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### Course: Macroeconomics > Unit 4

Lesson 8: Interest rates and the time value of money# Present value 2

More choices as to when you get your money. Created by Sal Khan.

## Want to join the conversation?

- Say I would like to see what the risk free rate is today. How can I find that out? is there a website or anything that can tell you the risk free rate today?(35 votes)
- We assume that US government bonds are risk free. So any day you want, Just google 30 day treasury rate or 30 year treasury rate. The risk free rate goes up the longer you have to lock up your money so pick your amount of time and google away!(12 votes)

- Why did you use a 5% interest rate? Is it some benchmark rate or was it just random?(7 votes)
- It was not random, but was arbitrary, just like the $100 was arbitrary. Sal chose a simple, round number to make it easy to see where everything goes when you do this kind of math.(26 votes)

- on the third choice, how come you did not consider the interest that the person could have earned with the 20 and 50 that were given to him. The person could have earned 5% on those two payments.(16 votes)
- The math works out to about the same. $20 at 5% for a year yields $21. Add the $50, and $71 at %5 yields $74.05. Add the $35, and you get $109.05, less than either of the other two options. (P.S. Thank you, Sal, for teaching me how to do that in my head while I watch your vids!)(39 votes)

- Why compute backwards to present value instead of computing forwards to get the final amount? Either way you want the one with the bigger result in the end it seems.(5 votes)
- They both work when trying to determine which is worth more, but someone might just want to know what something is worth today. Really it's no more difficult, you just divide instead of multiply.(11 votes)

- Lets say i just add up $20 + $50 + $35 = $105 on the third one, then I just take the PV(105), is that possible?(5 votes)
- Not quite- you'd need to take the interest you'll receive from the $20 and $50 into account.(10 votes)

- I'm missing a nuance here and maybe someone can help. For the last option it seems to me you are really comparing what you could earn on the money if you invested each yearly payment till year 2 vs. if you got it all year one and invested it for two years vs. getting a larger amount in year 2. To my mind the correct way to calculate the third option is to: 20*1.05=21, then (21+50)*1.05=74.55, then for year 2 (74.55+35)*1.05= $115.02. This is what you'd get if you invested the money yearly and it compounded, which is what you'd do in the real world. Can someone clarify?(2 votes)
- Close, Robert. $20*1.05+$50 ($71) will be what you'll have at year 1. Then $71*1.05+$35 ($109.55) will be what you'll have at year 2. You don't add the interest to the final amount ($35) because you're just getting that now.(8 votes)

- Why do we use PV to select the best deal as opposed to comparing the future value? For example if we had a choice 4 with the highest future value (FV) after two years among the choices, would we choose choice 1 with the highest PV or choice 4 with the highest FV?(2 votes)
- We normally choose the option with the highest present value. This is because we have no idea when in the future we are going to need the money. As a result, we just use present value to approximate that.(4 votes)

- at8:36, Sal divides the future payments by 1.05 to find the present value. But those values haven't been invested for that time, so it can't have accrued the 5% interest rate. So why do you have to divide by 1.05 to find the present value. wouldn't it still be the same value, since it hasn't been invested during that time?(3 votes)
- Present value is the value today of a payment made in the future. For example, if I am trying to sell you a municipal bond that matures in one year and bonds just like it are yielding 4.5% (assuming you are not going to get a coupon payment), then the value of that bond today (that is, what you SHOULD pay today for that bond) is $956.94. (Bonds usually have a face value of $1000). So for that payment that they are going to give you in a year, in order to make 4.5% on your money, you need to pay $956.94 for it today (and then hold it for one year at which point you get back $1,000). Make sense?(2 votes)

- What is the term for the 1.05 at around5:50? I know the rate is 5% but if i were to make a sort of equation, i would do Principle/1.05. Thanks :)(2 votes)
- Just wondering...how would the blowing up of the financial system play out? In what ways would that affect all of us?(2 votes)

