When to Break Golden Handcuffs: The Endowment Equation

Recently, a few of my friends have been going through something of an existential crisis. It's only natural as they're all just entering the unstructured real world-- I myself went through a mini-nihilism phase after graduation. One cure, I've found, is to set a lifelong goal, any goal, and work tirelessly to achieve it-- anything from becoming an olympic athlete to helping as many people as possible. A nihilist would say all goals are equally arbitrary-- but having one in spite of that fact is a sure-fire way to get your sense of purpose back in my book.

Naturally, the first step many people, including a lot of my friends, choose to take to achieve their goal is to acquire enough capital such that they don't have to worry about their finances anymore. Note that seeking early retirement isn't a goal everyone chooses to pursue and, indeed, it's not something I care about (I'm probably never retiring). But given that the initial goal of accruing "retirement money" is so common, I thought I'd write a little blurb to help make the abstract goal of retiring a little more concrete. The goal of this article is to help you compute exactly how much money you need to have in the bank before you can dedicate your life to a potentially revenue-negative venture (like sailing around the world or something like that).

I started wondering how much money one would need to have as an "endowment" before one can live off the interest indefinitely without having to worry about working anymore. There's the "4% rule," which states that one can spend 4% of an endowment every year indefinitely, without running out of money ever, but I wanted a more exact answer.

I frame the problem as follows:

• Start with an endowment of $Y
• You pick an amount $X that you want to "make" every year as your "income" off the endowment.
• How big does Y have to be so that you can sustain making $X adjusted for inflation every year, forever, while having your original endowment, Y, also keep up with inflation?

The equation looks as follows:

where we have:

• n = the number of years after "retirement" (ie the number of years you've been living off your endowment for)
• r_{inflation} = percent inflation every year (3% if you want to be conservative)
• r_{s&p} = percent return on the s&p500 (6% if you want to be conservative)
• (r_{s&p} - r_{inflation}) = excess returns on inflation
• Y = your initial endowment
• X = the amount of money you want to make as income off your endowment (eg 200k)

What this equation says is that after living off your endowment for n years, your excess returns on inflation (left-hand side) should be equal to your desired inflation-adjusted income (right-hand side).

Solving this equation, and adding X to account for your income the first year, yields the following simple relation:

What does this equation tell you? Well, effectively, it tells you how much money you need to have in the bank before you can comfortably live off the interest, taking inflation into account. Once you get to this point, you can have some fun fiddling around with the numbers.

Assuming inflation is 3% and the s&p returns 6% annually (both conservative numbers), here's how big your endowment needs to be to earn various levels of income without doing jack sh**:

• 100k/yr => endowment of ~3.54 MM
• 200k/yr => endowment of ~7.1   MM
• 500k/yr => endowment of ~17.7 MM
• 1M / yr  => endowment of ~35.4 MM

It's linear in the desired yearly income-- but it's still nice to see the numbers written down. At the end of the day, having 7.1MM in the bank makes you pretty much set for life.

Here's a plot you can check out that shows the endowment size against the desired annual income with mathematica code for you to play with if you so desire:

f[inflation_, sandpreturn_, X_] := X * (1 + (1 + inflation)/(sandpreturn - inflation))

Plot[f[.03, .06, x*1000]/1000000, {x, 0, 1000}, Frame -> True, FrameLabel -> { "Desired Annual Income (in thousands of dollars)", "Required Endowment (in $MM)"}]

f[.03, .06, 1000000]

LaTeX:

Y ( 1 + r_{inflation} )^n (r_{s\&p} - r_{inflation}) = X (1 + r_{inflation})^{n+1}

Y = X \left(1 + \frac{1 + r_{inflation}}{r_{s\&p } - r_{inflation}}\right)


So, in summary, if your goal is to retire as soon as possible, simply figure out the annual income you'd like to have in retirement and plug it into the formula to compute the amount of capital you'll need to have in order to retire securely. As I said, racing to retirement isn't for everyone-- but if it's for you, hopefully this post helps put a number in front of you.

Afterthought: Actually, the endowment from this equation is an overestimate, since when you die you still have the whole endowment intact. If instead you want the endowment to go to zero around when you die, I think you can save significantly less money. Maybe I'll compute this case in the future.