Market valuation and the 'two egg' problem
By Chan Akya

It is by now clear to all but the most blinkered of investors that market valuations are being driven by whole range of factors of which fundamentals are but one element. "Technical" factors or the impact of liquidity combined with behavioral finance models form a growing, if generally inexplicable, body of explanations for market swings.

To be sure, this isn't particularly different from five years ago, when a number of commentators kept talking about "technical" factors, leading to the general joke that anyone talking about "technical" explanations didn't know what was actually driving the market. What's different this time around is the sheer number of

people talking up the technical, even as market levels have swung ever higher (and the pendulum, ie volatility, swings even more).

All that said, it is also generally undeniable in looking at asset markets - be they in US stocks or London property - that a step function of sorts has been evident. In other words, after pretty much every stable move in markets, these asset markets have moved up a step, rather than gradually drift upwards.

It is these step-like movements in asset prices that are considered in this article. Take a look at any graph - from say Yahoo! Finance or Bloomberg for the S&P500 or London property prices; but be sure to use a five-year or longer window. Within the secular context of rising asset prices, you will see a series of movements ("steps") each of which will show a sharp rise, followed by a small plateau. A second feature that's visible is that each step is less steep than its preceding step. In other words, if the market jumps up 100 points in the first step, it then goes up 90 points in the next, 80 in the following and so on. [4]

Source: Yahoo! Finance section - SPX index. Used for illustration purposes only

It is the combination of these two observations that led me to consider a mathematical problem from a completely different area.

The 'two egg' construct
Away from the markets, let us consider some mathematical constructs that could help in the understanding of this problem. There are many formulations of the two egg problem, but the one we use is from the website datagenetics below (1,2,3).

"You are given two eggs, and access to a 100-story building. Both eggs are identical. The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that's it for that egg. If an egg breaks when dropped from floor n, then it would also have broken from any floor above that. If an egg survives a fall, then it will survive any fall shorter than that.

"The question is: What strategy should you adopt to minimize the number egg drops it takes to find the solution?. (And what is the worst case for the number of drops it will take?)"

I invite you to read the solution in detail, but in brief, what it deals with is a strategy of maximizing the number of floors whilst minimizing the number of drops; using the fact that with two eggs, one can freely experiment on the first egg ("destructive testing") whilst leaving the other egg for optimization. The solution is expressed as: n (n+1) / 2 >= 100

In effect, what this implies is that for a "two egg" construct involving a building of 100 floors, you're supposed to drop the first egg from the 14th floor (n) and if it doesn't break, then go to up 13 floors to number 27, and then 12 floors to 39 and so on.

When one thinks about this progressive rise - 14, 13, 12 - it becomes fairly apparent that factors similar to the ones in markets are at play.

The question then becomes, why?

What are the two eggs in the markets?
Consciously or more likely without realization, it appears that investors are pushing up the markets in ever smaller increments to figure out where the breaking point lies. Think about the following:
a. A focus on liquidity rather than fundamentals;
b. Nominal earnings and cash flows, rather than growth;
c. Overplaying of positive news and underplaying negative events;
d. A clear gap between observed inflation (e.g. commodities in 2010-11) and discounting rates;
e. Liberal mixing up of investment-led growth and consumption-led growth.

If one considers the market's two eggs to be "fundamentals" and "technicals", then the use of the "technical" egg to ride the market ever higher till its breaking point would be the optimal strategy from a pure mathematical perspective.

That is however, an oversimplification. One also needs to consider that in the markets, the two eggs are not identical, nor indeed are their strengths constant. To put that in context, let us think about how the strength of technical factors actually feed into fundamental valuation factors today, and also how quickly the change in technical factors can affect the market's fragility:
a. Liquidity is extended to assuming a free flow of credit as long as credit ratings of borrowers stack up;
b. Banks are assumed to have unlimited credit appetite because they will always be bailed out (that's what "whatever it takes" means);
c. Discounting rates for asset valuations are kept at minimal levels (that's what "QE", or quantitative easing, means);
d. There is no such thing as inflation - see "discounting rates" above;
e. Investor expectations remain anchored - ie bullish outlook is maintained.

As the market climbs ever higher, the probability of breakage rises; but equally, the level at which the next reset happens - ie the point below from which investors will restart the climb - also changes. Instead of a strict (n+1) construct, you're more likely - thanks to the intrinsic volatility of markets but also the interplay between fundamentals and technical factors - to see multiple steps downward rather than single steps.

Thus, a real-world application of the "two eggs" construct suggests far sharper downward moves than is being assumed by participants if and when there is a correction.

Notes:
1. A reference to one possible solution.
2. Another approach.
3. A variation that's applicable for single constraint problems in general.
4. Yahoo! Finance, "basic technical analysis"

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