Actuarially Fair Price of Insurance Policy

A Finance 4335 student asked me the following question during today’s office hours:

Q: “How do you find the actuarially fair price (premium) for an insurance policy?”

Here’s my answer to this question:

A: The actuarially fair price (premium) corresponds to the expected value of the insurance indemnity; the indemnity is the amount of coverage offered by an insurance policy. Under “full coverage”, 100% of the loss is indemnified, and in such a case, the actuarially fair premium is equal to the expected value of the loss distribution.

For what it’s worth, the concept of “actuarially fair” insurance prices/premiums, along with implications for the demand for insurance, is explained in two previously assigned readings (italics added for emphasis):,

  1. on page 4 of the Supply of Insurance reading (just prior to the section entitled “Example 2: Correlated Identically Distributed Losses), the following sentence appears, “A premium that is equal to the expected outcome is called an actuarially fair premium”;
  2. on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) reading, consider the following excerpt: “Ignoring transaction costs, an insurer charging a premium equal to expected loss would break even if it held a large portfolio of such policies. This premium could be called a fair premium or an actuarially fair premium, denoting that the premium is equal to the expected value of loss (sometimes called the actuarial value of the policy). The term fair is not construed in a normative sense; rather it is simply a reference point”; and
  3. on page 43 of Doherty, Chapter 2, in the first sentence of the first full paragraph: “We know from the Bernoulli principle that a risk averter will choose to fully insure at an actuarially fair premium.””

The Birthday Paradox: an interesting probability problem involving “statistically independent” events

During the statistics tutorial, we have discussed (among other things) the concept of statistical independence and focused attention on some important implications of statistical independence for the binomial probability distribution.

Here, I’d like to call everyone’s attention to an interesting (non-finance) probability problem related to statistical independence. Specifically, consider the so-called “Birthday Paradox”. The Birthday Paradox pertains to the probability that in a set of randomly chosen people, some pair of them will have the same birthday. Counter-intuitively, in a group of 23 randomly chosen people, there is slightly more than a 50% probability that some pair of them will both have been born on the same day.

To compute the probability that two people in a group of n people have the same birthday, we disregard variations in the distribution, such as leap years, twins, seasonal or weekday variations, and assume that the 365 possible birthdays are equally likely.[1] Thus, we assume that birth dates are statistically independent events. Consequently, the probability of two randomly chosen people not sharing the same birthday is 364/365. According to the combinatorial equation, the number of unique pairs in a group of n people is n!/(2!(n-2)!) = n(n-1)/2. Assuming a uniform distribution (i.e., that all dates are equally probable), this means that the probability that no pair in a group of n people shares the same birthday is equal to p(n) = (364/365)^[n(n-1)/2]. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability is p’(n) = 1 – (364/365)^[n(n-1)/2].

Given these assumptions, suppose that we are interested in determining how many randomly chosen people are needed in order for there to be a 50% probability that at least two persons share the same birthday. In other words, we are interested in finding the value of n which causes p(n) to equal 0.50. Therefore, 0.50 = (364/365)^[n(n-1)/2]; taking natural logs of both sides and rearranging, we obtain (ln 0.50)/(ln 364/365) = n(n-1)/2. Solving for n, we obtain 505.304 = n(n -1); therefore, n is approximately equal to 23.[2]

The following graph illustrates how the probability that a pair of people share the same birthday varies as the number of people in the sample increases:[1] It is worthwhile noting that real-life birthday distributions are not uniform since not all dates are equally likely. For example, in the northern hemisphere, many children are born in the summer, especially during the months of August and September. In the United States, many children are conceived around the holidays of Christmas and New Year’s Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g., a class in a school), this creates a tendency toward particular dates. Both of these factors tend to increase the chance of identical birth dates since a denser subset has more possible pairs (in the extreme case when everyone was born on three days of the week, there would obviously be many identical birthdays!).

[2]Note that since 54 students are enrolled in Finance 4335 at Baylor University this semester, this implies that the probability that two Fall 2021 Finance 4335 students share the same birthday is p’(54) = 1 – (364/365)^[54(53)/2] = 98.03%, although given footnote 1’s caveats, it’s likely that there may be several shared birthday pairs.

Origin of the “Power Rule”, and Visualizing Taylor polynomial approximations

This blog entry provides a helpful follow-up for a couple of calculus-related topics that we covered during the Mathematics Tutorial.

  1. See page 12 of the above-referenced lecture note. There, the equation for a parabola (y = {x^2}) appears, and the claim that dy/dx = 2x is corroborated by solving the following expression: \displaystyle\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \left[ {\frac{{{{(x + \Delta x)}^2} - {x^2}}}{{\Delta x}}} \right].In the 11-minute Khan Academy video at https://youtu.be/HEH_oKNLgUU, Sal Kahn takes on the solution of this problem in a very succinct and easy-to-comprehend fashion.
  2. On pp. 18-23 of the Mathematics Tutorial, I showed how y = ex can be approximated with a Taylor polynomial centered at x=0 for \delta x values ranging from -2 to +2.  In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn essentially replicates the work I did; the main difference between my example and Sal’s example is that Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0.  The important insight provided in both cases is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

On the ancient origin of the word “algorithm”

The August 31st assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) provides a succinct synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. Here’s a fascinating quote from page 33 which explains the ancient origin of the word “algorithm”:

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.

Calculus and Probability & Statistics recommendations…

Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with algebra, differential calculus, and probability & statistics, the second class meeting during the Spring 2021 semester will include a mathematics tutorial, and the third and fourth class meetings will cover probability & statistics. I know of no better online resource for brushing up on (or learning for the first time) these topics than the Khan Academy.

So here are my suggestions for Khan Academy videos that cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):

Finally, if your algebra skills are generally a bit on the rusty side, I would also recommend checking out the Khan Academy’s review of algebra.