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Basic Equation
Series Sum

# Financial Equation Derivation

The financial equation is derived in the following manner:

Start with the basic equation to find the balance or Present Value, PV[1], after one payment period. Note PV[1] is the Present Value after one payment and PV[0] is the initial Present Value. PV[0] will be shortened to just PV.

The interest due at the end of the first payment period is the original present value, PV, times the interest rate for the payment period plus the periodic payment times the interest rate for beginning of period payments:

ID[1] = PV * i + X * PMT * i = (PV + X * PMT) * i

The Present Value after one payment is the original Present Value with the periodic payment, PMT, and interest due, ID[1], added:

       PV[1] = PV + (PMT + ID[1])
       PV[1] = PV + (PMT + (PV + X * PMT) * i)
       PV[1] = PV * (1 + i) + PMT * (1 + Xi)

This equation works for all of the cash flow diagrams shown previously. The Present Value, money received or paid, is modified by a payment made at the beginning of a payment period and multiplied by the effective interest rate to compute the interest due during the payment period. The interest due is then added to the payment to obtain the amount to be added to the Present Value to compute the new Present Value.

For diagram 1): PV < 0, PMT == 0, PV[1] < 0
For diagram 2): PV == 0, PMT < 0, PV[1] < 0
For Diagram 3): PV > 0, PMT < 0, PV[1] >= 0 or PV[1] <= 0
For Diagram 4): PV < 0, PMT > 0, PV[1] <= 0 or PV[1] >= 0

X may be 0 or 1 for any diagram.

For the standard loan, PV is the money borrowed, PMT is the periodic payment to repay the loan, i is the effective interest rate agreed upon and FV is the residual loan amount after the agreed upon number of periodic payment periods. If the loan is fully paid off by the periodic payments, FV is zero, 0. If the loan is not completely paid off after the agreed upon number of payments, a balloon payment is necessary to completely pay off the loan. FV is then the amount of the needed balloon payment. For a loan in which the borrower pays only enough to repay the interest due during a payment period, interest only loan, the balloon payment is equal to the negative of PV.

To calculate the Present Value after the second payment period, the above calculation is applied iteratively to PV[1] to obtain PV[2]. In fact to calculate the Present Value after any payment period, PV[n], the above equation is applied iteratively to PV[n-1] as shown below.

       PV[2] = PV[1] + (PMT + (PV[1] + X * PMT) * i)
             = PV[1] * (1 + i) + PMT * (1 + iX)
             = (PV * (1 + i) + PMT * (1 + iX)) * (1 + i) + PMT * (1 + iX)
             = PV * (1 + i)^2 + PMT * (1 + iX) * (1 + i)
                              + PMT * (1 + iX)

Similarly, PV[3] is computed from PV[2] as:

       PV[3] = PV[2] + (PMT + (PV[2] + X * PMT) * i)
             = PV[2] * (1 + i) + PMT * (1 + iX)
             = (PV * (1 + i)^2 + PMT * (1 + iX) * (1 + i)
                               + PMT * (1+  iX)) * (1 + i)
                               + PMT * (1+  iX)
             = PV * (1 + i)^3 + PMT * (1 + iX) * (1 + i)^2
                              + PMT * (1 + iX) * (1 + i)
                              + PMT * (1 + iX)

And for the n'th payment, PV[n] is computed from PV[n-1] as:

       PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i)
       PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * (1 + i)^(n-1)
                              + PMT * (1 + iX) * (1 + i)^(n-2) +
                              .
                              .
                              .
                              + PMT * (1 + iX) * (1 + i)
                              + PMT * (1 + iX)
       PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^(n-1) + ... + (1 + i) + 1]

The formula for PV[n] can be proven using mathematical induction.

# Basic Financial Equation

As shown above, the basic financial transaction equation is simply:

       PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i)
             = PV[n-1] * (1 + i) + PMT * (1 + iX)
        for: n >= 1

relating the Present Value after n payments, PV[n] to the previous Present Value, PV[n-1].

# Series Sum

The sum of the finite series:

1 + k + (k^2) + (k^3) + ... + (k^n) = (1-k^(n+1))/(1-k)

as can be seen by the following. Let S(n) be the series sum. Then

S(n) - k * S(n) = 1 - k^(n+1)

and solving for S(n):

S(n) = (1-k^(n+1))/(1-k) = 1 + k + (k^2) + (k^3) + ... + (k^n)

Using this in the equation above for PV[n], we have:

       PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^(n-1) + ... + (1 + i) + 1]
             = PV * (1 + i)^n + PMT * (1 + iX) * [1 - (1 + i)^n]/[1 - (1 + i)]
             = PV * (1 + i)^n + PMT * (1 + iX) * [1 - (1 + i)^n]/[-i]
             = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^n - 1]/i

or:

       PV * (1 + i)^n + PMT * [(1 + i)^n - 1]/i - PV[n] = 0

If after n payments, the remaining balance is repaid as a lump sum, the lump sum is known as the Future Value, FV[n]. Since FV[n] is negative if paid and positive if received, FV[n] is the negative of PV[n].

