Geometric Shorts

Over the last month, very few assets have provided any return. Many strategies that weathered the downturn well included some type of short selling program. I don’t sell assets short, so I’ve leaned on other methods for survival.

But if short selling helps a portfolio during bear markets, why don’t I short? Well simply put, it’s not compatible with my investment strategy. Lets see why and in the process examine how a falling asset should move through time.

What is Short Selling?

Short selling (or shorting) a stock is simply selling something you don’t own. You borrow the security (i.e., a stock) from someone else, and then sell that security on the market to someone else getting cash back in return, say $100 per share. Your broker usually takes care of the borrowing part for you.

Now you still owe your broker the original security you borrowed, so at some time in the future, you have to buy the stock back from someone in the market. The hope is that the price has fallen since you sold the stock and there will be cash left over as a profit after paying back your “stock loan”. So if you could buy the stock back at $90, you would make $10 per share.

Instead of buying low and then selling high, you aim to flip the order, selling high and then buying low.

You make money when when the investment falls in price.

Why Would an Investment Fall in Price?

Two reasons:

1. Some assets will fall randomly, even if they should provide a positive return
2. The asset should fall in price, i.e. it’s expected to fall.

Betting on something to fall on pure chance doesn’t make much sense. It’s a bit like a casino game. You could win sometimes, but the odds are against you.

Therefore, I suspect many people short sell in isolation (not long-short) because they expect the asset to fall in price. However this is a problem if you view markets as efficient.

If the market allocates prices for assets properly, why would it ever price an asset to have a negative return?

Efficient Market Pricing

I believe that the market is mostly efficient. Therefore, you should expect every security to provide a positive return (on average) over some time frame or else no one would own it.¹ By this I mean:

1. The security is expected to gain in price over the short term or,
2. The security is expected to gain in price over the long term or,
3. Both

I think when most people think of security prices being priced efficiently and accurately, they automatically assume #3; it’s expected that the security will go up in the long term and the short term. I’m sure this is true most of the time. If you focus on the arithmetic return only, one would assume it’s true all the time.

Some people clearly have a long horizon for their investments. They don’t really care about the short term, and so they only focus on #2, an expected gain in the long term. This makes sense from an investing philosophy, but it doesn’t make sense to me from an efficient market.

If the asset is expected to fall in price over the short term, wouldn’t the market quickly drive the price down until it is expected to gain value again? I believe it would.

What I want to focus on is option #1, expected short term gains.

Short Term Gains, Potential Long Term Losses

I believe all assets have a positive arithmetic return.¹ This means over the short term, they are likely to provide a positive return. If the asset is likely to rise in the short term, it’s perfectly rational to include it in your portfolio even if you think it will fall later, the idea being you will sell it after the expected rise. The momentum investment philosophy is pretty much built around this idea.

The efficient market theory is therefore compatible with only expecting assets to rise in the short term. There is a strong reason to include the asset in the portfolio because the short term return is positive.²

However, the efficient market theory doesn’t decide the long term return.

A positive arithmetic return alone doesn’t tell you the expected long term return. You need to know the standard deviation (as well as the other moments for the distribution of returns) to understand the long term return.

At the shortest possible timeframe, the expected return is the arithmetic return. At infinite time, the long term return is the geometric return. In between that time, the “expected return” is somewhere between the geometric return and the arithmetic return. In “What is the Expected Return” I stated:

Between the very short term, and the extreme long term, the expected return must slowly move between the arithmetic and geometric return. I’ve shown the math to calculate this value with coin flips. In investment markets, the math is more complicated, but the overriding concept is the same. The longer the investment goes through time, the closer the expected returns will trend toward the geometric return.

Why Investments Fall In Price

Investments don’t fall in price because they have a negative arithmetic return. Investments fall in price because their standard deviation, their volatility, creates a negative geometric return through this equation:

Arithmetic Return – Volatility ² / 2 = Geometric Return

Over the very short term, everything should rise. Over the long term, rise or fall depends on the volatility.

An Example

Let’s pretend one month is the shortest possible trading time. This is obviously silly, but doing so keeps the return and standard deviation values relatable. The same logic works when the minimum “time” is a day or an hour or a minute or a second.

