Now This is Just an Estimate
Vol #135: June 29th, 2023
One way to finish a marathon is to begin running at the starting line, follow the course 26.2 miles, and then cross the finish line. Another way is to skip that irksome running part and get on the subway.
Rosie Ruiz might be the most famous athlete whose only official times are DQ. She skyrocketed to fame not because of her work at the commodities firm Metal Traders, but because she accidentally won the Boston Marathon.
Her athletic career began when she falsely claimed to be suffering from brain cancer to gain a late entry exemption to the NYC Marathon in 1979. Rather than loop through Central Park, she took the Lexington Ave express and posted a rather competitive time of 2:56:29 - good enough for 11th place.
Now having qualified for the 1980 Boston Marathon, she attempted to repeat her success by slinking through the crowd on Commonwealth Ave and jumping into the race. The only problem was she was about 25 minutes early.
Ruiz finished the race with a winning time of 2:31:56 - the fastest female in Boston’s history. The miscalculation that led to sudden glory also begot a flood of questions. Everything from her resting heart rate and absence of sweat stains to a suspicious lack of split recollection took everyone to the videotapes, where she was nowhere to be seen.
Being off by 25 minutes matters quite a bit here. For a three hour race that’s an estimation error of about 17%. Cheating in sports requires more precision than that, even in 1980.
In a lot of cases 17% is a very reasonable margin for error. If I asked you to estimate the population of a small town in Connecticut, the difference between 7,510 and 8,742 doesn’t feel significant at all. It’s actually probably very useful for whatever amateur demography you’re practicing.
How, and how well we estimate varies greatly depending on the use case. For most purposes outside athletics, it’s much more efficient to take the subway than it is to train for the kind of grueling race that killed its first participant.
Finance - be it personal budgets or pit trading - demands a bit more precision. But not as much as it often seems. The asymptote in the plot of effort versus accuracy plateaus at a very reasonable output level.
If you want to calculate the monthly payments on a house, there are a couple of ways to do it. My go to was always Excel. The PMT formula takes the interest rate, number of periods, and the present value of the loan, and voila. That will get you an exact figure without having to do the manual calculations with their pesky exponents.
On the go, instead of pulling up Zillow, you can flex your brain with a quick short cut. (h/t to @docmilanfar for posting this.)
Monthly Payment = (Principal + Interest ) / Number of Months.
Interest = (.5) * (Years) * (Principal) * (Interest Rate)
This estimation technique gets bonus points for being very intuitive. The monthly payment is the sum of paying back the principal, plus all the interest you owe, divided by the number of periods in the loan.
To determine that interest, it makes a linear approximation that on average you’ll be paying interest on half the balance. The line of your declining balance marks the hypotenuse, and the area of that triangle is your balance (y) over time (x). (½) * Base * Height gives you “total borrowed dollars” which you can multiply by the interest rate.
Interest is of course not linear, and the reality is you pay more than the linear estimate because in those starting years you’re not chipping away at much principal but just floating the cost of the loan. The space between the orange (compounded) and green (linear) curves shows how much greater the total cost is, ergo the monthly payment difference.
The estimate breaks down as you either a) increase the interest rate or b) increase the time. More juice, more time to make an impact - just like volatility. That orange line inflates outward. The below table shows the under-estimate between the precise calculation and the shortcut across different terms in years versus annual interest rates.
Though this estimate is less useful today than it was 18 months ago, you’re talking a low teens estimation error on a thirty year term. That’s cocktail party credible, but if you’re talking about a new house budget, Excel only takes another minute more.
There are a number of good short cuts for options traders specifically. The most popular is probably estimating the average daily move based on the implied volatility. Most of us don’t think in distributions, so an annualized variance isn’t intuitive. Dollar change today, that jibes a bit more.
Just take the IV percentage and divide it by 16, and you get the expected daily move in percentage terms, which can then be quickly converted into dollars. If SPY is trading at 12 vol right now, we’d expect daily moves of about .75%. The reason we use 16 is that it’s very close to the square root of 252, which is the number of trading days in the year, and volatility is proportional to the square root of time.
For a 60 vol stock the implied percentage move is only .02 percentage points different if you just short cut 16 for the square root function. Why bother wasting the extra punches on your calculator?
Estimating a trade’s delta also has more in common with mortgage talk over zinfandel than the stochastics and trinomial trees. Starting in mock class the urgency of getting your hedge was pounded into our brains. There were no explicit deltas written on the mock board, just bids and offers, so every put sold or call spread bought had to be hedged with your size times a best guess.
Sunny never barked about the difference between a 25^ or 30^ hedge, he was already moving the stock you missed trying to be unnecessarily precise. In the real world stock jumped because every other trader selling calls on the ticket was hedging their risk, and to the equity market markers a bunch of dumb limit orders hit their offers and it’s time to fade fast.
Trading is estimates all the way down. The precise option delta that your terminal gives you is just based on a guess about what implied volatility should be. Or what interest rates and carrying costs will be. And that model is only an approximation of the way real world asset prices behave. Look at three different brokers and you’ll get five different IVs.
Risk management also relies heavily on estimation. Accuracy here matters more than speed. From a capital allocation perspective, the more precisely you can estimate your risk, the more efficient you can be. Bank regulators will require certain capital ratios, but that’s not the same as putting your money where it will be most useful.
Popular trade sizing rubrics like the Kelly Criterion are nothing but a best guess. It’s easy to apply this to a coin flip, but no ones offering unlimited flips for $.95 to win $1.00. Any casino game like roulette that has known probabilities will necessarily present negative edge to participants and you shouldn’t play. Anything that possibly offers positive edge, you’re just taking a guess at what the true probability of a win will be. Choose wisely, and fractionalize your Kelly.
Options are priced the way they are, because they themselves are an estimate of future value. If you’re trying to come up with the value for something, start with the things you’re absolutely certain about - intrinsic value. Just like a company shouldn’t be worth less than its liquidation value, an in the money American style option has to be worth AT LEAST as much as if you exercised it right now and took the stock.
Everything on top of that is the extrinsic value - a finely tuned prediction that takes into account what this thing will be worth if stock zigs, zags, or zooms. All those cheap teenies you see on the board are just a reflection of probability and potential.
If you want to get meta, even the underlying valuation is just an estimate. If the stock market is a weighing machine in the long run, the short term voting through daily price action is just the ping ponging of estimation. The noise of variance is an artifact of the market’s estimation process.
Did the stock market lose precisely $1.527 trillion dollars in future discounted cash flows yesterday? Absolutely not. But over time the estimates converge on a platonic ideal of “correct” as news and information continually get digested.
We estimate for many reasons, but they all share one thing in common. We must understand what is, before we can forecast and act towards what will be.
