From e8e179fd8fb7a951037e7b825d6c18d091adcde3 Mon Sep 17 00:00:00 2001 From: Geert Janssens Date: Thu, 23 May 2024 15:03:50 +0200 Subject: [PATCH] Drop remainder of files in libgnucash/doc Relevant diagrams have been added in the wiki as png files. The html files with financial calculations are copyrighted by someone outside of the gnucash project. I'd rather not add that info to the wiki. Similar information should be easy to find on the internet. --- libgnucash/CMakeLists.txt | 1 - libgnucash/doc/CMakeLists.txt | 9 - libgnucash/doc/README | 4 - libgnucash/doc/constderv.html | 93 - libgnucash/doc/dia/components.dia | Bin 3346 -> 0 bytes libgnucash/doc/dia/structures-alt.dia | Bin 3442 -> 0 bytes libgnucash/doc/dia/structures.dia | Bin 3745 -> 0 bytes libgnucash/doc/finderv.html | 337 ---- libgnucash/doc/finutil.html | 2278 ------------------------- 9 files changed, 2722 deletions(-) delete mode 100644 libgnucash/doc/CMakeLists.txt delete mode 100644 libgnucash/doc/README delete mode 100644 libgnucash/doc/constderv.html delete mode 100644 libgnucash/doc/dia/components.dia delete mode 100644 libgnucash/doc/dia/structures-alt.dia delete mode 100644 libgnucash/doc/dia/structures.dia delete mode 100644 libgnucash/doc/finderv.html delete mode 100644 libgnucash/doc/finutil.html diff --git a/libgnucash/CMakeLists.txt b/libgnucash/CMakeLists.txt index a0ea47cf490..44dc1e5325d 100644 --- a/libgnucash/CMakeLists.txt +++ b/libgnucash/CMakeLists.txt @@ -4,7 +4,6 @@ add_subdirectory (app-utils) add_subdirectory (backend) add_subdirectory (core-utils) -add_subdirectory (doc) add_subdirectory (engine) add_subdirectory (gnc-module) add_subdirectory (quotes) diff --git a/libgnucash/doc/CMakeLists.txt b/libgnucash/doc/CMakeLists.txt deleted file mode 100644 index f816f92d1b8..00000000000 --- a/libgnucash/doc/CMakeLists.txt +++ /dev/null @@ -1,9 +0,0 @@ -set(doc_FILES - constderv.html - finderv.html - finutil.html - README - ) - -set_local_dist(doc_DIST_local CMakeLists.txt ${doc_FILES}) -set(doc_DIST ${doc_DIST_local} ${doc_design_DIST} PARENT_SCOPE) diff --git a/libgnucash/doc/README b/libgnucash/doc/README deleted file mode 100644 index ea663726e8c..00000000000 --- a/libgnucash/doc/README +++ /dev/null @@ -1,4 +0,0 @@ - - -Please note that there is additional documentation in various -source subdirectories: for example, src/engine/*.txt diff --git a/libgnucash/doc/constderv.html b/libgnucash/doc/constderv.html deleted file mode 100644 index 0dc58bbaa62..00000000000 --- a/libgnucash/doc/constderv.html +++ /dev/null @@ -1,93 +0,0 @@ - - - -Financial Equations Documentation - - - -Return -
-

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: substituting 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.
diff --git a/libgnucash/doc/dia/components.dia b/libgnucash/doc/dia/components.dia
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z&M9#Ny>-KOL$4r|vs~v34>VxoC|$hpQRLlG
-
-
-Financial Equations Documentation
-
-
-
-Return
-

-

-

-
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))]
-
        
-
-
      
-
    
-

-

-Return
-
diff --git a/libgnucash/doc/finutil.html b/libgnucash/doc/finutil.html
deleted file mode 100644
index ca8687e3bf2..00000000000
--- a/libgnucash/doc/finutil.html
+++ /dev/null
@@ -1,2278 +0,0 @@
-
-
-
-
-Financial Utility Documentation
-
-
-
-

-
Financial Transaction Utility

-
-

-

-
Financial Calculator

-
Time Value of Money

-  
Simple Interest

-  
Compound Interest

-  
Periodic Payments

-  
Financial Transactions

-
Standard Financial Conventions

-
Cash Flow Diagrams

-  
Appreciation

-  
Annuity

-  
Amortization

-  
Annuity

-
Interest

-  
Compounding Frequency

-  
Payment Frequency

-  
NAR to EIR for Discrete Interest Periods

-  
NAR to EIR for Continuous Interest

-  
Normal CF/PF Values

-  
EIR to NAR for Discrete Interest Periods

-  
EIR to NAR for Continuous Compounding

-
Financial Equation

-  
Financial Equation Derivation

-
Amortization Schedules

-  
Effective and Initial Payment Dates

-  
Effective Present Value

-  
Iterative Amortization Schedule

-  
Annual Summary

-  
Final Payment Calculation

-  
Amortization Cases

-  

-    
Constant Repayment to Principal, Original Data

-    
Constant Repayment to Principal, Delayed Repayment

-    
Original Data Schedule

-    
Recomputed Final Payment

-    
Recomputed Periodic Payment

-    
Recomputed Term

-  

-  
Amortization Schedule Display

-
Financial Calculator Usage

-  
Calculator Commands

-  
Calculator Input

-  
Calculator Functions

-  
User Defined Variables

-  
Rounding

-
Examples

-  
Simple Interest 

-  
Compound Interest

-  
Periodic Payment

-  
Conventional Mortgage

-  
Final Payment

-  
Conventional Mortgage Amortization Schedule - Annual Summary

-  
Conventional Mortgage Amortization Schedule - Periodic Payment Schedule

-  
Conventional Mortgage Amortization Schedule - Variable Advanced Payments

-  
Conventional Mortgage Amortization Schedule - Constant Advanced Payments

-  
Balloon Payment

-  
Canadian Mortgage

-  
European Mortgage

-  
Bi-weekly Savings

-  
Present Value - Annuity Due

-  
Effective Rate - 365/360 Basis

-  
Mortgage with "Points"

-  
Equivalent Payments

-  
Perpetuity - Continuous Compounding

-  
Investment Return

-  
Retirement Investment

-  
Property Values

-  
College Expenses

-  
Certificate of Deposit, Annual Percentage Yield

-
References

-

-

-
-
Financial Calculator

-
Financial Calculator
-
-
-
-
-
This is  a complete  financial computation  utility to  solve for  the five
-standard financial values: n, %i, PV, PMT and FV
-

-
-
In addition, four additional parameters may be specified:
-
    
-
  1. Compounding Frequency per year, CF. The number of times the interest is compounded
-during the year. The default is 12. The compounding frequency per year may be
-different from the Payment Frequency per year
-
-
  2. Payment Frequency per year, PF. The number of payments made in a year. Default is 12.
-
-
  3. Discrete or continuous compounding, disc. The default is discrete compounding.
-
-
  4. Payments may be at the beginning or end of the payment period, beg. The default is for
-payments to be made at the end of the payment period.
-

-
-
When an amortization schedule is desired, the financial transaction Effective Date, ED,
-and Initial Payment Date, IP, must also be entered.
-
-
Canadian  and  European  style  mortgages  can  be  handled  in  a simple,
-straight-forward manner. Standard financial sign conventions are used:
-
-

-
"Money paid out is Negative, Money received is Positive"
-

-
-
-
Time Value of Money

-
If you borrow money, you can expect  to pay rent or interest for its  use;
-conversely you expect to receive rent interest on money you loan or invest.
-When you rent property, equipment,  etc., rental payments are normal;  this
-is  also  true  when  renting  or  borrowing  money.  Therefore,  money  is
-considered to have a "time value". Money available now, has a greater value
-than money available at some future date because of its rental value or the
-interest that it can produce during the intervening period.
-
-
-
Simple Interest

-
If you loaned $800 to  a friend with an agreement  that at the end of  one
-year he would  would repay you  $896, the "time  value" you placed  on your
-$800 (principal) was $96 (interest) for  the one year period (term) of  the
-loan. This  relationship of  principal, interest,  and time (term) is  most
-frequently expressed as an Annual  Percentage Rate (APR). In this  case the
-APR  was  12.0%  [(96/800)*100].  This  example  illustrates the four basic
-factors involved  in a  simple interest  case. The  time period (one year),
-rate (12.0%  APR), present  value of  the principal  ($800) and  the future
-value of the principal including interest ($896).
-
-
-
Compound Interest

-
In many cases the interest charge is computed periodically during the term
-of  the  agreement.  For  example,  money  left  in a savings account earns
-interest that  is periodically  added to  the principal  and in  turn earns
-additional interest during succeeding periods. The accumulation of interest
-during  the  investment  period  represents  compound interest. If the loan
-agreement you  made with  your friend  had specified  a "compound  interest
-rate" of  12% (compounded  monthly) the  $800 principal  would have  earned
-$101.46 interest for the  one year period. The  value of the original  $800
-would be increased  by 1% the  first month to  $808 which in  turn would be
-increased  by  1%  to  816.08 the second month,  reaching a future value of
-$901.46 after the twelfth iteration. The monthly compounding of the nominal
-annual rate (NAR) of 12% produces an effective Annual Percentage Rate (APR)
-of 12.683% [(101.46/800)*100].  Interest may be  compounded at any  regular
-interval; annually, semiannually, monthly, weekly, daily, even continuously
-(a specification in some financial models).
-
-
-
Periodic Payments

