(Concert) Ticket Insurance: When to buy?

A brief look at the math

Photo by Aditya Chinchure on Unsplash

Hi! Long time no see. Having recently finished my Master’s thesis (🎉🎉🎉), I’ve suddenly got quite some time on my hands. This means I will hopefully be posting more often, so yay to that! 😄¹

You’re buying a ticket to a concert (or other big event), but you might not make it. The website offers insurance, so you’ll get back some money if you can’t make it. Should you buy it?²

Setup + TLDR

Let’s look at the setup:

• The ticket costs T.

• The insurance costs C.

• If you can’t make it, the insurance pays you M.

• The risk you won’t make it is R, which is between 0 and 1.

• You value the concert (subjectively) at V ≥ T.³

TL;DR: The ticket is worth insuring if C < RM. I.e., the insurance cost is less than the likelihood you won’t make it multiplied with the amount you would get back.

Note that this assumes the insurance will pay you for sure, given that you can’t make it. The alternative is discussed in the penultimate section.

An Example

• The insurance costs C=10€.

• You think there’s a 10% chance you won’t make it: R = 10% = 0.10.

• The ticket costs T=100€. The insurance would pay 50% of that, so M = 50% * 100€ = 50€.

Remember, insurance is worth it if C < RM. Let’s plug these values in. R*M = 0.10 * 50€ = 5€. Since the insurance costs 10€ which is greater than R*M = 5€, buying the insurance isn’t worth it.

The Math

To figure out when you should buy the insurance, we will compare the expected value (EV) of buying vs. not buying it. (We won’t simplify the terms at first, because a lot of the complextiy is shared between the EVs and will fall away in the comparison.)
Let’s start with the EV of not insuring⁴:

\(\begin{align*} \text{E}[\text{no insurance}] &= (1-R)(-T + V) + R(-T) \end{align*}\)

Computing the EV of insuring⁵:

\(\begin{align*} \text{E}[\text{insurance}] &= (1-R)(-T + V - C) + R(-T -C + M) \\ &= (1-R)(-T + V) + (1-R)(-C) + R(-T) + R(-C + M) \\ &= (1-R)(-T + V) + R(-T) + (1-R)(-C)+ R(-C + M) \end{align*}\)

Comparing the two EVs, we see that the first two terms are identical. So, insuring is worth it only if the last two terms of E[insurance] give it an edge:

\(\begin{align*} \text{E}[\text{insurance}] &> \text{E}[\text{no insurance}] \\ \Longleftrightarrow - (1-R)C + R(-C + M) &> 0 \\ \Longleftrightarrow -C + RC - RC + RM &> 0 \\ \Longleftrightarrow C &< RM \\ \end{align*}\)

Thus, insurance is worth it if C < RM, or alternatively, R > C/M. Interestingly, the formula depends on neither the ticket price T nor the value V, so you only have to focus on the risk R and the insurance’s input/output ratio C/M.

Let’s consider an example: If the insurance costs C=5€ and you think the chance you’ll miss out is R=10%, the insurance should pay at least M=C/R=50€. If you’re really unsure you’ll make it, say R=50%, then them paying you just M>10€ would be enough to warrant the purchase.

Considering Coverage Cases

Ticket insurance usually covers only some cases. If you can’t make if for reasons not covered, you won’t get anything. When is insuring warranted in this scenario?
Mathematically, it’s mostly the same. Let P denote the (conditional) probability of being paid given that you couldn’t make it. The resulting condition is: C < RPM.⁶

We’ll continue the example from above (C=5€, R=10%). Say you’re pretty unsure the insurance will actually cover your costs, given you won’t make it: P=50%. Then, in the case where they do pay, you should receive at least M=100€ to warrant the purchase.

Conclusion

Buying an insurance is worth it if C < RPM, so if

• the insurance is cheap,

• it pays back a lot,

• it’s likely you’ll won’t make it, and/or

• it’s likely you’ll get paid given that you coudn’t make it.

The ticket price is “irrelevant”.

Keep in mind that this is simplified in various ways. For example, if you can’t make it, you could sell your ticket instead of buying insurance. Also, I don’t know a lot about the real-world insurances offered for these kinds of events, so if you spot an error, please let me know.

In any case, I hope you’ve enjoyed this little ride and will find it helpful! 😊

1. Apparently, emojis can be in italics??? 😄🤯❤👍💡 WTH who came up with this??? I have no idea in what context this would be a desriable feature. LMK if you have any idea.

2. I’m talking of “you” here, because I can barely remember the last time I have been to a concert. In maths terms: Introvert + Loud Music + 1000000 People =

Photo by Usman Yousaf on Unsplash

3. Just interpret this as the maximum price you would be willing to pay for the ticket. It turns out not to matter in the end.

4. If you can make it, you pay the price T and experience some value V from the event. If you can’t, you’ve paid the price T and received nothing.

5. If you can make it, you pay the ticket T + insurance C and receive some value V. If you can’t, you pay the same, but get compensation M.

6. The M is simply replaced by PM everywhere. Structurally, only the last part of E[insurance] changes. It becomes:

\(\dots + R(-T-C + PM + (1-P)\cdot0)\)

So, given that you couldn’t go, you get M with probability P and nothing with probability (1-P).