Calculating savings for financial independence

0:00- [Instructor] We're told that Amina is 29 years old and wants to be financially independent by age 58. And then they give us all of these assumptions, how much she wants to spend when she's retired in today's dollars, where I think we're gonna have to think about how much money that will be based on inflation in 29 years when she's 58.

0:18They give us the inflation rate to assume the investment return. Nominal means we're not adjusting for inflation. This is just how much the dollars grow. Current savings. The financial independence target uses the 4% rule, which is really saying that 4% of her savings, at any given time, she could use 4% of it to live.

0:36So that 4% of her savings needs to be equivalent to what would be $32,000 today. I'm gonna deposit the same amount at the end of each year until age 58. What is the annual amount she must save to reach financial independence by age 58? So like always, pause the video if you're so inspired.

0:56Let's now work, break into this. So let's first think about, $32,000 per year, how much will that be when she's 58? Well, to think about that, that would be $32,000 grown by inflation for 29 years, so that's going to be times 1.02 to the 29th power. So that's how much her desired spending will be in, let's say, tomorrow's dollars or in dollars 29 years in the future.

1:33Now, the amount of money she needs to save, it needs to be an amount where 4% of that amount, in the future, in 29 years will amount to this. So if we take this amount, actually, let me just calculate it. Let me get a calculator here. So if I say 32,000 times 1.02 to the 29th power, to the 29th power, this is $32,000 today.

2:05If we assume 2% inflation compounding for 29 years, it's almost going to be $57,000 a year when she is 58. Now, as I said, this needs to be 4% 'cause she's going to be able to spend 4% of her savings every year. So this needs to be 4% of some amount. If you wanna figure out, what is this 4% of?

2:27You divide by 4%. So divided by 0.04 gets us to 1.4 million and something. $1.421 million. So let me just take this aside and write this down. So if I took, this is how much she's gonna spend a year in 29 years, and I am going to divide that by 0.04. This is how much she would have to save in 29 years in order for 4% of it to spit out how much he's going to spend.

3:02And we figured out that this is going to be equal to $1,420,675.75 if we round to the nearest cent. So now the question is, how can she save up that much? Well, my brain first thinks, "All right, she has this..." Before we even think about these annual payments she's going to make, she has this current savings of $2,000.

3:31And that money, you could just say, "Hey, that's gonna grow by 6% for the next 29 years." So if you just took $2,000 and grow that times 1.06 to the 29th power, that's how much her current savings would have grown. Now, if we wanna figure out how much her payments need to add up to, we could think about the total amount that she needs and then we could subtract from that what her current savings are going to be in 29 years.

4:01Let's actually figure that out. Let me get the calculator again. So here, that's what we got before. And now I'm going to subtract minus 2,000 times 1.06 to the 29th power. To the 29th power gives a little, a slightly smaller number, $1,409,838.98 or so, and now we need to figure out, what's going to be the payment plan?

4:35So we've already seen the formula, which we prove in some mathier videos on Khan Academy, that a future value is going to be equal to some payment, some payment times one plus your rate of growth to the number of periods to that power minus one divided by the percent that you're growing every year, so divided by r.

4:59So what we're need to solve for is this payment right over here. The future value we're going to solve for is this thing. And I have it written down, this future value is $1,409,838.98. That's the future value, and we can figure out all of this. R, we know that the investment return is 6% per year, so this is 0.06, or maybe I'll write it this way.

5:32This is 1.06. This is 0.06, and then we are compounding annually. This is a 6% growth per year, and we are doing this for 29 years so to the 29th power. So another way to think about this is, if I divide both sides of this equation by this number right over here.

5:57I know it might not look like a number. It's 1.06 to the 29th power minus one divided by 0.06. But if we divide both sides by that, we will be able to solve for the payment. And so if we were to just take your payment, and you could do it step by step, but I'm trying to retain the precision on my calculator, is going to be equal to that future value, 1,409,838.98 divided by all of this, or dividing by a fraction is the same thing as multiplying by the reciprocal.

6:29So that's times, if I just multiply by the reciprocal, that's 0.06 over, over 1.06 to the 29th power minus one. So now I can put all of that into my calculator and I'm going to get... I still have that original number. I'm gonna multiply it times 0.06 and then I'm gonna divide that by, let me open parentheses, 1.06 to the 29th power minus one, close parentheses.

7:06And I think we deserve something of a drum roll. We will get what that payment, that annual end-of-year payment needs to be for Amina. That equals $19,145.07. So based on the math I just did, I got... Let me write it over here. So my payment is equal to $19,145.07 if I round to the nearest penny.