[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:02.56,Default,,0000,0000,0000,,So let's generalize a bit\Nwhat we learned in the
Dialogue: 0,0:00:02.56,0:00:03.83,Default,,0000,0000,0000,,last presentation.
Dialogue: 0,0:00:03.83,0:00:07.28,Default,,0000,0000,0000,,Let's say I'm\Nborrowing P dollars.
Dialogue: 0,0:00:07.28,0:00:08.79,Default,,0000,0000,0000,,P dollars, that's what I\Nborrowed so that's my
Dialogue: 0,0:00:08.79,0:00:10.74,Default,,0000,0000,0000,,initial principal.
Dialogue: 0,0:00:10.74,0:00:14.73,Default,,0000,0000,0000,,So that's principal.
Dialogue: 0,0:00:14.73,0:00:17.07,Default,,0000,0000,0000,,r is equal to the rate,\Nthe interest rate that
Dialogue: 0,0:00:17.07,0:00:18.31,Default,,0000,0000,0000,,I'm borrowing at.
Dialogue: 0,0:00:18.31,0:00:22.60,Default,,0000,0000,0000,,We can also write that\Nas 100r%, right?
Dialogue: 0,0:00:22.60,0:00:24.37,Default,,0000,0000,0000,,And I'm going to borrow\Nit for-- well, I
Dialogue: 0,0:00:24.37,0:00:29.16,Default,,0000,0000,0000,,don't know-- t years.
Dialogue: 0,0:00:29.19,0:00:32.21,Default,,0000,0000,0000,,Let's see if we can come up\Nwith equations to figure out
Dialogue: 0,0:00:32.21,0:00:35.96,Default,,0000,0000,0000,,how much I'm going to owe at\Nthe end of t years using either
Dialogue: 0,0:00:35.96,0:00:38.17,Default,,0000,0000,0000,,simple or compound interest.
Dialogue: 0,0:00:38.17,0:00:41.45,Default,,0000,0000,0000,,So let's do simple first\Nbecause that's simple.
Dialogue: 0,0:00:41.45,0:00:48.46,Default,,0000,0000,0000,,So at time 0-- so let's make\Nthis the time axis-- how
Dialogue: 0,0:00:48.46,0:00:49.31,Default,,0000,0000,0000,,much am I going to owe?
Dialogue: 0,0:00:49.31,0:00:51.95,Default,,0000,0000,0000,,Well, that's right when I\Nborrow it, so if I paid
Dialogue: 0,0:00:51.95,0:00:55.22,Default,,0000,0000,0000,,it back immediately, I\Nwould just owe P, right?
Dialogue: 0,0:00:55.22,0:01:00.73,Default,,0000,0000,0000,,At time 1, I owe P plus the\Ninterest, plus you can kind of
Dialogue: 0,0:01:00.73,0:01:04.46,Default,,0000,0000,0000,,view it as the rent on that\Nmoney, and that's r times P.
Dialogue: 0,0:01:04.46,0:01:06.39,Default,,0000,0000,0000,,And that previously, in the\Nprevious example, in the
Dialogue: 0,0:01:06.39,0:01:07.90,Default,,0000,0000,0000,,previous video, was 10%.
Dialogue: 0,0:01:07.90,0:01:11.04,Default,,0000,0000,0000,,P was 100, so I had to pay $10\Nto borrow that money for a
Dialogue: 0,0:01:11.04,0:01:13.26,Default,,0000,0000,0000,,year, and I had to\Npay back $110.
Dialogue: 0,0:01:13.26,0:01:18.61,Default,,0000,0000,0000,,And this is the same thing\Nas P times 1 plus r, right?
Dialogue: 0,0:01:18.61,0:01:21.83,Default,,0000,0000,0000,,Because you could\Njust use 1P plus rP.
Dialogue: 0,0:01:21.83,0:01:24.08,Default,,0000,0000,0000,,And then after two years,\Nhow much do we owe?
