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Present value 4 (and discounted cash flow)

Discounted cash flows are a way of valuing a future stream of cash flows using a discount rate. In this video, we explore what is meant by a discount rate and how to calculated a discounted cash flow by expanding our analysis of present value. Created by Sal Khan.

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  • starky ultimate style avatar for user Luke
    Khan claims that with the last discount rate (5% for 2y, 1% for 1y) you would be best off going with the 3rd option because the PV of the 3rd option =$101.25 where PV of the 1st option =$100. We can apply all the same variables and find that the two year future value (FV) of the 3rd option =$20*1.05^2+$50*1.01+$35=$107.55, but the FV of the 1st option =$110.25. If we look at PV then the 3rd option is better but if we look at the FV then the 1st option is better. Shouldn't we decide based off the FV?
    (106 votes)
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    • blobby green style avatar for user JJ McCammon
      HankyUSA, this is a great question, but your (and the rest of the responders') logic is slightly off. Think about what you are really saying in your example:

      I will invest $20 at a 5% interest rate STARTING NOW
      I will invest $50 at a 1% interest rate STARTING NOW
      I will invest $35 at a 0% interest rate STARTING NOW

      Do you see the error? You are assuming a timeline that starts all of your investments at the current time. But of course that isn't true. You won't have access to the $50 for a whole year and the $35 for a whole two years. Therefore you can't use addition to simply sum $20, $50*1.01, and $35*(1.02^2) because $50 isn't the present value it's the FUTURE VALUE in one year's time. Similarly, $35 is what the value WILL BE in 2 years time. This conundrum is the entire reason for using the discounting method.

      The correct logic is to ask the question: How much money would I need today to have $50 in a year at a 1% interest rate. That is exactly the formula Sal gave ($50/1.01). And the same goes for $35 in two years at 2%.

      Another way to think about it is that the present value as Sal calculated is $101.25. Using the FV interest calculation given in a previous video we have (1.05)^2 multiplied by $101.25 (the present value of the investment) which gives us $111.63. Clearly more than the $110.25 in option 1. Hope this helps.
      (170 votes)
  • blobby green style avatar for user Matthew Rindelaub
    Why was inflation not included in the discount rate?
    (13 votes)
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  • leaf green style avatar for user khan.student
    Why does Sal do compound interest? When do you use compound interest?
    (8 votes)
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  • marcimus pink style avatar for user chum
    Towards the end where Sal says that Option 3 would be the best ( the one where he works out discount rate in 1st yr=1% and 2nd yr=5%) and says if you understand why this is actually better than Choice 2 you understand this very well......i can see that Choice 3 is better but don't understand why that choice is better :S can someone help me understand this?
    (11 votes)
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    • mr pink red style avatar for user Benjamin Black
      To understand why, it is helpful to also understand why option 2 is worse. You are offered $110 after 2 years, but the discount rate for 2 years is much bigger (5%) than it is for 1 year. The full amount has to be discounted at the higher rate, and you have to do it twice, to get the present value of $99.77. That 5% discount rate really eats away at the $110.

      In the third option, the PV is split three ways, Unlike in option 2, you are discounting only $35 at the higher rate, a fraction of the full amount. Then, you are discounting a much higher fraction of the total -- $50 -- at a much lower discount rate of 1%, so you "lose" less money from the $50 by discounting.
      (19 votes)
  • blobby green style avatar for user barney.freedman
    Is inflation and the value of money reflected in the exercise?
    (2 votes)
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    • duskpin ultimate style avatar for user Brian
      Inflation basically is the value of money, domestically at least. The exercise does NOT include those figures, sort of. If by value of money, you mean value of liquid assets, no. In some cases in life, it is more worth while to have $150 dollars today than $20,000 in 10 years. That sort of emotional/societal value is not included.

      Now what I think you're really trying to get at is inflation and the physical value of money. While it is not directly involved, it can easily be understood and inferred.

      Let's assume that in Country A, inflation is always rising by a steady 2%. All you have to do is adjust your discount rate (the gross interest rate). If you were going to make 5% a year on the deal, you will now be making 3%. This is the REAL interest rate (Gross adjusted for inflation, gives you the real buying power of the currency).

      Lets apply this to Sal's example:
      Instead of offering 1% the bank offered you 2%. But there was 1% inflation that year. You would use the 1.01 discount rate in the denominator. Although the bank advertises 2%, and you will receive 2%! Your money will be able to only purchase 1% more, because the average prices will rise as well.
      In CD number 2, the bank offered you 8%. But the was 1% inflation in year one and 2% inflation in year two! Your REAL interest rate is 5% that is the 8% adjusted for inflation.

      You really will go from having $100 to $108. But in year two because of inflation $108 dollars will only buy as much $105 dollars bought when you made the investment.

