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19.12.2023

The (Pricing) Nightmare before Christmas

The legend that is Santa Claus / Father Christmas / Kris Kringle goes back to the 4th century in what is now modern-day Türkiye. A monk called St Nicholas was renowned for his kindness and later known as the protector of children and sailors. In Western Christian countries, St Nicholas’ Day is celebrated on 6 December, the anniversary of his death.

In the UK, Father Christmas dates as far back as the 16th Century; the feast day of St Nicholas on the 6 December was moved to coincide with the celebration of Christmas Day and included Father Christmas as an emblem of good cheer.

The first reference to the Santa Claus we know today in the big red coat with his sleigh being pulled by reindeer was in 1821 when an illustrated children’s poem was published in New York, ‘Old Santeclaus with much Delight’.

The legend grew following the anonymous publication of the poem ‘A Visit from St. Nicholas’, better known today as ‘The Night before Christmas’ in 1823. It is largely credited for the more traditional legend we know today with ‘a right jolly old elf’ with ‘clothes all tarnished with ashes and soot’, twinkling eyes, merry dimples, and a beard ‘as white as snow’, a ‘miniature sleigh’ and ‘tiny reindeer’.

Whatever we call him, he’s a huge part of the popular culture of Christmas across the world, and he’s showing no sign of slowing down. However, despite his longevity and mystical powers, Santa is just like many of us; there are areas in his life both, personal and commercial, where he incurs a great deal of risk. As such, there are many areas where actuaries could be of use to him; be it pricing insurance for his grotto, for his reindeer, or simply business interruption insurance for when the elves inevitably go on strike.

Some of the questions this raises might be slightly silly. How do you price insurance against a magical reindeer’s shiny nose?

It might be difficult to get an exact answer, but pricing famous body parts is nothing new to actuaries. For example, in 2006, David Beckham’s legs were famously insured for £100 million. Rudolph might not be as profitable as Beckham, but he is essential in bringing joy to billions of children, so it would be conceivable to price such a thing.

But that is not the question we have set out to answer today. The silly question we are approaching, in as serious and logical a way as we possibly can is “How do you insure Santa’s sleigh?”. So, fellow actuaries, grab your hot chocolate, wrap up in something warm and join us in unravelling how we reached our totally serious answer.

Where to start?

We decided to use aviation insurance as our starting point, with the reasoning that planes:

  1. are commonly insured, and
  2. fly

When it comes to pricing an insurance contract for a plane there are many factors we need to take into consideration, such as: distance travelled, cargo and plane value, experience of the pilot, and so on. In our attempt to mirror these considerations with Santa’s sleigh, let us break down the numbers.

How far is he travelling?

To get the most accurate answer we would create a network of houses and find the shortest route between them. We would consider population density, distribution of belief in Santa, the curvature of the Earth, and many more things.

Due to materiality considerations, this is not what we did!

To estimate the number of houses he would have to visit, we assumed 25% of the Earth’s population of 8 billion are children (in this article defined as people 0-14 years of age, to match with most statistical measures), and that 35% of children believe in Santa. Around 30% are Christian, but Christmas is not just a Christian holiday anymore, it has become more secular over time, hence the extra 5%. We also have not discounted for attrition in the belief in Santa, as if Santa is delivering gifts and buying insurance, why would any child not believe in him? We used current fertility and household rates to estimate that on average there are 2.3 children per house.

Number of households = (8,000,000,000 × 0.25 × 0.35) / 2.3 = 304,000,000

To calculate the distance travelled between these houses on Christmas Eve, we imagined Earth as nearly 1.5 million 10km by 10km squares, (in total this would be equal to the landmass of the planet) , with houses normally distributed within each one. Finding Santa’s route is akin to a Travelling Salesman Problem (although Santa does not return to the original house within each square) and we have used a Greedy Algorithm to find a path for him.

On average, for this type of problem, this route is 25% longer than the shortest one possible. We have not adjusted our final answer to accommodate this as we assume that Santa also does not have the time or inclination to find the optimal route.

We modelled this in R to find an average distance between households. In each square we assume Santa starts at the one closest to the origin (this was chosen arbitrarily) and that he travels from this house to the next closest house and so on, until he reaches the end. We also assume he has come from a point an equal distance away on another square.

We then multiply by the number of squares to get a final distance travelled of and used the number of households to get an average distance travelled of over 140 million km. This is the same as travelling to the moon and back almost 200 times!

How much is the sleigh worth?

