**This is just a continuation of the first blogpost titled “Modeling a Risk-Averse Investor in the Stock Market Pt. 2”. **

**Finding an Upper Bound for our Heuristic:**

To approximate the upper bound, let’s take *J* –> ∞ and not include the investment# 1 case as it is a special case then we have:

Above, we shifted the index of r by 1. It now has to have an initial value of r = 2 insuring the first-rate equals ½.

We can make another table to intuitively understand what this represents:

ROI based upon the period we invested in (1/r) | Sum of each investment | |||||

Period 1 ROI | Period 2 ROI | Period 3 ROI | Period 4 ROI | Period J | ||

*SPECIAL CASE* | $1/1 | $1/1 | $1/1 | $1/1 | + … | 1+1+1+1 + … = $J |

Investment #1 | $1/2 | $1/4 | $1/8 | $1/16 | + … | ½ + ¼ + 1/8 + 1/16 + … = $2.00 |

Investment #2 | $1/3 | $1/9 | $1/27 | $1/81 | + … | $1/3 + $1/9 + $1/27 + …= $3/2 |

Investment #3 | $1/4 | $1/16 | $1/64 | $1/256 | + … | $1/4 = $4/3 |

Total sum of all investments ~$4.33 + $J |

**Table 2. **The table displays the *X *number of investments that are accruing value over an infinite *J* period. In our model, we had *X *= *J* hence setting *J* to infinity is an obvious over approximation. The shape produced by the cells is rectangular (*J* x *X*) where as in the true model it is triangular ~1/2(*J *x *X*).

So, we can clearly see that:

Investment #1 produces –> $J

Investment #2 produces –> $2/1

Investment $3 produces –> $3/2

Investment $4 produces –> $4/3

By induction, we can see the pattern for the sum of each investment # — barring the first investment — follows a nice pattern. Let’s remove the investment and shift the index of the investments by one hence.

Special case #0 produces –> $J

Investment #1 produces –> $2/1

Investment #2 produces –> $3/2

Investment $3 produces –> $4/3

Investment $4 produces –> $5/4

Hence for some large number of investments J we find:

From literature, we can conclude that this approximates to

Hence going back to our original function F(*X, J*) to find the sum of the investment we get:

**Finding the lower bound:**

I’ll make this inaccurate but simple. We will just say that *K* is the partial sum of the harmonic series. If you reference the tables this is obviously true. The series representation of the diagonal starting at cell (1,1) going to cell (*X*, *J*) hence the lower bound is:

**Relevance in real life:**

This model more accurately represents real life initial investment is not a mere $0.50. We can just solve this problem by multiplying our initial investment value by some constant *A *to get a large initial investment. This is the same as writing for the upper bound:

Or the lower bound:

Let’s say that A = 10,000 and the investing period goes for 10 years.,

We will have for the initial investment $0.50 * 10,000 = $5000. We can expect our return to be roughly bounded by: ($2.93)*10,000 < value of portfolio < ($12.93)*10,000

$29,300 < value of portfolio < $129,300

The lower bound is obviously grossly wrong and hence the minimum should be higher. Regarding the maximum return, it seems feasible that a conservative investment would yield a YOY return would of 38.4%. Even though this maximum boundary seems high relative to the standard 10-12% expected YOY, it isn’t unheard of. There are at least 100 ETFs — many of whom are publicly traded.

If we wanted a more realistic answer, then we need to include more parameters. We could either or both of the following fixes:

- Find a more accurate lower bound approximation and/or find the closed form solution.
- Multiple the series by (-1)^n giving us oscillations. This would represent volatility in that occurs in actual markets.

An example #2 would be:

Year 1: +$0.50

Year 2: $-0.25 + $0.33

Year 3: +$.125 – $0.11 + $0.25

Etc, etc.

Another interesting question would be what if the investments were sporadic. This could be model explicitly to have the stocks only bought in *prime* year investing 1/*prime* dollars per stock (i.e. 2, 3, 5, 7, 11 for a $1/2, $1/3, $1/5, $1/7, $11, respectively). You even theoretically add the cherry on top and have force model include oscillating too. This heuristic would be the most accurate yet the most hard to calculate – way out of the scope of this blog.

If you read this far, I congratulate you. If you read this far and understood what was said, give yourself a pat on the back and a round of applause at same exact time (it’s impossible to do that) for being a diligent reader. I will post a part III if I can provide a closed form solution to the equation at the top of this page, or present more economic relevance of the model.

~Ben