Kelly,

The amount, A(t), that $P becomes after t years at an interest rate of r% compounded m times per year is

A(t) = P(1+r/m)^mt.

In your example A(1) = 100000(1+.005/365)³⁶⁵ = $100501.25. This is a ridiculously low rate of interest and I suspect that who ever is offering you this is counting on you not being able to do the calculations. Beware!

Penny

Kelly wrote back

Dear Penny , Please explain the answer that You gave me regarding the amount of interest that I'll earn on $100K investment @.005% interest compounded daily... Please ... I thought that the interest compounded daily , could be more than $500.00 ? Am I "WRONG" ??? Please reply. Thankx. Kelly

Kelly,

The annual interest rate is 0.5%, which as a decimal is 0.005, so the daily interest rate is
 ^0.005/₃₆₅. Thus if you invest $100,000 at this rate for one day you receive, at the end of the day, what you invested

$1000,000

plus the interest

^0.005/₃₆₅ $100,000

In other words, at the end of the first day you receive

$100,000 + ^0.005/₃₆₅ $100,000 = $100,000 (1 + ^0.005/₃₆₅)

This is the amount you invest on the second day so at the end of the second day you have what you invested

$100,000 (1 + ^0.005/₃₆₅)

plus the interest

^0.005/₃₆₅ $100,000 (1 + ^0.005/₃₆₅)

which is

$100,000 (1 + ^0.005/₃₆₅) + ^0.005/₃₆₅ $100,000 (1 + ^0.005/₃₆₅)

= $100,000 (1 + ^0.005/₃₆₅) (1 + ^0.005/₃₆₅)

= $100,000 (1 + ^0.005/₃₆₅)²

This is the amount you invest on the third day so at the end of the third day you have

$100,000 (1 + ^0.005/₃₆₅)³

The pattern continues and hence at the end of one year you have

$100,000 (1 + ^0.005/₃₆₅)³⁶⁵

Penny

Hi Kelly,

Perhaps you have been offered a very strange investment indeed.

Normally, interest rates are quoted as annual interest rates. Penny has interpreted your situation that way and I believe this is almost certainly the case for your situation. There is however a slight chance that you really have been offered the rate of 0.5% accrued daily (rather than 1/365 of that).

If so, (and this is a big "if" that would have to be very clearly spelled out in the contract - or better yet, converted to the equivalent normal annual form), then your total at the end of one year would be:

A(t) = P(1+r/m)^mt

but here r is the rate per DAY, m is the number of times paid DAILY (1), and t is the number of DAYS, so

A(365) = 100 000(1+0.005)³⁶⁵ = $617 465

The equivalent annual (normal) rate of interest compounded daily is

$617 465 = 100 000(1+r/365)³⁶⁵

which works out to

r = 182.5 %

This is too good to be true - I'd certainly consult a lawyer and perhaps the police before I'd invest my money here.

Stephen La Rocque.>