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.

10 comments to Cost of a Pyramid

  • Simon

    Another good article, thanks. I have something I would like to get your opinion on, but I may be getting ahead of this post.

    I notice that in your example above you use a fixed dollar amount for each step – $2. It strikes me that this makes the pyramid very top heavy. What I mean by that is the difference in price movement that is required to trigger entry and exit at the top of the pyramid and at the bottom is very large.

    The movement between $11 and $9 is an 18% decline to trigger the buy and then a 22% increase to trigger the sell. But at the bottom of the pyramid between $1 and $3 it is a 67% decline and then a 200% increase.

    If you were to get to the bottom of your pyramid, then you would be looking for incredible increases in the share price to trigger your pyramid levels.

    I have been playing around with a fixed % for each step, for example if I had 5 levels as above, I could use 33% step and would have a pyramid that looked like this:

    Depth 0 – $11.00
    Depth 1 – $7.37
    Depth 2 – $4.94
    Depth 3 – $3.31
    Depth 4 – $2.22
    Depth 5 – $1.49

    This means that I buy on every 33% decline and sell on every 50% rally. It seems to me that this offers better protection to your capital if the price was to decline 90%, like some stocks did in the recent bear market, as you would be trading your way out of drawdown a lot quicker at the lower levels.

    I look forward to hearing your views on this.

    • Simon,

      Your observations are true indeed. I am fully aware of that constant step in dollar terms means variable step in terms of percentages. Conversely, constant step in terms of percentages means variable step in dollar terms. Although this is easy to state and these two approaches may appear very similar, they make a hell of a lot difference when it comes to application and performing corresponding calculations on associated pyramids. In the article that defined step, I mentioned that: “the benefit of variable step is smaller than the downside of having to handle complex pyramids whose step changes with price”.

      I’ve chosen to deal with step that is constant in dollar terms. As a consequence, such step varies in terms of percentages. The benefit is tremendous in terms of reducing complexity of calculations. Many aspects of Pyramid Investing that I still haven’t addressed yet are drastically easier to handle this way. Even during the daily application of pyramids I find very convenient to have the step fixed in dollar terms. Then I know where my next executions are without even looking at pyramid spreadsheets. I will leave you for now to trust me that reduced complexity is reason enough to apply the step in a way I presented it.

      On the other hand, what are the downsides of constant dollar step?

      I can state that in terms of profitability of pyramids, having a constant dollar step is NOT a downside. And profitability is the bottom line we definitely don’t want jeopardized.

      So it remains, as you’ve already noticed, the downside of the pyramid being “very top heavy”. That is true, but there is very easy way to fix it: use a constant dollar step pyramid, however increase the value of increment. This will affect the cost of your pyramid, so a few iterations and tweaks of other parameters will be necessary to bring the cost back to the original value. The result will be a pyramid whose weight has been increased towards the bottom.

      You also have to realize that the pyramid I used above as an example is not very realistic one. Practically, we want many more levels and the step expressed as a percentage of top is normally a single digit number. That leaves us with pyramids of 10 to 100 depths! Once you deal with such small steps, their percentage variation diminishes.

      Another thing I would like to point out is the way investment industry likes to express moves in stock prices: “Price of company XYZ dropped two points today”. I have to admit that this drives me crazy since many times I have no clue what the price of company XYZ is to begin with. Why not say that price dropped 0.1% or 10%? That would be self-sufficient. However, those who actually trade the stock daily and thus have the influence on price moves, think in terms of points, not percentages. They tend to get accustomed to certain “point moves”. When the stock melts 80%, their minds don’t keep up with new price and they still think in terms of old point moves that have suddenly become huge in percentage terms. As an example, I clearly remember fall of 2008 when most of my stocks were moving 20% most of the days!

      Therefore, even though it doesn’t seem logical, stocks tend to move more as they go down in price. That is exactly compensated in pyramids that have fixed dollar step. With a constant percentage step, you would be overwhelmed with entering orders if we get to anything similar to fall of 2008. On the other hand, nothing prevents you from inserting additional orders between two depths IF the stock melts and you are not happy with huge percentage difference between original depths. Set aside some capital for this very purpose that would be accessible only in a case of meltdown.

      In your example, you mention 200% rally as a huge one. And that is true if one experiences 200% rally from the top. However 200% rally after the stock melts is not unusual. Since general stock meltdown in fall of 2008 many stocks have gone 10 to 20 TIMES up. Those moves are measured in thousands of percents!

