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94 lines
2.3 KiB
HTML
94 lines
2.3 KiB
HTML
<HTML>
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<TITLE>Financial Equations Documentation</TITLE>
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<BODY>
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<A NAME="TOP"></A>
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<A HREF="./finutil.html#ConstOrigData">Return</A>
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<HR>
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<h1>Constant Repayment to Principal Equations Derivation</h1>
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<p>In this loan, each total payment is different, with each succeeding payment
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less than the preceding payment. Each payment is the total of the constant
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amount to the principal plus the interest for the period. The constant payment
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to the principal is computed as:
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<pre>
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C = -PV / N
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<p>Where PV is the loan amount to be repaid in N payments (periods). Thus the
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principal after the first payment is:
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<pre>
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PV[1] = PV[0] + C = PV + C
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</pre>
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<p>after the second payment, the principal is:
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<pre>
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PV[2] = PV[1] + C = PV[0] + 2C
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</pre>
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<p>In general, the remaining principal after n payments is:
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<pre>
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PV[n] = PV[0] + nC = PV + nC
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</pre>
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<p>If the effective interest per payment period is i, then the interest for the
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first payment is:
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<pre>
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I[1] = -i*PV[0]
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</pre>
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<p>and for the second:
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<pre>
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I[2] = -i * PV[1]
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</pre>
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<p>and in general, for the n'th payment the interest is:
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<pre>
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I[n] = -i * PV[n-1]
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= -i * (PV + (n - 1)C)
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</pre>
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<p>The total payment for any period, n, is:
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<pre>
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P[n] = C + I[n]
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= C - i * (PV + (n - 1)C)
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= C(1 + i) - i * (PV + nC)
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</pre>
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<p>The total interest paid to period n is:
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<pre>
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T[n] = I[1] + I[2] + I[3] + ... + I[n]
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T[n] = sum(j = 1 to n: I[j])
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T[n] = sum(j = 1 to n: -i * (PV + (j-1)C))
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T[n] = sum(j=1 to n: -i*PV) + sum(j=1 to n: iC) + sum(j=1 to n: -iCj)
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T[n] = -i*n*PV + i*n*C - i*C*sum(j=1 to n:j)
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sum(j=1 to n:j) = n(n+1)/2
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T[n] = -i*n*(PV + C) - i*C*n(n+1)/2
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T[n] = -i*n*(PV + (C*(n - 1)/2))
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</pre>
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<p>Note: substituting for C = -PV/N, in the equations for PV[n], I[n], P[n], and T[n]
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would give the following equations:
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<pre>
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PV[n] = PV*(1 - n/N)
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I[n] = -i*PV*(1 + N - n)/N
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P[n] = -i*PV*(2 + N - n)/N
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T[n] = -i*n*PV*(2*N - n + 1)/(2*N)
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</pre>
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<p>Using these equations for the calculations would eliminate the dependence
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on C, but only if C is always defined as above and would eliminate the
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possibility of another value for C. If the value of C was less than -PV/N
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then a balloon payment would be due at the final payment and this is a possible
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alternative for some people.
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