Dialogue: 0,0:01:24.08,0:01:28.19,Default,,0000,0000,0000,,Well, every year, we just\Npay another rP, right?
Dialogue: 0,0:01:28.19,0:01:30.86,Default,,0000,0000,0000,,In the previous example,\Nit was another $10.
Dialogue: 0,0:01:30.86,0:01:34.00,Default,,0000,0000,0000,,So if this is 10%, every\Nyear we just pay 10% of
Dialogue: 0,0:01:34.00,0:01:35.36,Default,,0000,0000,0000,,our original principal.
Dialogue: 0,0:01:35.36,0:01:38.73,Default,,0000,0000,0000,,So in year 2, we owe P plus\NrP-- that's what we owed in
Dialogue: 0,0:01:38.73,0:01:42.50,Default,,0000,0000,0000,,year 1-- and then another\NrP, so that equals
Dialogue: 0,0:01:42.50,0:01:45.35,Default,,0000,0000,0000,,P plus 1 plus 2r.
Dialogue: 0,0:01:45.35,0:01:47.72,Default,,0000,0000,0000,,And just take the P out,\Nand you get a 1 plus r
Dialogue: 0,0:01:47.72,0:01:49.84,Default,,0000,0000,0000,,plus r, so 1 plus 2r.
Dialogue: 0,0:01:49.84,0:01:54.77,Default,,0000,0000,0000,,And then in year 3, we'd owe\Nwhat we owed in year 2.
Dialogue: 0,0:01:54.77,0:02:00.33,Default,,0000,0000,0000,,So P plus rP plus rP, and then\Nwe just pay another rP, another
Dialogue: 0,0:02:00.33,0:02:03.83,Default,,0000,0000,0000,,say, you know, if r is 10%, or\N50% of our original principal,
Dialogue: 0,0:02:03.83,0:02:10.30,Default,,0000,0000,0000,,plus rP, and so that\Nequals P times 1 plus 3r.
Dialogue: 0,0:02:10.30,0:02:15.91,Default,,0000,0000,0000,,So after t years,\Nhow much do we owe?
Dialogue: 0,0:02:15.91,0:02:18.82,Default,,0000,0000,0000,,Well, it's our original\Nprincipal times 1 plus,
Dialogue: 0,0:02:18.82,0:02:22.33,Default,,0000,0000,0000,,and it'll be tr.
Dialogue: 0,0:02:22.33,0:02:25.92,Default,,0000,0000,0000,,So you can distribute this out\Nbecause every year we pay Pr,
Dialogue: 0,0:02:25.92,0:02:27.39,Default,,0000,0000,0000,,and there's going\Nto be t years.
Dialogue: 0,0:02:27.39,0:02:28.97,Default,,0000,0000,0000,,And so that's why\Nit makes sense.
Dialogue: 0,0:02:28.97,0:02:31.94,Default,,0000,0000,0000,,So if I were to say\NI'm borrowing-- let's
Dialogue: 0,0:02:31.94,0:02:33.41,Default,,0000,0000,0000,,do some numbers.
Dialogue: 0,0:02:33.41,0:02:35.46,Default,,0000,0000,0000,,You could work it out this way,\Nand I recommend you do it.
Dialogue: 0,0:02:35.46,0:02:37.10,Default,,0000,0000,0000,,You shouldn't just\Nmemorize formulas.
Dialogue: 0,0:02:37.10,0:02:45.82,Default,,0000,0000,0000,,If I were to borrow $50 at 15%\Nsimple interest for 15-- or
Dialogue: 0,0:02:45.82,0:02:50.70,Default,,0000,0000,0000,,let's say for 20 years, at the\Nend of the 20 years, I would
Dialogue: 0,0:02:50.70,0:03:04.00,Default,,0000,0000,0000,,owe $50 times 1 plus the\Ntime 20 times 0.15, right?
Dialogue: 0,0:03:04.00,0:03:08.96,Default,,0000,0000,0000,,And that's equal to $50 times 1\Nplus-- what's 20 times 0.15?