-
When money is loaned for longer  periods of time, it is customary  for the
-agreement to require the borrower  to make periodic payments to  the lender
-during the term of the loan. The payments may be only large enough to repay
-the interest,  with the  principal due  at the  end of  the loan period (an
-interest only loan), or large enough  to fully repay both the interest  and
-principal during the term of the loan (a fully amoritized loan). Many loans
-fall somewhere between, with payments that do not fully cover repayment  of
-both the principal and interest. These loans require a larger final payment
-(balloon)  to  complete  their  amortization.  Payments  may  occur  at the
-beginning or end of a payment period. If you and your friend had agreed  on
-monthly repayment of  the $800 loan  at 12% NAR  compounded monthly, twelve
-payments of $71.08 for a total of $852.96 would be required to amortize the
-loan. The $101.46  interest from the  annual plan is  more than the  $52.96
-under the monthly plan because under the monthly plan your friend would not
-have had the use of $800 for a full year.
-
-
-
Financial Transactions

-
The  above  paragraphs  introduce  the  basic  factors  that  govern  most
-financial  transactions;  the  time  period,  interest rate, present value,
-payments and  the future  value. In  addition, certain  conventions must be
-adhered to: the interest rate must be relative to the compounding frequency
-and payment periods, and the term must be expressed as the total number  of
-payments (or compounding periods if there are no payments). Loans,  leases,
-mortgages, annuities, savings plans, appreciation, and compound growth  are
-among the many financial problems that can be defined in these terms. Some
-transactions do not involve payments, but  all of the other factors play  a
-part in "time value of money" transactions. When any one of the five  (four
-- if no payments are involved)  factors is unknown, it can be  derived from
-formulas using the known factors.
-
-
-
Standard Financial Conventions

-
The Standard Financial Conventions are:
-
-

-
-
-
Cash Flow Diagrams

-
If payments are  a part of  the transaction, the  number of payments  must
-equal the number of periods (n).
-
-
Payments may be  represented as occurring  at the end  or beginning of  the
-periods.
-
-
Diagram  to  visualize  the  positive  and  negative cash flows (cash flow
-diagrams):
-
-
Amounts shown above the line are positive, received, and amounts shown below the
-line are negative, paid out.
-
-

-
-
Appreciation

-
Appreciation
-
Depreciation
-
Compound Growth
-
Savings Account
-
-                                                                A FV*
-          1   2   3   4   .   .   .   .   .   .   .   .   .   n |
- Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
-        |
-        V
-        PV
-

-
-
-
-
-

-
-
Annuity (series of payments)

-
Annuity (series of payments)
-
Pension Fund
-
Savings Plan
-
Sinking Fund
-
-
-     PV = 0                                                     A
-                                                                |
- Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
-        | 1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n
-        V   V   V   V   V   V   V   V   V   V   V   V   V   V
-       PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT
-

-
-
-
-
-

-
-
Amortization

-
Direct Reduction Loan
-
Mortgage (fully amortized)
-
-
-     PV ^
-        |                                                      FV=0
- Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
-          1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n |
-            V   V   V   V   V   V   V   V   V   V   V   V   V   V
-           PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT
-

-
-
-
-

-
-
Annuity

-
Annuity
-
Lease (with buy back or residual)*
-
Loan or Mortgage (with balloon)*
-
-                                                                A FV*
-           PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT  | +
-            A   A   A   A   A   A   A   A   A   A   A   A   A   A PMT
-          1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n |
- Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
-        |
-        V
-        PV
-

-
-
-
-

-
-
Interest

-
Before discussing the financial equation, we will discuss interest. Most
-financial transactions utilize a nominal interest rate, NAR, i.e., the interest
-rate per year. The NAR must be converted to the interest rate per payment
-period and the compounding accounted for before it can be used in computing
-an interest payment. After this conversion process, the interest used is the
-effective interest rate, EIR. In converting NAR to EIR, there are two concepts
-to discuss first, the Compounding Frequency and the Payment Frequency and
-whether the interest is compounded in discrete intervals or continuously.
-
-
-
Compounding Frequency

-
The compounding Frequency, CF, is simply the number of times per year, the
-monies in the financial transaction are compounded. In the U.S., monies
-are usually compounded daily on bank deposits, and monthly on loans. Sometimes
-long term deposits are compounded quarterly or weekly.
-
-
-
Payment Frequency

-
The Payment Frequency, PF, is simply how often during a year payments are
-made in the transaction. Payments are usually scheduled on a regular basis
-and can be made at the beginning or end of the payment period. If made at
-the beginning of the payment period, interest must be applied to the payment
-as well as any previous money paid or money still owed.
-
-
-
Normal CF/PF Values

-
Normal values for CF and PF are:
-

-
-
The  Compounding  Frequency per year,  CF,  need not be identical  to the
-Payment Frequency per year, PF. Also,
-Interest may be compounded in either discrete intervals or continuously
-compounded and payments may be made at the beginning of the payment period or at the
-end of the payment period.
-
-
CF and PF are defaulted to  12. The default is for discrete interest intervals
-and payments are defaulted to the end of the payment period.
-
-
When a solution  for n, PV,  PMT or FV  is required, the  nominal interest
-rate, i, must first be converted to the effective interest rate per payment
-period. This rate, ieff, is then used to compute the selected variable.  To
-convert i to ieff, the following expressions are used:
-
-
-
NAR to EIR for Discrete Interest Periods

-
To convert NAR to EIR for discrete interest periods:
-
-
ieff = (1 + i/CF)^(CF/PF) - 1
-
-
-
NAR to EIR for Continuous Compounding

-
to convert NAR to EIR for Continuous Compounding:
-
-
ieff = e^(i/PF) - 1 = exp(i/PF) - 1
-
-
When interest is computed, the computation produces the effective interest
-rate, ieff. This value must then be converted to the nominal interest rate.
-Function  _I  in the "fin.exp" utility returns  the  nominal  interest
-rate NOT the effective interest rate. ieff is converted to i using the following expressions:
-
-
-
EIR to NAR for Discrete Interest Periods

-
To convert EIR to NAR for discrete interest periods:
-
-
i = CF*([(1+ieff)^(PF/CF) - 1)
-
-
-
EIR to NAR for Continuous Compounding

-
To convert EIR to NAR for continuous compounding:
-
-
i = ln((1+ieff)^PF)
-
-
-
-
-
-
-
Financial Equation

-
NOTE: in the equations below for the financial transaction, all interest rates
-are the effective interest rate, ieff. The symbol will be shortned to just i.
-
-
The financial equation used to inter-relate n,i,PV,PMT and FV is:
-
-
1) PV*(1 + i)^n + PMT*(1 + iX)*[(1+i)^n - 1]/i + FV = 0
-
-
-   Where: X   == 0 for end of period payments, and
-          X   == 1 for beginning of period payments
-          n   == number of payment periods
-          i   == effective interest rate for payment period
-          PV  == Present Value
-          PMT == periodic payment
-          FV  == Future Value
-

-
-
-
-
Financial Equation Derivation

-
The derivation of the financial equation is contained in the
-Financial Equations
-section.
-
-
-
-
-
-
Amortization Schedules.

-
-
-
Effective and Initial Payment Dates

-
Financial Transactions have an effective Date, ED, and an Initial Payment
-Date, IP. ED may or may not be the same as IP, but IP is always the same
-or later than ED. Most financial transaction calculators assume that
-IP is equal to ED for beginning of period payments or at the end of the
-first payment period for end of period payments.
-
-
This is not always true. IP may be delayed for financial reasons such as cash
-flow or accounting calendar. The subsequent payments then follow the
-agreed upon periodicity.
-
-
-
Effective Present Value

-
Since money has a time value, the "delayed" IP
-must be accounted for. Computing an "Effective PV", pve, is one means of
-handling a delayed IP.
-
-
If
-
-
-ED_jdn == the Julian Day Number of ED, and
-IP_jdn == the Julian Day Number of IP
-

-
-
pve is computed as:
-
-
-   pve = pv*(1 + i)^(s*PF/d*CF)
-
-   Where: d = length of the payment period in days, and
-          s = IP_jdn - ED_jdn - d*(1 - X)
-

-
-
-
Iterative Amortization Schedule

-
Computing an amortization Schedule for a given financial transaction is
-simply applying the basic equation iteratively for each payment period:
-
-
-   PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i)
-         = PV[n-1] * (1 + i) + PMT * (1 + iX)
-    for n >= 1
-

-
-
At the end of each iteration, PV[n] is rounded to the nearest cent. For
-each payment period, the interest due may be computed separately as:
-
-
-   ID[n] = (PV[n-1] + X * PMT) * i
-

-
-
and rounded to the nearest cent. PV[n] then becomes:
-
-
-   PV[n] = PV[n-1] + PMT + ID[n]
-

-
-
-
Annual Summary

-
For those cases where a yearly summary only is desired, it is not necessary
-to compute each transaction for each payment period, rather the PV may be
-be computed for the beginning of each year, PV[yr], and the FV computed for
-the end of the year, FV[yr]. The interest paid during the year is the computed as:
-
-
-   ID[yr] = (NP * PMT) + PV[yr] + FV[yr]
-    where: NP == number of payments during year
-              == PF for a full year of payments
-

-
-
-
Final Payment Calculation

-
Since the final payment may not be equal to the periodic payment, the final
-payment must be computed separately as follows. Two derivations are given below
-for the final payment equation. Both derivations are given below since one or
-the other may be clearer to some readers. Both derivations are essentially
-the same, they just have different starting points. The first is the fastest to derive.
-
-
Note, for the purposes of computing an amortization table, the number of periodic
-payments is assumed to be an integral value. For most cases this is true, the two
-principles in any transaction usually agree upon a certain term or number of periodic
-payments. In some calculations, however, this may not hold. In all of the calculations
-below, n is assumed integral and in the gnucash implementation, the following calculation
-is performed to assure this fact:
-
-
-    n = int(n)
-

