses a discount rate of 6%. So it's going to cost us a million dollars today. But over the 10 year life of the cars, let's see. So we got 10 years, let's call this five. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. OK, so there we have our 10 years on a number line. Each year we're going to save $150,000 plus 150. Sorry, I have to write this 10 times plus 150 plus 150 plus 150 plus 150 plus 150 plus 150. And the question is, what's the value of saving this $150,000 a year over the next 10 years? And the answer is not 1.5 million, 150 times 10 is 1.5 million. But of course, that's future dollars. They need to be discounted to today. And so what we learned in previous videos is, well, if I want to discount this, I can just go, well, this 150 divided by 1 plus the discount rate, 1.06 to the 10 and do this one divided by 1.06 to the 9 divided by 1.06 to the 8 and so on. And add those all up and I can figure out what the discounted cash flows are going to be. But that's a pain in the neck, isn't it? So you could do this and get the right answer, but it's too much work and it's too tedious. So there's really three ways to solve here, and I'm going to go over two. So the way I'm not going to go over is these things called present value tables, which your professor might use. And if they do use, unfortunately, this isn't the video series for you. Although I do think you can, even if they give you present value tables, you can still solve it formulaically. And that's the first way we are going to solve it together. The second way we're going to solve it together is with the calculator. So of the three ways, we're not going to use tables. We are going to do it via the present value formula. And we're also going to do it in our financial calculator. And that's what we'll do in this video. So let's solve it. So some smart mathematician looked at this situation was like, OK, yes, I could brute force and calculate 10 present values and add them all together. But that seems like a lot of work. You know, why don't I solve for this mathematically? I don't know how they come up with this formula. They're much smarter people than me, but here's the formula they came up with. They said, OK, the present value of annuity equals the recurring cashflow amount. And it's got to be recurring. So in this case, it's $150,000 of recurring, you know, annual cashflow times. Now this is such an awkward construction, but you get used to it. Once you've done this like 10 times, you sort of get used to it. One minus one divided by one plus our discount rate. And our discount rate was, uh, what was it? 6%. So 1.06 is what goes in this bracket. 1.06, one plus the discount rate to the power of time T. This was 10 years, uh, all divided by 0.06. Okay. So now we just have to like crunch this down. Uh, I'll do it in the calculator. So one point. So I start with the exponents, right? There's sort of an order of operations, things going here. And, uh, uh, we start with this exponent, which is inside the bracket. So 1.06 to the power of 10. Then we want to go one divided by that number. Well, we have actually a one divided by that number button in our financial calculator. So I go one over X. So now what's inside this bracket is, you know, in here is 0.558. I'm going to just keep it in my calculator though. So I go one minus that number. The way I do this is I just go that number minus one, and then I hit the plus minus button. So one minus that number is 0.441. And then I divide by 0.06, divide by 0.06 and I get 7.36. So this big bracket is 7.3600. So I take 150,000 times. And then again, the value of that bracket, when you break it all out is 7.36. So 150,000 times 7.36 is 1 1 0 4 0 1 3 1 1 0 4 0 1 3. And okay. So we've kind of solved it, uh, 1 1 0 4 0 1 3. So compute the present value to determine if this is a worthwhile investment. It costs us a million dollars to generate $150,000 annually of positive cashflow from our cash savings on the, uh, uh, electrification of our cars. We're saving $1.1 million. So we're spending a million dollars to today to save us the present value of $1.1 million. Is it worthwhile? The answer is yes. Right. Overall, we are 104,013 better off if we switch the fleet to electric. Now, of course there could be a million little things we're not considering here, but just based on these predictions, we should do it. And actually you see lots of taxicab companies have done that just because it does, uh, may save them. They, they likely did similar math. Um, in any event, there we are. We're $104,000 better off if we go electric. So we've solved it. So we've done it the formula way. I'll show you how to do it within our calculator. And this isn't too complicated. If you've been using your calculator all along, if you don't have a calculator, you know, say goodbye, goodbye everybody. But if you do have a calculator, this is probably the way to do it. Once you get into PV of an annuity, a lot of the lump sum present values, I would just do them by hand. I think it's quicker. PV of an annuity is the one where I go, ah, you know, I think I'd rather just crunch the numbers within the calculator. Things can go wrong with the math we just did. Uh, so in your calculator, make sure you've cleared your memory. Second function, clear TVM, clear time value of money. Second function, clear work. I don't know what that does, but I always hit it just in case. Uh, and then let's sort of figure out what we got. And we have an, that's the number of periods. And the number of periods here is 10. It's a 10 year life of the car. I Y is our discount rate, which is six. We enter six as I Y PV. We don't know. That's what we're going to be solving for. Or do we know our PV? Well, no, we don't know our PV. Um, we know our payment is, uh, $150,000 flowing to us. 1 5 0 0 0 PMT. It didn't happen at the beginning of a period. We'll look at those. Those are called annuities due and FV is zero. These things go down to zero. So I'm going to compute the present value of those payments essentially. So I go compute PV and I get 1 1 0 4 0 1 3. Well, look at that. 1 1 0 4 0 1 3. It's the exact same answer. Let me put them beside each other. And there we have it. Uh, so the value of this is 1 1 0 4 0 1 3. If I have to pay a million dollars to get $1.1 million in value, I should do it. We've solved this using both the formula as well as our financial calculators. Thanks for watching as always. Thanks for sticking with me. Have a great day, everybody. I'll see you in the next video. Bye bye.