Module 1 / Lesson 2.3
Expected Value and Actuarial Fair Premiums
Learning guide
- Learning: Learn about the expected value and acturial fair premiums for insurance.
- Reflection: As you learn this material, make sure you understand how fair value is calculated and how it affects both buyer and insurer.
Recap and Motivation
In Lesson 2.2, we saw how insurance smooths wealth: you give up money in the good state to get money in the bad state. But before we decide if that trade is "worth it," we need to understand the math of averages.
Definition of Expected Value
Expected Value (E[V]) is just a fancy term for "weighted average." It is the value you expect to get on average. You are just adding up the value of each possible outcome, but weighting each one by its probability of happening. (Note that upper-case "P" refers to the premium of an insurance policy while lower-case "p" refers to the probability of an outcome, which is the "p" we care about here.) We use the notation "E[V] to refer to the expected value of something, so that E[number of heads] is the expected number of times a coin comes up heads, E[wealth] is what you expect your wealth to be, etc. "E[x]" is just our way of writing the expected value of "x", whatever "x" is.
Expected Value = E[V] = (Probability of Outcome A × Value if Outcome A) + (Probability of Outcome B × Value if Outcome B) + ...
Example: The Coin Flip (Two Flips)
Let's make this concrete. If you flip a coin twice, what is the Expected Value of the number of heads you will get?
To find this, we list every possible "branch" of history that could happen. There are 4 equally likely outcomes (25% chance each):
| Scenario | First Flip | Second Flip | Total Heads (Value) | Probability (p) |
|---|---|---|---|---|
| 1 | Tails | Tails | 0 | 25% (0.25) |
| 2 | Tails | Heads | 1 | 25% (0.25) |
| 3 | Heads | Tails | 1 | 25% (0.25) |
| 4 | Heads | Heads | 2 | 25% (0.25) |
The Calculation
To find the Expected Value, we sum up the value of every scenario, weighted by how likely it is to happen. Let's look at the expected number of heads if you flip a fair coin twice.
Step-by-Step:
Expected Value = E[number of heads] = p(0 Heads) × 0. + p(1 Head) × 1 + p(2 Heads) × 2
= (0.25 × 0) + (0.25 × 1) + (0.25 × 1) + (0.25 × 2)
= 0 + 0.25 + 0.25 + 0.50
= 1
Actuarially Fair Premiums for Insurance
How much should insurance cost? While there is no single "correct" answer, one benchmark is the "actuarially fair premium."
This is the premium where the insurance company (insurer) and the customer (insured) both break even ON AVERAGE.
- The premium (what the insured pays) is equal to the expected payout (what the insurer pays in claims).
- The insurer earns zero profit on average.
In the real world, insurers charge more than the fair premium to cover administrative costs (salaries, buildings) and earn a profit. But the fair premium is our mathematical starting point.
Calculating the Fair Premium
Consider a loss of size L that happens with probability p.
If we have Full Insurance (Indemnity = Loss), the insurer pays out L with probability p, and $0 with probability 1-p.
Expected Payout = E[Payout]= (p × L) + ((1-p) × 0) = p × L
Therefore, the Actuarially Fair Premium (Pfair) is simply:
Pfair = E[Payout] = p × L
Example: Expected Wealth and Fair Premiums
The Scenario
Recall our example from Lesson 2.1: You have $121. There is a risk of bike theft (Loss = $96).
Parameters:
- Initial Wealth (W0): $121
- Loss (L): $96
- Probability of Theft (p): 40/96 (approx 41.7%)
- Probability of No Theft (1-p): 56/96 (approx 58.3%)
- Full Insurance (I=L): I = $96
Outcomes Without Insurance:
- Good State (No Theft): You keep $121.
- Bad State (Theft): You have $121 - $96 = $25.
Using our formula Pfair = p × L:
Pfair = (40 / 96) × 96
Pfair = $40
You pay the $40 premium for sure. If a loss happens, you get fully reimbursed.
Wealth = $121 (Initial) - $40 (Premium) = $81 (Certainty)
Expected Wealth = E[W] = (PGood × $121) + (PBad × $25)
Expected Wealth = E[W] = ((56/96) × 121) + ((40/96) × 25)
Expected Wealth = E[W] = 70.58 + 10.42 = $81
The Puzzle: Why Buy Insurance if it Lowers Expected Wealth?
If you buy "fair" insurance in this example, your wealth becomes a certain $81. If you don't buy it, your average wealth is also $81.
Often, insurance companies charge a premium that is higher than the expected loss to cover their costs. This means your Expected Wealth with insurance is often lower than without it.
Here, you can see a video that explains the idea of expected value with several examples, and explains the "fair" price of several gambles (just like the actuarially fair price we learned about above.)
So, why do we buy it? To answer that, we need to move beyond dollars and look at utility (happiness). Next Lesson 2.4 explores this idea.