Present Value 3 What happens when we change the discount rate?
Present Value 3
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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.
- 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.
- The present value of this is -- it increased by $6 just by
- the discount rate going down by 3%.
- 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.
- 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.
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