If I Had Money, I Wouldn't Need Money
Vol #67 - January 28th, 2022
When there’s red on the screens, one of the most frustrating nuggets of truth to hear is, “it’s a good time for compounding.”
This platitude is a weak salve for the down market blues. Pain is in the present, and that suite upgrade on your vacation is fully paid for by last week’s high water mark.
But it is true. Dollar cost averaging is a compounding strategy that consistently puts money in the market at whatever the prevailing valuation might be. You buy both the dips and the rips, but you’re always buying a little bit more when the asset is cheaper, letting the magic of compounding work.
The mechanics are relatively simple, but since it defies intuition, compounding is much harder in practice. It’s not just the psychological biases that are working against us, compound interest is one of those arithmetic anomalies that leaves even the most adroit on their arse.
This phenomenon seems to be particularly exacerbated by the fact that we’ve been in a persistently low interest rate environment - rates are powerful, but a lot less so at fractions of a percent. Our interest rate muscles have been dulled by the soothing wash of quantitative easing.
I’ve been trading stocks for 20 years and depending on where you draw the lines, I’m on the tail end of a generation that saw real interest rates, or the bleeding edge of one who only knows how to refinance their mortgages lower every other year.
When I first started market making in options, we used to relish the end of the month short interest report. Old school accounting meant this balance only accrued once a month, for a nice PnL boost on the last day. Now, with interest rates barely above zero, there are no longer short stock rebates - most short stocks now cost you money.*
Once finance integrated the interest calculations into our daily balances, they didn’t seem quite so significant. While this wasn’t explicitly due to compounding, interest has a way of sneaking up on you. Whether it’s a credit card, savings account, or stock balance, the effects of interest are both powerful and quirky.
In DeFi, there are plenty of places offering real interest rate opportunities. Billions of dollars are earning rates between 8-18%. Even 8% per annum is eye popping to anyone who was born after Volcker.
Now let’s put that 18% to good use. If you can scrounge up $100,000 to put to work, and have the confidence to bridge it over to an algorithm, that’s an “easy” $18,000 a year - more than a nice PnL boost. But what if you want to take monthly withdrawals on it?
If you take monthly withdrawals on an 18% APY, you’re only getting $1,388 a month - quite a few coffees less than the $1500 it feels like you should get from $18,000 / 12 months. That difference adds up to over $1300 a year - practically an extra month’s return.
If you can keep your money compounding, it will keep working hard for you. The big catch is, you can’t touch it. That means no panic sales when the market goes down, and making sure your withdrawal timing takes into account the loss of interest.
Mos Def quips on a track about life questions the circular logic of getting an ID card. We can swap ID for money here: “Why do I need money to make money? If I had money, I wouldn't need money.”
Compounding is even more powerful the higher the rate of return. At current big bank savings rates, the fractional basis point doesn’t mean much- even compounded. But move that to a place where dollars are demanded like DeFi, or where risk is rewarded over the long term like equity markets, and it becomes exponentially more significant.
If you want to get really wonky with your compounding arithmetic, there’s Euler’s number. This is the mathematical constant that can be conceived of as the compounding asymptote - applying a small change an infinite number of times. Hit F5 on the web browser as fast as you can - the answer is 2.7182-sh. That’s the maximum multiple that infinite compounding can add to your yield.
This works the opposite way with debt. Paying down your mortgage regularly, with a dollop of extra principal on top can dramatically reduce your debt cost and horizon. This is why banks limit the number of times you can recast your mortgage each year and usually charge a nominal fee.
All of these compounding experiments exist in a frictionless bubble. In addition to the practical obstacles, transaction fees present a drag on returns in an even less intuitive way.
As soon as you introduce a kernel of cost into the equation, the compounding curve starts to bend in a different direction. As I always like to do with options pricing, take the example to an extreme. If your incremental yield for a given interval is $1, and it costs $1 to compound (i.e. reinvest profits) then you’re simply treading water.
With a savings account, interest will automatically compound, so there’s no need to worry. But reinvesting stock market dividends, or yield that accrues separately from your principal has some friction. Whether it’s brokerage fees or the cost of gas, there is almost always some transaction cost.
At what point does the cost and benefit curve maximize?
Let’s take our 18% APY, and assume it costs $1 to reinvest our profits. At the far right, you only reinvest once, and end up with exactly $118,000. On the far left, you’re compounding every 12 hours. For this amount of money and transaction cost, the sweet spot is compounding every 9 days.
If all of the sudden we make that transaction fee $10, reinvesting profits any more frequently than every three days actually makes you a net loser to once a year, despite the additional time that yield is invested. It also shifts the optimal frequency of compounding out to three times less often at every 28 days.
This is hard to wrap your head around, and it usually helps to put something together in a spreadsheet (or borrow someone else’s). If you’re reading this on your iPhone, you can also turn your screen sideways in the calculator tool, and have fun playing with this.
Type in 1.18, hit x^y, and then 3. That’s the value of 18% compounded over 3 years.
Happy compounding.
_____
*Customers are natural buyers of puts and sellers of calls, because they tend to be long stock. Taking the other side of this, market makers hedge long calls and short puts with short stock. Short stock meant you borrowed stock, sold it, and sat on a cash balance. This cash balance actually earned a positive interest rate!
With more sophisticated customers most stocks a market maker ends up short are going to be “negative rate” - you have to pay to borrow. This is a function of the low base interest rate, but more so due to supply and demand considerations. When lots of people want to short a stock, they can either borrow it to sell it (complicated), or more easily go to the options market to buy puts. Market makers hedge their short puts with short stock, and all of this nets to high demand for stock to borrow, and thus a higher cost.