Learn about Present Value 2
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Khan Academy Presents: More choices as to when you get your money.

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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 $100.00 so today, you’ll get a $100.00. Choice two is that not in one year but in two years. So, let’s say this is your year one and now this is year two. Actually, I want to give you three choices that will really hopefully hit things home. So actually, let me scoot this choice too over to the left. So choice two, I am willing to give you, let’s say $110.00 in two years. So, not in one year, in two years, I am going to give you a $110.00. And so, I’ll circle in magenta when you actually get your payment. And then choice three is going to be fascinating. I am making this up on the fly as I go. Choice three, I am going to pay you $20.00 today. I am going to pay you $50.00 in one year. So, let’s see. That’s 70. Let me make this so it’s close. And then I am going to pay you $35.00 in year three. So, all of these are payments. I want to differentiation 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 am as risk-free as a federal government and they had opposed on the previous present value where someone talked about if the federal government really that safe. And this is the point. The federal government when if borrows from you. Let’s say it borrows a $100.00 and it promises to pay it in year, it will give you that $100.00. The risk is what is that $100.00 worth? Because they may 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 that but the U.S. government has never defaulted. It inflated its currency, so that’s kind of a roundabout way of defaulting but is 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 and once again, let’s say that risk-free I could lend it to the federal government at 5%. And it does not matter over what 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 of simplicity, the government will pay you 5% whether you give them the money for one year, whether you give them the money for two years or whether you give them the money for three years. So, if I had a $100.00, what would that be worth in one year? We figured that out already. It is 100 × 1.05, so that’s $105.00. And if you got another 5%, so the government is giving you 5% per year. It would be a 105 × 1.05. And what is that? So, I have 105 × 1.05 = $110.25. So, that is the value in two years. So immediately, without even doing any present value, we see that you’ll actually be better off in two years if you were to take the money now and just lend it to the government because the government risk-free will give you a $110.25 in two years while I’m only willing to give you a $110.00. So, that’s all fair and good but the whole topic is what we’re to solve is present values so let’s take everything in today’s money and to take this $110.00 and say, what is that worth today? We can just discount it backwards by the same method. So, $110.00 in two years; what is its one year value? Well, you take a $110.00 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 ÷ 1.05. And then, to get its present value, its value today, you divide that by 1.05 again. If I were to divide by 1.05 again, what do I get? I divide it by 1.05 and then I divide it by 1.05 again, I am dividing by 1.052. And what is that equal? And I am writing this in purpose because I want to get used to this notation because this is what all of our present values and our discounted cash flow; this type of dividing by one 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 two years out. So, let’s do that. Well, let’s just do that. 110/1.052 = $99.77. So, it equals 99.77. So once again, we have verified by taking the present value of a $110.00 in two years to today. If we assume of 5% discount rate and this discount rate is where all of the fudge factors 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 a simplification, we’re assuming a risk-free discount rate. But a present value based on that, you get $99.77 and you say “Wow!” Yeah, this really isn’t as good as this. I would rather have a $100.00 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 $20.00 today; well, that is just $20.00. What’s the present value of $50.00 in one year? So, +$50 ÷ 1.05, that is the present value of the $50.00 because one year out. And then I want the present value of the $35.00, so that’s +$35 divided by what? It is two years out, so you have to discount it twice; divide it by 1.052 just like we did here. So, let’s figure out what that present value is because notice, I am just adding up the present values of each of those payments. So, the present value of the $20.00 payment is $20.00 plus the present value of $50.00 payment. Well, that is just 50 ÷ 1.05 plus the present value of our $35.00 payment. And it is two years out, so we discount it by our discount rate twice. So, it’s divided by 1.052 and then, that is equal to $99.37. So now, we can make a very good comparison between the three options. This might have been confusing before. 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 the years A, B and C and I’ll pay you that near as B, C, D.” And you’re like, “Well, how do I compare if that’s really a good value?” Well, this is how you compare it. You present value of the payments and you say, “Well, what is that worth to me today?” And here we did that and we said, “Well, actually choice number one is the best deal.” And it just depended on how the mathematics worked out. If I’d lower 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 am out of time and I’ll see you in the next video.