## Video transcript

Now I'll give you a slightly
more complicated choice between two payment options. Both of them are good, because
in either case you're getting money. So choice one. Today I will give you $100. I'll circle the payment when
you get it in magenta. So today you get $100. Choice two. And I'll try to write this
choice a little bit neater. Choice two is that not in
1 year, but in 2 years. So let's say this is year 1. And now this is year 2. Actually I'm going to give
you three choices. That'll really hopefully
hit things home. So actually let me scoot this
choice two over to the left. Back to green. So now I'm back in business. So choice two, I am willing to
give you, let's say, oh I don't know, $110 in 2 years. So not in 1 year. In 2 years I'm going
to give you $110. And so I'll circle in
magenta when you actually get your payment. And then choice three . And choice three is going
to be fascinating. I've done it in a slightly
different shade of green. Choice three, I am going to pay
you-- I'm making this up on the fly as I go-- I'm going
to pay you $20 today. I'm going to pay you
$50 in 1 year. That's $70. Let me make this
so it's close. And then I'm going to pay you,
I don't know, $35 in year 3. So all of these are payments. I want to differentiate between
the actual dollar payments and the
present values. And just for the sake of
simplicity, let's assume that I am guaranteed. I am the safest person
available. If the world exists, if the sun
does not supernova, I will be paying you this
amount of money. So I'm as risk-free as the
federal government. And I had a post on the previous
present value, where someone talked about,
well is the federal government really that safe? And this is the point. The federal government, when
it borrows from you $100. Let's say it borrows $100
and it promises to pay it in a year. It'll give you that $100. The risk is, what is
that $100 worth? Because they might inflate
the currency to death. Anyway, I won't go into
that right now. Let's just go back to this
present value problem. And actually sometimes
governments do default on debt. But the U.S. government
has never defaulted. It has inflated its currency. So that's kind of a round
about way of defaulting. But its never actually said,
I will not pay you. Because if that happened, our
entire financial system would blow up and we would all be
living off the land again. Anyway, back to this problem. Enough commentary from Sal. So let's just compare choice
one and choice two again. And once again let's say that
risk-free, I could put money, I could lend it to the federal
government at 5%. Risk-free rate is 5%. And for the sake of simplicity--
in the next video I will make that assumption less
simple-- but for the sake simplicity, the government will
pay you 5% whether you give them the money for 1 year,
whether you give them the money for 2 years, or
whether you give them the money for 3 years, right? So if I had $100, what would
that be worth in 1 year? We figured that out already. It's 100 times 1.05. So that's $105. And then if you got
another 5%? So the government is giving
you 5% per year. It would be 105 times 1.05. And what is that? So I have 105 times 1.05,
which equals $110.25. So that is the value
in 2 years. So immediately, without even
doing any present value, we see that you'll actually be
better off in 2 years if you were to take the money now
and just lend it to the government. Because the government,
risk-free, will give you $110.25 in 2 years, while I'm
only willing to give you $110. So that's all fair and good. But the whole topic, what we're
trying to solve, is present value. So let's take everything
in today's money. And to take this $110 and say
what is that worth today, we can just discount it backwards
by the same method, right? So $110 in 2 years, what
is its 1-year value? Well, you take $110 and
you divide it by 1.05. You're just doing the reverse. And then you get some
number here. Well that number you get
is 110 divided by 1.05. And then to get its present
value, its value today, you divide that by 1.05 again. So you get 110 divided. If I were to divide by 1.05
again what do I get? I divide by 1.05, and then
I divide by 1.05 again. I'm dividing by 1.05 squared. And what does that equal? And I'm writing this on purpose,
because I want to get you used to this notation. Because this is what all of
our present values and our discounted cash flow, this type
of dividing by 1 plus the discount rate to the power of
however many years out, this is what all of that's
based on. And that's all we're doing
though, we're just dividing by 1.05 twice because we're
2 years out. So let's do that. 110 divided by 1.05 squared
is equal to $99.77. So once again we have verified,
by taking the present value of $110 in 2
years to today, that its present value-- if we assume
a 5% discount rate. And this discount rate, this
is where all of the fudge factor occurs in finance. You can tweak that discount
rate and make a few assumptions in discount
rate and pretty much assume anything. But right now, for
simplification, we're assuming a risk-free discount rate. But when the present value is
based on that, you get $99.77. You say, wow, yeah, this really
isn't as good as this. I would rather have $100 today
than $99.77 today. Now this is interesting. Choice number three. How do we look at this? Well what we do is, we
present value each of the payments, right? So the present value of $20
today, well that's just $20. What's the present value
of $50 in 1 year? Well the present value of that
is going to be-- so plus $50 divided by 1.05, right-- that's
the present value of the $50, because it's
1 year out. And then I want the present
value of the $35. So that's plus $35 divided by
what-- it's 2 years out, right, so you have to
discount it twice-- divided by 1.05 squared. Just like we did here. So let's figure out what
that present value is. Notice I'm just adding up the
present values of each of those payments. Get out my virtual TI-85. Let's see, so the present value
of the $20 payment is $20, plus the present value
of the $50 payment. Well that's just 50 divided by
1.05, plus the present value of our $35 payment. 35 divided by-- and it's 2 years
out, so we discount by our discount rate twice-- so
it's divided by 1.05 squared. And then that is equal to--
we'll round it-- $99.37. So now we can make a very
good comparison between the three options. This might have been
confusing before. You know, you have this
guy coming up to you. And this guy is usually in
the form of some type of retirement plan or insurance
company, where they say, hey, you pay me this for years a,
b, and c, and I'll pay you that in years b, c, and d. And you're like, boy, how do I
compare if that's really a good value? Well this is how
you compare it. You present value all of the
payments and you say well what is that worth to me today. And here we did that. We said well actually choice
number one is the best deal. And it just depended on how
the mathematics work out. If I lowered the discount rate,
if this discount rate is lower, it might have changed
the outcomes. And maybe I'll actually do that
in the next video, just to show you how important
the discount rate is. Anyway I'm out of time,
and I'll see you in the next video.