      I know this is actually complex. So I'll check back in a few days to see if I need to re explain anything.
      (5 votes)
  • leaf green style avatar for user khan.student
    Why would you get a higher interest rate if you locked up your money longer? Is this beneficial for the banks or beneficial for the customers? This concept doesn't make sense to me.
    (1 vote)
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    • blobby green style avatar for user lsethhill
      When a lending institution is planning its cash flows putting the flexibility in their control as opposed to yours is beneficial to them so they offer you a benefit (higher interest rate) to entice you to give them that flexibility. Leave behind Sals Govt. Bonds storyboard for a second and think of a bank. A bank may wish to improve its balance sheet by increasing the volume of car loans it offers. This will require the bank to make loans with terms of 3 -5 years. One of the ways they might raise the money needed to make these loans is to sell CD’s. CD’s are similar to bonds in that they are a way consumers and institutions can lend banks money. If the bank issues a bunch of 1 year CD’s it’s not in a very good position to make those auto loans. However if it issues a bunch of 5 year CD’s and keeps selling those year after year they can increase the number of auto loans they are able to sell. So it makes sense for them to offer you an incentive to keep that in the bank longer. For another answer to your question watch the video on Fractional Reserve Banking.
      (6 votes)
  • blobby green style avatar for user wattabigail
    Can I just clarify that if you agree to 'lock in' your money for two years you get 5% interest for both years 1 and 2? Not just 1% in the first year then 5% in the second? Why is this? What would happen if you just choose to lend it to the government for 1 year? Would you only receive $101 and not receive anything in year 2 or would it be 1% over the two years?
    (3 votes)
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  • blobby green style avatar for user Ulises Fernandez
    Hello Sal, in this video when you are calculating PV backwards from the FV you always choose the highest PV as the best option. This is seen in the summary given at the 6-minute mark. However, if you are calculating the PV backwards from the highest FV wouldn't you want your PV to be the lowest possible so that you have to invest the lowest amount possible and yet have high returns? Meaning you want to have the widest gap between the PV and FV since that would indicate a greater return.
    (3 votes)
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    • blobby green style avatar for user pjacademics
      Thank you! This is what I'm stuck on and reviewing the comments for some clarification. PV, as I've learned in my textbook, asks "how much do I need to pay today to get $X in the future?" so the lower the PV in this case, the better! It has been years since Sal made this video; hoping either he or someone in his team can respond to this!
      (1 vote)
  • starky seed style avatar for user Clyde Mazie
    What is the different between Interest rate and Discount rate?
    (3 votes)
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  • leaf green style avatar for user 1ofakind
    I am confused on FV of option 1 in scenario 3:

    my understanding is it's given 2 different rate 1% year 1 and 5% year 2 so the calculation of FV should be 100*(1.01) for year 1 = 101 then 101*(1.05) for year 2 = 106.05 instead of taking 100*(1.05)^2 = 110.25

    could someone please explain?
    (2 votes)
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    • leaf green style avatar for user 1ofakind
      Just want to point out if the video is inaccurate, Sal should make a correction to avoid people learning in mistake. For option 1 in scenario 3, the FV2 at the end of Year 2 should be using the formula below:

      FV1 = PV0(1 + i) which i = 1%, PV0 = 100 here
      FV2 = FV1(1 + i) which I = 5%, PV1 = 101 here
      (2 votes)

