Knot Theory Meets Private Equity: The Takahashi-Alexander Model Explained

During their recent episode, Taylor, Carlisle, Ben Beneche discussed Knot Theory Meets Private Equity: The Takahashi-Alexander Model Explained. Here’s an excerpt from the episode:

Jake: All right. So, for today’s veggie segment, in mathematics, there’s this Knot theory which, like K-N-O- T, like, actually studying knots and the mathematics behind knots. It’s really these loops that are in three-dimensional space that they don’t intersect with themselves. So, think of it as taking a piece of string, and tying into a loop and then joining the ends, so it can’t be undone. There’s actually mathematical theories behind all this.

The main goal is really to classify and understand different types of knots, and studying their properties and how they relate to each other. There’s these things about knots that don’t change even when they’re stretched and twisted. These are then called knot invariants, like not changing, basically. And so, for example, the number of crossings within a knot is an invariant and counts how many times it crosses itself.

Another important invariant is this Alexander polynomial, which, it’s like this mathematical tool that helps you distinguish between different knots. Keep that name in the back of your mind for a minute longer. Another thing is this thing called Takahashi manifolds, which are a special concept in knot theory, named after Michio Takahashi. These are three dimensional spaces created from knots where they have this thing called Dehn, D-E-H-N, surgery. This is a process where you cut a knot, and then reglue it back together in a new way to form different 3D spaces and unique properties.

I don’t understand any of this stuff. I’m just [Tobias laughs] reading off the teleprompter, [laughs] I’m just kidding. But there’s actually some practical applications from all this, surprisingly. So, in theoretical physics, this knot theory helps understand these complex phenomena like quantum entanglement, and how particles become interconnected in strange ways. And then, also studying knots has actually led to improvements in how we allocate resources in complex systems, like distribution networks. There’s really abstract concepts, but they actually can help in logistical planning.

And so, here’s where things get a little bit weird and coincidental, okay? Remember those names, the Takahashi manifolds and the Alexander polynomials? Okay. Well, in the world of like large capital allocation like endowments, there’s a model that’s used to help forecast cash flows for private equity and venture capital. The name of that is the Takahashi-Alexander model. It was developed in 2001 by Dean Takahashi and Seth Alexander at Yale University.

No relation to these mathematical knot people. So, what are the odds of that? I found that to be quite striking. Yale, of course, was famous for David Swensen’s trailblazing asset allocation models that he did. Takahashi and Alexander were both Swenson acolytes. Alexander now runs MIT’s endowment.

So, what’s this Takahashi Alexander model in the endowment world? It’s a framework for estimating future cash flows and valuations in private equity and venture capital portfolios. It can be really difficult for an allocator to plan, because the money is typically called in chunks over the life, let’s say, the first five years of a 10-year fund. And then, the next five years after that, the money’s then distributed after these businesses are liquidated.

So, in the simplest terms, the model helps you figure out how much money you’ll need to invest upfront and rolling as the investment period runs and then when can you expect to get it back? The analogy might be like if you’re planning a road trip and you need to know how much gas you’ll need to buy along the route, these are the capital calls. And then, how many snacks you can afford to buy along the way, and these are your distributions. You don’t want to end up stranded in Barstow with no gas money. So, the Takahashi-Alexander model helps you plan out this liquidity scheduling.

So, as George Box famously said, “All models are wrong, but some are still useful.” The TA model, it relies heavily on assumptions about the future contributions and distribution. So, if you’re wrong about those, then the model’s going to give you pretty crap in, crap out, as the saying goes. And of course, it can lead to overconfidence in your predictions, because it gives you a single point estimate outcome and not really a range of probabilities.

So, of course, people have taken that now and run like Monte Carlo simulations, lots of historical data inputted that helps you create a range of possible outcomes instead of just one single prediction. That makes it a little bit more of a probabilistic approach, which is probably an improvement. Now, let’s see if we can wed knot theory and cash flow planning into some unholy matrimony.

Tobias: Land it, JT.

Jake: I’m going to try. So, a lot of the key lessons remain consistent between these twos, like simplicity, like thinking about a simple knot or a simple cash flow planning can lead to incredible complexity. Interconnectedness matters. Models, you have to consider uncertainty in them. Simple loops can form very complex shapes, just like a straightforward asset allocation model can actually create very intricate patterns of cash flow modeling, and then, just like these knot invariants that we talked about help maintain stability, like they’re a common thing that don’t change.

These models should have principles in them that can stay effective even when the market conditions are changing. So, probably, give that 6 out of 10 on the sticking the landing, but [Tobias laughs] I just couldn’t believe that there’s this Takahashi-Alexander, like they’re both things in both knot math, which is like, who the hell’s even never looked at knot math before? And then, also an obscure endowment model system. How’d that happen?

Tobias: I think the Takahashi manifold in my Toyota has broken down. I’m taking it in for-

Jake: Take it in for repairs?

Tobias: -check up tomorrow. The Takahashi manifold again. It’s always the Takahashi manifold.

Jake: [laughs]

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