What I want to do in this video is talk a little bit about compounding interest, and then have a little bit of a discussion of a way to quickly, kind of an approximate way to figure out how quickly something compounds, and then we'll actually see how good of an
So just as a review, let's say I'm running some type of a bank, and I tell you that I am offering 10% interest that compounds annually.
舉個例子來複習一下，假設我經營一家銀行，我告訴你我提供年複利 10% 的利率。
00:32
That's usually not the case in a real bank.
在真實的銀行中通常不是這樣運作的。
00:35
You would probably compound continuously, but I'm just going to keep it a simple example, compounding annually.
你可能會選擇連續複利，但我這裡只用一個簡單的例子，採用年複利。
00:40
There are other videos on compounding continuously.
關於連續複利，網路上有其他影片可以參考。
00:43
This makes the math a little simpler, and all that means is that, let's say today, today, you deposit $100 in that bank account.
這樣數學會簡單一些，意思是說，假設今天你在這個銀行帳戶存入 100 美元。
00:53
If we wait one year, and you just keep that in the bank account, then you'll have your $100, you'll have your $100, plus 10% on your $100 deposit. 10% of $100 is going to be another $10.
So after a year, you're going to have $110, so you can just say I added 10% to the 100, 100, and then after two years, or a year after that first year, after two years, you're going to get 10%, not just on the $100, you're going to get 10% on the $110.
That was, you can imagine, you're depositing entering your second year, and then you get plus 10% on that, not 10% on your initial deposit, that's why we say it compounds.
You get interest on the interest from previous years, so 110 plus now $11.
你從前幾年的利息中又賺取利息，所以 110 美元再加上現在的 11 美元。
01:56
So every year, the amount of interest we're getting, if we don't withdraw anything, goes up, so now we have $121, and I could just keep doing that, and the general way to figure out how much you have after, let's say, n years, is you multiply it, so let's say my
original, I'll use a little bit of algebra here, so let's say this is my original deposit, or my principal, however you want to view it, after x years, so after one year, you would just multiply it to get to this number right here, you multiply it by 1.1, actually let
原始，我會在這裡使用一點代數，所以假設這是我的原始存款，或稱為本金，你想怎麼看都行，經過 x 年後，所以經過一年後，你只需要乘以它來得到這裡的這個數字，你乘以 1.1，實際上讓
02:32
me do it this way, I don't want to be too abstract, so just to get the math here, so to get to this number right here, we just multiply it, that number right there is 100 times 1 plus 10%, or you could say 1.1, now this number right here is going to be this
110 times 1.1 again, so it's this, it's the 100 times 1.1, which was this number right there, and now we're going to multiply that times 1.1 again, and remember, where does the 1.1 come from? 1.1, 1.1 is the same thing as 100% plus another 10%, right, that's what
we're getting, we have 100% of our original deposit plus another 10%, so we're multiplying by 1.1, here we're doing that twice, we multiply by 1.1 twice, so after three years, how much
100 times 1.1 to the 1.1 to the third power, after n years, now we're getting a little abstract here, we're going to have 100 times 1.1 to the nth power, and you can imagine this
錢？100 乘以 1.1 的三次方，經過 n 年後，現在我們變得有點抽象了，我們會有 100 乘以 1.1 的 n 次方，你可以想像這
03:47
is not easy to calculate, and if this was all the situation where we're dealing with 10%, if we're dealing a world with, let's say it's 7%, so let's say this is a different reality here, where we have 7% compounding annual interest, then after 1 year, we would have 100 times,
instead of 1.1, it'd be 100% plus 7%, or 1.07, after, let's skip, let's go to 3 years, after 3 years, I could do 2 in between, it'd be 100 times 1.07 to the third power, 1.07 times
itself three times, after n years, it'd be 1.07 to the nth power, so I think you get the sense here that, although the idea is reasonably simple, to actually calculate compounding interest is actually pretty difficult, and even more, let's say I would ask you, how long does it take, how long does it take to double your money? So if you were to just use this
自身三倍，經過 n 年後，會是 1.07 的 n 次方，所以我認為你這裡可以理解到，雖然概念相當簡單，但實際要計算複利其實相當困難，而且更甚者，假設我問你，要花多久時間，要花多久時間才能讓你的錢翻倍？所以如果你只是用這個
04:58
math right here, you'd have to say, gee, I would have to double my money, I would have to start with $100, and I'm going to multiply that times, let's say whatever, let's say it's a 10% interest, 1.1, or 1.10, depending on how you want to view it, to the x is equal
這裡的數學，你得要說，哎呀，我得要讓我的錢翻倍，我得要從 100 美元開始，然後我要乘上，假設不管多少，假設是 10% 的利率，1.1，或 1.10，取決於你怎麼看，的 x 次方等於
05:15
to, well, I'm going to double my money, so it's going to have to equal to $200, and now I'm going to have to solve for x, and I'm going to have to do some logarithms here, and you can divide both sides by 100, you get 1.1 to the x is equal to 2, I just divided both sides by 100, and then you could take the logarithm of both sides base 1.1, and
you get x, and I'm showing you that this is complicated on purpose, and if any of this is confusing, there's multiple videos on how to solve these, you get x is equal to log base 1.1 of 2, and most of us cannot do this in our head, so although the idea is simple,
你會得到 x，而我故意向你展示這個很複雜，如果這裡有任何令人困惑的地方，有很多影片在教如何解這些，你會得到 x 等於以 1.1 為底 2 的對數，而我們大多數人沒辦法在腦中計算，所以雖然概念很簡單，
05:51
how long will it take for me to double my money to actually solve it to get the exact answer is not an easy thing to do, you can just keep, if you have a simple calculator, you can kind of keep incrementing the number of years until you get a number that's close, but no
straightforward way to do it, and this is with 10%, if we're doing it with 9.3%, it just becomes even, even more difficult. So what I'm going to do in the next video is I'm going to explain something called the rule of 72, which is an approximate way to figure out how long to
answer this question, how long does it take to double your, I forgot the word, the most important word, how long does it take to double your money, and we'll see how good of an approximation it is in that next video.