-
-
    
-
  1. final_pmt == final payment @ payment n
-
    From the basic financial equation derived above:
-
-
    -       PV[n] = PV[n-1]*(1 + i) + final_pmt * (1 + iX), i == effective interest rate
-
    
-
-
    solving for final_pmt, we have:
-
    NOTE: FV[n] = -PV[n], for any n
-
-
    -       final_pmt * (1 + iX) = PV[n] - PV[n-1]*(1 + i)
-                            = FV[n-1]*(1 + i) - FV[n]
-       final_pmt = FV[n-1]*(1+i)/(1 + iX) - FV[n]/(1 + iX)
-
-       final_pmt = FV[n-1]*(1 + i) - FV[n], for X == 0, end of period payments
-                 = FV[n-1] - FV[n]/(1 + i), for X == 1, beginning of period payments
-
    
-
-
  2. final_pmt == final payment @ payment n
-
-
    -       i[n] == interest due @ payment n
-       i[n] = (PV[n-1] + X * final_pmt) * i, i == effective interest rate
-            = (X * final_pmt - FV[n]) * i
-
    
-
-
    Now the final payment is the sum of the interest due, plus the present value
-at the next to last payment plus any residual future value after the last payment:
-
-
    -       final_pmt = -i[n] - PV[n-1] - FV[n]
-                 = FV[n-1] - i[n] - FV[n]
-                 = FV[n-1] - (X *final_pmt - FV[n-1])*i - FV[n]
-                 = FV[n-1]*(1 + i) - X*final_pmt*i - FV[n]
-
    
-
-
    solving for final_pmt:
-
-
    -       final_pmt*(1 + iX) = FV[n-1]*(1 + i) - FV[n]
-       final_pmt = FV[n-1]*(1 + i)/(1 + iX) - FV[n]/(1 + iX)
-
-       final_pmt = FV[n-1]*(1 + i) - FV[n], for X == 0, end of period payments
-                 = FV[n-1] - FV[n]/(1 + i), for X == 1, beginning of period payments
-
    
-

-
-
-
-
-
Amortization Cases

-
-
The amortization schedule is computed for six different situations:
-
-
    
-
-
    Constant Repayment to Principal, Original Data
    
-
  1. In a constant repayment to principal loan, each payment varies. A constant amount
-is applied to the principal for each payment, usually equal to the originating present value
-divided by the number of repayment periods, and the interest for the payment period is
-added to the constant principal payment. The derivation of the equation for this type
-is contained in the Constant Repayment Equations section. This
-case computes the amortization schedule with the original loan data and a constant repayment
-to principal.
-
-
-
    Constant Repayment to Principal, Delayed Repayment
    
-
  2. In a constant repayment to principal loan, each payment varies. A constant amount
-is applied to the principal for each payment, usually equal to the originating present value
-divided by the number of repayment periods, and the interest for the payment period is
-added to the constant principal payment. The derivation of the equation for this type
-is contained in the Constant Repayment Equations section. This
-case computes the amortization schedule with the delayed loan data and a constant repayment
-to principal.
-
-
-
    Original Data Schedule
    
-
  3. The original financial data is used. This ignores any possible agjustment to
-the Present value due to any delay in the initial payment. This is quite
-common in mortgages where end of period payments are used and the first
-payment is scheduled for the end of the first whole period, i.e., any
-partial payment period from ED to the beginning of the next payment period
-is ignored.
-
-
-
    Recomputed Final Payment
    
-
  4. The original periodic payment is used, the Present Value is adjusted for the
-delayed Initial Payment. The total number of payments remains the same. The
-final payment is adjusted to bring the balance into agreement with the
-agreed upon final Future Value.
-
-
-
    Recomputed Periodic Payment
    
-
  5. A new periodic payment is computed based upon the adjusted Present Value, the
-originally agreed upon number of total payments and the agreed upon Future Value.
-The new periodic payments are computed to minimize the final payment in accordance
-with the Future Value after the last payment.
-
-
-
    Recomputed Term
    
-
  6. The original periodic payment is retained and a new number of total payments is computed
-based upon the adjusted Present Value and the agreed upon Future Value.
-

-
-
-
Amortization Schedule Display

-
The amortization schedule may be computed and displayed in three manners:
-
-
    
-
  1. The payment#, interest paid, principal paid and remaining PV for each payment period
-are computed and displayed.
-
    At the end of each year a summary is computed and displayed
-and the total interest paid is displayed at the end.
-
-
  2. A summary is computed and displayed for each year. The interest paid during the
-year is computed and displayed as well as the remaining balance at years end.
-
    The total interest paid is displayed at the end.
-
-
  3. An amortization schedule is computed and displayed for a common method of
-advanced payment of principal.
-
    In this amortization schedule, the principal for the
-next payment is computed and added into the current payment. This method will
-cut the number of total payments in half and will cut the interest paid almost
-in half.
-
    For mortgages, this method of prepayment has the advantage of keeping
-the total payments small during the initial payment periods
-The payments grow until the last payment period when presumably the borrower
-can afford larger payments.
-

-
-
-
NOTE: For Payment Frequencies, PF, semi-monthly or less, i.e., PF == 12 or PF == 24,
-a 360 day calendar year and 30 day month are used. For Payment Frequencies, PF,
-greater than semi-monthly, PF > 24, the actual number of days per year and per payment
-period are used. The actual values are computed using the built-in 'jdn' function
-
-
-
-
Financial Calculator Usage

-
the Financial Calculator is run as a QTAwk utility. If input is to be interactive and
-from the keyboard, do not specify any input files on the command line. The financial
-calcutlator reads all input from the standard input file. The calculator is started
-as:
-
-
-QTAwk -f fin.exp
-

-
-
The calculator will clear the display screen and display a two screen help:
-
-
-Financial Calculator
-Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-To compute Loan Quantities:
-N ==> to compute # payment periods from i, pv, pmt, fv
-_N(i,pv,pmt,fv,CF,PF,disc,bep) ==> to compute # payment periods
-I ==> to compute nominal interest rate from n, pv, pmt, fv, CF, PF, disc, bep
-_I(n,pv,pmt,fv,CF,PF,disc,bep) ==> to compute interest
-PV ==> to compute Present Value from n, i, pmt, fv, CF, PF, disc, bep
-_PV(n,i,pmt,fv) ==> to compute Present Value
-PMT ==> to compute Payment from n, i, pv, fv, CF, PF, disc, bep
-_PMT(n,i,pv,fv,CF,PF,disc,bep) ==> to compute Payment
-FV ==> to compute Future Value from n, i, pv, pmt, CF, PF, disc, bep
-_FV(n,i,pv,pmt,CF,PF,disc,bep) ==> to compute Future Value
-Press Any Key to Continue
-

-
-
The first screen displays the calculator commands which are available. Press any key
-and a second screen displays the variables defined by the calculator and which must be
-set by the user to use the financial calculator functions.
-
-
-[Aa](mort)? to Compute Amortization Schedule
-[Cc](ls)? to Clear Screen
-[Dd](efault)? to Re-Initialize
-[Hh](elp) to Display This Help
-[Qq](uit)? to Quit
-[Ss](tatus)? to Display Status of Computations
-[Uu](ser) Display User Defined Variables
-
-Variables to set:
-n    == number of periodic payments
-i    == interest per compouding interval
-pv   == present value
-pmt  == periodic payment
-fv   == future value
-disc == TRUE/FALSE == discrete/continuous compounding
-bep  == TRUE/FALSE == beginning of period/end of period payments
-CF   == compounding frequency per year
-PF   == payment frequency per year
-
-ED   == effective date of transaction, mm/dd/yyyy
-IP   == initial payment date of transaction, mm/dd/yyyy
-

-
-
-
Calculator Commands

-
The financial calculator commands available are listed above and below.
-
-
Note that the first letter of the command is all that is necessary to activate the
-desired function.
-
-
    
-
  1. [Aa](mort)? to Compute Amortization Schedule
-
    After all financial variables have been defined as well as the transaction dates,
-the amortization schedule can be computed for all financial transactions in which
-one would make sense.
-
  2. [Cc](ls)? to Clear Screen
-
    This command clears the screen and displays the copyright.
-
  3. [Dd](efault)? to Re-Initialize
-
    This command re-initializes all calculator variables to their start-up values.
-
  4. [Hh](elp) to Display This Help
-
    This command is used to display the start-up help screens at any time.
-
  5. [Qq](uit)? to Quit
-
    When the calculator is used interactively from the keyboard, this command allows
-the user to terminate the calculator session.
-
  6. [Ss](tatus)? to Display Status of Computations
-
    This command displays the status of the calculator variables. A typical status display
-would be:
-
-
    -Financial Calculator
-Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved
-Current Financial Calculator Status:
-Compounding Frequency: (CF) 12
-Payment     Frequency: (PF) 12
-Compounding: Discrete (disc = TRUE)
-Payments: End of Period (bep = FALSE)
-Number of Payment Periods (n): 360              (Years: 30)
-Nominal Annual Interest Rate (i): 7.25
-  Effective Interest Rate Per Payment Period: 0.00604167
-Present Value (pv): 233,350.00
-Periodic Payment (pmt): -1,591.86
-Future Value (fv): 0.00
-Effective       Date: Tue Jun 04 00:00:00 1996(2450239)
-Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
-<>
-
    
