# Answer

**Analysis of BoB CD Investment**

Depositing $1000 into a BoB CD causes you to owe the BoB *another *$1000 at maturity. In other words, you lose *twice *your original principal. It is indeed the deal of a lifetime, but not for you.

What follows is the explanation, complete with an animation depicting this novel financial instrument. Picture your balance on the complex plane. Your balance starts on the positive real axis. After the first second, a tiny bit of imaginary interest accrues, nudging your balance upwards off the real axis. The picture below shows this first step. The green dot represents the initial balance; the blue dot represents the balance after compounding:

The picture greatly exaggerates the angle θ. Under BoB's per-second compounding, the actual angle is about 6.6 millionths of a degree. The next picture shows the balance, as a purple dot, after two seconds, again with an exaggerated angle.

Each compounding rotates your balance about the origin, and pushes it away from the origin ever so slightly. For now, consider just the rotation. Each rotation is tiny, but they are many, about 27 million under the BoB's terms. The net rotation is a full 180 degrees.Your money ends rotated onto the **negative** real axis! The video below shows the rotation of one dollar invested in a BoB CD.

You could recover your money by waiting through another 180 degrees of rotation. But the BoB CD offering clearly states that the CD is non-renewable. You *did *read the terms carefully, right? The BoB is most scrupulous about following through on contract terms.

But for argument's sake, how much would you have if you could renew? Note that in the diagrams, the hypotenuse of each triangle is longer than the long leg of the triangle. So your money doesn't just go around in circles, is also spirals outward ever so slightly. You might hope to gain a lot here. After all, with real interest rates, more frequent compounding gets you more interest, so per-second compounding sounds great. Alas with imaginary interest rates, more frequent compounding decreases the spiral growth! In the limit of infinitely frequent compounding, your money runs around in a circle! After one term your CD would be worth exactly $-1000, which you can compute using the standard formula for continuous compounding and the famous identity *e*^{πi }= -1:

$1000×*e*^{(100π)(0.01i)} = $1000×*e*^{πi} = $1000×(-1) = $-1000

With continuous compounding and a renewal for a second term (in violation of the contract), you would get back to exactly $1000.

However, BoB is compounding per second, not continuously, so there would be a tiny bit bit of push outwards. Thus there would be some real interest to be gained, if you somehow managed to renew the BoB in spite of the offering terms. How much? About about 0**.**04 cents. That 4 hundredths of a cent, a sliver of chump change, evaporates after rounding. You *did *read that part of the terms too, right?

Most of the financial press moved on to more pressing news, though a few outlets reported the denouement: