Pyramid Investing

Cost of a Pyramid

Elementary measures

So far, we have defined number of different terms used in Pyramid Investing: some price-related terms such as Top, Bottom and Step or amount-related terms such as Base and Increment. Values of these terms represent elementary measures of a pyramid. These terms as a group, that is combination of their values, uniquely define any arbitrary pyramid. In other words, we can say that any pyramid is fully determined by values of the following terms:

Top
Bottom
Step
Base and
Increment

If the value of any one of these terms is missing, the pyramid is undetermined. The only “redundant” term in this group may be the Bottom and only because we assume bottom is zero unless specified differently.

Many times, we will refer to a specific pyramid and list the values of its Top, Step, Base and Increment. No matter how big the pyramid is or how many discrete levels it has, these four elementary measures are all we need to uniquely describe the pyramid in question. Elementary measures determine position and shape of a pyramid.

Derived measures

Beside elementary measures, we make use of other measures that are derived from elementary ones. The process of deriving actually involves some simple calculations. For example, the price at each Depth is derived from Top and Step. The formula to calculate Price (n) that corresponds to Depth n is:
Price (n) = Top − n × Step

Similarly, the amount of shares bought at certain Depth n is derived from Base and Increment:
Size (n) = Base + (n−1) × Increment

I would like to note that the last formula does not hold in general case since Size (n) is not uniquely defined for arbitrary Depth n due to dynamics of trading. However, for the sake of this article, we will assume that price descends in a straight line from top to the bottom. In such a case, the formula above holds true and we can freely make use of it.

We can further derive dollar value of Trade that occurs at Depth n, which is calculated from corresponding price and the amount of shares traded at that level:
Trade (n) = Price (n) × Size (n)
Trade (n) = [Top − n × Step] × [Base + (n−1) × Increment]

Even though these formulas involve some basic mathematics, their calculation is rather tedious if we attempt to do it on a piece of paper. However, with a help of any spreadsheet software, results for multiple levels can be obtained instantaneously.

Cost of a pyramid

Most frequently used derived term or measure is cost of a pyramid. Cost of a pyramid is sum of all trades across all pyramid levels. In simple terms, cost of a pyramid represents the amount of money we would need to buy the whole pyramid or the amount of dollars we would employ if the price of underlying stock went to zero. We sincerely hope that our underlying stock will never go to zero, but nevertheless the cost of a pyramid represents ultimate amount of money to be used for a given pyramid in the most extreme scenario.

In terms of risk, cost of a pyramid is the maximum amount of money we can possibly lose applying a given pyramid. It is always good that the risk is limited and easy to calculate. It may not be so easy to accept this risk but we certainly have a way to deal with it. Besides, who said that investing is risk-free endeavor anyway?

Cost of a pyramid is very important measure and we heavily rely on it in the process of Pyramid Investing planning. As I mentioned elsewhere, Pyramid Investing consists of meticulous planning and robotical (virtually emotion-free) execution of the plan. Cost of a pyramid is also unavoidably exploited in calculation of Pyramid Investing returns. We will address each of these issues separately. For now, it is important to understand how to calculate the cost for a given pyramid.

Example of pyramid cost calculation

So let’s assume that the pyramid we want to calculate the cost of is given by the following elementary measures:

Top = $11
Step = $2
Base = 10sh
Increment = 5sh

First, we need to calculate price for each pyramid Depth:
Price (n) = Top − n × Step

Price (1) = $11 − 1 × $2 = $9
Price (2) = $11 − 2 × $2 = $7
Price (3) = $11 − 3 × $2 = $5
Price (4) = $11 − 4 × $2 = $3
Price (5) = $11 − 5 × $2 = $1
Price (6) = $11 − 6 × $2 = −$1

In this process, we increase the Depth and calculate corresponding price. Although we say Bottom is zero, practically only non-zero and positive prices make sense. Therefore we repeat the process until we hit either zero or some negative price. We stop there and we do not include that last calculation. In our example, pyramid has five depths and actual Bottom is $1.

For the sake of accuracy, I would like to point out that, strictly looking, the unit for Top, Step and Price is $/sh (dollar per share). We use $ sign only for the reasons of simplicity.

