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Current time:0:00Total duration:7:42

in the last video we figured out what is the present value of these three different payment timing choices if we had a five percent risk-free rate and if these payments were risk-free you can instead of coming from you can almost view of that this is some type of government program where they're asking you to choose which of these three payment streams for the government do you want and so we'll use the same rate that the government would pay you is if you lent them money and that's given by the the Treasury rate and in the first case we assumed a 5% Treasury rate and if you watch the first present value video I think you understand why compounding going forward is the same thing as discounting that rate by going backwards right if you want to know how much $100 is in a year from now you multiply that times 1 plus the disc one plus the 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 would divided 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 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 for all for dependent no matter how long the how far away the money is given to you and and you'll see what I mean because I'll change that assumption in a second but if we assume that the that the risk-free rate is 5% then the present value of the hundred dollars today well that was just $100 one hundred and ten dollars in two years we got that by doing 110 divided by 1.05 squared right you divided by 1.05 there and then you get divided by 1.05 again and then you get we got ninety nine dollars and seventy seven cents ninety-nine point seven seven 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 twenty 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 dollars and 36 cents and that's that's the 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 a second best choice number three was third best fair enough now what happens and you might want to after I posed the question you might want to think about it before I show you the answer what happens if I don't assume a five percent discount rate what happens if I assume PV let's assume a two percent discount rate this is just my notation what is the present value of these if I assume a two percent risk-free rate or two percent discount rate well the hundred dollars I'm getting that today so that's still worth $100 you can even do that as one hundred let me do that in a more vibrant color as 100 divided by one point O two to the zeroth power because we're getting it today but that's just one point O two divided by one which is just a hundred dollars right a hundred dollars today what's the present value it's a hundred dollars now what's the 105 doll what's 110 dollars two years out going to be worth so this is interesting when the interest rate goes down right went from five percent to two percent I'm going to be dividing by a smaller number right 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 now let's figure out what that is so if I take $110 and I divide it by one point O two squared right just got it twice I get one hundred five dollars and seventy two cents 105 and 72 cents right oh and how did I get that that was equal to I'm doing it in Reverse here but that was equal to 110 divided by 1.0 two squared and our intuition was correct right just by the interest rate going from five percent to two percent the present value of this payment two years out right it's in year three but it's two years out actually I should real able 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 call this year two so this is now so you could call this year zero right this is year one and this is year two anyway the present value of this is 105 it increased by six dollars just by the discount rate going down by three percent 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 it's present values 20 plus 50 divided by 1.0 to plus 35 divided by 1.0 to squared see what this ends up 20 plus 50 divided by 1.02 plus 35 divided by 1.0 two squared one hundred and two dollars and sixty six cents so this is equal to 102 dollars and sixty six cents 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 lower the interest rate and now choice number two is the best followed by choice number three followed by choice number one so it almost you know choice number one was the best when we had a 5% discount rate not 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 lower the interest rate than choice number three did write this its present value went from ninety nine dollars and seventy seven cents to one hundred five dollars and seventy set so it's almost six dollars while here it only improved by less than three dollars right so why is that well when you lower the interest rate right the terms that are using that discount rate the most benefit the most so that all of this payment was two years out right so it benefited the most by decreasing the the the discount rate right the one point oh two squared it changed this value the most these payments are spread out right 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 right 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 in the next video