--- /dev/null
+[za3k](/) > finances > retirement math, part 2: retiring for infinity years
+
+I retired at 31, which is pretty unusual. And I get asked about it a fair deal (and even more people want to ask but are too polite). So I wanted to go ahead and answer "how did you retire early?". The basic answer is
+ - I made a bunch of money
+ - I don't spend much
+ - Math
+
+Even reasonable people people imagine the wrong specifics for all three, though. I'll walk you through it. We'll start with the math, which is the most surprising to most people.
+
+Here's how people *imagine* the math will work.
+
+Imagine you're currently 20. You plan to live until 80, then die [^1]. You make $60K/year, and spend $50K/year.
+
+That's enough to retire at 65. How can you improve that to retiring at 30?
+
+There are two common approaches. The more common assumption is the "high income", or "fixed expenses" method:
+- You spend $50K/year.
+- You want to retire from 30 to 80, for 50 years.
+- You need $50K/year x 50 years = $2500 for retirement. That's $2.5 million.
+- Therefore, to make that by retirement you need to make $2500K / 10 years = $250K/year.
+- So you need to increase your income to $250/year.
+
+The second common approach is the "low expenses", or "fixed income" method:
+- You make $60K/year.
+- You will retire in 10 years.
+- When you retire, you will have made $60K/year x 10 years = $600K.
+- You need that to last you your remaining life, from 20 to 80. That's 60 years.
+- You need to spend $600K / 60 years = $10K/year.
+- So you need to decrease your expenses to $10K/year.
+
+So you can increase your income, or decrease your spending, or do some combination of both. This basic insight is correct.
+
+But all the numbers are *wrong*. The math is *incorrect*. If you retire at 65, it's a *little wrong*, but if you retire at 30 it's *wildly wrong*. A new factor dominates when retiring at 30 or 40.
+
+[>>retire for infinity years](/articles/retirement_math1)
+
+[^1]: Not, you know, that people *plan* to die at 80. They don't circle a date in their calendar. But you have to know how long you'll be retired to plan financially. Actually, living to 100 can turn into a financial problem, because you've run out of savings. This is part of how insurance [got started](https://en.wikipedia.org/wiki/Tontine).
--- /dev/null
+[za3k](/) > finances > retirement math, part 2: retiring for infinity years
+
+In [part one](/articles/retirement_math1), I walked you through how people imagine the math works.
+
+We imagined you made $60K/year, and spent $50K/year. You're 20, want to retire at 30, and plan to live to 80.
+In part 1, we incorrectly concluded that you need to either:
+- increase your income to $250K/year
+- OR decrease your expenses to $10K/year
+
+The instinct is right, but the numbers are wrong. To explain why, let me ask you an odd question. How much would you need to save to retire not until you're 80, but until you're 8,000? 80,000[^2]? You need to save up thousands of times more, right? No, because the math was *wrong*. You can do the above and you'll still be retired at 33.
+
+### The math of retiring forever
+
+- Suppose you make $60K a year, and you spend $20K a year. That means you're saving $40K per year.
+
+- After 12.5 years, you will have saved up $500K.
+- Assuming 4%[^3] stock market returns on that $500K, you are now making $20K/year from stock market investments.
+- You are spending as much as you're making, so you're living on your investments.
+- This means that you can now, in theory, stop working forever and live on your investments.
+- Not retire for 10 years, not retire for 20 years--*forever*. If you live to 80,000[^2] you'll be fine, as long as the stock market continues to make about 4% a year.
+
+That's right, money makes you money. If you live cheap and invest even below-average, you can retire forever at the age of 32.5.
+
+Our original math said you can retire until the age of 80 at 30 using the same strategy. We can retire forever with 2.5 years extra? Is that right? Partly it is! But mostly the original math was incorrect. Our math now takes investing into account. As long as you want to live for infinity years (if you plan to die, read on).
+
+### But what about me?
+
+Well, what if you're not making $60K and spending $20K? I'll walk you through the general formula. You need to know three numbers
+
+- Your yearly income (**I**)
+- Your yearly expenses (**E**). *And*, these need to stay the same after you retire!
+- How much you expect to make on the stock market, which I'm just going to set for you, at 4%[^3] yearly. Sometimes you'll see 0.04, sometimes 25[^4].
+
+The rough[^5] estimate is that you can retire in: **25 x expenses / (income - expenses)** years.
+
+Working through our example, that's = 25 * 20K / (60K/y - 20K/y) = 500K / 40K per year = 12.5 years. Imaginary-use is 20, so that works out to 32.5 years old. Checks out.
+
+Okay, but what if... the stock market takes a hit?
+
+[>> up and down](/retirement_math3.md)
+
+[^2]: Which I am 100% in favor of. Get cracking on that science.
