Understanding Edge - Moontower

Summary: Even if an edge is small (1%) but its REAL you can build on it, as over time law of large numbers will work in your favoir.

But it's messy, and you never know what your real edge is, and if its small (or even not that small) it will take a lot of trials and a lot of time to VERIFY that its not just noise. Of course, the larger the SD the longer it will take.

Luck vs skill very difficult

When distribution is skewed, have to be careful using straddles. A skewed distribution will have a "cheap" straddle if you use it to imply overall standard deviation.

In my indoctrination into trading the term “edge” was equated to the bookie’s “vig” or a casino’s “house edge”. This makes a sense since I started in this business as a market maker. The interview questions I faced were focused on mathematical expectation or expected value. For example, if someone offered you a game that pays you the number that comes up on a single die, what would you pay to play? The weighted average payout of the game is $3.50. So if you can pay $3 to play, you’d make $.50 in theoretical profit. Of course, you could still lose if you roll a 1 or 2, but if you could do this every day, you’d earn 14% ($.50/$3.50) in the long run.

If you can price accurately and manage risk competently, you can crystallize the edge as surely as the Wynn will take your money.

In this post, I will share:

First let’s define fair value. I will decompose it into 2 concepts.

With the idea of fair value established, we can begin exploring the nature of edge with a familiar toy model — the coin flip.

Imagine a coin flip game. Call the toss correctly, make $1, otherwise lose $1. Let’s pretend you could predict the coin flip with 50.5% accuracy. Sweet.

The expected value of playing the game is 1% because your payoff is equal to .505 * $1 – .495 *$1

1% edge on this coin flip doesn’t seem like much. The .02 Sharpe ratio is a laughable signal to noise ratio. But as we increase N from 1 flip to many, the binomial distribution can be closely approximated by the familiar Gaussian curve [Taleb, spare my window, I’ll address reality later].

Look closely. The Sharpe ratio increases with N. Specifically, it increases at the rate of √N.

Why? Because the edge or numerator grows linearly with N while the denominator, or vol, only increases at √N. This property of edge is the foundation of trading and gambling. With enough trials, victory is nearly guaranteed. With a 1% edge on a coin flip, you are 90% certain you will be up money after 4,000 trades. So if you have 10 traders making 20 trades each business day, in one month you are more than 90% certain you are winning. In one year, you can’t lose.

Let’s look at the math in reverse. In Excel, we can use Norm.INV() to find what return corresponds to a desired probability for a given EV and vol. Let’s say we want to be 95% certain we make money. In math language we are interested in the point where the 5th percentile return of the CDF is equal to 0.

We want to ask Excel:

How many trials do I need to have so that my Sharpe ratio sets my 5th-percentile return to zero?

To do this let’s standardize the vol to 1. The equation we need to solve is:

NORM.INV(5%, EV, 1) = 0

To solve for EV we use Excel’s goalseek function. We find EV = 1.645

Since we standardized the vol to 1, then we have discovered that at a Sharpe ratio of 1.645 (again Sharpe is EV/vol), the 5th percentile return is 0. That is the Sharpe ratio we need to be 95% certain we make money.

Remember that having 1% edge on a single coin flip only has a Sharpe of .02

But as we increase N, the Sharpe increases by √N :

SR of 1 trial x N/√N = SRN .02 x N/√N = 1.645 N = 6,764

If we flip the coin 6,764 times, we are 95% sure we will make money even though we have a tiny edge on a volatile bet.

Let’s recap in English what we did here:

Let’s look at the relationship between a single bet Sharpe to how many trials we need to be 95% certain we win.

This table highlights another important point: by increasing the Sharpe per bet by an order of magnitude (ie from 1% to 10%) we cut the required number of trials by 2 orders of magnitude (27,055 to 271).

Think about that. The improvement in Sharpe leads to a quadratic reduction in trials needed to maintain the same risk/reward for the series of bets.