-
  7. [Uu](ser) Display User Defined Variables
-
    If any variables have been defined by the user, this command displays their names and
-values.
-

-
-
-
Calculator Input

-
The calculator displays an input prompt whenever it is waiting for input
-from the keyboard. The input prompt is simply <>. The desired
-input is typed at the keyboard and the enter key pressed. The result of calculating the
-value of the input line is then displayed by the calculator. For example, if the user wanted
-to set the value of the nominal interest in the calculator to 6.25, the following line would be
-input to the calculator:
-
-
i=6.25.
-
-
A semi-colon at the end of the input is optional.
-The line as seen on the display with the calculator input prompt would be:
-
-
-<>i = 6.25
-    6.25
-

-
-
Note that the calculator displays the value of the result, 6.25 in this case.
-
-
The calculator is controlled by setting the calculator variables to the desired values
-and "executing" the calculator functions to derive the values for the unknown
-variables. For example, for a conventional home mortgage for $233,350.00 with a thirty year
-term, nominal annual rate of 7.25%, n, i, pv and fv are known:
-
-
-n == 360 == 12 * 30
-i == 7.25
-pv= 233350
-fv = 0
-

-
-
The payments to completely pay off the mortgage with the 360 periodic payments is desired.
-To compute the desired periodic payment value, the PMT function is used. Since the
-function has no defined arguments, in invoking the function no arguments are specified. The
-complete session to input the desired values and calculate the periodic payment value would
-appear as:
-
-
-<>n=30*12
-        360
-<>i=7.25
-        7.25
-<>pv=233350
-        233,350
-<>PMT
-        -1,591.86
-

-
-
Note that the input may contain computations, n=30*12. In addition, any QTAwk
-built-in function may be specified and any functions defined in the financial calculator.
-This can be handy for computing intermediate values or other results from the results of
-the calculator.
-
-
Note that the output of the PMT function is rounded to the nearest cent. Over the
-thirty year term of the payback, the rounding will affect the last payment. To determine
-the balance due, fv, after 359 payment have been made, decrement n by 1 and compute the
-future value:
-
-
-<>n-=1
-        359
-<>FV
-        -1,580.20
-<>n+=1
-        360
-<>FV
-        2.12
-<>
-

-
-
The future value after 359 payments is less than the periodic payment and a full final payment
-will overpay the loan. The final FV computation with n restored to 360 shows an overpayment
-of 2.12.
-
-
-
Calculator Functions

-
The calculator functions:
-
-
-N
-I
-PV
-PMT
-FV
-

-
-
can be used to calculate the variable with the corresponding lower case name, using the
-values of the other four calculator variables which have already been set. In addition, the
-calculator functions:
-
-
-_N(i,pv,pmt,fv,CF,PF,disc,bep)
-_I(n,pv,pmt,fv,CF,PF,disc,bep)
-_PV(n,i,pmt,fv,CF,PF,disc,bep)
-_PMT(n,i,pv,fv,CF,PF,disc,bep)
-_FV(n,i,pv,pmt,CF,PF,disc,bep)
-

-
-
can be used to compute the value of the corresponding quantity for any specified value
-of the input arguments.
-
-
There are three differences between the functions N, I, PV, PMT, FV and the
-functions
-_N(i,pv,pmt,fv,CF,PF,disc,bep), _I(n,pv,pmt,fv,CF,PF,disc,bep), _PV(n,i,pmt,fv,CF,PF,disc,bep),
-_PMT(n,i,pv,fv,CF,PF,disc,bep), _FV(n,i,pv,pmt,CF,PF,disc,bep).
-
    
-
  1. The first set of functions take no arguments and
-use the calculator variables, n, i, pv, pmt, fv, CF, PF, disc
-and bep to compute the desired value. The second set of functions use the values passed in
-the function arguments. The first set of functions call the second set with the necessary
-arguments.
-
  2. The first set of functions round the computed value returned by the call to the second set
-of functions to the nearest cent. The second set of functions perform no rounding.
-
  3. The first set of functions set the calculator variables with the corresponding lower case name
-to the value computed. The second set of functions set no global variable values.
-

-
-
-
User Defined Variables

-
User defined variables may be defined and their values set to a desired qunatity. For example,
-to save computation results before re-initializing the calculator to obtain other results. If
-the user desired to compare the periodic payments necessary to fully pay the conventional
-mortgage cited above, the payment computed above could be saved in the variable
-end_pmt, the payments set to beginning of period payments and the new payment
-computed. The new value could be set into the variable beg_pmt. The two payments
-could then be viewed with the u command. The difference could then be computed
-between the two payment methods:
-
-
-<>n=30*12
-        360
-<>i=7.25
-        7.25
-<>pv=233350
-        233,350
-<>PMT
-        -1,591.86
-<>end_pmt=pmt
-        -1,591.86
-<>bep=1
-        1
-<>PMT
-        -1,582.30
-<>beg_pmt=pmt
-        -1,582.30
-<>u
-
-Financial Calculator
-Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-Current Financial Calculator Status:
-User Defined Variables:
-end_pmt == -1,591.86
-beg_pmt == -1,582.30
-<>beg_pmt-end_pmt
-        9.56
-<>
-

-
-
The financial calculator is thus a true calculator and can be used for computations
-desired by the user beyond those performed by the functions of the utility.
-
-
-
Rounding

-
Note that the output of the calculator is rounded to the nearest cent for floating
-point values. Sometimes the full accuracy of the value is desired. This can be obtained
-by redefing the calculator variable ofmt to the string "%.15g". You might want to
-save the current value in a user variable for resetting. For example in the above
-conventional mortgage, the exact value of the periodic payment can be displayed as:
-
-
-<>sofmt=ofmt
-        "%.2f"
-<>ofmt="%.15g"
-        "%.15g"
-<>pmt=_PMT(n,i,pv,fv,CF,PF,disc,bep)
-        -1,591.85834951112
-<>ofmt=sofmt
-        "%.2f"
-<>
-

-
-
Note that the current value of the output format string, ofmt, has been
-saved in the variable, sofmt, and later restored.
-
-
-
Examples

-
-
-
-
-
-
-
-
-
-
-
Simple Interest 

-
Simple Interest
-
 Find the annual simple interest rate (%) for an $800 loan to be repayed at the
- end of one year with a single payment of $896.
-
- <>d
- <>CF=PF=1
-         1
- <>n=1
-         1
- <>pv=-800
-         -800
- <>fv=896
-         896
- <>I
-         12.00
-

-
-
-
Compound Interest

-
Compound Interest
-
Find the future value of $800 after one year at a nominal rate of 12%
- compounded monthly. No payments are specified, so the payment frequency is
- set equal to the compounding frequency at the default values.
-
- <>d
- <>n=12
-         12
- <>i=12
-         12
- <>pv=-800
-         -800
- <>FV
-          901.46
-

-
-
-
Periodic Payment

-
Periodic Payment
-
Find the monthly end-of-period payment required to fully amortize the loan
- in Example 2. A fully amortized loan has a future value of zero.
-
- <>fv=0
-        0
- <>PMT
-        71.08
-

-
-
-
Conventional Mortgage

-
Conventional Mortgage
-
Find the number of monthly payments necessary to fully amortize a loan of
- $100,000 at a nominal rate of 13.25% compounded monthly, if monthly end-of-period
- payments of $1125.75 are made.
-
- <>d
- <>i=13.25
-         13.25
- <>pv=100000
-         100,000
- <>pmt=-1125.75
-         -1,125.75
- <>N
-         360.10
-

-
-
-
Final Payment

-
Final Payment
-
Using the data in the above example, find the amount of the final payment if n is
-changed to 360. The final payment will be equal to the regular payment plus
-any balance, future value, remaining at the end of period number 360.
-
- <>n=int(n)
-        360
- <>FV
-        -108.87
- <>pmt+fv
-        -1,234.62
-

-
-
-
Conventional Mortgage Amortization Schedule - Annual Summary

-
Conventional Mortgage Amortization Schedule - Annual Summary
-
Using the data from the loan in the previous example, compute the amortization
-schedule when the
-Effective date of the loan is June 6, 1996 and the initial payment is
-made on August 1, 1996. Ignore any change in the PV due to the delayed
-initial payment caused by the partial payment period from June 6 to July 1.
-
-
- <>ED=6/6/1996
- Effective Date set: (2450241) Thu Jun 06 00:00:00 1996
- <>IP=8/1/96
- Initial Payment Date set: (2450297) Thu Aug 01 00:00:00 1996
- <>a
-   Effective       Date: Thu Jun 06 00:00:00 1996
-   Initial Payment Date: Thu Aug 01 00:00:00 1996
-   The amortization options are:
-   The Old Present Value (pv)     was: 100,000.00
-   The Old Periodic Payment (pmt) was: -1,125.75
-   The Old Future  Value (fv)     was: -108.87
-   1: Amortize with Original Transaction Values
-       and final payment: -1,125.75
-
-   The New Present Value (pve)  is:  100,919.30
-   The New Periodic Payment (pmt) is:  -1,136.10
-   2: Amortize with Original Periodic Payment
-       and final payment: -49,023.68
-   3: Amortize with New Periodic Payment
-       and final payment: -1,132.57
-   4: Amortize with Original Periodic Payment,
-       new number of total payments (n): 417
-       and final payment: -2,090.27
-
-   Enter choice 1, 2, 3 or 4: <>
-

-
-
Press '1' to choose option 1:
-
-
-    Amortization Schedule:
-   Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
-   Enter choice y, p or a:
-   <>
-

-
-
Press 'y' for an annual summary:
-
-
-   Enter Filename for Amortization Schedule.
-     (null string uses Standard Output):
-

