Suppose someone wants to invest in the stock market, how would you approach modeling an individual investing in this asset class?
The particular asset class at hand here is stocks. We will make intuitive assumptions about the stock market and how a risk-averse individual operates in the stock market. Then, we will try to transform said assumptions to make a simple heuristic. This heuristic will then generate a numerical value which we will describe in terms being between an upper and lower boundary. Let’s make the assumptions
Creating the Assumptions:
- ROI on stocks to diminish overtime due to increasing market capitalization. Growth in stock markets is essentially logarithmic and this model can be approximated by the ln(x).
- The total amount of principle is expected invested diminishes overtime as a risk-averse person doesn’t want to invest in one type of asset.
- A risk averse individual will invest in less risky stocks as they get older hence less return per investment made
Heuristic – Rules Based upon Assumptions:
- You invest in one unique stock per year
- You invest a dollar value exactly inversely proportional to the year that you invested. For example, on year 3 you invest $(1/3) or $0.33 into the stock market. On year 4, you invest $(1/4) or $0.25 into the stock market. On year 5, you would invest $(1/5) or $0.20 into the stock market
- Each individual stock invested in has a diminish ROI of exactly *(1/ initial value) per year.
That is a mouth-full so let’s use an example to illustrate this. Suppose an investor is in the third year of investing. That means the newest stock invested in would have an initial value of $(1/3) à 33 cents. At the end of year four, the return on the stock would be = initial value + initial value *(1/year invested) = $(1/3) + $(1/3)(1/3) =$.44
Since we know from rule A. we invested in a stock in year two then the value of year 2’s stock would $(1/2) + $(1/2)(1/2) + $(1/2)(1/2)(1/2) = $0.875
Stock #2 has exactly one year more of diminished return as it was bought exactly one year before Stock #3.
*Side note: We are ignoring the quantity of shares the investor has in a particular stock. This seems okay as in real life there are many stocks with a diverse price range. Low-priced stocks at low risk levels exist along with high priced stocks at high risk levels. They are uncommon, but they exist.
With these conditions we satisfy assumptions #1, #2, #3. We are investing less money per unique stock with every succeeding year (#2). The next stock invested in the succeeding year will have diminished returns (#3). The stocks growth is related to the multiple of year invested in and hence grows geometrically over time (#1).
Intuition for Rules:
- The first thing to address is the rate of change. A geometric rate of change is logical way to model the stock market as market capitalization follows a logarithmic progression and hence a diminishing ROI. A geometric rate of change summed up overtime closely models ln(x) assuming this is not an unfathomable time frame.
- The second point addresses diminishing initial investments overtime. That’s right we are talking about the principal sum invested in. The fact that I chose the principal amount per investment to diminish overtime at 1/x is somewhat arbitrary. If this is supposed to be an easy investment plan to follow, investing 1/(investment number) seems easy to remember and use in practice. It is perfectly fine for someone to invest per time period at a linear rate e.g. ($1 for the first investment, $0.9 for the second investment, $0.8 for the third investment, etc.). To me, it’s more sensible to invest more money earlier and less later as to take more risk when you are younger and less risk when you are older.
- The fact that the rate of the ROI diminishes seems fair as people generally tend to become more conservative when they get older. Hence people will not only invest less but invest in less risky stock options. Modeling the rate of ROI diminishing proportional to the principal investment therefore makes sense in this context.
Example Case Applying Rules:
We can make a table out of this to better visualize the problem. Let’s make the table as a function of two variables as we have two things changing per year: the money made per year per stock and the total number of investments we own.
Suppose we applied the heuristic for four years, then we can approximate what the total sum of our portfolio would be.
|*Profit based upon the period we invested in*||Sum of each investment|
|Year 1 profit||Year 2 profit||Year 3 profit||Year 4 profit|
|Stock #1||$1/1||$1/1||$1/1||$1/1||$1+$1+$1+$1 = $4.00|
|Stock #2||$0||$1/2||$1/4||$1/8||$1/2 + $1/8 = $ 0.875|
|Stock #3||$0||$0||$1/3||$1/9||$1/3 + $1/9 = $0.44|
|Stock #4||$0||$0||$0||$1/4||$1/4 = $0.25|
|Total sum of all investments = $5.565|
Table 1. Summing the total net value of our investments after 4 years. For every new year we receive a return on previously bought stocks, and we add one new stock to our collection. The returns are a function of the total year (J) that passed along with the number of investments invested in (X) where X = J.
Let’s first analyze the table by starting at the first the first investment at the first period of ROI. Going down the first column, we see that there are many zeros. These zeros represent that no money has been invested into the stock yet. Notice that if we exclude the zeros, the geometric pattern of our trend looks triangular – which is highlighted for clarity.
In non-empty cells, the change per cell is just the previous cell’s denominator minus one. This follows our heuristic as we stated the initial money put down on the investment is the reciprocal of the year we invested (e.g. on year #2 we invested (1/year #) à (1/year 2) à (1/2) à $0.50).
Going across any row, we are getting diminished returns by exactly the amount we invest in. Exclude cases where $0 transforms to the principal investment. Technically, no rate can be multiplied by $0 to produce a non-zero value the next year. This table is just a convention.We can write this trend in mathematical notation where transformation where X = J.
Mathematical Representation of Example Table:
F(X,J) represents the total sum seen on the right column of the table. While I can’t give an exact closed form solution at this time, I can provide a rough upper bound for when we invest in over large number of stocks over a large number of years. For now, take that equation to be true. I’ll try to prove it along with a closed form solution in a future post. If you’re talking with a friend and this blogpost and they are interested in how you derived that equation, handwave wave at them then walk away (as I am handwaving away this proof).