continuous compounding

I've got a couple of friends who were professional mathematicians. I know very well that I'm not in that class, but I have used mathematics to solve problems from time to time. I also know where to look to find worked-out mathematical solutions. It's a tedious search. Happily Spehro Pefhany has done it for us.

Happily for YOU I think you mean. ;->

Let's calculate the effective APR for RichD's notional 2% APR continuously compounded for a one year term. Let:

P = $1.00 r = 0.02 t = 1

then A = Pe^(rt) = 1.00e^(0.02) = 1.02020134 so the effective APR over a period of one year is 0.02020134 and the effective APR over a ten year term becomes 0.0221402758.

We're in an era where central bankers speak of Negative Interest Rate Policy (NIRP) with a straight face. (Yield, in the form of a bond flip to a Greater Fool, allows a NIRP bond investor to recover principle lost to negative interest.)

Plug in a notional -0.5% APR over a one year term into the formula:

A = 1.00e^(-0.005) = 0.995012479

So, a NIRP investor loses about half a cent in one year, or about a nickel over ten years. Not a big bite. Unless, of course, you're a fund manager with one hundred million dollars on deposit. Then you lose half a million per year.

Danke,

So a mortgage schedule with constant payments and compounded monthly is a monthly pmnt schedule of:

Pmnt= P x i/ [ 1 - (1/(1+i))^N)

Pmnt= monthly P= original principal i= monthly interest N= term of loan on months

Total interest paid N x Pmnt - P , gets big.

What strategy, other than abrupt prepayment, minimizes total interest and loss, assuming additional money used to supplement payment earns interest at 2 x i?

yowzer!

As expected, most here dismissed this as a boring student math problem. A few nerds took up the cudgels. Hail fellows!

But nobody got the point: it's the pilfering coin minter, of last week, in disguise! That one was presented in discrete form: n boxes, one coin per box, etc. Also random and probabilistic. This one presents itself as deterministic and continuous. At the end of the day, same same!

The lesson is that a problem might be more or less soluble, depending on presentation, or language. But I suppose every hominid on this rock understands that -

So, open it: did you ever face a tough nut to crack, then you rotated it, shone a different light, and the clouds parted? Share your war stories, grampa -

um, yes, but as the teacher admonished: "Show your work"

Specifically, what is e?

Rate of change of principal is dp/dt = i*p

dp/dt*(1/p) = i

integrate both sides to get

ln(p(t) = i*t + C

take exponential of each side

p(t) = e^(i*t)*K

K = e^C is the initial amount.

As this is an electronic ng there shouldn't be anyone here who is not familiar with the base of the natural logarithms 2.7182818...

e^(j*pi) = -1

You saying that doesn't make it so. Either demonstrate it analytically or go take a hike.

Give us a break...