Present Value 3

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Present Value 3발음듣기

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Start In the last video, we figured out what is the present value of these three different payment timing choices.발음듣기

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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?발음듣기

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And so we'll use the same rate that the government would pay you, if you lent them money.발음듣기

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And that's given by the treasury rate.발음듣기

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And in the first case we assumed a 5% treasury rate.발음듣기

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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.발음듣기

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If you want to know how much $100 is a year from now, you multiply that times one plus the interest rate, right?발음듣기

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So if it's 5%, you multiply that times 1.05.발음듣기

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If you're taking $110 and going a year back, you divide by 1.05.발음듣기

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So it's just the same operation.발음듣기

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You're just going forward or back.발음듣기

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Forward is multiplication, backwards is division.발음듣기

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But anyway, the result that we got in the last video is that the present value - let me do this in a different color.발음듣기

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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.발음듣기

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And you'll see what I mean because I'll change that assumption in a second.발음듣기

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But if we assume that the risk-free rate is 5%, then the present value of $100 today, well that was just $100.발음듣기

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$110 in two years, we got that by doing 110 divided by 1.05 squared, right?발음듣기

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You divide by 1.05 there, and then you divide by 1.05 again.발음듣기

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And then you get $99.77. I don't want to run out of too much space.발음듣기

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I could have probably done this whole thing a little bit bigger.발음듣기

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And then choice number three. How did we get that?발음듣기

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Well, we said - let me do that in a different color.발음듣기

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That was the present value of the $20 today, plus $50 in one year, divided by that, discounted to the present day.발음듣기

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So divided by 1.05 plus $35 divided by 1.05 squared.발음듣기

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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.발음듣기

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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.발음듣기

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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?발음듣기

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What happens if I assume a 2% discount rate?발음듣기

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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?발음듣기

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Well $100, I'm getting that today, so that's still worth $100.발음듣기

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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.발음듣기

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But that's just 1.02 divided by 1, which is just $100. $100 today.발음듣기

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What's the present value? It's $100.발음듣기

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Now what's the $110 two years out going to be worth?발음듣기

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So this is interesting. When the interest rate goes down, from 5% to 2%, I'm going to be dividing by a smaller number.발음듣기

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1.02 squared is a smaller number than 1.05 squared.발음듣기

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So the present value of this payment should go up.발음듣기

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Interesting. This is something to keep in mind for later, when we start thinking about bonds.발음듣기

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When you lower the interest rate, the present value of this future payment goes up.발음듣기

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And it just falls out of the math.발음듣기

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You're discounting by a smaller number.발음듣기

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Let's figure out what that is.발음듣기

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So if I take $110 and I divide it by 1.02 squared, right?발음듣기

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Discounted twice. I get $105.72. Oh, and how did I get that?발음듣기

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That was equal to - I'm doing it in reverse here - that was equal to 110 divided by 1.02 squared.발음듣기

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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.발음듣기

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Actually I should re-label this.발음듣기

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I should call this now, the present.발음듣기

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I should call this year one.발음듣기

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I was calling this year two, one year out.발음듣기

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But I think that makes it confusing.발음듣기

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I called this year two, so this is now.발음듣기

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So you could call this year zero.발음듣기

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This is year one. And this is year two. Anyway.발음듣기

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The present value of this is - it increased by $6 just by the discount rate going down by 3%.발음듣기

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Fascinating. Now let's see what happens to choice number three.발음듣기

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Choice number three, the $20 today, the $20 now, well that's just worth $20.발음듣기

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Its present value is 20 plus 50 divided by 1.02, plus the 35 divided by 1.02 squared.발음듣기

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Let's see what this adds up.발음듣기

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20 plus 50 divided by 1.02 plus 35 divided by 1.02 squared. $102.66.발음듣기

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Now there's a couple of really interesting things.발음듣기

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And this is a really good time to kind of let it all sink in.발음듣기

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All of a sudden we lowered the interest rate.발음듣기

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And now choice number two is the best, followed by choice number three, followed by choice number one.발음듣기

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So it almost - choice number one was the best when we had a 5% discount rate.발음듣기

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Now at a 2% discount rate, choice number two is all of a sudden the best.발음듣기

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And there's something else interesting here.발음듣기

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Choice number two improved by a lot more when we lowered the interest rate, than choice number three did.발음듣기

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Its present value went from $99.77 to $105.70, so it's almost $6.발음듣기

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While here it only improved by less than $3, right?발음듣기

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So why is that? Well, when you lower the interest rate, the terms that are using that discount rate the most, benefit the most.발음듣기

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So all of this payment was two years out, right?발음듣기

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So it benefited the most by decreasing the discount rate, the 1.02 squared.발음듣기

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It changed this value the most.발음듣기

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These payments are spread out. Only some of its payment is two years out.발음듣기

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Then some of its payment is one year out, and that's going to benefit less.발음듣기

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And then some of its payment is today.발음듣기

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So it will benefit, because you are discounting some of the cash payments.발음듣기

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But it's going to benefit by less.발음듣기

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Anyway, I'll leave you there in this video.발음듣기

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And in the next video, we're going to see what happens when we have different discount rates for different amounts of time.발음듣기

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See you in the next video.발음듣기

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