-
-
Press enter to display output on screen:
-
-
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (359): -1,125.75
-  Final payment (# 360): -1,125.75
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Year      Interest   Ending Balance
-  1996     -5,518.42       -99,889.67
-  1997    -13,218.14       -99,598.81
-  1998    -13,177.17       -99,266.98
-  1999    -13,130.43       -98,888.41
-  2000    -13,077.11       -98,456.52
-  2001    -13,016.28       -97,963.80
-  2002    -12,946.88       -97,401.68
-  2003    -12,867.70       -96,760.38
-  2004    -12,777.38       -96,028.76
-  2005    -12,674.33       -95,194.09
-  2006    -12,556.76       -94,241.85
-  2007    -12,422.64       -93,155.49
-  2008    -12,269.63       -91,916.12
-  2009    -12,095.06       -90,502.18
-  2010    -11,895.91       -88,889.09
-  2011    -11,668.70       -87,048.79
-  2012    -11,409.50       -84,949.29
-  2013    -11,113.78       -82,554.07
-  2014    -10,776.41       -79,821.48
-  2015    -10,391.53       -76,704.01
-  2016     -9,952.43       -73,147.44
-  2017     -9,451.49       -69,089.93
-  2018     -8,879.99       -64,460.92
-  2019     -8,227.99       -59,179.91
-  2020     -7,484.16       -53,155.07
-  2021     -6,635.56       -46,281.63
-  2022     -5,667.43       -38,440.06
-  2023     -4,562.94       -29,494.00
-  2024     -3,302.89       -19,287.89
-  2025     -1,865.36        -7,644.25
-  2026       -236.00          -108.87
-
-  Total Interest: -305,270.00
-

-
-
 NOTE: The amortization table leaves the FV as it was when the amortization
-function was entered. Thus, a balance of 108.87 is due at the end of the
-table. To completely pay the loan, set fv to 0.0:
-
-<>fv=0
-    0
-<>a
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  The amortization options are:
-  The Old Present Value (pv)     was: 100,000.00
-  The Old Periodic Payment (pmt) was: -1,125.75
-  The Old Future  Value (fv)     was: 0.00
-  1: Amortize with Original Transaction Values
-      and final payment: -1,234.62
-
-  The New Present Value (pve)  is:  100,919.30
-  The New Periodic Payment (pmt) is:  -1,136.12
-  2: Amortize with Original Periodic Payment
-      and final payment: -49,132.55
-  3: Amortize with New Periodic Payment
-      and final payment: -1,148.90
-  4: Amortize with Original Periodic Payment,
-      new number of total payments (n): 417
-      and final payment: -2,199.14
-
-  Enter choice 1, 2, 3 or 4: <>
-

-
-
Press '1' for option 1:
-
-
-    Amortization Schedule:
-   Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
-   Enter choice y, p or a:
-   <>
-

-
-
Press 'y' for annual summary:
-
-
-   Enter Filename for Amortization Schedule.
-     (null string uses Standard Output):
-

-
-
Press enter to display output on screen:
-
-
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (359): -1,125.75
-  Final payment (# 360): -1,234.62
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Year      Interest   Ending Balance
-  1996     -5,518.42       -99,889.67
-  1997    -13,218.14       -99,598.81
-  1998    -13,177.17       -99,266.98
-  1999    -13,130.43       -98,888.41
-  2000    -13,077.11       -98,456.52
-  2001    -13,016.28       -97,963.80
-  2002    -12,946.88       -97,401.68
-  2003    -12,867.70       -96,760.38
-  2004    -12,777.38       -96,028.76
-  2005    -12,674.33       -95,194.09
-  2006    -12,556.76       -94,241.85
-  2007    -12,422.64       -93,155.49
-  2008    -12,269.63       -91,916.12
-  2009    -12,095.06       -90,502.18
-  2010    -11,895.91       -88,889.09
-  2011    -11,668.70       -87,048.79
-  2012    -11,409.50       -84,949.29
-  2013    -11,113.78       -82,554.07
-  2014    -10,776.41       -79,821.48
-  2015    -10,391.53       -76,704.01
-  2016     -9,952.43       -73,147.44
-  2017     -9,451.49       -69,089.93
-  2018     -8,879.99       -64,460.92
-  2019     -8,227.99       -59,179.91
-  2020     -7,484.16       -53,155.07
-  2021     -6,635.56       -46,281.63
-  2022     -5,667.43       -38,440.06
-  2023     -4,562.94       -29,494.00
-  2024     -3,302.89       -19,287.89
-  2025     -1,865.36        -7,644.25
-  2026       -344.87             0.00
-
-  Total Interest: -305,378.87
-

-
-
Note that now the final payment differs from the periodic payment and
-the loan has been fully paid off.
-
-
-
Conventional Mortgage Amortization Schedule - Periodic Payment Schedule

-
Conventional Mortgage Amortization Schedule  - Periodic Payment Schedule
-
Using the loan in the previous example, compute the amortization table and display the
-results for each payment period.
-As in example 6, ignore any increase in the PV due to the
-delayed IP.
-
-
-<>
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (359): -1,125.75
-  Final payment (# 360): -1,234.62
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Pmt#       Interest      Principal        Balance
-     1      -1,104.17         -21.58     -99,978.42
-     2      -1,103.93         -21.82     -99,956.60
-     3      -1,103.69         -22.06     -99,934.54
-     4      -1,103.44         -22.31     -99,912.23
-     5      -1,103.20         -22.55     -99,889.68
-  Summary for 1996:
-    Interest  Paid: -5,518.43
-    Principal Paid: -110.32
-    Year Ending Balance: -99,889.68
-    Sum of Interest Paid: -5,518.43
-  Pmt#       Interest      Principal        Balance
-     6      -1,102.95         -22.80     -99,866.88
-     7      -1,102.70         -23.05     -99,843.83
-     8      -1,102.44         -23.31     -99,820.52
-     9      -1,102.18         -23.57     -99,796.95
-    10      -1,101.92         -23.83     -99,773.12
-    11      -1,101.66         -24.09     -99,749.03
-    12      -1,101.40         -24.35     -99,724.68
-    13      -1,101.13         -24.62     -99,700.06
-    14      -1,100.85         -24.90     -99,675.16
-    15      -1,100.58         -25.17     -99,649.99
-    16      -1,100.30         -25.45     -99,624.54
-    17      -1,100.02         -25.73     -99,598.81
-  Summary for 1997:
-    Interest  Paid: -13,218.13
-    Principal Paid: -290.87
-    Year Ending Balance: -99,598.81
-    Sum of Interest Paid: -18,736.56
-  Pmt#       Interest      Principal        Balance
-    18      -1,099.74         -26.01     -99,572.80
-    19      -1,099.45         -26.30     -99,546.50
-    .
-    .
-    .
-   346        -171.99        -953.76     -14,622.84
-   347        -161.46        -964.29     -13,658.55
-   348        -150.81        -974.94     -12,683.61
-   349        -140.05        -985.70     -11,697.91
-   350        -129.16        -996.59     -10,701.32
-   351        -118.16      -1,007.59      -9,693.73
-   352        -107.03      -1,018.72      -8,675.01
-   353         -95.79      -1,029.96      -7,645.05
-  Summary for 2025:
-    Interest  Paid: -1,865.45
-    Principal Paid: -11,643.55
-    Year Ending Balance: -7,645.05
-    Sum of Interest Paid: -305,034.80
-  Pmt#       Interest      Principal        Balance
-   354         -84.41      -1,041.34      -6,603.71
-   355         -72.92      -1,052.83      -5,550.88
-   356         -61.29      -1,064.46      -4,486.42
-   357         -49.54      -1,076.21      -3,410.21
-   358         -37.65      -1,088.10      -2,322.11
-   359         -25.64      -1,100.11      -1,222.00
-  Final Payment (360): -1,235.49
-   360         -13.49      -1,222.00           0.00
-  Summary for 2026:
-    Interest  Paid: -344.94
-    Principal Paid: -7,645.05
-
-  Total Interest: -305,379.74
-

-
-
The complete amortization table can be viewed in the
-Periodic Amortization Schedule for this loan.
-
-
You will notice several differences between this amortization schedule and the
-Annual Summary Schedule. The Periodic Payment Schedule lists the interest paid for
-each payment as well as the principal paid and the remaining balance to be repaid.
-At the end of each year an annual summary is printed. At the end of the table the
-total interest is printed as in the Annual Summary Schedule.
-
-
You will notice that the total interest output at the end of the Periodic Payment
-Schedule differs slightly from the total interest output at the end of the Annual Summary
-Schedule:
-
-
Total Interest for Periodic Payment Schedule:
-
-  Total Interest: -305,379.74
-

-
-
Total Interest for Annual Summary Schedule:
-
-
-  Total Interest: -305,378.87
-

-
-
The difference in total interest is due to the rounding of all quantities at
-each periodic payment. The Total Interest paid shown in the Periodic Payment
-Schedule will be the more accurate since all quantities exchanged in a financial
-transaction will be done to the nearest cent.
-
-
-
Conventional Mortgage Schedule - Variable Advanced Payments

