Calculating the Expected Value of Credit Cards

Expected value is a term I use often, and the idea is that you want to analyze the costs, risks, and benefits of a situation and take the path that will give you positive expected value. 

For example, let's say we want to flip a coin and the cost to play is $1. If your guess is correct, then you win $3. If your guess is false, then you win $0. 

Coin Flip (Fair Coin)

Cost to play = $1

  • If your guess = true
    • then you win $3
  • If your guess = false
    •      then you win $0

Expected Value (EV)

EV = Outcomes - Cost
EV = Outcome #1 ($3*50%) + OC2 ($0*50%) - Cost ($1)
EV = $1.50 + $0 - $1
EV = $0.50 = Positive 

There's a 50/50 chance that you'll win, so the positive expected value is $0.50. One coin flip isn't representative of probability, so you would have to flip the coin a few more times.

Another example of probability is rolling a fair dice. The cost to play is $100, and if your guess is correct, then you win $700. If your guess is false, then you win $0. Let's assume if you roll a 6 on the dice, you win $700. Any other number is a loss.

Roll Dice (Fair Dice)                                                                

Cost to play = $100

If your guess = true
     then you win $700
If your guess = false
     then you win $0

Expected Value

EV = Outcomes - Cost
EV = OC1 ($700 * 1/6) + OC2 ($0)... + OC6 ($0) - Cost ($100)
EV = $117 + $0 - $100
EV = $17 = Positive

Assuming you aren't risk adverse and you need $100, then it makes sense to roll the dice because you're getting positive expected value. Again, you would need to roll the dice more than one time to get a real representation of probability. 

Probability ties into credit cards because the expected value depends on your lifestyle and spending habits. 

For example, the American Express Platinum card has a $550 annual fee, but most people won't apply for it because of the high fee.

Amex Platinum (Expected Value)                                                

Cost = $550

(Some) Benefits

a) $200 travel credit
b) $200 Uber/UberEats Credit
c) Global Entry / TSA Precheck
d) Lounge Access

Expected Value

= $200 + $200 - $550 + $100 (GE)
= -$50 = Negative

If you're able to only use the $200 travel credit, $200 Uber credit, and $100 Global Entry/TSA PreCheck credit, then your expected value for the Amex Platinum is -$50.

However, if you're someone who buys a meal or snack at the airport, then the lounge access would help you get positive expected value from the card since you won't have to purchase food anymore.

Even if you don't get the Amex Platinum card, I recommend applying for cards that give you benefits that compliment your lifestyle. For example, if you travel internationally a few times a year, I recommend getting a card that gives you the $100 Global Entry/TSA PreCheck credit, especially if you were planning on paying out of pocket for the benefit anyway. 

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