Business Math

Cassandra is repaying an installment loan of $3,500 with 20 equal monthly
payments of $196 each. what is the annual percentage rate of the loan?

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  1. The standard formula for installment amount A, for n periods with interest per period i, and present value (principal) P is
    A=P((1+i)^n)i/((1+i)^n-1)
    where
    n=20,
    A=196
    P=3500
    substitute to get
    196=3500i(1+i)^20/((1+i)^20-1)

    I do not believe there is an explicit formula to solve for i, so we can solve it by trial and error or by iteration.

    Rearrange to give
    f(i)=196-3500i(1+i)^20/((1+i)^20-1)
    and we look for i that makes f(i)=0.
    Make a table:
    f(0.005)=11.67
    f(0.01)=2.04
    f(0.015)=-7.86
    Interpolating, we try
    f(0.011)=0.0878
    f(0.0111)=-0.1086
    So give a final try of
    f(0.011045)=-0.0005...
    The interest per period is 1.1045%
    Annual interest
    =1.1045*12=13.254% approx.

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