CF 3-13A – Perpetuities – Corporate Finance

Let's examine problem three 13, a it's a perpetuity question. Good news is the math of perpetuities is less complicated than much of the math we've been doing. The present value of perpetuity is, uh, the cashflow, the recurring cashflow divided by the discount rate. That's it. So, uh, comparing it to an annuity formula, fairly simple. Uh, let's jump into the question. You want to start a scholarship that will award the winning student with a $2,000 with $2,000 each year. The scholarship will be awarded to a new student every year in perpetuity. The first scholarship will be awarded one year from now. And the discount rate is 7%. How much do you need to donate today to fund the scholarship? Okay. So it's 2000 divided by 0.07 2000 divided by 0.07 it's 28, 571, 28, 571. That's how much you would have to donate today. And the idea here is if you donate exactly 28, 571, and it grows by 7%. Well, let's do some math. 28, 571 times 0.07. Look at that. You get $2,000. So you can just let the money grow by $2,000. Then skim the $2,000 off the top to pay the scholarship. And then you're back to 28, 571, and it's going to grow another $2,000. And the idea is you can just do this in perpetuity. So that's what makes a perpetuity a perpetuity. Um, not that you'd ever want to do this, but to enter a perpetuity into a financial calculator, again, the formula is so simple, but I know some folks will come to rely on their financial calculators. Here's how you do it. Uh, you just go, uh, 2000 as your PMT recurring payment. You go seven as your IY, uh, you go, uh, no, this is the tricky part. N is forever. So I just put like a billion, you know, a billion years from now is your N and then you compute PV 2, 8, 5, 7, 1. You can see 2, 8, 5, 7, 1. It's the same answer. So just wanted to show that, you know, you could do this in a financial calculator, you just put a billion as the timeframe and it works out to the same. Um, okay. Suppose you wanted the scholarship to commence five years from today. And the first payout to be five years from today. It means you've got a discount. So, so it's the same formula. So, okay, let me read the question. Suppose you wish for the scholarship to commence five years from today. How much would you need to donate today to fund the scholarship? So remember we need 2, 8, 5, 7, 1 at the beginning of this year. So we pay it out at the end of the year. Cause it's going to grow by 2000 and that's what makes this a perpetuity. So we actually need this 2, 8, 5, 7, 1 to be in place four years from now in order to start paying out the perpetuity five years from now, so I need 2, 8, 5, 7, 1 four years from now. How much money do I need to put away today? That's going to grow at 7% to get to 2, 8, 5, 7, 1 four years from now. That's the question. And here's the math. You go 2, 8, 5, 7, 1 divided by 1.07 to the fourth, right? It's just a present value of a lump sum. So, uh, it's, you know, this formula, right? Uh, PV times one plus R to the T equals FV. Well, we know our FV. We want 2, 8, 5, 7, 1 to be in place four years from now, not five, because again, it's got to grow to, uh, grow by 2000, uh, to pay the scholarship at the end of the year, um, okay. So in this circumstance, we know our FV is 2, 8, 5, 7, 1. We know one plus R to the T is 1.07 to the fourth, and we don't know our PV. That's how much we got to put away today. So that's how we sort of rearrange to get to here. 1.07 to the power of four is 1.31. I'm going to go one over X times 2, 8, 5, 7, 1 to do the math there. Oh, no, I messed up 1.07 to the four. One over X times 2, 8, 5, 7, 1 wonder what I clicked wrong in my calculator. There, there we go. 21, 7 97. So I need to put away 21, 7 97 today. It's going to grow in some, you know, investment account for, uh, four years. At the start of year five, I'll have 21, uh, 28, 5 71 put away. It'll grow by two grand. We'll trim the two grand off the top, pay our first student and do that in perpetuity. There we have it. We've solved 313 A as always. I hope these videos are helping. And if they're helping you, I hope you'll consider helping me by hitting one of those two buttons. Thanks for watching. Have a great day. Bye for now.