When you ask anyone how much Santa’s sleigh costs, they’ll tell you ‘Nothing – it’s on the house’.

To come up with an actual answer, we opted for a more familiar benchmark to calculate the value of the sleigh – the value of a Boeing 747. With each of these aircraft valued at over £330 million, the question arises: how does the sleigh measure up in comparison?

To bridge this valuation gap, we have decided to use the factor of weight , which leads us to yet another question of how much will the sleigh weigh? To simplify our calculations, we will consider only the sleigh and presents and not include reindeer weight or fuel. Santa will be carrying over 700 million presents (if each child receives exactly one gift), and we are assuming each present weighs 1kg.

Generally, Boeing 747s have a maximum weight on take-off of twice their own weight, so by this logic , the sleigh will also weigh 700,000,000kg. This is equivalent to 3,750 Boeing 747s!

Total cost = 3,750 × 330,000,000 = £1,240,000,000,000

How much are the presents worth?

You will be relieved to hear this part is simple! We are assuming Santa is delivering 700 million gifts, each worth £10 so our total cargo is worth £7,000,000,000.

What are the chances of the sleigh crashing?

The chance of a regular plane crashing is roughly 1 in 12 million and 96% of these crashes happen at take-off or landing. In our model, we are assuming Santa is taking off and landing just once at his own home and throwing the presents through the chimneys whilst airborne, so this is less of a concern for us. The crucial part is the 4% of crashes that transpire in mid-air. Typically, the average plane journey is 1,000km and since Santa is travelling 140,000 times further, we are increasing his probability of crashing by this factor.

Chance of crash = 0.000000083 × 0.96 + (1 – (1 – 0.000000083 × 0.04) ^ 140,000) = 0.00046

Any discounts?

With Santa boasting an impressive 202 years of sleigh-piloting, he has a combined total of nearly 10,000 hours of flight hours, elevating him to the esteemed rank of an ‘experienced’ pilot. Santa also has an enviable track record of almost no claims, save for the minor mishap immortalised in the 2003 film Elf, where he had a run-in with Central Park. However, since the incident was minor and a full 2 decades ago, we couple this with his extensive experience to secure him a 25% discount on his insurance quote.

It is not all bad news for Santa, his seasoned expertise enables him to qualify for some substantial discounts.

Accounting for time differences, Santa has a 31-hour window in which he can deliver his presents on Christmas Eve. Additionally, not all countries expect a delivery this day, for example, he visits Hungary and Slovenia in early December and Russia, Georgia, and Ukraine in early January. These different delivery days help him to clock in at least an additional 18 hours of flight experience a year.

What is the final number?

Final hull insurance   = Cost of sleigh × Chance of crashing × Discount
= 1,240,000,000,000 × 0.00046 × 0.75
≈ 428,000,000
Final cargo insurance = Cost of cargo × Chance of crashing × Discount
= 7,000,000,000 × 0.00046 × 0.75
≈ 2,420,000

We have calculated that safeguarding the sleigh itself would cost a whopping £428,000,000. On top of this, Santa cannot risk all those precious presents going unprotected – so we have also factored in another £2,420,000 for cargo insurance.

Our final cost for Santa’s insurance is nearly half a billion pounds which seems like quite a hefty sum, but this is just the baseline coverage. If Santa wants to protect against the Grinch’s thieving tendencies, insure Rudolph and Co. with key person insurance, or cover potential mishaps causing injury or property damage, those policies would jingle up an even pricier total . On top of this, our quote also does not include any loading for profit, so he can expect the final price to be even higher!

Sources:

https://www.history.com/topics/christmas/santa-claus#the-legend-of-st-nicholas-the-real-santa-claus
https://www.historic-uk.com/CultureUK/St-Nicholas-Day/
https://en.wikipedia.org/wiki/Santa_Claus
https://www.english-heritage.org.uk/christmas/the-history-of-father-christmas/#:~:text=Historians%20think%20that%20Santa%20Claus,appeared%20in%20a%20mummer’s%20play.
https://knaviation.net/boeing-747-specs/
https://www.worldometers.info/world-population/
https://ourworldindata.org/fertility-rate
https://data.worldbank.org/indicator/SP.POP.0014.TO.ZS
https://www.aerotime.aero/articles/29762-how-fast-santa-must-be-during-christmas-eve
https://simpleflying.com/boeing-747-value-2022/#:~:text=According%20to%20Boeing’s%20current%20list,slightly%20higher%20at%20%24419.2%20million

Georgina Williams, Patrick Vieira and Aileen Clements

December 2023