*If something has a given chance of happening this year, what is the likelihood that it will happen tomorrow?*

In a recent valuation engagement we ran across the following scenario: given the probability of an event occurring in the next year, what is the likelihood of the same event occurring on any given day? The assumption dealt with the chance of a company’s stock price reaching a future, pre-designated threshold price. If the future threshold price were achieved, the terms of the equity grant triggered the exercise of options, thereby resulting in dilution.

The client estimated (and provided the assumption) that its stock had a 20 percent likelihood of reaching the threshold in the following year. Clearly, this type of forward-looking assumption is speculative and subjective, but in the absence of any stronger basis this is nonetheless an accepted practice for many types of financial modeling and valuation exercises. The issue we wrestled with was not the validity of the assumption, which is beyond the scope of this post, but simply how to correctly break down the annual probability for use in a daily forecasted financial model. The value of an outstanding class of stock, which was the overall purpose of our engagement, would be directly impacted by when and if a dilution event were to occur.

Specifically, the question we needed to address was, given the assumption that a stock has a 20 percent chance of reaching a threshold in the next year, what is the daily probability of this event occurring? We assumed 252 trading days in a year for the exercise.

Probability of event occurring in the next year: P_{s} 20{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646}

Discrete chances for the event to occur: N 252 days

Daily probability of the event occurring: D_{s} ?

A first instinct might be to assume that a stock has a 20 percent chance of reaching the threshold every day. This is wrong – a daily probability of 20 percent would lead to nearly a 100 percent annualized probability, as there is an element of compounding that needs to be accounted for when considering the total probability of multiple events. We needed a formula that produces a smaller daily probability.

After recognizing that “D_{s}” of 20 percent vastly exaggerates the yearly probability, it may be natural to consider a daily probability of P_{s} / N = .20 / 252 = 0.079 percent. However, this method of dividing the total likelihood by the number of periods is inaccurate and understates the daily probability. It would roughly approximate the daily probability if our initial assumption, P_{s}, was a 20 percent chance of the event occurring ONCE. In our model we were only interested if the event occurred. Whether the event happened once or every day made no difference, and this probability excluded any potential for the event to occur more than once, thereby underestimating the daily probability.

Financial analysis often requires the consideration of annual rates (interest rates, costs of capital, etc.) over periods of less than a full year. Our scenario shares a similarity with rate calculations in that a percentage cannot be simply divided by the number of periods to come up with a daily/monthly/quarterly figure. With a loan charging 20{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646} interest annually, the quarterly effective rate is not 5 percent, as this would ignore compounding and overstate the period rate. This thought process can be appealing; a rate looks similar to a probability and we already established that there is an element of compounding that needs to be considered in our probability calculation. So we try the formula:

*Er* = ( 1 + *R *)^( 1 / *N *) – 1

Where:

Er = effective rate for period or periodic rate

R = annual rate

N = number of periods

It is tempting to apply this formula to our scenario, substituting annual probability for the annual interest rate. In fact, we were encouraged by an auditor to do exactly this. If we apply this formula to our example in calculating the daily probability:

*Er* = ( 1 + 20{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646} )^( 1 / 252 ) – 1 = 0.072{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646}

It is clear after a quick comparison, however, that this formula under-predicts the daily probability of an event occurring. Recognizing that our “P over N” approach underestimates the true probability with a 0.079 percent probability, yet provides a higher result than the effective rate formula, 0.072 percent, we conclude that this approach is flawed.

Ultimately, we applied Jacob Bernoulli’s research in probability and statistics. The basis of his work focused on a repeating binomial model that only considers two options- success and failure. Retracing the steps of his experiment allowed us to find a usable formula for calculating daily probability.

Success: P_{s} = .20

Failure: P_{f} = 1 – P_{s} = .80

With P_{f} representing the annual probability of the event NOT occurring.

The probability of the event NOT occurring would look like the following:

P_{f} = .8 = (D_{f})(D_{f})(D_{f}) … (D_{f})

With D_{f}, the daily probability of the event not occurring, repeating as a factor as many times as there are discrete opportunities for the event to occur. In our scenario, there are 252 trading days.

.8 = (D_{f})^{252}

Rearranging, we get:

D_{f} = (.8)^{1/252}

Any scenario which is not a failure (the event not occurring on each of the 252 chances) would be a success scenario (the event occurring one or more times). Knowing that *something* will occur each day, either success or failure:

D_{f} + D_{s} = 1

D_{s} = 1 – D_{f}

Or,

D_{s} = 1 – (.8)^{1/252} = .089{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646}

And, more generally:

D_{s} = 1 – (1 – P_{s}) ^{1 / N}

Therefore, if we know the probability (P_{s}) over a longer period of time, we can calculate the probability of success (D_{s})in each discrete period within that horizon (N). We used the daily output to probability-weight the impact of dilution on the class of stock we valued, but this formula provides meaningful results in a variety of scenarios when reducing the horizon for any time-dependent probabilities.

1. This formula requires the simplifying assumption that daily stock prices are independent of each other.