Video transcript

So far, we've been assuming that the discount rate is the same thing, no matter how long of a period we're talking about. But we know if you go to the bank and you say, hey, bank, I want to essentially invest in a one-year CD, they'll say, oh, OK, one-year CD will give you 2%. And you're like, well, what if we give you the money for two years? So you can keep our money, locked in for even longer. They'll say, oh, then we'll give you a little bit more interest, because we have more flexibility. For two years, we don't have to worry about paying you. So instead of giving you 2%, we'll give you 7%, because we get to keep your money for two years. And maybe if you say, well, you know, I actually don't even need my money for 10 years, so let me give you the money for 10 years. They'll say oh, 10 years, if we get to keep your money, we'll give you 12%. So in general-- and this tends to be the case, although it's not always the case-- the longer that you defer your money, or the longer you lock up the money, the higher an interest rate you get. So the same thing is true when you're doing a discount rate. Oftentimes you want to discount a payment two years out by a higher value than something that's only one year out. So how do you do that? So let's say the risk-free rate, if you were to go out and get a government bond-- the one-year rate, let's say that they're only giving you 1%. But let's say that the two-year rate, they'll give you 5%. So what does that mean? Well, let's take the example. So that means you could take that $100 and essentially lend it to the federal government, and in a year they'll give you 1% on it. So that these are annual rates. So 1%, 1.01 times 100, that's just $101, right? Fair enough. Now your other option is, you could lock it in. You could lend it to the federal government for two years and not see your money. And they say, oh, then we're going to give you 5% a year. So then you're going to go 5% a year. So how much do you end up with in two years? Well, remember, this is an annual rate. These are always quoted in annual rates. So if you're getting 5% a year, that's going to be equal to-- let's do it on the calculator. That's going to be 100-- after one year you're going to get 1.05, and after two years you're going to get 1.05. Or you can view that as 100 times 1.05 squared. So you'd have $110.25. So you already see, not even doing any present value, this is actually-- you can almost view this as a future value calculation. If you take a future value, you already know that this option is better than this option, when you have these varying interest rates. But anyway, the whole topic of this is to talk about present value, so let's do that. So in this circumstance, what is the present value of the $110? Well, actually, what is the present value of the $100? Well, we always know that. That's easy. That is $100. Present value of $100 today is $100. What is the present value of the $110? So we take $110, and we're going to use the two-year rate, and discount twice. And that makes sense, because essentially you're deferring your money for two years. You're not going to get anything, even a year from now. So you're deferring your money for two years. So you divide it by 1-- so it's a 5% rate, 1.05 squared. And then that is equal to-- I think that was our first problem, right? So I'll just do it again. 110 divided by 1.05 squared. That's equal to $99.77, right? That was our first problem. And now this one is interesting. The $20 you get today-- and this is a side note. It's very important when you're doing this, when they talk about year one, or year zero, just make sure-- is that today, is that a year from now? Because if it's a year from now, you'd have to discount it by the one-year interest rate. If it's today, you don't discount it. So anyway, I clarified that. I was a little ambiguous about that in the last two videos, but I clarified it. The $20 is now. So the present value of something given you today, is the value of it. So it's $20 plus $50. Now $50, what do we use? Do we use the one-year rate or the two-year rate? Well of course, we use the one-year rate, because you're not deferring the pleasure of that $50 for two years. You're actually getting it in one year. So plus $50 divided by the one-year rate. Divided by 1.01. Plus $35 divided by the two-year rate-- but this is an annual rate, so you have to discount it twice-- divided by 1.05 squared. Let's get the TI-85 out. So you get 20 plus 50 divided by 1.01, plus 35 divided by 1.05 squared, is equal to $101.25. So notice, the actual payment streams I did not change in any of the three scenarios. And let me just draw a line between them, because I got a little bit messy. So that was scenario one. This is scenario two. And this is scenario three. But in scenario one, because we used a 5% discount rate for all-- you could say, I don't want to use fancy words-- but for all durations out we used a 5% discount rate. We saw that choice number one was the best. But then if the discount rate were to change-- if we were to change our assumption. If we had a 2% rate, for whatever reason, we could lend money to the federal government in the form of buying bonds from them-- we could lend the federal government two years over any time period at 2%. Then all of a sudden, choice two became the best option. And then finally, if we had this kind of-- and this is the most realistic scenario, and even though the math is fairly simple, we're actually doing something fairly sophisticated here. When I had a different discount rate for my one year out cash flows and my two year out cash flows, and it was these exact numbers. I had to play with the numbers to get the right result. Then all of a sudden choice three was the best option. I'll leave it to you-- I want you to think about why this was better for choice three than it was for choice two. And if you really understand that, then I think you are starting to have a lot of intuition about present values. And frankly, what we're learning here is a discounted cash flow. What is a discounted cash flow? I'm giving you a stream of cash flows. $20 now, $50 a year from now, $35 in two years. And you are essentially discounting them back to get today's present value. So when someone says, you know, I can use Excel to do a discounted cash flow, that's all they're doing. They're making some assumption about the discount rates. And they're just using this fairly straightforward mathematics to get the present value of those future cash flows. But it's a very powerful technique. Because if you were to take-- if you're good at Excel, and you were to say, oh, I have a business. And based on my assumptions, in year one, right now, this business gives me $20. The next year it's going to give $50. The year after that it's $35. And this risk-free is the big assumption. But if it was risk-free, you could discount it like that. You'd say, if these are the interest rates, this business is worth $101.25. That's what I'm willing to pay for it. Or, I'm neutral. If I could get it for $90, that's a good deal for me. That's all a discounted cash flow is. But the big learning from this is how dependent the present value of future payments are on your discount rate assumption. The discount rate assumption is everything in finance. And this is where finance really diverges from a lot of other fields, especially the sciences. There really is no correct answer. It's all assumption driven. All of these discounted cash flows, and all these models, they're really just to help you understand the dynamics of things. And frankly-- and this happens a lot in the real world of finance-- if you ever become an analyst at an investment bank, you'll probably do this yourself. But you can almost justify any present value, by picking the right discount rate. And actually the whole topic of, how do you decide on the right discount rate? Because we assumed risk-free. Everything is risk-free. You're guaranteed these payments. But we know in the real world, if you're investing in pets.com and they tell you that they're going to pay these cash flows to you, that's not risk-free. There's some risk implicit in that. So actually, most of finance, and most of portfolio theory, and modern finance, is based on figuring out that discount rate. And that is the crux of everything, because as we see, that completely changes which of these options is the best. But anyway, I don't want to confuse you too much. What you have already is a very powerful tool. If you can think of a discount rate, you can make a very rational comparison between three, or ten, or whatever different types of payments. And this is actually really useful. You don't realize how many things in the world are like this. These college payment schemes where you pay some company $25 a year for 20 years, and then in year 21 they're willing to pay for your college tuition, or your kids' college tuition. You could figure out with that really is worth, how much money are they making off of you, by taking a discounted cash flow. And of course if you're paying out, these become negative numbers. And when they pay you, it becomes a positive number. Anyway. Maybe I'll do that in a couple of videos, because I think that's a fairly useful thing to be able to analyze. See you in the next video.