In our last post we talked about the importance of probability in the way the world works. One can apply probability to arrive at an estimation of a company's market cap. For instance, one interpretation of a company's current market value is that it is a reflection of the market's view as to the ** probability-weighted discounted present value of future cash flows to the company's shareholders**. For example, in very simplistic terms, the market cap of AAPL is currently around $500 billion. One scenario (out of many) that

*could*lead the market to arrive at this figure is the following, in which the market views APPL as having a:

- 20% chance of being worth $300 billion as the discounted present value of all future cash flows to shareholders or 20% x $300bn = $60bn expected value
- 40% chance of being worth $400 billion as the discounted present value of all future cash flows to shareholders or 40% x $400bn = $160bn expected value
- 40% chance of being worth $700 billion as the discounted present value of all future cash flows to shareholders or 40% x $700bn = $280bn expected value

The total of all expected values is $60bn + $160bn + $280bn = **$500bn in market cap or probability-weighted discounted PV of all future cash flows to shareholders**. In other words, of all the future possibilities, there is a 20% chance that AAPL's bear case scenario plays out where it's worth $300bn; a 40% chance that the base case occurs where it's worth $400bn; and a 40% chance that the upside case occurs where it's worth $700bn. When you add up the expected values from all the likely scenarios (until the probabilities add up to 100%), you arrive at a probability-weighted discounted present value of future cash flows to AAPL's shareholders of $500bn.

The same concept of **probability-weighted outcomes** applies in insurance, as well. Except in the case of insurance, the cash flows are flowing in the opposite direction in the form of losses suffered by the company. The math is exactly the same, but instead of valuing cash flows generated by the company for its shareholders, we are valuing the cash flows lost by the company via claims payments to its policyholders. It's expected equity value in the former and expected obligation or liability value in the latter.

We are convinced that Buffett thinks of risk, be it via investing or via underwriting, using the same approach grounded in probability and expected values.

Above is a simple example of Trump who, during 1 year, can experience NO LOSS or a LOSS of -$100. Trump has ONLY two possible scenarios, each with a probability of 50%, that can occur during the year. When you multiply the 50% probability to the “Loss/Person” and “Premium/Person”, you end up with a probability-weighted expected “Loss/Person” and “Premium/Person” – the difference is the probability-weighted expected underwriting profit (loss) per person for the year.

As one can see, if we assume the insurance company is charging Trump a premium of $50 in both scenarios, scenario 1 with NO LOSS results in the company profiting $50 from underwriting. In scenario 2, with a -$100 LOSS, netted against the $50 premium charged to Trump, an underwriting loss of -$50. If one probability weights (with 50% probability) each of the scenarios, the expected underwriting profit of each scenario yields $25 and -$25, respectively. Adding them together, we get an expected value of $0 or break-even underwriting profit.

The problem in this case is that these expected values are “expected” over many trials. What we have here is a single coin flip where you’ll either gain $50 or lose -$50. If you could only make this coin flip once a year, is this a coin flip you would make? I wouldn’t. Even if the premiums were higher, the degree of gain and loss would be fairly wide.

The only way to make this a relatively safe coin flip is to charge a premium of $100, to be covered in the case there is a loss of $100. If there isn’t, then a profit of $100 is made.

But what happens when we add more policyholders?

You’ll notice above that the expected underwriting profit per person is still $0 on an aggregated basis, but the range of expected values has tightened from a max profit of $12.50 to a max loss of -$12.50, which is considerably a much more palatable set of outcomes vs. the previous situation where the max expected profit was $25 and the max loss was -$25. In other words, the law of large numbers has allowed the set of outcomes to wrap tighter and tighter around the net expected value of $0 of underwriting profit. See what happens when we add even more policyholders:

The more policyholders are in the pool, the more scenarios of LOSS and NO LOSS combinations there are, such that the probability of the worst case scenario of ALL policyholders experiencing a LOSS (max expected underwriting loss) becomes a smaller and smaller percentage of the increasingly larger and larger set of outcomes. Hence, as we see above, when there are 16 potential scenarios among 4 different policyholders, the maximum expected underwriting loss is -$3.13 with maximum expected underwriting profit of $3.13, both of which had been probability-weighted by 6.25% or 1 out of 16 outcomes.