Arbitraging Anthropic

If you wanted to purchase an annual subscription of Claude Pro you have two options; either pay twenty dollars at the beginning of each month for 12 months, or pay two hundred up front. Implicitly, this structure is similar to a car loan in the sense that you can choose between paying a smaller nominal amount for the product now ($200), or a larger nominal amount over the course of twelve payments ($240). In a loan, the difference between the principal and total amount paid is the interest, and the rate which makes the series of payments equal to the principal is the interest rate. In our case, we’re going to be solving for the monthly interest rate, which tells you how much a dollar at the start of the month is worth at the end of the month. For example, if the interest rate is 2%, a dollar at the start of the month grows to $1.02 by the end. A dollar n-periods in the future is therefore worth \(1 \cdot (1.02)^n\) Flipping it around, we can bring future dollars back to present-day dollars through discounting, or dividing by one plus the interest rate. A future dollar, discounted by n-periods, is therefore worth \(1 \cdot (\frac{1}{1.02})^n = 1 \cdot (0.9804)^n\) The term \(\frac{1}{1+i}\) is sometimes termed \(v\), or the present value factor, to emphasize it’s what you multiply by to bring a future dollar back to the present. With that in mind, we can ask the question; what is the interest rate implied by these payment options?

Let \(v\) be the present value factor corresponding to the interest rate of \(i\). Each successive payment is discounted once more by \(v\), so the entire series is:

\(200 = 20 + 20v + 20v^2 ... 20v^{10} + 20v^{11}\)

\(200 = 20 \cdot \frac{1-v^{12}}{1-v}\)

This can’t be solved algebraically, so using a calculator: \(v = 0.966 = \frac{1}{1+i}\), \(i = 3.5\%\)

Multiplying by 12 months in a year gives us an APR of 42%, which is considerably higher than most credit cards. What this means practically is that if you’re liquidity constrained and want to keep the same monthly payment structure; you can do so while paying less then Anthropic charges. At a 24.5% APR (my credit card rate), you could put that $200 on the card and pay only $18.58 at the start of each month thereby arbitraging this internal rate to get the best of both worlds. A cheaper total outlay ($240 vs $223.96), and monthly payments.