The arithmetic return of the asset for our example will be 1% per month. The standard deviation will be 15% per month. Therefore the geometric return is:

1% – 1/2 x 15%² = -0.125%

A slight negative geometric return.

Since the arithmetic return is 1%, what’s the expected return after one month?

It is 1% (the arithmetic return) because there is only one period.

That was easy. But what about 2 months?

Compound Growth Rate Return

Here it gets trickier and we have a choice to make. If we just use the average expected price in the future (arithmetic return), we would predict this asset to continue upward forever. But as we know, at infinite time, the value of the asset will fall zero due to the negative geometric return. So this is clearly the wrong tactic.

Instead, let’s base our “expectation” on the average compound growth rate (geometric return), and expand this compound growth rate over the number of periods in question. After two periods this leads to the expectation of:

“Expected” Return After 2 months = Arithmetic Return x Geometric Return

(1+.01) x (1-.00125) = 1.0087 = 0.87%

The math here is very similar to, and slightly simpler than the effect I discussed in the “What is the Expected Return.” Here I showed how a portfolio of assets with individual negative geometric returns will grow to start, and then bend over and fall back towards their geometric returns.

The equation for “expected” return based on the average compound growth rate of a single asset through multiple periods is conceptually easier to grasp than it is for a portfolio. It is³:

“Expected” Return = Arithmetic Return x Geometric Return^(t-1)

where t is the number periods. The average compound growth rate at time t is the “expected” return^(1/t). For our example the trajectory of the “expected” return looks as follows:

The returns start at the arithmetic return. Unlike the portfolio however, it rolls over after only one period and and then falls due to the negative geometric return. The line looks straight over a few periods, but it actually curves flatter over time.

For our example, the returns are positive up until the 10th month. So even though the asset will ultimately fall due to the negative geometric return, you should still expect it to rise at the start.

Simply put, if you knew the arithmetic return and you knew the geometric return, you could predict how long it should take for a “short” to start making money.⁴

Similarly, the compound growth rate, which is what this analysis is based around, starts at the arithmetic return. But then falls toward the geometric return as time increases.

When the geometric return is positive, the same math applies. That trajectory just goes upward, getting closer and closer to the geometric return as time goes on. This is the same idea I’ve outlined in “The Most Misunderstood Force in the Universe“. The compound growth rate starts at the arithmetic return and then falls towards the geometric return through time.

Seems Like You’re Supporting Short Selling…

Yes it does. And I could see investors trying to use this philosophy to potentially find shorting candidates. But it doesn’t fit with my philosophy because I rebalance regularly.

As discussed here, rebalancing moves the returns from the geometric return toward the arithmetic return. Because of this, I rebalance frequently to drive the returns upward. This is great for a portfolio, but suicide for a short position.

Under the framework above, the short position needs time for the asset to fall in price. By rebalancing I’m not giving the portfolio “time”. I’m cutting it off and trying to drive the return of the assets toward the arithmetic returns. It’s driving asset returns up, and away from potential losses and toward gains.

When you sell an asset short, you want to realize the geometric return.

Shorting and rebalancing therefore are counterproductive.

A Rise Before the Fall.

I’ve discussed three key points in the post.

• First, you can believe an asset will fall in price over time within the confines of efficient market security pricing.
• Secondly, I’ve shown the math to estimate the time it should take for the short to play out within that view.
• Third, I’ve explained how frequent rebalancing doesn’t mesh well with short selling in this framework.

In following posts, I will expand this thought process to other areas such as long-short strategies.

But for now, just remember even if you expect an asset to fall over time, the first move of every investment should be up before volatility takes over.

1-You can make a strong case that an asset with a negative correlation could be expected to fall in price. This is more true now that the risk free rate is back at zero. But for now, let’s leave this issue out of the discussion, as it will only take from the key points.

2-As an example for most of the month of March, the S&P 500 projected as a negative geometric return in my system. But the portfolio still called for it’s inclusion because the arithmetic return was positive.

3-This equation is a slight simplification that is much easier to conceptually understand. I’ve tried to derive the real equation for the average compound growth rate as:

To find the “expected” return over the period, raise this value to the number of periods. I will explain this equation in a future post.

*That equation may only be approximately correct. Here is another equation from this paper which is also an approximation. They use in where I used t.

4-It’s obviously very difficult, maybe impossible, to “know” these values.