Now, we calculate the amount of shares bought at each Depth n:
Size (n) = Base + (n−1) × Increment

Size (1) = 10 + (1−1) × 5 = 10sh
Size (2) = 10 + (2−1) × 5 = 15sh
Size (3) = 10 + (3−1) × 5 = 20sh
Size (4) = 10 + (4−1) × 5 = 25sh
Size (5) = 10 + (5−1) × 5 = 30sh

Then, we multiply price and amount of shares for each Depth n to obtain corresponding Trade dollar amount:
Trade (n) = Price (n) × Size (n)

Trade (1) = $9 × 10sh = $90
Trade (2) = $7 × 15sh = $105
Trade (3) = $5 × 20sh = $100
Trade (4) = $3 × 25sh = $75
Trade (5) = $1 × 30sh = $30

Finally, we sum up all trades to obtain the Cost of our pyramid:
Cost = ∑ Trade (n), where n = 1,…,N
Cost = Trade (1) + Trade (2) + Trade (3) + Trade (4) + Trade (5)

Cost = $90 + $105 + $100 + $75 + $30
Cost = $400

Therefore, the Cost of our pyramid is $400. If we change any of initial elementary measures, subsequent pyramid gets to be different and that may result in corresponding Cost to change. This is not to say that two different pyramids cannot have the same cost.

Cost is usually given

In practice, instead of calculating cost for a given pyramid, we usually have a cost that is given. Therefore, our task is to design a pyramid that fits the given cost. This is normally an iterative process. Besides trying to satisfy the given cost, we also strive to take care of other aspects of a pyramid at the same time. This process of designing a pyramid is also called shaping the pyramid. In future articles, we will dedicate considerable attention to the process of pyramid shaping. We will also address various criteria we want to satisfy (or at least make the trade off) during this process.

Talking about the cost of a pyramid, we can also say that some pyramids are “costlier” than others. Of course, pyramids that have larger cost are “more expensive”. This means that we need more investing capital to properly cover more expensive pyramids.

Pyramid in a spreadsheet form

Graphical representation of pyramids is good for initial understanding of terms and their relationships. However, practical use of pyramids involves more abstract representation that is also more suitable for our investing purposes. We normally turn to spreadsheets that also help us in number crunching activities. I extensively use Excel spreadsheets to help me in daily Pyramid Investing efforts.

The following figure shows the above-mentioned pyramid in a spreadsheet form:

Fig 1. Example of a Pyramid in a spreadsheet form

This is only a snapshot (screen capture) of a spreadsheet pyramid and therefore it is not interactive. Actual spreadsheet contains above-mentioned formulas in corresponding cells and is capable of calculating values based on given elementary measures of a pyramid. Cost of a pyramid is highlighted in yellow.

Deeper Price Oscillation

Depth of basic oscillation is one step

Earlier in this category, we introduced Basic Price Oscillation and defined exact actions that are to be performed along the price path. Price change that corresponds to the basic price oscillation spans exactly one price step. In other words, price starts from one discreet price level and falls one step lower to the next adjacent discreet level before it rises up again. We can say that price moves one step into the depth. While performing this oscillation, price never falls more than one step and thus the oscillation is considered basic.

Depth of deeper oscillation is two steps or more

Although basic price oscillation happens quite often in reality, it is only a special case of a deeper price oscillation where price experiences larger swing that spans two or more steps. Price drops deeper than within the basic price oscillation and moves two or more steps into the depth.

Please note that since price and the way it moves is out of our control, we can obtain the equivalent result by reducing the step size of the pyramid. Therefore, the same absolute price move can span more than one step if the step is smaller. Thus for the same price swing, instead of basic price oscillation, we may be acting according to deeper price oscillation.

Interesting fact about deeper price oscillation is that the rules of pyramidal trades kick in. Since price moves at least two steps in depth, then at least two buy trades occur. Similarly, at least two sell trades occur as the price moves back up. I would like to point out that we still haven’t defined the term offset and thus we have equal number of buy and sell trades for a given symmetrical price oscillation. When we introduce offset in selling, levels at which sells occur will shift accordingly.

Since there are two or more trades in each direction, they have certain relationship. Naturally, the relationship is pyramidal. Pyramidal trades imply ever increasing buy amounts as the price drops lower and ever increasing sell amounts as the price rises higher. An example of deeper price oscillation that spans four steps is shown in Figure 1.

Fig 1. Deeper Price Oscillation

We can observe that blue rectangle widths increase as we buy into larger weakness and red rectangle widths increase as we sell into larger strength. But, let’s break apart deeper price oscillation and look closer at each portion thereof.

Downward price movement (buying)

We have already mentioned earlier that in Pyramid Investing we exclusively buy price weakness. Considering the price is at the top when oscillation begins (depth = 0), price expresses ever larger weakness with each new discreet price level reached on the way down. The idea is to buy into weakness. Also as the price tanks, the size of price weakness relative to the top is getting larger. The larger the price weakness, the larger our commitment to buy the underlying stock. Buying weakness is always difficult and feeling of fear is usually in our way. However, buying weakness is the most proper action we can do. Courage of buying weakness is always handsomely rewarded.