-
Conventional Mortgage Schedule - Variable Advanced Payments
-
Again using the loan in the previous examples, compute the amortization table using
-the advanced payment
-option to prepay the loan. As in the previous example, ignore any increase in the PV due to the
-delayed IP.
-
-
-
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  The amortization options are:
-  The Old Present Value (pv)     was: 100,000.00
-  The Old Periodic Payment (pmt) was: -1,125.75
-  The Old Future  Value (fv)     was: 0.00
-  1: Amortize with Original Transaction Values
-      and final payment: -1,234.62
-
-  The New Present Value (pve)  is:  100,919.30
-  The New Periodic Payment (pmt) is:  -1,136.12
-  2: Amortize with Original Periodic Payment
-      and final payment: -49,132.55
-  3: Amortize with New Periodic Payment
-      and final payment: -1,148.90
-  4: Amortize with Original Periodic Payment,
-      new number of total payments (n): 417
-      and final payment: -2,199.14
-
-  Enter choice 1, 2, 3 or 4: <>
-

-
-
Press 1 for option 1:
-
-
-   Amortization Schedule:
-  Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
-  Enter choice y, p or a:
-  <>
-

-
-
Press a for the Advanced Payment Option:
-
-
-  Enter Filename for Amortization Schedule.
-    (null string uses Standard Output):
-

-
-
Press enter to display output on screen:
-
-
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (359): -1,125.75
-  Final payment (# 360): -1,234.62
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Advanced Prepayment Amortization
-  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-     1    -1,104.17       -21.58       -21.82    -1,147.57   -99,956.60
-     2    -1,103.69       -22.06       -22.31    -1,148.06   -99,912.23
-     3    -1,103.20       -22.55       -22.80    -1,148.55   -99,866.88
-     4    -1,102.70       -23.05       -23.31    -1,149.06   -99,820.52
-     5    -1,102.18       -23.57       -23.83    -1,149.58   -99,773.12
-  Summary for 1996:
-    Interest  Paid: -5,515.94
-    Principal Paid: -226.88
-    Year Ending Balance: -99,773.12
-    Sum of Interest Paid: -5,515.94
-  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-     6    -1,101.66       -24.09       -24.35    -1,150.10   -99,724.68
-     7    -1,101.13       -24.62       -24.90    -1,150.65   -99,675.16
-     8    -1,100.58       -25.17       -25.45    -1,151.20   -99,624.54
-     9    -1,100.02       -25.73       -26.01    -1,151.76   -99,572.80
-    10    -1,099.45       -26.30       -26.59    -1,152.34   -99,519.91
-    11    -1,098.87       -26.88       -27.18    -1,152.93   -99,465.85
-    12    -1,098.27       -27.48       -27.78    -1,153.53   -99,410.59
-    13    -1,097.66       -28.09       -28.40    -1,154.15   -99,354.10
-    14    -1,097.03       -28.72       -29.03    -1,154.78   -99,296.35
-    15    -1,096.40       -29.35       -29.68    -1,155.43   -99,237.32
-    16    -1,095.75       -30.00       -30.34    -1,156.09   -99,176.98
-    17    -1,095.08       -30.67       -31.01    -1,156.76   -99,115.30
-  Summary for 1997:
-    Interest  Paid: -13,181.90
-    Principal Paid: -657.82
-    Year Ending Balance: -99,115.30
-    Sum of Interest Paid: -18,697.84
-  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-    18    -1,094.40       -31.35       -31.70    -1,157.45   -99,052.25
-    19    -1,093.70       -32.05       -32.40    -1,158.15   -98,987.80
-    20    -1,092.99       -32.76       -33.12    -1,158.87   -98,921.92
-    .
-    .
-    .
-   167      -298.87      -826.88      -836.01    -1,961.76   -25,404.90
-   168      -280.51      -845.24      -854.57    -1,980.32   -23,705.09
-   169      -261.74      -864.01      -873.55    -1,999.30   -21,967.53
-   170      -242.56      -883.19      -892.94    -2,018.69   -20,191.40
-   171      -222.95      -902.80      -912.77    -2,038.52   -18,375.83
-   172      -202.90      -922.85      -933.04    -2,058.79   -16,519.94
-   173      -182.41      -943.34      -953.76    -2,079.51   -14,622.84
-  Summary for 2010:
-    Interest  Paid: -3,448.07
-    Principal Paid: -20,232.96
-    Year Ending Balance: -14,622.84
-    Sum of Interest Paid: -152,300.57
-  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-   174      -161.46      -964.29      -974.94    -2,100.69   -12,683.61
-   175      -140.05      -985.70      -996.59    -2,122.34   -10,701.32
-   176      -118.16    -1,007.59    -1,018.72    -2,144.47    -8,675.01
-   177       -95.79    -1,029.96    -1,041.34    -2,167.09    -6,603.71
-   178       -72.92    -1,052.83    -1,064.46    -2,190.21    -4,486.42
-   179       -49.54    -1,076.21    -1,088.10    -2,213.85    -2,322.11
-   180       -25.64    -1,100.11    -1,222.00    -2,347.75         0.00
-  Summary for 2011:
-    Interest  Paid: -663.56
-    Principal Paid: -14,622.84
-
-  Total Interest: -152,964.13
-

-
-
The complete amortization table can be viewed in the
-Advanced Payment Amortization Schedule for this loan.
-
-
This schedule has added two columns over the Periodic Payment Schedule in Example 7. Namely,
-Prepay and the Total Pmt columns. The Prepay column is the
-amount of the loan prepayment for the period. The Total Pmt column is the sum
-of the periodic payment and the Prepayment. Note that both the Prepay and the
-Total Pmt quantities increase with each period.
-
-
-
Conventional Mortgage Schedule - Constant Advanced Payments

-
Conventional Mortgage Schedule - Constant Advanced Payments
-
Using the loan in the previous examples, compute the amortization table using
-another payment option for repaying a loan ahead of schedule and reducing the interest
-paid, constant repayments at each periodic payment. Suppose a constant $100.00 is paid
-towards the principal with each periodic payment. How many payments are needed to fully payoff
-the loan and what is the total interest paid?
-
-
As in the previous example, ignore any increase in the PV due to the
-delayed IP.
-
-
There are two ways to compute the amortization table for this type of prepayment option.
-In the first method, set the variable 'FP' to the amount of the monthly prepayment.
-
-
-<>FP=-100
-  -100
-<>a
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  The amortization options are:
-  The Old Present Value (pv)     was: 100,000.00
-  The Old Periodic Payment (pmt) was: -1,125.75
-  The Old Future  Value (fv)     was: 0.00
-  1: Amortize with Original Transaction Values
-      and final payment: -1,234.62
-
-  The New Present Value (pve)  is:  100,919.30
-  The New Periodic Payment (pmt) is:  -1,136.12
-  2: Amortize with Original Periodic Payment
-      and final payment: -49,132.55
-  3: Amortize with New Periodic Payment
-      and final payment: -1,148.90
-  4: Amortize with Original Periodic Payment,
-      new number of total payments (n): 417
-      and final payment: -2,199.14
-
-  Enter choice 1, 2, 3 or 4: <>
-

-
-
Press 1 for option 1:
-
-
-   Amortization Schedule:
-  Yearly, y, per Payment, p, Advanced Payment, a, or Fixed Prepayment, f, Amortization
-  Enter choice y, p, a or f:
-  <>
-

-
-
Press f for the Fixed Prepayment schedule.
-
-
-  Enter Filename for Amortization Schedule.
-    (null string uses Standard Output):
-

