Bankroll Growth from Repeated Fixed-Fraction Investments

Summary

The note asks how to calculate bankroll growth when the same fraction of the current bankroll is invested in a sequence of opportunities with specified payoff multipliers. Its worked example starts with a bankroll of $100, invests 10% at each step, and applies multipliers of 1, 0.9, and 2 in order. The displayed arithmetic reaches a final balance of $108.28.

For a fully invested bankroll, the note recognizes that successive multipliers can be combined as a product. It asks for the analogous expression when only a fraction is invested each time, but supplies no general formula or answer. The example illustrates that the invested amount changes with the bankroll after each outcome, so order and reinvestment assumptions matter. It does not discuss uncertainty, transaction costs, or risk; its scope is deterministic sequential arithmetic.

Key ideas

• Each stake is calculated as a fraction of the bankroll available at that step.
• The bankroll retains the uninvested portion while the stake receives the stated payoff multiplier.
• With repeated fixed-fraction investments, each outcome changes the base for the next stake.
• The note gives a worked sequence but does not provide a general closed-form expression.

Tags

# Compounding with fixed fraction of bankroll


# Compounding with fixed fraction of bankroll












Suppose you have $100

But you invest 10% of in in each investment with payoffs (multipliers). that are as follows;

`1, 0.9, 2`

Investing 10% into 1:

```
10% of 100 is $10, Balance  ($100 - $10) = $90

$10 * 1 = $10

Bankroll is now $90 + $10 = $100
```

Investing another 10% in to 0.9:

```
10% of $100 is $10, Balance is (100 - 10) = $90

$10 * 0.9 = $9

Bankroll is now $90 + $9 = $99
```

Investing the last 10% of bankroll into 2:

```
10% of $99 is $9.9, Balance is $99 -  $9.9 = $89.1

$9.9 * 2 = $19.8

Total Bankroll is $19.8 + $89.1 = $108.28
```

I know compound interest has a simple formula where you can plug in variables and see the compounding over x years.

My question is.

Is there a simple formula were you can plug in variables of the multiples and the fraction of the portfolio invested in each sequence to find out how much one would end up with this. In this case $108.28

I can do this in code using loops, but I wanted to know if there was a mathematically elegant solution.

For a fraction of 100% the solution is simple,

`You just need to multiply 100 x 1 x 0.9 x 2 = $180` which is a product series.

What is the solution for a fraction of 10%?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.