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# Constant Repayment to Principal Equations Derivation

In this loan, each total payment is different, with each succeeding payment less than the preceding payment. Each payment is the total of the constant amount to the principal plus the interest for the period. The constant payment to the principal is computed as:

            C = -PV / N

    Where PV is the loan amount to be repaid in N payments (periods). Thus the
    principal after the first payment is:

        PV[1] = PV[0] + C = PV + C

    after the second payment, the principal is:

        PV[2] = PV[1] + C = PV[0] + 2C

    In general, the remaining principal after n payments is:

        PV[n] = PV[0] + nC = PV + nC

    If the effective interest per payment period is i, then the interest for the
    first payment is:

        I[1] = -i*PV[0]

    and for the second:

        I[2] = -i * PV[1]

    and in general, for the n'th payment the interest is:

        I[n] = -i * PV[n-1]
             = -i * (PV + (n - 1)C)

    The total payment for any period, n, is:

        P[n] = C + I[n]
             = C - i * (PV + (n - 1)C)
             = C(1 + i) - i * (PV + nC)

    The total interest paid to period n is:

        T[n] = I[1] + I[2] + I[3] + ... + I[n]
        T[n] = sum(j = 1 to n: I[j])
        T[n] = sum(j = 1 to n: -i * (PV + (j-1)C))
        T[n] = sum(j=1 to n: -i*PV) + sum(j=1 to n: iC) + sum(j=1 to n: -iCj)
        T[n] = -i*n*PV + i*n*C - i*C*sum(j=1 to n:j)
            sum(j=1 to n:j) = n(n+1)/2
        T[n] = -i*n*(PV + C) - i*C*n(n+1)/2
        T[n] = -i*n*(PV + (C*(n - 1)/2))

    Note: substituing for C = -PV/N, in the equations for PV[n], I[n], P[n], and T[n]
    would give the following equations:

        PV[n] = PV*(1 - n/N)
        I[n]  = -i*PV*(1 + N - n)/N
        P[n]  = -i*PV*(2 + N - n)/N
        T[n]  = -i*n*PV*(2*N - n + 1)/(2*N)

    Using these equations for the calculations would eliminate the dependence
    on C, but only if C is always defined as above and would eliminate the
    possibility of another value for C. If the value of C was less than -PV/N
    then a balloon payment would be due at the final payment and this is a possible
    alternative for some people.