Dialogue: 0,0:03:08.96,0:03:11.22,Default,,0000,0000,0000,,That's 3, right?
Dialogue: 0,0:03:11.22,0:03:12.06,Default,,0000,0000,0000,,Right.
Dialogue: 0,0:03:12.06,0:03:17.55,Default,,0000,0000,0000,,So it's 50 times 4, which\Nis equal to $200 to
Dialogue: 0,0:03:17.55,0:03:18.74,Default,,0000,0000,0000,,borrow it for 20 years.
Dialogue: 0,0:03:18.74,0:03:22.92,Default,,0000,0000,0000,,So $50 at 15% for 20\Nyears results in a $200
Dialogue: 0,0:03:22.92,0:03:24.70,Default,,0000,0000,0000,,payment at the end.
Dialogue: 0,0:03:24.70,0:03:27.01,Default,,0000,0000,0000,,So this was simple\Ninterest, and this was
Dialogue: 0,0:03:27.01,0:03:28.37,Default,,0000,0000,0000,,the formula for it.
Dialogue: 0,0:03:28.37,0:03:32.56,Default,,0000,0000,0000,,Let's see if we can do the same\Nthing with compound interest.
Dialogue: 0,0:03:32.56,0:03:39.11,Default,,0000,0000,0000,,Let me erase all this.
Dialogue: 0,0:03:39.11,0:03:42.80,Default,,0000,0000,0000,,That's not how I\Nwanted to erase it.
Dialogue: 0,0:03:42.80,0:03:48.20,Default,,0000,0000,0000,,There we go.
Dialogue: 0,0:03:48.20,0:03:53.43,Default,,0000,0000,0000,,OK, so with compound interest,\Nin year 1, it's the same thing,
Dialogue: 0,0:03:53.43,0:03:55.02,Default,,0000,0000,0000,,really, as simple interest, and\Nwe saw that in the
Dialogue: 0,0:03:55.02,0:03:55.82,Default,,0000,0000,0000,,previous video.
Dialogue: 0,0:03:55.82,0:04:04.81,Default,,0000,0000,0000,,I owe P plus, and now the rate\Ntimes P, and that equals
Dialogue: 0,0:04:04.81,0:04:08.19,Default,,0000,0000,0000,,P times 1 plus r.
Dialogue: 0,0:04:08.19,0:04:09.45,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:04:09.45,0:04:12.81,Default,,0000,0000,0000,,Now year 2 is where compound\Nand simple interest diverge.
Dialogue: 0,0:04:12.81,0:04:14.82,Default,,0000,0000,0000,,In simple interest, we would\Njust pay another rP, and
Dialogue: 0,0:04:14.82,0:04:17.17,Default,,0000,0000,0000,,it becomes 1 plus 2r.
Dialogue: 0,0:04:17.17,0:04:19.19,Default,,0000,0000,0000,,In compound interest,\Nthis becomes the new
Dialogue: 0,0:04:19.19,0:04:22.01,Default,,0000,0000,0000,,principal, right?
Dialogue: 0,0:04:22.01,0:04:25.05,Default,,0000,0000,0000,,So if this is the new\Nprincipal, we are going to pay
Dialogue: 0,0:04:25.05,0:04:28.37,Default,,0000,0000,0000,,1 plus r times this, right?
Dialogue: 0,0:04:28.37,0:04:29.82,Default,,0000,0000,0000,,Our original principal was P.
Dialogue: 0,0:04:29.82,0:04:35.00,Default,,0000,0000,0000,,After one year, we paid 1 plus\Nr times the original principal
Dialogue: 0,0:04:35.00,0:04:38.27,Default,,0000,0000,0000,,times 1 plus r rate.
Dialogue: 0,0:04:38.27,0:04:42.52,Default,,0000,0000,0000,,So to go into year 2, we're\Ngoing to pay what we owed at
Dialogue: 0,0:04:42.52,0:04:47.64,Default,,0000,0000,0000,,the end of year 1, which is P\Ntimes 1 plus r, and then we're
Dialogue: 0,0:04:47.64,0:04:49.64,Default,,0000,0000,0000,,going to grow that\Nby r percent.
