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 February 5th, 2008, 07:10 AM #1 Member   Joined: Sep 2007 Posts: 30 Thanks: 0 Solving for exponential rate of return I wish to calculate the rate of return for an investment account with periodic investments. I would also like the rate of return to be the exponential rate so that number of times compounded is infinite. So an account balance=B after time=T at a rate=R will be worth B*exp(T*R). Suppose for a time period of one year there are 2 deposits (D1, D2) into an account with starting balance B. At the end of the year B has been invested for 1 year, D1 has been invested for T1 of the year and D2 has been invested for T2 of the year. The equation for the balance at the end of the year for rate=R is: B*exp(R)+D1*exp(T1*R)+D2*exp(T2*R) If I know my account balance is K at the end of the year how do I solve the equation : K=B*exp(R)+D1*exp(T1*R)+D2*exp(T2*R) for R I currently use the solver program in excel but I would assume there is some financial equation that would solve for R. February 5th, 2008, 07:42 AM #2 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms There's no obvious solution -- likely no closed form solution exists, depending on your choice of T1 and T2. You can see this by letting E = exp(R). Then K = B*E + D1*E^T1 + D2*E^T2 and so you have a fractional power equation. I'd use a generic solver like bisection, Newton's method, or the secant method. February 21st, 2008, 06:02 PM   #3
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Re: Solving for exponential rate of return

Quote:
 Originally Posted by RFurball I wish to calculate the rate of return for an investment account with periodic investments. I would also like the rate of return to be the exponential rate so that number of times compounded is infinite. So an account balance=B after time=T at a rate=R will be worth B*exp(T*R). Suppose for a time period of one year there are 2 deposits (D1, D2) into an account with starting balance B. At the end of the year B has been invested for 1 year, D1 has been invested for T1 of the year and D2 has been invested for T2 of the year. The equation for the balance at the end of the year for rate=R is: B*exp(R)+D1*exp(T1*R)+D2*exp(T2*R) If I know my account balance is K at the end of the year how do I solve the equation : K=B*exp(R)+D1*exp(T1*R)+D2*exp(T2*R) for R I currently use the solver program in excel but I would assume there is some financial equation that would solve for R.
What you are calculating is the continuous internal rate of return (IRR). There is no closed form, but there is an Excel IRR function that iterates attempted solutions until it finds one. The function returns the annualized one year return so you would have to convert it to continous by ln(1+IRR) Tags exponential, rate, return, solving ,

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