      One more statement to make: Pyramid Investing is full of trade offs.
      You mentioned:

      This means that I buy on every 33% decline and sell on every 50% rally. It seems to me that this offers better protection to your capital if the price was to decline 90%…

      You would definitely have better protection of capital (and much more profits) in case of a 90% decline. But think what such a pyramid would be trading 99% of the time when you DON’T have 90% decline? You would most likely have only a trade or two in a year. I bet you wouldn’t be happy with that either. There is a fine line for every investor where one gets to be happy in both the ordinary times and turbulent times. Being prepared for every scenario means finding that sweet spot that suits your investment personality – finding that perfect trade off where you are likely to fare reasonably well no matter what happens. Pyramid Investing is a concept that allows you to achieve that.

  • Once you’ve decided on a top price at which you’re content to be all-out, a bottom price at which you are content to be all-in, then you can either use log or linear step sizing.

    A simple log stochastic measure can identify the appropriate amount of exposure for log based step sizes

    ( log(current) – log(bottom) ) / ( log(top) – log(bottom) )

    for linear steps just use the straight stochastic

    ( current – bottom ) / ( top – bottom )

    The easiest way to manage an account is to baseline to cash reserve rather than stock value. For example if I were investing $10,000 in the Dow and decided to set top=20,000 and bottom = 5,000 when the current was Dow 10,000

    (log(10000) – log(5000) ) / ( log(20000) – log(5000)) = 0.5 (50% cash reserve or $5000 cash indicated).

    If the Dow rose to 12000 the log stoch = 0.63 (63% cash reserve indicated), so if the total fund value was $11000 at that time = $6946 cash reserve indicated ($4054 stock value implied).

    The tricky part is working out what top and bottom settings to use. Too narrow and you’ll end up all-in or all-out too quickly, too wide and you potentially miss out on rebalance benefits. Generally being all-out quickly isn’t as bad as being all-in too quickly so you want to focus more on deeper downside cover (bottom) than upside range cover (top). That will generally mean that you’ll have relatively small amounts of stock exposure (and large cash reserves), which in turn means that you might look to run multiple ladders (as I call them) ideally with low/inverse correlations in the underline holdings such that as one might be calling upon cash reserves the other might be adding to cash reserves.

    Long dated treasuries and a stock index are one such somewhat inverse correlation pair. Check out TLT and Dow for instance.

    • Clive,

      I like your formulas and mathematical approach to defining cash reserve size. I think that approach can easily be programmed and successfully utilized within some software trading platform. I would also tend to lean toward log formula since that one gives more weight to lower prices which resembles pyramids more closely. Linear formula is analogous to what I term “skyscraper”.

      The Pyramid Investing approach is more numeric and discrete. It is also nonlinear. However, one aspect where your formulae would need tweaking is trading dynamics. In the article above, dynamics is neglected. In Pyramid Investing dynamics, trade size is determined on a relative basis with respect to the nearest points of price turns. I would imagine, in a trading system using your formula, trading would need to “jump” from one formula to another with different parameters, pretty much each time the price turns and trade occurs. That can be done of course. My point is, Pyramid Investing in not only nonlinear but there are points of discontinuity that would need to be taken care of.

      In other words, in Pyramid Investing, there is no unique relationship between the price and let’s say cash reserve. That complicates things and necessitates more than a single formula to be utilized throughout the trading range. Depending on the path and history of price moves, certain price can be associated with more than one amount of cash reserve. This concept of non-uniqueness or relativity is a powerful risk/reward managing mechanism that adds a great value to Pyramid Investing and I think is worth the increased complexity.

      I completely agree with your analysis of top/bottom setting and I think your conclusions are valid. The only problem I have is using bonds in the current financial environment. However, I do utilize the same seesaw effect by means of bear (short) ETF’s. One only needs to be careful to skew the long to short ratio appropriately during trending markets.

  • Here’s a couple of postings and calculators that I posted on investors hub a couple of years back that you might find to be of interest

    http://investorshub.advfn.com/boards/read_msg.aspx?message_id=31982126

    http://investorshub.advfn.com/boards/read_msg.aspx?message_id=34991336

    More recently I prefer the log stochastic calculation method that I outlined in my previous posting as the means to identify current indicated exposure amounts. Restricting trades to occur at Point and Figure reversals or on other technical indicators is a reasonable choice IMO, coupled with running multiple Ladders against a range of low correlated underline investments (stocks, long dated treasuries and gold as per Harry Browne’s Permanent Portfolio set is a reasonable set as you’ll also likely have some cash as well when using multiple ladders against those assets and therefore enjoy the low draw-downs that the Permanent Portfolio enjoys).

    Another way I’ve used Ladder is to dynamically adjust the top and bottom values over time in alignment with Bollinger bands or other multiples of standard deviations and/or support/resistance levels.

    ZigZag charts can also be of use when looking to set top and bottom values.

    Coupled with stop-loss management and given a period of high zigzag price action you can churn out some nice profits in a relatively low risk manner even when the underline investments share price ends at an amount below the start dates price.

