Thread by @choffstein

Over time, I've developed a few foundational mental frameworks with respect to investing.

These aren't pithy philosophies nor are they about trade ideas.

Rather, they are concepts that have helped me re-frame my thinking, develop better intuition, and make better decisions.

I want to share two that I wish I internalized earlier in my career.

Over time, I'd like to add to this list, but I'd also welcome others to share their own.

1️⃣ "IT'S LONG/SHORT PORTFOLIOS ALL THE WAY DOWN."

Every portfolio, and every portfolio decision, can be decomposed into being long something and being short something else.

Sounds trivial. But it's a powerful framework.

👆 Let's start with a simple example: an active, long-only stock portfolio.

We can decompose this portfolio into two pieces.

1. The market-cap weighted benchmark.
2. A long/short portfolio where longs capture our relative over-weight positions and shorts our underweights.

👆 Let’s normalize the long/short such that each leg equals 100% notional and call it our “active bets.”

We'll call the normalization scaling factor our “active share.”

We now have:

Active Portfolio = Market Portfolio + Active Share x Active Bets

👆 Let's say we're charging 100bp to manage this portfolio. Since β is basically free today, that means we need:

Return of (Active Share x Active Bets) > 100bp

or

Return of Active Bets > 100bp / Active Share

to actually create value for a client.

👆 Which means that the smaller our Active Share, the higher our hurdle rate!

But it's clear that high Active Share doesn't mean the manager is going to add value: the Active Bets are where they create value!

👆 But this only begins to scratch the surface of this mental model.

e.g. With rebalancing, we're selling one thing to buy another!

(Rebalance timing luck (jii.pm-research.com/content/10/1/2…) exist because you're suddenly overlaying your old portfolio with a long/short portfolio!)

👆 Want to better understand how different steps of a portfolio process contribute to its return?

Think of each step of the construction as layering on a new long/short portfolio to the prior layer.

Isolate those long/short portfolios and track their performance.

👆"It's long/short portfolios all the way down" has been one of the most useful mental models I've developed in my career.

It completely changed my frame of reference when analyzing portfolio decisions and helped me better isolate true drivers of returns.

2️⃣ JENSEN'S INEQUALITY

f(E[X]) ≠ E[f(X)]

Or, in English, "a function applied to the expectation does not necessarily equal the expectation of the function."

Come again?

👆 Let's start with a simple example.

f(X) is our function. It's the blue line.

For simplicity, let's assume X ∈ {a, b}.

Note how (f(a) + f(b)) / 2 > f((a + b) / 2).

👆 "Corey, what in the world does this have to do with investing?"

Consider this example: What's the difference between a portfolio built from stocks ranked on a blend of factor scores and a blend of portfolios each each built from stocks ranked upon a single factor?

👆This is a classic f(E[X]) versus E[f(X)] question.

"f" is our portfolio construction function. X is our factor signals.

The first method blends the signals (E[X]) and then builds the portfolio. The second method builds individual portfolios (f(X)) and then blends them.

👆 Jensen's Inequality won't tell you which is better. Just that *they're different.*

Here's another simple example (I come across this with tactical signals all the time):

- You have 10 buy (+1) / sell (-1) signals.
- F(x) says to be long if X > 0 or short otherwise.

👆 So how does f(E[X]) differ from E[f(X)]?

Let's say 6 of the signals are on and 4 are off.

Case 1: our average score is 0.2, so we go 100% long.

Case 2: 6 sub-portfolios are long and 4 are short, leading us to net 20% long exposure.

👆 Why is this so important?

Because non-linearities show up everywhere in portfolio construction.

Portfolio optimization? Non-linear.
Maximum or minimum position sizes? Non-linear.
Rank-based cut-offs? Non-linear.

👆 The more non-linear the function, the greater the wedge.

This also helps us understand why certain constraints can reduce this wedge!

These two ideas – "It's long/short portfolios all the way down" and "Jensen's Inequality" – have had a bigger impact on my understanding and intuition around portfolio construction and design than just about anything else.

So I hope you find use in them too.

I’m gonna go ahead and tag @KrisAbdelmessih here. I bet he can add a useful mental model or two. And then he can tag someone else.