Dialogue: 0,0:04:49.64,0:04:53.24,Default,,0000,0000,0000,,So we're going to multiply\Nthat again times 1 plus r.
Dialogue: 0,0:04:58.04,0:05:02.90,Default,,0000,0000,0000,,And so that equals P\Ntimes 1 plus r squared.
Dialogue: 0,0:05:02.90,0:05:04.95,Default,,0000,0000,0000,,So the way you could think\Nabout it, in simple interest,
Dialogue: 0,0:05:04.95,0:05:09.17,Default,,0000,0000,0000,,every year we added a Pr.
Dialogue: 0,0:05:09.17,0:05:12.33,Default,,0000,0000,0000,,In simple interest, we\Nadded plus Pr every year.
Dialogue: 0,0:05:12.33,0:05:16.76,Default,,0000,0000,0000,,So if this was $50 and this is\N15%, every year we're adding
Dialogue: 0,0:05:16.76,0:05:19.84,Default,,0000,0000,0000,,$3-- we're adding--\Nwhat was that?
Dialogue: 0,0:05:19.84,0:05:20.46,Default,,0000,0000,0000,,50%.
Dialogue: 0,0:05:20.46,0:05:23.52,Default,,0000,0000,0000,,We're adding $7.50 in interest,\Nwhere P is the principal,
Dialogue: 0,0:05:23.52,0:05:24.56,Default,,0000,0000,0000,,r is the rate.
Dialogue: 0,0:05:24.56,0:05:27.48,Default,,0000,0000,0000,,In compound interest, every\Nyear we're multiplying the
Dialogue: 0,0:05:27.48,0:05:31.68,Default,,0000,0000,0000,,principal times 1 plus\Nthe rate, right?
Dialogue: 0,0:05:31.68,0:05:33.93,Default,,0000,0000,0000,,So if we go to year 3,\Nwe're going to multiply
Dialogue: 0,0:05:33.93,0:05:35.23,Default,,0000,0000,0000,,this times 1 plus r.
Dialogue: 0,0:05:35.23,0:05:39.09,Default,,0000,0000,0000,,So year 3 is P times 1\Nplus r to the third.
Dialogue: 0,0:05:39.09,0:05:42.16,Default,,0000,0000,0000,,So year t is going to be\Nprincipal times 1 plus
Dialogue: 0,0:05:42.16,0:05:45.24,Default,,0000,0000,0000,,r to the t-th power.
Dialogue: 0,0:05:45.24,0:05:47.98,Default,,0000,0000,0000,,And so let's see\Nthat same example.
Dialogue: 0,0:05:47.98,0:05:50.87,Default,,0000,0000,0000,,We owe $200 in this example\Nwith simple interest.
Dialogue: 0,0:05:50.87,0:05:53.19,Default,,0000,0000,0000,,Let's see what we owe\Nin compound interest.
Dialogue: 0,0:05:53.19,0:05:59.21,Default,,0000,0000,0000,,The principal is $50.
Dialogue: 0,0:05:59.21,0:06:00.64,Default,,0000,0000,0000,,1 plus-- and what's the rate?
Dialogue: 0,0:06:00.64,0:06:02.69,Default,,0000,0000,0000,,0.15.
Dialogue: 0,0:06:02.69,0:06:06.18,Default,,0000,0000,0000,,And we're borrowing\Nit for 20 years.
Dialogue: 0,0:06:06.18,0:06:14.91,Default,,0000,0000,0000,,So this is equal to 50 times\N1.15 to the 20th power.
Dialogue: 0,0:06:14.91,0:06:18.07,Default,,0000,0000,0000,,I know you can't read that,\Nbut let me see what I can
Dialogue: 0,0:06:18.07,0:06:20.68,Default,,0000,0000,0000,,do about the 20th power.
