Liquidity Preference as Behavior Towards Risk

The question Roy left open

Roy left the disaster threshold as something the investor works out alone. Where d comes from had no answer; it was a subjective number.

Tobin does not answer that question. He removes it.

If the market holds an asset whose return is certain — a government bond, a deposit — the zero point of the comparison is not invented. The market supplies it. The threshold stops being a preference and becomes a price.

But that is not the real break.

What the paper is actually about

This has to be said up front: Tobin's paper is not about portfolio theory, it is about the demand for money. The title says so.

The question it asks is this: why do people hold cash that pays no interest? Keynes's theory of liquidity preference explained it but had no micro foundation. Tobin builds that foundation — holding cash is not a preference, it is what avoiding risk looks like.

What follows below is a by-product of that setup. Presenting the paper as though it were only the separation theorem would misrepresent it.

But this is not a weakness; it is the story itself. One of the cornerstones of portfolio theory came out of a paper on monetary theory.

Adding a risk-free asset

You hold two things: an asset with a certain return (rf) and a risky portfolio P. You put a share w of your wealth on the risky side and leave the rest in cash.

The expected return:

The risk:

The symbols keep the same meaning throughout the page:

Look at both lines: each is linear in w. Eliminate w and write one in terms of the other, and μ becomes an affine function of σ.

So in the (σ, μ) plane these portfolios lie on a straight line. Markowitz's curve has not gone anywhere, but the best attainable set is no longer that curve — it is the line drawn from rf to it.

Separation

Which line? The steepest one, because for a given risk it returns the most. The steepest line is the one tangent to the frontier.

And the point of tangency is unique. It is the same place regardless of where you want to stand.

The figure below has two bars. Pull the dial to all cash, then take it out to leverage. The lower bar moves the whole way. The upper one does not budge.

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Figure parameters

Share in the risky portfolio0.60Risk-free rate%2.0Number of assets6Average correlation0.30

That upper bar is the composition of the risky portfolio. Its stillness is not a drawing choice; it is the theorem.

Two questions, two separate answers

Portfolio selection splits in two, and the halves do not look at each other:

Which risky portfolio? One answer, the same for everyone. The point of tangency does not depend on where you choose to stand; what fixes it is the market — μ, Σ and rf. And it can be computed:

One note: that closed form solves the problem where short selling is allowed. The frontier on this page carries no shorts — in Tobin's setup the investor divides wealth between cash and risky assets and the shares do not go negative. Where the two coincide, that is, where every weight the formula returns is positive, they give the same portfolio; where they do not, what is drawn is always the constrained one.

How much risk? That answer is entirely yours. But it is a single number: w.

A cautious investor and a bold one hold the same risky portfolio. What differs is how much money they put into it. Change the share and watch the composition stay put.

Leverage

The line does not end at the point of tangency; it carries on past it. That stretch is leverage: borrowing at rf and putting the proceeds into the risky portfolio.

When all your wealth sits on the risky side, w = 1 and you are at the tangency. Wanting more means borrowing; w goes above one and your cash position turns negative.

One note: Tobin's own setup turns on the question of how much wealth stays in cash, and there cash does not go negative. Extending the line to the right is an extension that has since become standard; it is in the figure because the answer to "how much risk" is only fully visible that way.

Where the assumption breaks

So far there has been a single rf. In reality there are two: the interest you receive on deposits is below the interest you pay on credit.

Once they part, the line stops being straight. One tangent from the lending rate, another from the borrowing rate, and the frontier itself between them — a stretch where nothing is lent and nothing is borrowed.

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Figure parameters

Gap between the two rates%4.0Lending rate%1.0Number of assets6Average correlation0.30

In that middle region the separation theorem weakens: investors with different appetites no longer hold the same risky portfolio. Open the gap and the region grows; close it and the single line returns.

That closing behaviour matters: Tobin's assumption is not wrong, it is a special case.

What changed

In Markowitz everyone stood at a different place on the curve. The answer to "which portfolio is good" was a curve, and where you stood on it was left to you.

In Tobin everyone holds the same portfolio. The only thing that differs is the amount.

This is the largest simplification in the story since Markowitz. The proportions among risky assets become independent of anyone's appetite for risk.

What comes next

Now take one more step.

If everyone in the market holds the same risky portfolio, what can that portfolio be? The sum of everyone's risky holdings is every risky asset in existence. So that common portfolio has to be the market itself.

Once that step is taken the question changes: no longer "which portfolio should I hold" but "what do prices look like if everyone behaves this way". William Sharpe asks exactly that in 1964.

That is the next item in the series.

The closed-form tangency portfolio was compared against the constrained numerical solution on 1,200 randomly generated valid inputs — those where the closed form carries no short position, so that both sides are solving the same problem. The largest relative deviation was 5.77e-15, against a tolerance of 1e-9.