Solving for RateDate: 8/8/96 at 14:59:25 From: Anonymous Subject: Solving for Rate If P=V*R/(1-(1+R)^(-n)) How is this simplified to solve for "R"? Date: 8/9/96 at 8:10:26 From: Doctor Anthony Subject: Re: Solving for Rate Depending on the value of n (is it a positive integer?), you cannot in general get R explicitly as a function of P and V. i.e. you cannot swing the equation into the form R = f(P,V). The easiest approach for this type of problem is to express the equation in the form R = f(R) and then use iterative methods to find R. This will work if f'(R)<1 near the root. An alternative method is Newton-Raphson. (See any standard textbook on the calculus). If you have particular values of P, V and n, I will happily solve for R if you send me the information. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 8/9/96 at 15:26:2 From: Anonymous Subject: Re: Solving for Rate Yes, "N" would always be a positive integer. What I'm trying to do is determine the interest rate if the loan amount, term and monthly payments are known ("N" is the monthly term). More precisely, to determine the effective interest rate on the lease plans offered by car dealerships. The example I've been playing with is a $12,000.00 loan for 24 months with a payment of $548.22. The rate should be 9.0% and I thought the equation for determining the payment {(P=V*R/(1-(1+R)^(-N)) where P is payment, V is loan amount, R is rate and N is term} could be converted to solve for rate if all other variables were known. (Since it is a lease, the effective loan amount would be the assumed current cash price less the locked in purchase price at the end of the lease.) Thanks for any assistance. Date: 8/17/96 at 16:45:18 From: Doctor Anthony Subject: Re: Solving for Rate If we look at the actual equation, it is, (assuming R = 9 percent per year = 0.75 percent per month) P{1 + 1.0075 + 1.0075^2 + ...+ 1.0075^23} = V{1.0075}^24 P{1.0075^24 - 1}/0.0075 = V{1.0075}^24 Now at this point, we assume that P and V are known, but the percentage rate is unknown. So we shall replace 1.0075 in this expression by x, and also enter the known values of P and V. We get 548.22{x^24 - 1} = (x-1)*12000*x^24 548.22x^24 - 548.22 = 12000x^25 - 12000x^24 12548.22x^24 - 12000x^25 = 548.22 If you now try x=1.0075 in this expression, you will find that it gives the exact solution. So the information given in the question is consistent. You will observe, too, that an equation of the 25th degree does not allow x to be expressed explicitly in terms of P and V. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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