# Calculating Cost of Death Risk

# Background

I’m planning a day-trip and trying to weight my travel options. I want to include the following costs:

- Direct $ cost
- lost productivity (“opportunity cost” ???)
- chance of death

I normalize everything to dollars to make them more comparable/tangible.

TL;DR: check out my spreadsheet. I decided to bus.

# Cost Calculations

## Direct Costs

This is easy to calculate, e.g. cost of bus/plane tickets, cost of gas, etc. Factoring in vehicle depreciation was the trickiest part - since I have a 12-year old vehicle with 200k miles, I just roughly estimated $0.10 / mile, which is on the low end of typical depreciation estimates. As you’ll see, this didn’t end up making a difference since my car has such bad mileage that it’s cheaper to rent a car anyway.

## Cost of lost productivity

This is also pretty easy to calculate. Since this is happening on a weekend and my weekends are usually just me sleeping most of the day anyway, I valued my time at a measly $5 / hour. I also give multipliers for how “productive” I could be on a bus vs plane.

## Cost of death

This is an interesting one. First I calculate chance of death, then the expected value of days of my life lost, then convert to dollars.

### Chance of death

First calculate the chance of death. This is easy to do at a first-order approximation - passenger cars have ~1.5 deaths / billion vehicle miles according to NHTSA. Buses are purportedly 50 times safer than passenger cars, and planes 750 times safer. More accurate calculations are tricky for a number of reasons:

- actually a lot of the deaths due to cars are e.g. pedestrian deaths, not passenger deaths
- car safety ratings matter
- my driving abilities matter - I give a 1.5 multiplier since I consider myself a below-average driver
- time of day matters
- geographical location matters (e.g. easy highway vs busy metro city)
- non-fatal injury is not accounted but does matter

In any case, I just use the first-pass approximation.

### Expected value of days lost

We could analyze this 2 ways:

- what is the cost of going on the trip in the first place
- assuming I am definitely going on the trip, what’s the cost of death for different options?

I’m pretty sure these both turn out to be very nearly equivalent. Let me explain why.

#### Justification of equivalence

First, option 2 is easy to calculate: just multiply the (probability of death) by the (sum of all future joys) to figure out how much joy you’ve lost by exposing yourself to death risk.

Option 1 is more interesting. The formula should be something like:

\[E[\text{no trip}] - E[\text{yes trip}]\]where \(E[\cdot]\) denotes the expected value. “no trip” means I never went on the trip, while “yes trip” means I did go on the trip, and took a given mode of transportation. Expanding out \(E[\text{yes trip}]\),

\[E[\text{yes trip}] = P(\text{not dying})V(\text{not dying}) + P(\text{dying})V(\text{dying})\]where \(P(\cdot)\) denotes the probability of an event occuring, and \(V(\cdot)\) denotes the value/joy derived from that event occurring. Both “not dying” and “dying” are assuming we did go on the trip. Note that obviously \(V(\text{dying}) = 0\) since we will never achieve any joy ever again by dying.

\[V(\text{not dying}) = V(\text{no trip}) + V(\text{the trip itself})\]here, \(V(\text{the trip itself})\) denotes the joy I derive from going on the trip. Then,

\[\begin{align*} E[\text{yes trip}] &= [1-P(\text{dying})][V(\text{no trip}) + V(\text{the trip itself})] + P(\text{dying})\cdot 0 \\ &= 1\cdot V(\text{no trip}) - P(\text{dying}) V(\text{no trip}) + (1-P(\text{dying})) V(\text{the trip itself}) \\ &= E[\text{no trip}] - P(\text{dying}) V(\text{no trip}) + (1-P(\text{dying})) V(\text{the trip itself}) \\ &\approx E[\text{no trip}] - P(\text{dying}) V(\text{no trip}) + V(\text{the trip itself}) \end{align*}\]So the cost ( cost = -value ) of going on the trip is:

\[E[\text{no trip}] - E[\text{yes trip}] = P(\text{dying}) V(\text{no trip}) - V(\text{the trip itself})\]To do a quick check to see if the trip is worth it in the first place, we calculate \(P(\text{dying}) V(\text{no trip})\) and see if it’s reasonable. To compare different transportation methods, the \(V(\text{the trip itself})\) term will cancel and we just compare \(P(\text{dying}) V(\text{no trip})\) for the different methods. So in other words, the value of we need to calculate is \(P(\text{dying}) V(\text{no trip})\), or in other words, (probability of death) \(\cdot\) (sum of all future joys), which is exactly the same as Eq. \eqref{eq:cost_to_chance_of_death} in option 2. (QED)

#### Evaluation

In section Chance of death we already calculated \(P(\text{dying})\) so now we just need to calculate \(V(\text{no trip}) := (\text{sum of all future joys})\).

This depends heavily on both the probability of death vs age curve (e.g. “# of lives” column of an Actuarial Life Table) and also the utility of a year of life at a given age (presumably younger years are more valuable than older years).

I just make a 0th order approximation that when you multiply these two together and average over all future years, 1 year is approximately worth 0.5 “units” where I define a “unit” to be the value of one year at my current age. My logic is that when multiplying together the death curve with the joy/utility curve, it will be something like a line that starts at 1 right now and ends at 0 at age 75, so the average will be 0.5.

At age 25 and assuming life expectancy 75, that’s

\[(50 \text{ years}) \cdot (0.5 \text{ joy-years/year}) \cdot (365 \text{ days / year}) = 9125 \text{ joy-days}\]Finally, to convert to dollars, I make the rough approximation that the value of 1 day is worth $400. This is based on the fact that I would work for about $50-100/hour for 40 hours / week; and, as a rational being, that is “worth it” to me, so that must be a reasonable estimate of my day’s worth. That makes $2000-4000 / week = $286-571 / day, so I say roughly $400 is reasonable. Note that if I thought my day was worth more, I would work fewer hours; and if I thought my day was worth less, I would work more hours. There’s some subtlety here that I’m missing but in any case, I think it’s a reasonable estimate.

Then I arrive at the figure,

\[(\text{sum of all future joys}) \approx (9125 \text{ joy-days}) \cdot ($400 / \text{joy-day}) = $3.65 \text{ million}\]Then the cost of traveling due to chance of death according to Eq. \eqref{eq:cost_to_chance_of_death} is

\[\boxed{P(\text{dying}) \cdot ($3.65 \text{ million})}\]# Final Cost Tables

## Transportation

Direct Cost | Productivity cost | Chance of Death | Expected cost of lost joy-days | $-equivalent joy-day cost | |
---|---|---|---|---|---|

Car | $254.82 | $63.33 | 0.0000171 | 0.1560375 | $62.42 |

Rental Car | $174.60 | $63.33 | 0.00002565 | 0.23405625 | $93.62 |

Bus | $80.00 | $23.75 | 0.000000342 | 0.00312075 | $1.25 |

Plane | $400.00 | $30.00 | 0.0000000228 | 0.00020805 | $0.08 |

## Lodging

LODGING | Direct Cost |
---|---|

Airbnb | $68.50 |

Hotel | $87.50 |

## Final Options

Direct Cost | Cost (including cost of death) | Cost (including cost of productivity + death) | |
---|---|---|---|

Rental + Sleep in car | $174.60 | $268.22 | $331.56 |

Car + Sleep in car | $254.82 | $317.24 | $380.57 |

Bus + AirBnB | $148.50 | $149.75 | $173.50 |