Expert: Paul Klarreich - 10/11/2006

Question
Hi

I have one question, maybe you can help me?

1. "What interest rate is required for an investment with continuously compounded interest to double in 6 years?"

Thank you

Answer
Pit Asks in Category Advanced Math ...
 
Subject:  Conic section
Private:  no
 
Question:  Hi

I have one question, maybe you can help me?

1. "What interest rate is required for an investment with continuously compounded interest to double in 6 years?"

Thank you
..........................................
Hi, Pit,

I'm not sure at what level you're asking this question, or what it has to do with the conic sections, but here's an answer:

Assume you deposit $1 at a six-yearly interest rate of r.  
That means 7% annual gives 42% in 6 years, or r = 0.42]

At simple interest, you get:

A = 1(1 + r)

Compounded four times:

A = 1(1 + r/4)^4

Compounded k times:

A = 1(1 + r/k)^k   << Formula 1.

Now we want to make use of a known value:

lim    (1 + 1/m)^m = e, the base of natural logarithms.
m->inf

formula 1 has an extra thing in it, the r.  So we substitute:

r/k = 1/m,  so k = mr

A = 1(1 + 1/m)^mr

and do some algebra of exponents:

A = 1[ (1 + 1/m)^m ]^r

Now the limit (you did say compounding continously, right?) as m -> inf is:

A = 1 e^r

Now we are ready.  We want the amount to double, so A = 2.

2 = e^r,  so  r = ln 2 = 0.69314718055994530941723212145818 (from my Windows calculator)

But that is in six years, so the annual rate is 1/6 of that,

r(annual) = ln 2 / 6 = 0.11552453009332421823620535357636

Or, about  11.55% per annum.