Learn about Present Value 3 Video

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Learn about Present Value 3

Learn about Present Value 3 In the last video, we figured out what is the present value of these different payment timing choices if we had a 5% risk-free rate and if these payments were risk-free. Instead from coming from me, you can almost view that this is some type of government program where they are asking you to choose, which of these three payments streams for the government do you want? And so, it will use the same rate that the government would pay you if you lent them money. And that is given by the treasury rate. In the first case, we assumed of 5% of treasury rate. And if you watched the first present value video, I think you understand why compounding going forward is the same this discounting that rate by going backwards. If you want to know how much a $100 is a year from now, you multiply that times 1 plus the interest rate. So, it’s 5%, you multiply that times 1.05. If you’re taking a $110 and going a year back, you would 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. And I’ll introduce my notation. The present value if we assume of 5% rate, no matter how far away the money is given to you. And you’ll see what I mean because I’ll change that assumption a second. But if we assume that the risk-free rate is 5%, then the present value of the $100 today, well that was just a $100. $110 in two years, we got that by doing a 110 ÷ 1.052. You divide it by 1.05 there and then you divide it by 1.05 again. And then we got $99.77. And then choice number three, how did we get that? Well, we said that was the present value of the $20 today +$50 in one year ÷ by that discounted to the present day. So, divided by 1.05 + $35 ÷ 1.052 and we had gotten $99.36. And 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. After I pose a 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 PV? Let’s 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 2% discount rate? Well $100, I am getting that today. So, that is still worth a $100. You can even view that as a 100 ÷ 1.020 because we’re getting it today but that is just 1.02 ÷ 1 which is just a $100. A $100 today, what’s the present value? It’s a $100. Now, what’s the $110 two years out going to be worth? So, this is interesting. When the interest rate goes down, it went from 5% to 2%. I am going to be dividing by a smaller number. 1.022 is a smaller number than 1.052. 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. And let’s figure out what that is. So, if I take a $110 and I divide it by 1.022, discounted twice, I get a $105.72. And how did I get that? I am doing it in reverse here but that was equal to a 110 ÷ 1.022. And our intuition was correct. Just by the interest rate going from 5% to 2%, the present value of this payment 2 years out, it’s in year 3 but it is 2 years out. Actually I should re-label this. I should call this now the present. I should call this year 1. I was calling this year 2, 1 year out but I think that makes it confusing. I’ll call this year 2. So, this is now. So, you could call this year 0. This is year 1 and this is year 2. Anyway, the present value of this is 105. 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 $29 today or the $20 now, well that is just worth $20. Its present value is 20 + 50 ÷ 1.02 + 35 ÷ 1.022. Let’s see what this ends up, 20 + 50 ÷ 1.02 + 35 ÷ 1.022. $102.66 so, this is equal to $102.66. Now, there are a couple of really interesting things and this is a really good time to kind of let it all sink it. All of a sudden, we lower the interest rate and now, choice number two is the best followed by choice number three followed by choice number one. Choice number one was the best when we had a 5% discount rate. Now, the 2% discount rate, choice number two is all of a sudden the best. And there is something else interesting there. 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 a $105.70. So, it’s almost $6 while here, it only improved by less than $3. 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 these payment was two years out. So, it benefited the most by decreasing the discount rate, the 1.022. It changed this value the most. These payments are spread out. Only some of its payment is 2 years out then some of it payment is 1 year out and that is going to benefit less. And then some of these payments 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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Learn about Present Value 3

In the last video, we figured out what is the present value of these different payment timing choices if we had a 5% risk-free rate and if these payments were risk-free. Instead from coming from me, you can almost view that this is some type of government program where they are asking you to choose, which of these three payments streams for the government do you want? And so, it will use the same rate that the government would pay you if you lent them money. And that is given by the treasury rate.

In the first case, we assumed of 5% of treasury rate. And if you watched the first present value video, I think you understand why compounding going forward is the same this discounting that rate by going backwards. If you want to know how much a $100 is a year from now, you multiply that times 1 plus the interest rate. So, it’s 5%, you multiply that times 1.05. If you’re taking a $110 and going a year back, you would 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. And I’ll introduce my notation. The present value if we assume of 5% rate, no matter how far away the money is given to you. And you’ll see what I mean because I’ll change that assumption a second. But if we assume that the risk-free rate is 5%, then the present value of the $100 today, well that was just a $100. $110 in two years, we got that by doing a 110 ÷ 1.052. You divide it by 1.05 there and then you divide it by 1.05 again. And then we got $99.77.

And then choice number three, how did we get that? Well, we said that was the present value of the $20 today +$50 in one year ÷ by that discounted to the present day. So, divided by 1.05 + $35 ÷ 1.052 and we had gotten $99.36. And 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.

After I pose a 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 PV? Let’s 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 2% discount rate? Well $100, I am getting that today. So, that is still worth a $100. You can even view that as a 100 ÷ 1.020 because we’re getting it today but that is just 1.02 ÷ 1 which is just a $100. A $100 today, what’s the present value? It’s a $100.

Now, what’s the $110 two years out going to be worth? So, this is interesting. When the interest rate goes down, it went from 5% to 2%. I am going to be dividing by a smaller number. 1.022 is a smaller number than 1.052. 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. And let’s figure out what that is.

So, if I take a $110 and I divide it by 1.022, discounted twice, I get a $105.72. And how did I get that? I am doing it in reverse here but that was equal to a 110 ÷ 1.022. And our intuition was correct. Just by the interest rate going from 5% to 2%, the present value of this payment 2 years out, it’s in year 3 but it is 2 years out. Actually I should re-label this. I should call this now the present. I should call this year 1. I was calling this year 2, 1 year out but I think that makes it confusing. I’ll call this year 2. So, this is now. So, you could call this year 0. This is year 1 and this is year 2.

Anyway, the present value of this is 105. 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 $29 today or the $20 now, well that is just worth $20. Its present value is 20 + 50 ÷ 1.02 + 35 ÷ 1.022. Let’s see what this ends up, 20 + 50 ÷ 1.02 + 35 ÷ 1.022. $102.66 so, this is equal to $102.66.

Now, there are a couple of really interesting things and this is a really good time to kind of let it all sink it. All of a sudden, we lower the interest rate and now, choice number two is the best followed by choice number three followed by choice number one. Choice number one was the best when we had a 5% discount rate. Now, the 2% discount rate, choice number two is all of a sudden the best. And there is something else interesting there. 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 a $105.70. So, it’s almost $6 while here, it only improved by less than $3. 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 these payment was two years out. So, it benefited the most by decreasing the discount rate, the 1.022. It changed this value the most. These payments are spread out. Only some of its payment is 2 years out then some of it payment is 1 year out and that is going to benefit less. And then some of these payments 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.

Transcription by: Scribe4you Transcription Services