Setting: FV[n] = -PV[n]

Since n is assumed to be the last payment, FV[n] will be shortened to simply FV for the last payment period.

       PV*(1 + i)^n + PMT*(1 + iX)*[(1 + i)^n - 1]/i + FV = 0

Up to this point, we have said nothing about the value of PMT. PMT can be any value mutually agreed upon by the lender and the borrower. From the equation for PV[1]:

       PV[1] = PV + (PMT + (PV + X * PMT) * i),

Several things can be said about PMT.

1. If PMT = -(PV * i), and X = 0 (end of period payments):

   The payment is exactly equal to the interest due and PV[1] = PV. In this case, the borrower must make larger future payments to reduce the balance due, or make a single payment, after some agreed upon number of payments, with PMT = -PV to completely pay off the loan. This is an interest only payment with a balloon payment at the end.

2. If |PMT| < |PV * i|, and X = 0 and PV > 0

   The payment is insufficient to cover even the interest charged and the balance due grows

3. If |PMT| > |PV * i|, and X = 0 and PV > 0

   The payment is sufficient to cover the interest charged with a residual amount to be applied to reduce the balance due. The larger the residual amount, the faster the loan is repaid. For most mortgages or other loans made today, the lender and borrower agree upon a certain number of repayment periods and the interest to be charged per payment period. The interest may be multiplied by 12 and stated as an annual interest rate. Then the lender and borrower want to compute a periodic payment, PMT, which will reduce the balance due to zero after the agreed upon number of payments have been made. If N is the agreed upon number of periodic payments, then we want to use:

             PV * (1 + i)^N + PMT*(1 +iX)*[(1 + i)^N - 1]/i + FV = 0

   with FV = 0 to compute PMT:

             PMT = -[PV * i * (1 + i)^(N - X)]/[(1 + i)^N - 1]

   The value of PMT computed will reduce the balance due to zero after N periodic payments. Note that this is strictly true only if PMT is not rounded to the nearest cent as is the usual case since it is hard to pay fractional cents. Rounding PMT to the nearest cent has an effect on the FV after N payments. If PMT is rounded up, then the final Nth payment will be smaller than PMT since the periodic PMTs have paid down the principal faster than the exact solution. If PMT is rounded down, then the final Nth payment will be larger than the periodic PMTs since the periodic PMTs have paid down the principal slower than the exact solution.

With a simple alegebraic re-arrangement, The financial Equation becomes:

      2) [PV + PMT*(1 + iX)/i][(1 + i)^n - 1] + PV + FV = 0

or

      3) (PV + C)*A + PV + FV = 0

where:

      4) A = (1 + i)^n - 1

      5) B = (1 + iX)/i

      6) C = PMT*B

The form of equation 3) simplifies the calculation procedure for all five variables, which are readily solved as follows:

      7) n = ln[(C - FV)/(C + PV)]/ln((1 + i)

      8) PV = -[FV + A*C]/(A + 1)

      9) PMT = -[FV + PV*(A + 1)]/[A*B]

     10) FV = -[PV + A*(PV + C)]

Equations 4), 5) and 6) are computed by the functions in the `"fin.exp"` utility:
`_A`
`_B`
`_C`

respectively. Equations 7), 8), 9) and 10) are computed by functions:
`_N`
`_PV`
`_PMT`
`_FV`

respectively.

The solution for interest is broken into two cases:

1. PMT == 0

   Equation 3) can be solved exactly for i:

              i = [FV/PV]^(1/n) - 1

2. PMT != 0

   Since equation 3) cannot be solved explicitly for i in this case, an iterative technique must be employed. Newton's method, using exact expressions for the function of i and its derivative, are employed. The expressions are:

        12) i[k+1] = i[k] - f(i[k])/f'(i[k])
              where: i[k+1] == (k+1)st iteration of i
                     i[k]   == kth iteration of i
              and:

        13) f(i) = A*(PV+C) + PV + FV

        14) f'(i) = n*D*(PV+C) - (A*C)/i

        15) D = (1 + i)^(n-1) = (A+1)/(1+i)

   To start the iterative solution for i, an initial guess must be made for the value of i. The closer this guess is to the actual value, the fewer iterations will have to be made, and the greater the probability that the required solution will be obtained. The initial guess for i is obtained as follows:

   1. PV case, PMT*FV >= 0

                          | n*PMT + PV + FV |
           16)     i[0] = | ----------------|
                          |      n*PV       |

                        = abs[(n*PMT + PV + FV)/(n*PV)]

   2. FV case, PMT*FV < 0
      1. PV != 0

                                 |      FV - n*PMT           |
              17)         i[0] = |---------------------------|
                                 | 3*[PMT*(n-1)^2 + PV - FV] |

                               = abs[(FV-n*PMT)/(3*(PMT*(n-1)^2+PV-FV))]

      2. PV == 0

                                 |      FV + n*PMT           |
              18)         i[0] = |---------------------------|
                                 | 3*[PMT*(n-1)^2 + PV - FV] |

                               = abs[(FV+n*PMT)/(3*(PMT*(n-1)^2+PV-FV))]

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