Is Hearthstone’s Heroic Tavern Brawl worth it?

TL;DR: If you’re a scratch player, Heroic Tavern Brawl has a slightly higher expected value than purchasing an equivalent amount of packs in the store. However, it has far less value than purchasing the same number of packs via gold.

My little brothers introduced me to Blizzard’s Hearthstone about a year ago. It and Factorio are the only games I play these days. If you have no idea what I’m talking about, then click here.

Today, Blizzard announced a new week-long mode called Heroic Tavern Brawl. For an entrance fee of $9.99 (or 1,000 gold), players are allowed to construct a deck and play in a makeshift tournament until they achieve either 12 wins or 3 losses. The more you win, the better the rewards.

Is it worth playing? I crunch the numbers in this Excel Book and explain below the fold. If you download the file, you can customize the calculation to your own situation.

Continue reading →

Grand Theft Autocorrelation

Michael-Counting-Money-16_RGBIn my last post, I discussed the concept of correlation. In my free time since then, I’ve been playing a lot of Grand Theft Auto V. It’s time to merge these noble pursuits.

As you may know, GTA V includes an online stock market that allows players to invest their ill-gotten gains in fictitious companies. Naturally, a Reddit user has created an updating database of Stock Market prices. I play on a 360, so I’ve analyzed the Xbox prices. I’ve developed a strategy that will earn money in the long-run, but first let’s do some learning.

As we previously discussed, correlation is a measure of how well the highs of one series line up with the highs of another series. Autocorrelation is a measure of how well highs of a series line up with the highs of the previous observation of the same series. Continue reading →

Cruising with Minimal Wallet Bruising

How fast should you drive on long road trips?

One common piece of financial advice is to drive slowly. Per, “gas mileage usually decreases rapidly at speeds above 50 mph. You can assume that each 5 mph you drive over 50 mph is like paying an additional $0.25 per gallon for gas.”

That’s a big deal, and it can lead to some significant savings. However, this piece of advice ignores a giant cost of driving slowly: it takes more time. Therefore, I’ve created back-of-the-envelope cost estimates at various speeds. This is part of my ongoing quest to optimize literally everything.

Without further ado, here’s a graph of the total cost of driving 500 miles at a sustained cruising speed. I’ve assumed leisure time is worth $30 per hour, and I’ve used the same fuel assumptions specified at the above link. With these parameters, the cost of the entire trip is minimized at 95 miles per hour.


All of these can be adjusted in the workbook attached below. For instance, changing the value of leisure time to $10 per hour changes the optimal speed to 67 miles per hour.


I was somewhat surprised to see the cost of time completely dominate the cost of fuel at the interstate speed limit. Fuel is expensive, but you can always buy more of it.

Amusingly, one can use Excel’s goalseek function to back into the value of time of the losers you pass in the right hand lane. Someone driving at 55 miles per hour is implicitly valuing their time at $5.64 an hour. As a persistent leadfoot, I was comforted by this analysis. However, I can already hear my mother begging me to exercise caution. Those who drive quickly are more likely to get into an accident, and that’s worth considering.

Fortunately, I’m not the only person to ask this question. Wikipedia led me to a discussion of the Solomon Curve, a graphical representation of driving risk. Apparently, the log of crash risk has a quadratic relationship with driving speed. Drive either above or below the median speed, and your risk of crashing exponentially increases.


Table 7 of the original paper presented a fairly detailed risk of injury and death depending on the driving speed and the time of day (day vs night). I used this data, the aforementioned relationship and two regressions to develop risk curves for day-time and night-time driving. This is old data, and someone else could probably improve the results by looking at more contemporary sources. However, it suits my back-of-the-napkin purposes.

To translate the risks of crashes into the expected value of crashes, I had to assume costs for both injuries and deaths. For injuries, I assumed $50,000. After insurance, that’s probably a little high, but it’s good enough for our purposes.

The next assumption I had to make is slightly more complicated. How does one value the threat of losing their life? What cost do you assign to daylights, sunsets, midnights and cups of coffee forever squandered in a driver’s overzealous haste to get to the next red light? Sure, in 1999, the EPA estimated a life’s value at $5.5 million while calculating the value of air regulations, but just because I’m an economist doesn’t mean I’m a monster. Who am I to blindly annoint a dollar value to an anonymous other’s very existence? It would go against everything I hold dear to say lives are worth $5.5 million each.