Inverting the logic:

If the risk/reward of your bet is halved, you need to bet 4x as many times for the strategy to maintain the same overall risk/reward.

The domain of many individual bets fits more under the umbrella of trading. For investing, we tend to think of the annual Sharpe ratios of investing styles or asset classes. Without looking this up, I’d guess that the SP500 has a long term Sharpe ratio of about .40. I’m estimating an 8% annual return divided by 20% vol.

We can use the same math we did above to see how many years we’d need to invest to be 95% certain we did not lose money in nominal terms. Turns out the answer is 17 years. The table below finds the number of years for other combinations of expected return and volatility.

Years Required to Be 95% Sure of Profit

Bell curves are great to build intuition but they are not reality. We can’t really be 95% sure we’ll make money by holding stocks for a generation because the historically sampled returns and volatilities are just that — sampled. We don’t know what the actual distributions are. Fat tails, skew, other moments I don’t even know about.

We can use a highly skewed bet to demonstrate how volatility can distort our impression of risk. This renders the Sharpe ratio useless in highly skewed scenarios.

Consider 2 stocks, both are fairly priced at $100. We’ll call them Balanced Corp and Skewed Corp.

Using the bimodal distribution we find that the stocks have the same volatility. However, they would have different straddle prices if there were options listed on them.

(It’s a good exercise for the reader to use what we know about expected value to manually compute the call and put prices).

So here we have 2 stocks with the same true volatility but different straddle prices if we compute them via expected value. Of course, we would not use B-S for a stock that was discontinuous and was going to magically open at one of 2 prices in a year. But this does show how the effect of a strong skew would suppress the value of a straddle for a given level of volatility.

This is actually more intuitive than it appears. FX carry is a highly skewed trade that might exhibit minimal vol on a daily basis. The volatility imputed by the straddle understates the risk because it derives most of its value from the behavior of daily moves, where the risk of a jump will be better reflected in the cost of OTM options. In the above case, the Balanced Corp 90 put is worthless while the 90 put on Skewed Corp is worth $2 (10% of the time it finishes $20 in-the-money).

So if you use straddle prices to impute volatilities which are then used to calibrate Sharpe ratios, you may be understating the risk of highly skewed assets. Your risk/reward ratio is actually overstated which means it will take far more trials to realize your edge, assuming you actually have any. And remember how diabolical the math is…if your Sharpe ratio is overstated by 2x (let’s say you think it’s .8 and it’s actually .4), then you need 4x the number of trades to maintain the same assumptions about making or losing money. How would you feel if you found at the long-run for your given strategy wasn’t 10 years, but 40?

Self-aware investors and traders are always questioning their edge. Evaluating a track record or doing post-mortems on your own strategies requires being able to handicap the true distribution of your trades. The more Gaussian they look (for example if you play limit poker instead of no-limit) the easier it is to ascertain the strength of your edge statistically. You can tell the difference between bad run vs a change in the quality of your edge. Some runs would be almost impossible if your edge was real.

Edge is scarce. When we prospect for it, we should expect to mostly find fool’s gold. There are many reasons for this.

On skew

While both high volatility or high skew make it harder to determine if you have an edge statistically, skew is especially tricky. It is hard to see without liquid option surfaces. Here’s an intuitive way to see how skew distorts reality. Imagine finding a video poker machine that didn’t show its payoff table. Under the hood, it gives slightly worse payoffs on a pair of Jacks or better, but offered a billion to one on the Royal Flush. You could play that machine for days or even weeks and never realize you had massively positive EV.

On sample size

On luck vs skill

The importance of edge

Edge in the real world is nebulous

Firms with provable edges are not trying to raise money. If it’s provable it does not need more eyeballs on it. The epistemological status of edges that are trying to raise money is unknown. Many will never get the sample size to prove it. Asset management is the vitamin industry. It sells noise as signal. It sells placebos. There will always be one edge that never goes out of style — marketing.

True mathematical edge is hard to find.

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