Fig. 2. Downward price movement - Price decline

Figure 2 shows the down slope of deeper price oscillation. Blue rectangles represent buy trades. Width of each blue rectangle represents the amount of shares bought at corresponding discreet price level. Number of additional shares bought at each depth is shown next to the rectangle. Of course, the numbers shown here are just as an example and the actual amounts bought get determined by particular pyramid in use. Also, larger font size is used for larger amounts to visually emphasize importance of pyramidal buys. Rectangles are framed into a pyramid for clearer distinction and easier shape discrimination.

Price bottoming and turning direction

At some point that is unknown to us, price bottoms and turns around. It then proceeds in an upward direction. Take a look at the Figure 3 and observe some important details related to the price bottoming.

Fig. 3. Price bottoming and turning direction

As the price was moving down, we already made our last buy. Soon after, price bottomed. Turning point is only slightly lower then the price of our last buy and certainly less lower than the step size. Otherwise we would have bought the next lower discreet price level. Since price turned around, that level was never reached. We can pretty much say that we bought the exact bottom, especially when our pyramid applies small step size.

Price turned around and quickly reached discreet price level of our last buy. An important rule can be reiterated here: Trade at certain price level disqualifies that price level at least until next trade occurs at some other price level. This may sound complicated but it only means no action is performed at the level we already made purchase at. On the other hand, if we would to sell at this level, it would only mean zero profit and incurred trade commissions, which of course doesn’t make any sense for us to do.

Therefore, we patiently wait for the price to move one step higher. Again, no offset is used and thus one step higher is all the move price needs to make before we make a sell. That is our first sell. Please note that the amount of first sell is significantly smaller than the amount of last buy.

Upward price movement (selling)

After the price bottomed, it began ascending. We have already mentioned earlier that in Pyramid Investing we exclusively sell price strength. Considering the price was at the bottom, price expresses ever larger strength with each new discreet price level reached on the way up. The idea is to sell into strength. Also as the price soars, the size of price strength relative to the bottom is getting larger. The larger the price strength, the larger our determination to sell the underlying stock. Selling strength is always difficult and feeling of greed is usually in our way. However, selling strength is an important professional action that we have to undertake.

Fig. 4. Upward price movement - Price ascent

Figure 4 shows the up slope of deeper price oscillation. Red rectangles represent sell trades. Width of each red rectangle represents the amount of shares sold at corresponding discreet price level. Number of additional shares sold at each level is shown next to the rectangle. Also, larger font size is used for larger amounts to visually emphasize importance of pyramidal sells. Rectangles are framed into a pyramid for clearer distinction and easier shape discrimination.

Symmetry

Taking a distant view of deeper price oscillation, we can observe certain symmetry between actions that occur along the downward and upward price moves. Take a look at Figure 5 and observe that buy and sell pyramids are exactly the same, except inverted (or flipped). Conclusion is: Only one pyramid is enough to determine both buying and selling.

Fig. 5. Symmetry of buy and sell pyramids

Vertical symmetry is a consequence of equidistant buys and sells as the price moves down and up. All buys and sells are exactly one step apart. Horizontal symmetry is a consequence of no intended accumulation of shares upon executed deeper price oscillation. In other words, if we add up all the amounts of shares purchased it is exactly the same as the amount of all shares sold. In our example, we buy 10+20+30+40=100 shares. Then we sell 10+20+30+40=100 shares. After completion of price oscillation, we are left with zero shares (and whole bunch of profits).

The approach of symmetry works perfectly in a sideways market. However, during trending markets we intentionally introduce asymmetry. We will address ways to create asymmetry in future articles. Profit considerations comprise topic for itself and that will be addressed in future articles as well.

Hourglass

Now, let’s broaden our Pyramid Investing terminology with some new terms. But before, let’s rearrange our buy and sell rectangles so they are center symmetrical. In addition, let’s move the sell pyramid on top of buy pyramid (see Figure 6). Now we lost the price association part, however the shape obtained resembles hourglass. Therefore, when we want to point out the nature of changing amounts of buys and sells as we trade in and out of certain stock, we tend to refer to hourglass. We even derived the verb “hourglassing” that stands for process of acquiring and disposing shares in an hourglass way.

Fig. 6. Hourglassing or Pyramiding

Take a look at the address bar in your web browser – you will see a little hourglass icon that was certainly not chosen by accident. The same hourglass icon appears in your favorites or bookmarks to give you association with Pyramid Investing.

Another term that is equivalent to hourglassing is “pyramiding”. We tend to say: Pyramiding in and out of position. Pyramiding refers to buying and selling shares in a pyramidal way or following principles of Pyramid Investing.