-
-
Press enter to display output on screen:
-
-
-Amortization Table
-Effective       Date: Thu Jun  6 00:00:00 1996
-Initial Payment Date: Thu Aug  1 00:00:00 1996
-Compounding Frequency per year: 12
-Payment     Frequency per year: 12
-Compounding: Discrete
-Payments: End of Period
-Payments (359): -1,125.75
-Final payment (# 360): -1,234.62
-Nominal Annual Interest Rate: 13.25
-  Effective Interest Rate Per Payment Period: 0.0110417
-Present Value: 100,000.00
-Advanced Prepayment Amortization - fixed prepayment: -100.00
-Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-   1    -1,104.17       -21.58      -100.00    -1,225.75   -99,878.42
-   2    -1,102.82       -22.93      -100.00    -1,225.75   -99,755.49
-   3    -1,101.47       -24.28      -100.00    -1,225.75   -99,631.21
-   4    -1,100.09       -25.66      -100.00    -1,225.75   -99,505.55
-   5    -1,098.71       -27.04      -100.00    -1,225.75   -99,378.51
-Summary for 1996:
-  Interest  Paid: -5,507.26
-  Principal Paid: -621.49
-  Year Ending Balance: -99,378.51
-  Sum of Interest Paid: -5,507.26
-Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
-   6    -1,097.30       -28.45      -100.00    -1,225.75   -99,250.06
-   7    -1,095.89       -29.86      -100.00    -1,225.75   -99,120.20
-   8    -1,094.45       -31.30      -100.00    -1,225.75   -98,988.90
-   9    -1,093.00       -32.75      -100.00    -1,225.75   -98,856.15
-  10    -1,091.54       -34.21      -100.00    -1,225.75   -98,721.94
-  11    -1,090.05       -35.70      -100.00    -1,225.75   -98,586.24
-  12    -1,088.56       -37.19      -100.00    -1,225.75   -98,449.05
-  13    -1,087.04       -38.71      -100.00    -1,225.75   -98,310.34
-  14    -1,085.51       -40.24      -100.00    -1,225.75   -98,170.10
-  15    -1,083.96       -41.79      -100.00    -1,225.75   -98,028.31
-  16    -1,082.40       -43.35      -100.00    -1,225.75   -97,884.96
-  17    -1,080.81       -44.94      -100.00    -1,225.75   -97,740.02
-Summary for 1997:
-  Interest  Paid: -13,070.51
-  Principal Paid: -1,638.49
-  Year Ending Balance: -97,740.02
-  Sum of Interest Paid: -18,577.77
-.
-.
-.
-
-Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
- 186      -298.60      -827.15      -100.00    -1,225.75   -26,115.84
- 187      -288.36      -837.39      -100.00    -1,225.75   -25,178.45
- 188      -278.01      -847.74      -100.00    -1,225.75   -24,230.71
- 189      -267.55      -858.20      -100.00    -1,225.75   -23,272.51
- 190      -256.97      -868.78      -100.00    -1,225.75   -22,303.73
- 191      -246.27      -879.48      -100.00    -1,225.75   -21,324.25
- 192      -235.46      -890.29      -100.00    -1,225.75   -20,333.96
- 193      -224.52      -901.23      -100.00    -1,225.75   -19,332.73
- 194      -213.47      -912.28      -100.00    -1,225.75   -18,320.45
- 195      -202.29      -923.46      -100.00    -1,225.75   -17,296.99
- 196      -190.99      -934.76      -100.00    -1,225.75   -16,262.23
- 197      -179.56      -946.19      -100.00    -1,225.75   -15,216.04
-Summary for 2012:
-  Interest  Paid: -2,882.05
-  Principal Paid: -11,826.95
-  Year Ending Balance: -15,216.04
-  Sum of Interest Paid: -156,688.79
-Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
- 198      -168.01      -957.74      -100.00    -1,225.75   -14,158.30
- 199      -156.33      -969.42      -100.00    -1,225.75   -13,088.88
- 200      -144.52      -981.23      -100.00    -1,225.75   -12,007.65
- 201      -132.58      -993.17      -100.00    -1,225.75   -10,914.48
- 202      -120.51    -1,005.24      -100.00    -1,225.75    -9,809.24
- 203      -108.31    -1,017.44      -100.00    -1,225.75    -8,691.80
- 204       -95.97    -1,029.78      -100.00    -1,225.75    -7,562.02
- 205       -83.50    -1,042.25      -100.00    -1,225.75    -6,419.77
- 206       -70.88    -1,054.87      -100.00    -1,225.75    -5,264.90
- 207       -58.13    -1,067.62      -100.00    -1,225.75    -4,097.28
- 208       -45.24    -1,080.51      -100.00    -1,225.75    -2,916.77
- 209       -32.21    -1,093.54      -100.00    -1,225.75    -1,723.23
-Summary for 2013:
-  Interest  Paid: -1,216.19
-  Principal Paid: -13,492.81
-  Year Ending Balance: -1,723.23
-  Sum of Interest Paid: -157,904.98
-Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
- 210       -19.03    -1,106.72      -100.00    -1,225.75      -516.51
- 211        -5.70      -516.51         0.00      -522.21         0.00
-
-Total Interest: 157,929.71
-
-

-
-
In the second method, the periodic payment is increased by 100. With this method,
-the annual summary table can also be computed.
-
-
-<>s
-Financial Calculator
-Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-Current Financial Calculator Status:
-Compounding Frequency: (CF) 12
-Payment     Frequency: (PF) 12
-Compounding: Discrete (disc = TRUE)
-Payments: End of Period (bep = FALSE)
-Number of Payment Periods (n): 360              (Years: 30)
-Nominal Annual Interest Rate (i): 13.25
-  Effective Interest Rate Per Payment Period: 0.0110417
-Present Value (pv): 100,000.00
-Periodic Payment (pmt): -1,125.75
-Future Value (fv): 0.00
-Effective       Date: Thu Jun 06 00:00:00 1996(2450241)
-Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
-<>pmt-=100
-        -1,225.75
-<>N
-        210.42
-<>
-

-
-
Thus, the loan will now be fully repaid in 210 full payments and a partial payment
-as illustrated in the previous table.
-To get the total interest paid, display the Annual Summary Amortization Schedule:
-
-
-Effective       Date: Thu Jun 06 00:00:00 1996
-Initial Payment Date: Thu Aug 01 00:00:00 1996
-The amortization options are:
-The Old Present Value (pv)     was: 100,000.00
-The Old Periodic Payment (pmt) was: -1,225.75
-The Old Future  Value (fv)     was: 0.00
-1: Amortize with Original Transaction Values
-    and final payment: -1,742.55
-
-The New Present Value (pve)  is:  100,919.30
-The New Periodic Payment (pmt) is:  -1,237.02
-2: Amortize with Original Periodic Payment
-    and final payment: -10,967.39
-3: Amortize with New Periodic Payment
-    and final payment: -1,757.20
-4: Amortize with Original Periodic Payment,
-    new number of total payments (n): 218
-    and final payment: -1,668.45
-
-Enter choice 1, 2, 3 or 4: <>
-

-
-
Press '1' for option 1:
-
-
- Amortization Schedule:
-Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
-Enter choice y, p or a:
-<>
-

-
-
Press 'y' for an annual Summary
-
-
-Enter Filename for Amortization Schedule.
-  (null string uses Standard Output):
-

-
-
Press enter to display the summary on the screen:
-
-
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (209): -1,225.75
-  Final payment (# 210): -1,742.55
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Year      Interest   Ending Balance
-  1996     -5,507.26       -99,378.51
-  1997    -13,070.52       -97,740.03
-  1998    -12,839.74       -95,870.77
-  1999    -12,576.45       -93,738.22
-  2000    -12,276.08       -91,305.30
-  2001    -11,933.40       -88,529.70
-  2002    -11,542.46       -85,363.16
-  2003    -11,096.45       -81,750.61
-  2004    -10,587.62       -77,629.23
-  2005    -10,007.12       -72,927.35
-  2006     -9,344.86       -67,563.21
-  2007     -8,589.32       -61,443.53
-  2008     -7,727.36       -54,461.89
-  2009     -6,744.00       -46,496.89
-  2010     -5,622.13       -37,410.02
-  2011     -4,342.24       -27,043.26
-  2012     -2,882.08       -15,216.34
-  2013     -1,216.25        -1,723.59
-  2014        -18.96             0.00
-
-  Total Interest: -157,924.30
-

-
-
From the last line the Total interest has been decreased from $305,379.74 to
-$157,924.30.
-
-
We can also ask how much of a constant repayment would be necessary to  fully
-repay the loan in 15 years and what would be the total interest paid?
-
-
-  <>n=12*15
-          180
-  <>opmt=pmt
-          -1,125.75
-  <>PMT
-          -1,281.74
-  <>pmt-opmt
-          -155.99
-

-
-
Thus, a constant advanced repayment per periodic payment of $155.99 would fully
-amortize the loan in 15 years.
-
-
-  <>a
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  The amortization options are:
-  The Old Present Value (pv)     was: 100,000.00
-  The Old Periodic Payment (pmt) was: -1,281.74
-  The Old Future  Value (fv)     was: 0.00
-  1: Amortize with Original Transaction Values
-      and final payment: -1,279.73
-
-  The New Present Value (pve)  is:  100,919.30
-  The New Periodic Payment (pmt) is:  -1,293.52
-  2: Amortize with Original Periodic Payment
-      and final payment: -7,915.43
-  3: Amortize with New Periodic Payment
-      and final payment: -1,293.20
-  4: Amortize with Original Periodic Payment,
-      new number of total payments (n): 185
-      and final payment: -1,738.05
-
-  Enter choice 1, 2, 3 or 4: <>
-

-
-
Press '1' for option 1:
-
-
-   Amortization Schedule:
-  Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
-  Enter choice y, p or a:
-  <>
-

-
-
Press 'y' for an annual Summary
-
-
-  Amortization Table
-  Effective       Date: Thu Jun 06 00:00:00 1996
-  Initial Payment Date: Thu Aug 01 00:00:00 1996
-  Compounding Frequency per year: 12
-  Payment     Frequency per year: 12
-  Compounding: Discrete
-  Payments: End of Period
-  Payments (179): -1,281.74
-  Final payment (# 180): -1,279.73
-  Nominal Annual Interest Rate: 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value: 100,000.00
-  Year      Interest   Ending Balance
-  1996     -5,501.01       -99,092.31
-  1997    -12,987.86       -96,699.29
-  1998    -12,650.80       -93,969.21
-  1999    -12,266.27       -90,854.60
-  2000    -11,827.58       -87,301.30
-  2001    -11,327.09       -83,247.51
-  2002    -10,756.12       -78,622.75
-  2003    -10,104.72       -73,346.59
-  2004     -9,361.57       -67,327.28
-  2005     -8,513.75       -60,460.15
-  2006     -7,546.51       -52,625.78
-  2007     -6,443.04       -43,687.94
-  2008     -5,184.14       -33,491.20
-  2009     -3,747.93       -21,858.25
-  2010     -2,109.42        -8,586.79
-  2011       -383.38             0.00
-
-  Total Interest: -130,711.19
-

-
-
The toral interest is reduced to $130,711.19. This compares to:
-
-
    
-
  1. $130,711.19 - Fixed prepayment $155.99/period, 15 year term
-
  2. $152,964.13 - Variable Advanced Repayment, 15 year term
-
  3. $305,379.74 - no prepayment, 30 year term
-

-
-
-
Balloon Payment

-
Balloon Payment
-
On long term loans, small changes in the periodic payments can generate
-large changes in the future value. If the monthly payment in the previous example is
-rounded down to $1125, how much additional (balloon) payment will be due
-with the final regular payment.
-
-  <>s
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  Current Financial Calculator Status:
-  Compounding Frequency: (CF) 12
-  Payment     Frequency: (PF) 12
-  Compounding: Discrete (disc = TRUE)
-  Payments: End of Period (bep = FALSE)
-  Number of Payment Periods (n): 180              (Years: 15)
-  Nominal Annual Interest Rate (i): 13.25
-    Effective Interest Rate Per Payment Period: 0.0110417
-  Present Value (pv): 100,000.00
-  Periodic Payment (pmt): -1,281.74
-  Future Value (fv): 0.00
-  Effective       Date: Thu Jun 06 00:00:00 1996(2450241)
-  Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
-  <>n=360
-          360
-  <>pmt=-1125
-          -1,125
-  <>FV
-          -3,579.99
-  <>
-

