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  • 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.

In the last video, we figured out what is the present value

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A2 初級

現值 3 (Present Value 3)

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    kellylin007 發佈於 2021 年 01 月 14 日
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