Safety First and the Holding of Assets

What it asks differently

There is an unjustified assumption in Markowitz's E-V rule. It never says why the investor should care about precisely the mean and the variance; the rule simply takes this as given.

Roy never asks that question. He starts somewhere else entirely: what the investor really fears is not fluctuation but disaster. Let my return not fall below this level.

Same year, same journal, neither aware of the other. And the place he arrives at is a point on Markowitz's curve.

Avoiding disaster

Roy's setup is not a preference parameter but a different definition of the problem. The investor fixes a threshold — d — and wants to minimise the probability that the return falls below it.

The threshold comes from outside. The minimum savings needed for retirement, an instalment on a debt, the level at which capital must be preserved. Roy does not discuss where it comes from; he leaves it as something the investor already knows.

The appeal of the setup is that it asks for no abstract quantity such as risk tolerance. "How risk averse are you" has no answer; "what level can you not fall below" does.

Without knowing the distribution

But an obstacle appears immediately. The probability distribution of the return is unknown. Normal, skewed, fat-tailed — unknown.

At this point Roy proceeds assuming nothing about the distribution. Chebyshev's inequality gives him an upper bound:

The symbols keep the same meaning throughout the page:

That the bound is loose is clear from the start: since no distribution is assumed, it speaks for the worst distribution one could imagine. But it costs nothing either — it beats assuming a wrong distribution and trusting it.

The reduction

Look at the right-hand side. σ² over (μ − d)². Making that expression small means making its reciprocal large; and since taking a square root does not change the ordering, it reduces to maximising:

The whole page rests on this ratio. And from here on it is geometry.

In the (σ, μ) plane a portfolio is a point. This ratio is the slope of the line drawn from (0, d) to that portfolio:

So Roy's problem becomes this: among the lines that start at the threshold and reach the attainable portfolios, which is the steepest? The steepest line is the one tangent to the frontier, and the point of tangency is the portfolio we are after.

Pull the threshold down and the line steepens, the point of tangency sliding left — less risk. Pull it up and the line flattens, the point moving right. If the threshold climbs above the highest attainable return, no line is left to draw: no portfolio clears it, and Chebyshev has nothing to say.

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Figure parameters

Disaster threshold d%-5.0Number of assets8Average correlation0.30Volatility spread0.8

How loose the bound is

Here is where Chebyshev is not free. The bound sits far above the actual probability; in most settings it even climbs above one, and "the probability is less than one" is no guarantee at all.

For comparison, suppose the return is normally distributed — an assumption Roy does not make. Then the probability of disaster can be computed directly. The distance between the two curves is the price of giving up knowledge of the distribution.

Yet both curves bottom out at the same portfolio. This is no coincidence: both are decreasing functions of the ratio (μ − d)/σ, so they order the portfolios in the same way. The only thing that changes is the probability assigned to the chosen portfolio.

What Roy does is not to estimate the probability correctly but to choose the right portfolio. Even when the bound empties out — that is, says no number at all — the ordering survives.

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Figure parameters

Disaster threshold d%-5.0Number of assets8Average correlation0.30

Meeting Markowitz

Now lay the two pages on top of each other.

Markowitz says "risk is variance" and sets up a trade-off between expected return and variance. Roy says "risk is disaster" and minimises the probability of falling below a threshold. They do not start from a shared assumption; their definitions of the problem differ too.

Even so, Roy's solution is a point on Markowitz's efficient frontier. The reason is simple and in fact unavoidable: for a given σ, the ratio (μ − d)/σ grows with μ. So no portfolio below the frontier can ever be chosen — above it there is always one that returns more at the same risk.

Different motivation, same geometry. What Roy adds is a rule that says where on the curve to stand: in Markowitz that was left to the reader.

A familiar ratio

Look once more at the expression being maximised. Subtract a threshold from the expected return, divide by the standard deviation. Put the risk-free rate in place of the threshold and this expression is the Sharpe ratio.

In 1952. Twelve years before Sharpe's 1964 work, derived on a completely different rationale — avoiding disaster.

Years later Markowitz wrote that Roy too had a claim to the founding of portfolio theory. Of the two independent answers given to the same question in the same year, one gave the field its name; the other stayed in a footnote for a long time.

What comes next

An open question sits where Roy leaves off: where does d come from?

The paper leaves it as the investor's subjective threshold. But if the market holds an asset whose return is certain — a government bond, a deposit — that asset removes the need to invent a threshold. If you can put part of your money there, that is the natural zero point of the comparison.

This idea rebuilds portfolio selection: the choice among risky assets and the decision of how much risk to take come apart. James Tobin will do exactly this in 1958.

That is the next item in the series.

The closed-form solution for the point of tangency was compared against an independent derivative-free search on 1,500 randomly generated valid inputs. The largest relative deviation was 2.58e-15, against a tolerance of 1e-9.