What Happens to the CAPM When You Cannot Borrow Freely: Black's Zero-Beta Model

The original CAPM contains an assumption nobody really believes: that any investor can borrow unlimited money at the same risk-free rate the US Treasury pays. In reality, your broker charges you more than the Treasury rate, many funds are forbidden from using leverage at all, and pension mandates routinely ban borrowing outright.

Fischer Black asked the obvious question: if you delete that assumption, what survives? His answer, published in 1972, is elegant. The CAPM's structure survives almost intact. What changes is one number, and that single change explains one of the model's most persistent empirical embarrassments.

The problem: the beta-return line is too flat

When people tested the CAPM in the 1960s and 70s, they kept finding the same distortion. Yes, riskier stocks earned more on average. But the relationship was flatter than theory demanded. Low-beta stocks, the sleepy, defensive ones, earned more than the CAPM predicted. High-beta stocks, the exciting ones, earned less. The line had the right slope sign but the wrong tilt, and it kept showing up in study after study, including Black's own work with Jensen and Scholes.

This is not a rounding error. It is a systematic bias in exactly one direction, which usually means the model is missing something structural.

The key idea, via analogy

Imagine a restaurant where the only way to get more calories is to order richer dishes, because the kitchen refuses to serve you double portions. If you are very hungry, you cannot just ask for two servings of salad. You are forced to order the fatty dish.

Now think about what that does to prices. Because hungry customers cannot scale up the sensible option, they bid aggressively for the rich dishes. The rich dishes become overpriced and the salads become cheap.

That is exactly Black's mechanism. An investor who wants aggressive returns would ideally borrow money and buy a levered position in a sensible, diversified portfolio. If borrowing is banned or expensive, that route is closed. So the aggressive investor does the next best thing: buys naturally high-beta stocks instead, which are the built-in levered exposure. Demand for high-beta stocks is therefore inflated by everyone who wanted leverage and could not get it. Inflated demand means high prices, and high prices mean low future returns.

The mirror image holds for low-beta stocks: nobody wants them, because you cannot lever them up into something exciting, so they sit unloved and cheap, and cheap means high future returns.

Black then works out the equilibrium properly. Without a risk-free asset to anchor the line, the role of "the safe rate" is taken over by a portfolio he calls the zero-beta portfolio: a combination of risky assets constructed to have no co-movement with the market at all. Expected returns still line up along a straight line against beta, but that line now starts at the zero-beta portfolio's return rather than at the Treasury rate. And crucially, the zero-beta rate comes out higher than the risk-free rate. A line that starts higher and passes through the market portfolio is, necessarily, a flatter line.

That is the whole trick. The empirical flatness that plagued the CAPM is not an anomaly at all under Black's version. It is the prediction.

Why it mattered

• It rescued the CAPM's logic while explaining its failure. Rather than throwing out the model, Black showed that its most notorious empirical defect follows from relaxing one unrealistic assumption. That is about as good as theory repair gets.
• It is the intellectual foundation of low-volatility investing. If boring stocks are systematically underpriced because leverage is constrained, then buying boring stocks and levering them modestly should beat the market on a risk-adjusted basis. The entire betting-against-beta literature, and a great many real low-vol funds, descend from this insight. Black himself later co-authored work on precisely this strategy.
• It gave testers a more honest null hypothesis. Black, Jensen and Scholes used the zero-beta version as the benchmark in their empirical work, and the two-factor structure it implies became the standard way to test the model against real data.
• It reframes leverage as a priced constraint. The deep lesson is that restrictions on how investors can act leave fingerprints in prices. Any binding constraint on a large group of investors becomes an opportunity for whoever is not constrained.

The honest limitations

• The zero-beta portfolio is not observable either. You cannot buy it off the shelf; you have to estimate it, which means the model swaps one unmeasurable anchor for another. Roll's critique applies with full force here too.
• Constrained leverage is one explanation among several. The flat beta line could also come from behavioral demand for lottery-like stocks, from benchmark-relative fund managers chasing beta, or from plain mismeasurement. Black's story is clean but not proven to be the only one operating.
• The low-vol edge is crowded now. Once "buy boring stocks, add leverage" became a famous, packaged product, the discount on boring stocks compressed. Low-volatility strategies have had long stretches of disappointment since.
• It still assumes shared beliefs and one period. The paper fixes the borrowing assumption and leaves the CAPM's other heroic assumptions untouched.

The one-line takeaway

Black showed that if investors cannot freely borrow, they buy high-beta stocks as a substitute for leverage, which bids up risky stocks and leaves safe ones cheap, flattening the risk-return line exactly as the data has always shown and turning the CAPM's biggest embarrassment into its own prediction.