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Solving for Rate

Date: 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!   

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!   
Associated Topics:
High School Basic Algebra
High School Calculus
High School Interest

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