# Present valueÂ 3

## Video transcript

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.