    With fixed ladders you’ll never sell stock at a loss excepting if the stock price goes to zero, so generally its best to run Ladder against index funds/ETF’s for their greater resilience to going broke.

    • Thanks for your comment. I would like to add the following (rough) list of investment instruments in order of risk of going broke (going off the board, going to zero) sorted by lowest risk first:

        - Gold (Au)
        - Food commodities (wheat, corn, soybean, sugar…)
        - Energy commodities (oil and natural gas)
        - Currencies (Dollar, Euro…)
        - Major market indices (Dow, S&P 500, China)
        - ETF’s (dozens of stocks combined)
        - Individual stocks
        - Bonds
  • In other words, in Pyramid Investing, there is no unique relationship between the price and let’s say cash reserve. That complicates things and necessitates more than a single formula to be utilized throughout the trading range.

    Thanks for the replies and Email PI Engineer.

    With the Ladder (log stochastic) approach you can adjust the amount allocated dynamically up/down as seen fit which scales the trade sizes up/down as desired.

    That may involve increasing if you want to trade more the further the price deviates from a perceived mean-reversion price level, or decreasing if the perceived risk is seen to have risen.

    The top, bottom and actual trade price levels can similarly be adjusted dynamically i.e. perhaps to varying Bollinger Band distances and/or support/resistance price levels.

    Thanks again. Clive.

  • Here’s a back-test spreadsheet for Ladder applied to the Dow from 1929 onwards

    http://www.jfholdings.pwp.blueyonder.co.uk/ladder_dow_1929.zip

    In cells F6, F7 and F8 you can tune the TOP and BOTTOM settings to standard deviation distances from the mean, use a straight log stochastic (x) or inverse log stochastic (1-x) exposure amount value, and/or opt to use 10 month moving average based timing on-top (similar to 200 day moving average based timing)

    Cells L6 and N6 show the results, based on comparing the Dow performance had the same amount of average stock exposure been held constantly throughout as that of the average stock exposure that the Ladder averaged over the total test period.

  • I suspect that Pyramid/Ladder like investing overall doesn’t add value generally. Over some periods it will, over other periods it won’t.

    More ‘rebalance benefit’ appears to be available for consistent capture using correlations.

    Here’s a random price generator that’s based around Harry Browne’s Permanent Portfolio allocations

    http://www.jfholdings.pwp.blueyonder.co.uk/random_PP.zip

    Tweak the standard deviation and mean yearly values for each of the components and then run the 254 year test (takes a minute or so to run), do that for a range of mean and standard deviations and you’ll find that generally for realistic mean and standard deviations values there is some rebalance benefit generated (around 1.5% p.a. on average when you compare the un-rebalanced totals with yearly rebalanced totals).

    That benefit however takes time to arise (decade or more potentially) and requires full market cycles. The choice of stocks, long dated treasuries, gold and cash help reduce that cycle time.

    When you have assets that each achieve similar real returns, but do so in a zigzag low/no correlated manner then simpler constant ratio rebalancing (back to 25% weightings in the Permanent Portfolio’s case) when the collective set generally holds value (low volatility in total value) will add value over time.

    The approach I’ve opted for therefore is to blend Permanent Portfolio (as a form of virtual cash type investment) with more speculative ‘stock’ investments (recently I had been using Mebane Faber’s Quantitative Model but currently I’m out on a Sell in May investment vacation but have in mind to use Mebane Faber’s Major Asset Rotation Model upon returning back into the markets in September).

    Another option that might be worth considering is Decision Moose but as that is highly concentrated you need to make sure to blend it with something like a Permanent Portfolio in order to be more diversified/reduce risk.

    Just my opinion.

    Best regards. Clive.

  • I should have left a link to my own web page that outlines a UK permanent portfolio: http://www.jfholdings.pwp.blueyonder.co.uk/

    There’s also some historic US Permanent Portfolio performance data on Craig’s blog over at:
    http://crawlingroad.com/blog/2008/12/22/permanent-portfolio-historical-returns/

    Treat the Permanent Portfolio as ‘CASH’ and blend that with ‘STOCK’ exposure invested in perhaps something like Mebane Faber’s Quantitative or Asset Rotation (there’s a blog that simplifies the process at http://taaforthemasses.blogspot.com/ ) and potentially you’ll do OK with not too much volatility along the way.

    Of particular note is how the Permanent Portfolio endures relatively low draw-downs. Also that whilst more recent performance appears to be good, over other periods it does lag (that’s when the other ‘STOCK’ part will tend to pull up the total rewards).

    I’m content with around a 50/50 split of both (with around 25% of total funds in gold and/or foreign currencies for domestic currency crisis risk protection). Others are happier with low total volatility and run just the Permanent Portfolio alone.

    Sorry about posting so much, hope its more of a help than an annoyance.

    Regards. Clive.

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