-
-
-
Canadian Mortgage

-
Canadian Mortgage
-
A "Canadian Mortgage" is defined with semi-annual compunding, CF == 2,
-and monthly payments, PF == 12.
-
-
Find the monthly end-of-period payment necessary to fully amortize a 25 year
-$85,000 loan at 11% compounded semi-annually.
-
- <>d
- <>CF=2
-         2
- <>n=300
-         300
- <>i=11
-         11
- <>pv=85000
-         85,000
- <>PMT
-         -818.15
-

-
-
-

-
European Mortgage
-
The "effective annual rate (EAR)" is used in some countries (especially
- in Europe) in lieu of the nominal rate commonly used in the United States
- and Canada. For a 30 year $90,000 mortgage at 14% (EAR), compute the monthly
- end-of-period payments. When using an EAR, the compounding frequency is
- set to 1.
-
- <>d
- <>CF=1
-         1
- <>n=30*12
-         360
- <>i=14
-         14
- <>pv=90000
-         90,000
- <>PMT
-         -1,007.88
-

-
-
-
Bi-weekly Savings

-
Bi-weekly Savings
-
Compute the future value, fv, of bi-weekly savings of $100 for 3 years at a
- nominal annual rate of 5.5% compounded daily. (Set payment to
- beginning-of-period, bep = TRUE)
-
- <>d
- <>bep=TRUE
-         1
- <>CF=365
-         365
- <>PF=26
-         26
- <>n=3*26
-         78
- <>i=5.5
-         5.50
- <>pmt=-100
-         -100
- <>FV
-         8,489.32
-

-
-
-
Present Value - Annuity Due

-
Present Value - Annuity Due
-
What is the present value of $500 to be received at the beginning of each
- quarter over a 10 year period if money is being discounted at 10% nominal
- annual rate compounded monthly?
-
- <>d
- <>bep=TRUE
-         1
- <>PF=4
-         4
- <>n=4*10
-         40
- <>i=10
-         10
- <>pmt=500
-         500
- <>PV
-         -12,822.64
-

-
-
-
Effective Rate - 365/360 Basis

-
Effective Rate - 365/360 Basis
-
Compute the effective annual rate (%APR) for a nominal annual rate of 12%
- compounded on a 365/360 basis used by some Savings & Loan Associations.
-
- <>d
- <>n=365
-         365
- <>CF=365
-         365
- <>PF=360
-         360
- <>i=12
-         12
- <>pv=-100
-         -100
- <>FV
-         112.94
- <>fv+pv
-         12.94
-

-
-
-
Certificate of Deposit, Annual Percentage Yield

-
Certificate of Deposit, Annual Percentage Yield
-
Most, if not all banks have started stating return rates on Certificates of Deposit, CDs, as
-an Annual Percentage Yoild, APY, and the nominal annual interest. For example, a bank will advertise
-a CD with a 18 month term, an APY of 5.20% and a nominal rate of 5.00. What values of CF and PF will
-are being used?
-
-
- <>d
- <>n=365
-         365
- <>CF=PF=365
-         365
- <>i=5
-         5
- <>pv=-100
-         -100
- <>FV
-         105.13
- <>CF=PF=360
-         360
- <>fv+pv
-         -5.20
-

-
-
Mortgage with "Points"

-
Mortgage with "Points"
-
What is the true APR of a 30 year, $75,000 loan at a nominal rate of 13.25%
- compounded monthly, with monthly end-of-period payments, if 3 "points"
- are charged? The pv must be reduced by the dollar value of the points
- and/or any lenders fees to establish an effective pv. Because payments remain
- the same, the true APR will be higher than the nominal rate. Note, first
- compute the payments on the pv of the loan amount.
-
-  <>n=30*12
-          360
-  <>i=13.25
-          13.25
-  <>pv=75000
-          75,000
-  <>PMT
-          -844.33
-  <>pv-=pv*0.03
-          72,750.00
-  <>I
-          13.69
-  <>
-

-
-
-
Equivalent Payments

-
Equivalent Payments
-
Find the equivalent monthly payment required to amortize a 20 year $40,000
-loan at 10.5% nominal annual rate compounded monthly, with 10 annual
-payments of $5029.71 remaining. Compute the pv of the remaining annual
-payments, then change n, the number of periods, and the payment frequency,
-PF, to a monthly basis and compute the equivalent monthly pmt.
-
- <>d
- <>PF=1
-         1
- <>n=10
-         10
- <>i=10.5
-         10.50
- <>pmt=-5029.71
-         -5,029.71
- <>PV
-         29,595.88
- <>PF=12
-         12
- <>n=120
-         120
- <>PMT
-         -399.35
-

-
-
-
Perpetuity - Continuous Compounding

-
Perpetuity - Continuous Compounding
-
If you can purchase a single payment annuity with an initial investment of
- $60,000 that will be invested at 15% nominal annual rate compounded
- continuously, what is the maximum monthly return you can receive without
- reducing the $60,000 principal? If the principal is not disturbed, the
- payments can go on indefinitely (a perpetuity). Note that the term,n, of
- a perpetuity is immaterial. It can be any non-zero value.
-
- <>d
- <>disc=FALSE
-         0
- <>n=12
-         12
- <>CF=1
-         1
- <>i=15
-         15
- <>fv=60000
-         60,000
- <>pv=-60000
-         -60,000
- <>PMT
-         754.71
-

-
-
-
Investment Return

-
Investment Return
-
A development company is purchasing an investment property with an annual net cash
-flow of $25,000.00. The expected holding period for the property is 10 years with an estimated
-selling price of $850,000.00 at that time. If the company is to realize a 15% yield on the
-investment, what is the maximum price they can pay for the property today?
-
-
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  <>CF=PF=1
-          1
-  <>n=10
-          10
-  <>i=15
-          15
-  <>pmt=25000
-          25,000
-  <>fv=850000
-          850,000
-  <>PV
-          -335,576.22
-

-
-
So the maximum purchase price today would be $335,576.22 to achieve the desired yield.
-
-
-
Retirement Investment

-
Retirement Investment
-
You wish to retire in 20 years and wish to deposit a lump sum amount in an account
-today which will grow to $100,000.00, earning 6.5% interest compounded semi-annually.
-How much do you need to deposit?
-
-
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  <>CF=PF=2
-          2
-  <>n=2*20
-          40
-  <>i=6.5
-          6.50
-  <>fv=100000
-          100,000
-  <>PV
-          -27,822.59
-

-
-
If you were to make semi-annual deposits of $600.00, how much would you need to deposit today?
-
-
-  <>pmt=-600
-          -600
-  <>PV
-          -14,497.53
-

-
-
If you were to make monthly deposits of $100.00?
-
-
-  <>PF=12
-          12
-  <>n=20*12
-          240
-  <>pmt=-100
-          -100
-  <>PV
-          -14,318.21
-

-
-
-
Property Values

-
Property Values
-
Property values in an area you are considering moving to are declining at the rate
-of 2.35% annually. What will property presently appraised at $155,500.00 be worth in 10 years
-if the trend continues?
-
-
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  <>CF=PF=1
-          1
-  <>n=10
-          10
-  <>i=-2.35
-          -2.35
-  <>pv=155500
-          155,500
-  <>FV
-          -122,589.39
-

-
-
-
College Expenses

-
College Expenses
-
You and your spouse are planning for your child's college expenses. Your child
-will be entering college in 15 years. You expect that college expenses at that time
-will amount to $25,000.00 per year or about $2,100.00/month. If the child withdrew
-the expenses from a bank account monthly paying 6% compounded on a daily basis (using
-360 days/year), how much must you deposit in the account at the start of the  four
-years?
-
-
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  <>CF=360
-          360
-  <>PF=12
-          12
-  <>n=12*4
-          48
-  <>i=6
-          6
-  <>pmt=2100
-          2,100
-  <>PV
-          -89,393.32
-

-
-
Your next problem is how to accumulate the money by the time the child starts college.
-You have a $50,000.00 paid-up insurance policy for your child that has a cash value
-of $6,500.00. It is accumulating annual dividends of $1,200 earning 6.75% compounded monthly.
-What will be the cash value of the policy in 15 years?
-
-
-  <>college_fund=-pv
-          89,393.32
-  <>d
-  <>PF=1
-          1
-  <>n=20
-          20
-  <>i=6.75
-          6.75
-  <>pmt=1200
-          1,200
-  <>FV
-          -48,995.19
-  <>insurance=-fv+6500
-          55,495.19
-  <>college_fund-insurance
-          33,898.13
-

-
-
The paid-up insurance cash value and dividends will provide $55,495.19 of the amount
-necessary, leaving $33,898.13 to accumulate in savings. Making monthly payments into
-a savings account paying 4.5% compounded daily, what level of monthly payments would be
-needed?
-
-
-  Financial Calculator
-  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
-  <>d
-  <>CF=360
-          360
-  <>n=PF*15
-          180
-  <>i=4.5
-          4.50
-  <>fv=college_fund - insurance
-          33,898.13
-  <>PMT
-          -132.11
-

-
-
-
References

-

-PPC ROM User's Manual
-
pages 148 - 164
-

-

-TOP
-
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