Dialogue: 0,0:06:20.68,0:06:28.26,Default,,0000,0000,0000,,Let me use my Excel and\Nclear all of this.
Dialogue: 0,0:06:28.26,0:06:31.84,Default,,0000,0000,0000,,Actually, I should just use my\Nmouse instead of the pen tool
Dialogue: 0,0:06:31.84,0:06:34.95,Default,,0000,0000,0000,,to the clear everything.
Dialogue: 0,0:06:34.95,0:06:36.77,Default,,0000,0000,0000,,OK, so let me just\Npick a random point.
Dialogue: 0,0:06:36.77,0:06:42.22,Default,,0000,0000,0000,,So I just want to-- plus 1.15\Nto the 20th power, and you
Dialogue: 0,0:06:42.22,0:06:46.94,Default,,0000,0000,0000,,could use any calculator:\N16.37, let's say.
Dialogue: 0,0:06:46.94,0:06:55.46,Default,,0000,0000,0000,,So this equals 50 times 16.37.
Dialogue: 0,0:06:55.46,0:06:58.17,Default,,0000,0000,0000,,And what's 50 times that?
Dialogue: 0,0:06:58.17,0:07:08.56,Default,,0000,0000,0000,,Plus 50 times that: $818.
Dialogue: 0,0:07:08.56,0:07:11.78,Default,,0000,0000,0000,,So you've now realized that if\Nsomeone's giving you a loan and
Dialogue: 0,0:07:11.78,0:07:14.32,Default,,0000,0000,0000,,they say, oh, yeah, I'll lend\Nyou-- you need a 20-year loan?
Dialogue: 0,0:07:14.32,0:07:16.34,Default,,0000,0000,0000,,I'm going to lend\Nit to you at 15%.
Dialogue: 0,0:07:16.34,0:07:19.84,Default,,0000,0000,0000,,It's pretty important to\Nclarify whether they're going
Dialogue: 0,0:07:19.84,0:07:24.40,Default,,0000,0000,0000,,to charge you 15% interest at\Nsimple interest or
Dialogue: 0,0:07:24.40,0:07:25.87,Default,,0000,0000,0000,,compound interest.
Dialogue: 0,0:07:25.87,0:07:28.77,Default,,0000,0000,0000,,Because with compound interest,\Nyou're going to end up paying--
Dialogue: 0,0:07:28.77,0:07:31.90,Default,,0000,0000,0000,,I mean, look at this: just to\Nborrow $50, you're going to
Dialogue: 0,0:07:31.90,0:07:36.18,Default,,0000,0000,0000,,be paying $618 more than if\Nthis was simple interest.
Dialogue: 0,0:07:36.18,0:07:40.48,Default,,0000,0000,0000,,Unfortunately, in the real\Nworld, most of it is
Dialogue: 0,0:07:40.48,0:07:41.69,Default,,0000,0000,0000,,compound interest.
Dialogue: 0,0:07:41.69,0:07:44.25,Default,,0000,0000,0000,,And not only is it compounding,\Nbut they don't even just
Dialogue: 0,0:07:44.25,0:07:46.17,Default,,0000,0000,0000,,compound it every year and they\Ndon't even just compound it
Dialogue: 0,0:07:46.17,0:07:48.81,Default,,0000,0000,0000,,every six months, they actually\Ncompound it continuously.
Dialogue: 0,0:07:48.81,0:07:50.83,Default,,0000,0000,0000,,And so you should watch the\Nnext several videos on
Dialogue: 0,0:07:50.83,0:07:53.75,Default,,0000,0000,0000,,continuously compounding\Ninterest, and then you'll
Dialogue: 0,0:07:53.75,0:07:57.19,Default,,0000,0000,0000,,actually start to learn\Nabout the magic of e.
Dialogue: 0,0:07:57.19,0:08:01.20,Default,,0000,0000,0000,,Anyway, I'll see you\Nall in the next video.