+[^3]: This is very conservative. Since it's founding in 1957, the S&P 500 on average made 7% yearly over inflation. You can ajust the numbers if you want.
+[^4]: Why 25? 25 is (1/0.04). You need 25 times any amount to make it at 4% per year. Want $100 per year? Invest $2500 at 4% per year.
+[^5]: The rough estimate assumes you don't invest until you retire. This is mostly because the real answer involves a *lot* more math. If you invest all your money the whole time, you can retire in exactly (not roughly) 25 * ln(1+E/(I-E)) years. Good luck! You may need a calculator.
+ For anyone who wants the derivation:
+ ```
+ > S(0)=0
+ > S'(t)=0.04*S(t)+I-E
+ ...differential equations is too hard, ask Wolfram Alpha...
+ > S(t)=25*(I-E)*(e^0.04t-1)
+ ...high school algebra, solve for S(t)=E/0.04...
+ t=25 * ln(1+E/(I-E))
+```
--- /dev/null
+[za3k](/) > finances > retirement math, part 3: up and down
+
+In [part one](/articles/retirement_math1), I walked you through how people imagine the math works to retire early. We need to make a ton of money, or spend almost nothing.
+In [part two](/articles/retirement_math2), I talked through retiring *forever*, using the real math. It only took a little longer than retiring until 80 with the fake math.
+
+So we're good now, right? Imaginary-us saves up 500K in the stock market, makes $20K/year, and can retire forever?
+
+Well, there is a caveat. You need an additional safety margin, even if the stock market doesn't collapse. You probably want to retire in 15 years, not 12.5.
+
+- Imagine your balance was "only" 400K, rather than 500K. Now you make 16K investment income each year. You still spend $20K each year. So you're now losing $4K per year! 25 years later your balance hits 300K, and you're losing $8k per year. It "snowballs" down exponentially until your savings hit zero, although fairly slowly at first (40 years to go broke).
+- Imagine your balance was all the way up at $600K. You make $24K investment income each year. You still spend $20K each year. So now you're making $4K per year. 25 years later your balance hits 700K, and you're making $8K per year. It "snowballs" up exponentially instead, and you just make more and more money indefinitely. Also slowly at first.
+- So there's this "magic" line at 500K. If you're above $500K, you expect your savings to grow forever with no work. If you're below $500K, you expect it to shrink forever unless you get a job.
+- The 4% per year from the stock market is an average. Stocks don't actually make 4% per year--they make 20%, 1%, lose money, whatever. So in practice, your bank balance is going up and down the whole time. That means if you're close to $500K, whether it snowballs up or down is basically random based on the first couple years. You want to have more like $600K instead, so you definitely snowball up, even if the stocks do badly for a year or two.
+- There's another advantage to "snowballing" up. If you start making $25K, you can spend $22K now and still make money. The rich literally get richer with no work. Another reason to aim a little high--you want to enjoy retirement!
+
+To summarize, if you live on $20K/year, and work at $60K/year, you can retire indefinitely in theory, after 12.5. In practice, add a safety margin. Now you're secure against random stock market problems, and you make *more money* over time.
+
+Okay, but you don't want to retire forever. You're not immortal[^2]. Can you retire even earlier? Yes.
+
+[why we aren't all broke](/retirment_math4.md)
+
+[^2]: Which I am 100% in favor of. Get cracking on that science.
--- /dev/null
+[za3k](/) > finances > retirement math, part 4: save money by dying
+
+In [part one](/articles/retirement_math1), I walked you through how people imagine the math works to retire early. We need to make a ton of money, or spend almost nothing.
+In [part two](/articles/retirement_math2), I talked through retiring *forever*, using the real math. It only took a little longer than retiring until 80 with the fake math.
+In [part three](/articles/retirement_math3), I explained about *snowballing up* and *snowballing down*, and why you might want to save more than the bare minimum.
+
+In this part, we'll talk about how people don't live forever, except for you.
+
+From [part two](/articles/retirement_math2), our rough estimate formula to retire was that it took: **25 * expenses / (income - expenses)** years. Let's plug in some numbers.
+
+Now let's say you make $50K/year, and spend $40K/year. The same math says it will take you 25 * 40K / (50K-40K) = 1,000K / 10K = 100 years to retire. You will never be able to retire indefinitely; you'll die first. That is a huge difference from previous examples. What gives? That's a pretty typical person, right? Why is everyone not starving on the street at 80? Let's take a quick digression.
+
+### Normal retirement
+In practice of course people do spend a lot of what they make and then retire. There are many reasons.
+- The formulas assume you're going to retire indefinitely. Actually, you only need enough money until you die. It's a 2.2X difference, if you retire at 65. That's a 55% discount for being mortal. We'll talk about the specifics below.
