How to calculate how much profit a loan of a certain risk level (based on dice rolls) is likely to make?

I'm working on a board game and am trying to check the balance of a mechanic. Specifically, I'd like to calculate the profit a player is likely to make by giving out a loan. Each loan has four values:

  • Principal: Initial amount the player has to forfeit
  • Payment: amount that the player can collect every turn
  • Length: the number of turns the loan lasts for
  • Risk: explained below
For each loan, the player will roll one six sided die every turn. If it lands on a value equal to or greater than the risk value (between 2-5) on the loan card, the player can collect an amount equal to the payment value. If the die roll is less than the risk value, the player discards that loan card. The length value determines the maximum number of payments you can collect, but it can be less if you roll poorly.

I need a formula I can use to help with balancing these loans. The end value should be how much profit the player will get on average if they give out the loan. It should take into account the fact that a higher risk level reduces how many payments the player will be able to collect.

Any thoughts on how to go about calculating this? Thanks in advance!
  • If you roll below the risk value, you will never collect on that loan again, right? That’s what you mean when you say it’s discarded?

  • Yes, that is correct.

  • I noticed a small mistake. The first term should be to the power of n, not n-1. Here's an updated demos link https://www.desmos.com/calculator/qi2icuicua

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  • Thank you for your answer! I don't have a strong math background, so I do have some questions about how to interpret the formula. 1. From my understanding the ∗ symbol after P, i, and L indicates that those values should be multiplied by what's in the parentheses, correct? 2. How is the value i calculated? I'm trying to convert your formula into one useable in Google Sheets, and am confused about what I should be subtracting with 1.

  • 3. If I'm correct about question 1, what is the order of operations for the part starting at "i *" up until "+ L"? Should i be multiplied by the values calculated within the parentheses that immediately follow it first, with that value then being multiplied by the parentheses before "+ L"?

  • Yes, * mean multiply. The value i refers to the index of the summation. I don't know how to do summation in Excel. Maybe you should ask that as a separate question with a separate bounty.

  • Also, I'm not sure if you saw my earlier comment up above, but the final formula has a mistake. The first exponent should be i, not i-1

  • Okay, thank you for all your help!

The answer is accepted.