Present value 3

In the last video, we figured out what is the present value of these three different payment timing choices. If we had a 5% risk-free rate, and if these payments were risk-free, instead of coming from -- you can almost view them as some type of government program, where they're asking you to choose which of these three payment streams from the government do you want? And so we'll use the same rate that the government would pay you, if you lent them money. And that's given by the treasury rate. And in the first case we assumed a 5% treasury rate. And if you watched the first present value video, I think you understand why compounding going forward is the same thing as discounting that rate by going backwards. If you want to know how much $100 is a year from now, you multiply that times one plus the interest rate, right? So if it's 5%, you multiply that times 1.05. If you're taking $110 and going a year back, you divide by 1.05. So it's just the same operation. You're just going forward or back. Forward is multiplication, backwards is division. But anyway, the result that we got in the last video is that the present value -- let me do this in a different color. And I'll introduce my notation. The present value, if we assume a 5% rate, no matter how long-- how far away the money is given to you. And you'll see what I mean because I'll change that assumption in a second. But if we assume that the risk-free rate is 5%, then the present value of $100 today, well that was just $100. $110 in two years, we got that by doing 110 divided by 1.05 squared, right? You divide by 1.05 there, and then you divide by 1.05 again. And then you get $99.77. I don't want to run out of too much space. I could have probably done this whole thing a little bit bigger. And then choice number three. How did we get that? Well, we said -- let me do that in a different color. That was the present value of the $20 today, plus $50 in one year, divided by that, discounted to the present day. So divided by 1.05 plus $35 divided by 1.05 squared. And we had gotten $99.36. And that's what that should be worth to you today, if you assume that these payments are risk-free, and you use a 5% discount rate. Fair enough. And based on these calculations, choice number one was the best, choice number two was second best, choice number three was third best. Fair enough. Now what happens -- after I pose the question, you might want to think about it before I show you the answer -- what happens if I don't assume a 5% discount rate? What happens if I assume a 2% discount rate? This is just my notation. What is the present value of these if I assume a 2% risk-free rate, or a 2% discount rate? Well $100, I'm getting that today, so that's still worth $100. You could even do that as -- let me do that in a more vibrant color -- as 100 divided by 1.02 to the 0 power, because we're getting it today. But that's just 1.02 divided by 1, which is just $100. $100 today. What's the present value? It's $100. Now what's the $110 two years out going to be worth? So this is interesting. When the interest rate goes down, from 5% to 2%, I'm going to be dividing by a smaller number. 1.02 squared is a smaller number than 1.05 squared. So the present value of this payment should go up. Interesting. This is something to keep in mind for later, when we start thinking about bonds. When you lower the interest rate, the present value of this future payment goes up. And it just falls out of the math. You're discounting by a smaller number. Let's figure out what that is. So if I take $110 and I divide it by 1.02 squared, right? Discounted twice. I get $105.72. Oh, and how did I get that? That was equal to -- I'm doing it in reverse here -- that was equal to 110 divided by 1.02 squared. And our intuition was correct. Just by the interest rate going from 5% to 2%, the present value of this payment two years out -- it's in year three, but it's two years out. Actually I should re-label this. I should call this now, the present. I should call this year one. I was calling this year two, one year out. But I think that makes it confusing. I called this year two, so this is now. So you could call this year zero. This is year one. And this is year two. Anyway. The present value of this is -- it increased by $6 just by the discount rate going down by 3%. Fascinating. Now let's see what happens to choice number three. Choice number three, the $20 today, the $20 now, well that's just worth $20. Its present value is 20 plus 50 divided by 1.02, plus the 35 divided by 1.02 squared. Let's see what this adds up. 20 plus 50 divided by 1.02 plus 35 divided by 1.02 squared. $102.66. Now there's a couple of really interesting things. And this is a really good time to kind of let it all sink in. All of a sudden we lowered the interest rate. And now choice number two is the best, followed by choice number three, followed by choice number one. So it almost -- choice number one was the best when we had a 5% discount rate. Now at a 2% discount rate, choice number two is all of a sudden the best. And there's something else interesting here. Choice number two improved by a lot more when we lowered the interest rate, than choice number three did. Its present value went from $99.77 to $105.70, so it's almost $6. While here it only improved by less than $3, right? So why is that? Well, when you lower the interest rate, the terms that are using that discount rate the most, benefit the most. So all of this payment was two years out, right? So it benefited the most by decreasing the discount rate, the 1.02 squared. It changed this value the most. These payments are spread out. Only some of its payment is two years out. Then some of its payment is one year out, and that's going to benefit less. And then some of its payment is today. So it will benefit, because you are discounting some of the cash payments. But it's going to benefit by less. Anyway, I'll leave you there in this video. And in the next video, we're going to see what happens when we have different discount rates for different amounts of time. See you in the next video.