An interest rate question extending Kate Nowak’s rate post

Earlier in the week Kate Nowak wrote a neat post about rates. The perspective in the post (in my words) is coming from writing curriculum materials for 6th grade math:

Here’s an alternate perspective on the same (or at least similar) issue that I encountered at work this week.

Suppose I ask you to play the following game:

(1) You pay me $2 today.
(2) I’ll then select an integer from 1 to 10 at random (uniformly)
(3) At the end of year 1 you pay me $1, and if my random number was 1 I’ll pay you $10 and the game stops. If my number wasn’t 1 we’ll meet again next year.
(4) In general, at the end of year n, you’ll pay me $1 and if the random number I picked was n, the game stops.

The interest rate question relating to this games is this: What is your expected rate of return for playing my little game?

Here are two different ways to think about it:

(1) Internal rate of return

You’ll see an expected set of cash flows that look something like this:

Screen Shot 2016-01-23 at 11.09.40 AM

The “internal rate of return” on those cash flows is about 12%, so you might say (and I think that many people would be quite comfortable saying) that your expected rate of return playing my game is about 12%.

(2) Accounting for the costs and the investment returns differently

One possible objection to the internal rate of return calculation is that your cash outflows are really part of your investment in the game and so are quite different than the investment return. In fact, to play the game all the way through, in addition to the $2, you need to be sure that you have access to $10 over time to play.

So, you might prefer to discount your cash outflows at a less risky rate – I’ve picked 4% just for example purposes – and discount the inflows (the investment returns) at a risky rate to measure your return. That calculation looks something like this:

Screen Shot 2016-01-23 at 11.14.43 AM

Using this method the expected investment return you’ll get for paying $2 to play my games is more like 8% per annum.

So, what is the correct way to think about the rate of return for playing my game?

I think the rate of return question here is pretty interesting to think about and gives a real life example of the things that Nowak is thinking about writing 6th grade curriculum.

Revisiting the Surreal Numbers

I bought a copy of Donald Knuth’s Surreal Numbers recently:

Reading it yesterday gave me the idea to revisit the surreal numbers with the boys today. Our previous look at the surreal numbers was inspired by Jim Propp’s off-the-charts-excellent post:

Life of Games

Although I hope today’s project stands on its own, our previous projects inspired by that post are here if you are interested:

Walking down the path to the surreal numbers Part 1

Walking down the path to the surreal numbers Part 2

Today I started by reminding the kids about the game “checker stacks” that Propp describes in his post and then playing through a few simple examples:

Next up was our first challenge – finding the value of a red + blue checker stack. The boys determine that the value is negative fairly quickly. After a bit of investigating they determine that the value is -1/2. I think this is a wonderful example of what kids doing math looks like:

Now we moved to a problem that is a bit more challenging – a blue – red – red stack. This is a more difficult investigation, but the boys spend about 10 minutes exploring and experimenting. Over the course of the next two videos they determine that the value is 1/4.

Again, I think this is a really nice example of what kids doing math looks like. I’d love to try out this investigation with a larger group of kids:

There are lots of directions to go with this investigation – tomorrow I’d like to explore the “deep blue,” “deep red,” and “deep purple” checkers discussed at the end of Propp’s post. I’m excited to see if kids can understand the ideas that come up with these special checkers.