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.

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