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

- n == number of payment periods
- %i == nominal interest rate, NAR, charged
- PV == Present Value of money
- PMT == Periodic Payment
- FV == Future Value of money

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:

- Money RECEIVED is a POSITIVE value and is represented by an arrow above the line
- Money PAID OUT is a NEGATIVE value and is represented by an arrow below the line.

# 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:

- 1 == annual
- 2 == semi-annual
- 3 == tri-annual
- 4 == quaterly
- 6 == bi-monthly
- 12 == monthly
- 24 == semi-monthly
- 26 == bi-weekly
- 52 == weekly
- 360 == daily
- 365 == daily

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

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