First, we must adjust it for inflation. In 2013 dollars, one life is worth roughly $7.7 million. Moving on…

Due to the quadratic and exponential nature of speed and collisions, the right side of the graph quickly balloons. With the original assumptions, the back of my envelope recommends going 65 miles per hour in the day and 60 miles per hour at night. For my non-American readers, that’s roughly 105 kmh and 97 kmh, respectively. For my pirate readers, that’s 56 and 52 knots, respectively.

The day chart is provided below. Please note that the x-axis has been restricted to 90 miles per hour, unlike the prior graphs.


Of course, all of this assumes you have a radar detector, an average driving ability and an average love of life. The Solomon Curve is anchored on the speed of your fellow travelers, not the speed of an old data set. Take everything into account, and don’t be afraid to disregard

My model can be downloaded here:

Total Cost of Trip

Edit to add: A friend of mine let me know another potential ending. The other possible moral of this story is that everyone should drive at 95 miles per hour. Balloonfoots are imposing a negative externality. How did I miss that?

Housing: Buy or Rent Calculator Part 2 of 2

In 2000, billionaire Nicolas “the homeless billionaire” Berggruen traded his New York apartment and private island for continual visits to 5-star hotels. This was likely a lifestyle rather than an investing choice, but it can be seen as an extreme version of the analysis I started in my last post on buying or renting.

When I first set about analyzing the issue, I wanted to give a definitive answer in favor of one side or another. However, there are too many factors at play. I’ve created a flexible excel model that allows your inputs to compare the payoff from buying and renting. Download it by clicking here:

Buy vs. Rent Calculator 10.26.12

The model compares the projected value of two portfolios:

  1. House equity — House Value less long-term debt.
  2. Investment portfolio — The expense saved from renting instead of buying is instead contributed into a stock market portfolio.
All assumptions can be adjusted on the third sheet. Sheet 2 gives a detailed look at the cash flows on all projected months. The first sheet graphically shows the value of each portfolio over time.
By default, I have a fifteen year mortgage analyzed for 30 years. With the default assumptions, the stock market portfolio can easily pay for the saved rent after the house is paid off. However, the appreciated value of the house makes the first portfolio worth far more. With the default specification, “buy” beats “rent” under every time horizon. However, if your individual situation merits different assumptions, then the forecast may change.
If you play with the model, you’ll quickly realize that it is extremely sensitive to changes in the house appreciation rate. If you are confident you’ll experience capital gains, then buying is usually a solid investment. However, you shouldn’t rush into a home purchase just because an excel model showed that buying was marginally better. If you get stuck in a bad location and are unable to sell the house for a profit, then the decades of reduced liquidity were for nothing.

Housing: Buy or Rent Calculator Part 1 of 2

Many Americans allocate the majority of their household’s assets into their home. Deciding whether to rent or buy isn’t just a lifestyle choice, it’s an investing decision that should be approached with as much caution as choosing the right stock.

The primary benefit of buying is that you retain the principal of your mortgage payments. All rent money is sacrificed to the landlord. Additionally, you might profit from any appreciation in housing prices.

Renting, however, is not without its benefits. Renting requires no down payment and is usually substantially cheaper per month, even before considering the additional expenses of home ownership (maintenance, insurance, HOA fees, property taxes, etc). Assuming one was financially secure enough to afford the house, the additional savings from renting can instead be placed into another investment, such as the stock market or a personal business, from which one might expect a higher return.

While higher returns from the market aren’t guaranteed, as 2008 showed us, neither is house price appreciation. Even in normal times when house prices are steadily marching upwards, house prices in certain areas may be drastically declining. Furthermore, while the median house sold has increased in price over the past five decades, the median house sold has also become larger, newer, more modern and more comfortable. Individual houses owned by a single owner become larger and more modern only after large capital expenditures, and they never become newer.

But even if the returns of the stock market might be higher with less overhead, that doesn’t mean that one will make more money from investing in the stock market. Real estate is not only an investment, it is a leveraged investment. A 2 percent annual house appreciation is applied not just to the down-payment but to the entire value of the house. Most Americans cannot achieve that level of leverage in the stock market. As long as the house value increases, it is a definite benefit.

Considering all of these factors, is home ownership a worthwhile investment? This is a quantitative question that I can’t answer with qualitative arguments. In my next post, I’ll share an excel workbook that you can customize with your specific situation.