+- People just retire. They're going to, frankly whether they can afford it or not. Some people retire at 65 because they're told they can or should, or they feel they deserve it. Even past that point, others can't work starting at a certain age.
+- Some people just go broke. Some go back to work, some starve on the street when they can't. It does happen. You can retire when it was a bad idea.
+- Beyond that, some people realize they don't have enough money, and cut back significantly. The imaginary people in my head do this when they're running out of money, not immediately when they retire. Cutting back may or may not fix the problem.
+- Most people in the US have special retirement accounts, which they must or are heavily encouraged to pay into. Employers may "match" retirement money (paying an equal share), or it may have special untaxed status. Usually you can't take money out of a retirement account until you're 65, to force people to actually use them for retirement. The account may also be designed so you can't take all the money out at once even after that point.
+- The USA has governement programs to help. Medicare provides free health care for almost every old person. Social Security sends monthly checks to old retired people, although rarely enough to live on comfortably.
+- We have social support programs. Families often support older relatives. Churches and other social groups can offer help too (as well as for broke and poor people).
+
+If you're retiring at 30, almost **none** of the above apply to you. You need to get by on savings, no external support. You don't get a 55% mortality discount on retirement, you get a measly 15% discount. You probably don't have much in your standard retirement account, and even if you do, you can't access it for another 35 years. You don't get medicare (though you *may* get medicaid). You don't get social security. Expect no social support. If people see a healthy 50-year old go broke from poor retirement planning, they should tell you to get a job. I would.
+
+### Mortality discounts
+
+I mentioned we get a "mortality discount". That is, to retire for 50 years is easier than retiring forever. Makes sense.
+
+Most people guess the math is, you need to save up **expenses x years**. The actual correct equation is **25 x expenses x (1 - e^(-0.04 * years you want to retire))**. They're pretty different.
+
+Some example numbers:
+- To retire for 15 years (at 65), you need 11 times your yearly expenses (not 15 times). That's a 24% discount over the simple math.
+- To retire for 25 years (at 55), you need 16 times your yearly expenses. That's a 36% discount.
+- To retire for 35 years (at 45), you need 19 times your yearly expenses. That's a 46% discount.
+- To retire for 45 years (at 35), you need 21 times your yearly expenses. That's a 53% discount.
+- To retire for 55 years (at 25), you need 22 times your yearly expenses. That's a 60% discount.
+- To retire at any age and live forever, you need 25 times your yearly expenses. Technically, that's a 100% discount, because the simple math says we need infinity dollars.
+
+In other words, if you can retire early, you need *way less money* to do it. That's a good reason all on its own.
+
+[>> just give me a calculator](/retirement_math5)
--- /dev/null
+[za3k](/) > finances > retirement math, part 5: the math
+
+In [part one](/articles/retirement_math1), I walked you through how people imagine the math works to retire early. We need to make a ton of money, or spend almost nothing.
+In [part two](/articles/retirement_math2), I talked through retiring *forever*, using the real math. It only took a little longer than retiring until 80 with the fake math.
+In [part three](/articles/retirement_math3), I explained about *snowballing up* and *snowballing down*, and why you might want to save more than the bare minimum.
+In [part four](/articles/retirement_math4), I explained to the dumber readers that retiring for less time than forever, costs less money.
+
+Okay, we're done now, right? Yes. We're done. Now we're going to tie everything together.
+
+In part two, we revealed the rough formula to retire forever. **Y = 25 x E / (I - E)**. We used a very convervative stock market return of 4%. Easy-peasy.
+
+Well actually, in the footnote of part two, we mentioned the exact formula: **Y = 25 * ln(1 + E/(I-E))**. Uh, that's a little worse, logarithms. Hmm.
+
+In part three, we added a fudge factor. We want to save up 20% or 50% extra. We won't actually factor that in. Do it yourself.
+
+From part four, we learned that to retire for Y years, we need **25 x E x (1 - e^(-0.04 * (80 - retirement_age)))** dollars, not **25 x E** like we claimed in part two. This is starting to look a little gross. Exponents?
+
+Combining our two equations, we get: **Y = 25 x ln(1 + E/(I-E)) x (1-e^(-.0.04 * (80-20-Y)))**. That's not looking... great.
+
+Actually, let's take out that rate of 4%, too. Someone could want to adjust that. **Y = (1/r) x ln(1 + E/(I-E)) x (1-e^(-r * (80-20-Y)))**.
+
+And we can add variables for the two ages. Maybe we're not 20, or we don't plan to die at 80. We only use the difference (60 years), so we add **L** for life-expectancy: **Y = (1/r) x ln(1 + E/(I-E)) x (1-e^(-r * (L-Y)))**
+
+We'll just solve for... well, uh. That looks kind of like a hot pile of garbage, huh? Five variables. Maybe we just back away slowly.
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