Allow” s generalise a little bit
what we learned in the last discussion. Allow” s say I ‘ m. loaning P bucks. P dollars, that ‘ s what I.
obtained so that” s my initial principal. That” s principal.
r amounts to the price,. the interest price that I ‘ m borrowing at.
We can also compose that. as 100r%, right?’And I ‘ m going to
obtain. it for– well, I wear ‘ t know– t’years. Allow ‘ s see if we can come up. with equations to figure out how much I” m mosting likely to owe at
. the end of t years making use of either compound or easy interest. So let” s do straightforward initially.
because that” s easy. At time 0– so allow” s make. this the time axis– just how much am I going to owe? Well, that” s right when I.
borrow it, so if I paid it back immediatelyRight away I.
would just would certainly P? Sometimes 1, I owe P plus the.
interest, plus you can kind of view it as the rent on that.
money, which” s r times P.And that formerly, in the.
previous instance, in the previous video clip, was 10%. P was 100, so I needed to pay $10.
to obtain that cash for a year, and I had to.
pay back $110. And this coincides point.
as P times 1 plus r, right? Because you could.
just use 1P plus rP. And after that after two years,.
just how a lot do we owe? Well, every year, we simply.
Pay an additional rP? In the previous example,.
it was one more $10. If this is 10%, every.
year we simply pay 10% of our initial principal. In year 2, we owe P plus.
rP– that” s what we owed in year 1– and after that an additional.
rP, to make sure that equals P plus 1 plus 2r.
And just take the P out,.
and you obtain a 1 plus r plus r, so 1 plus 2r. And after that in year 3, we” d owe. what we owed in year 2.
P plus rP plus rP, and then. we just pay another rP, another say, you know, if r is 10 %, or.
50% of our original principal, plus rP, therefore that. equates to P times 1 plus 3r.’So after t years,.
just how much do we owe? Well, it” s our initial. major times 1 plus, and
it ‘ ll be tr. So you can disperse this out. because every year we pay Public relations, and
there ‘ s going. to be t’years.
And so that ‘ s why. I ‘ m borrowing– let
‘ s do some numbers.You might function it out this way,.
You shouldn” t simply. If I were to borrow $50 at 15%.
easy interest for 15– or let” s say for 20 years, at the. end of the two decades, I would owe$ 50 times 1 plus the. Time 20 times 0.15? And that ‘ s equal to’$ 50 times 1. plus–‘what ‘ s 20 times 0.15?’That ‘ s 3? Right. So it ‘ s 50 times 4, which.
is equal to $200 to borrow it for two decades.
$ 50 at 15 %for 20. years results in a $200 repayment at the end. This was easy.
interest, and this was the formula for it. If we can do the same, allow ‘ s see.
point with compound rate of interest. Let me erase all this. That ‘ s not
exactly how I. wished to erase it’. There we go. OK, so with compound rate of interest,.
in year 1, it ‘ s the same thing, really, as simple passion, and. we saw that in the previous video clip. I owe P plus, and currently the rate. times P, which equals P times 1 plus r. Fair enough. Now year 2 is where substance.
and straightforward rate of interest deviate. In basic interest, we would certainly.
just pay one more rP, and it comes to be 1 plus 2r. In compound interest,. This ends up being the new principal? So if this is the brand-new. Principal, we are going to pay 1 plus r times this? Our initial principal was P.
After one year, we paid 1 plus. r times the initial primary times 1 plus r rate.
To go right into year 2, we ‘ re. The means you might believe.
concerning it, in basic rate of interest, annually we added a Public relations. In basic passion, we.
added plus Pr every year. So if this was $50 and this is.
15%, yearly we” re adding$ 3– we ‘ re adding–. what was that? 50 %.
We ‘ re adding $7.50 in interest,. where P is the principal, r is
the price’. In substance interest, every. year we ‘ re multiplying the principal times 1 plus.
If we go to year 3,. Year 3 is P times 1.
plus r to the 3rd. Year t is going to be. primary times 1 plus r to the t-th power. And so let ‘ s see.
that very same instance. We owe’$ 200 in this example.
with straightforward rate of interest. Allow ‘ s see what’we owe. in substance rate of interest. The principal is$ 50.
1 plus– and what ‘ s the rate? 0.15. And we ‘ re borrowing.
it for twenty years. This is equal’to 50 times.
1.15 to the 20th power. I recognize you can ‘ t read that,. however allow me see what I can do concerning the 20th power. Allow me utilize my Excel and. clear every one of this.
Really, I ought to just use my. mouse instead of the pen tool to the clear whatever. OK, so allow me just.
pick an arbitrary point. So I simply desire to– plus 1.15.
to the 20th power, and you could make use of any kind of calculator:.
16.37, let” s say. And what ‘ s 50 times that?
somebody” s providing you a loan and they state, oh, yeah, I” ll lend. you– you require a 20-year lending? I” m mosting likely to provide. it’to you at 15 %.
It ‘ s rather crucial to. clarify whether they” re mosting likely to bill you 15% passion at.
easy passion or compound interest.Because with compound passion,. you ‘ re mosting likely to wind up paying– I suggest, take a look at this: just to. obtain$ 50, you ‘ re mosting likely to be paying $618 greater than if. this was straightforward interest. In the real. world, a lot of it is compound interest. And not just is it compounding,. yet they wear ‘ t also simply worsen it annually and they.
put on” t even just compound it every 6 months, they actually.
substance it continuously. Therefore you should see the.
next several videos on continually intensifying.
passion, and afterwards you” ll in fact begin to learn.
concerning the magic of e.Anyway, I
‘ ll see you.
all in the next video clip.
Allow” s state I ‘ m. borrowing P dollars. It ‘ s 50 times 4, which.
15%, every year we” re adding$ 3– we ‘ re including–. 1 plus– and what ‘ s the price? And what ‘ s 50 times that?