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:
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"
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:
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.
A FV*
1 2 3 4 . . . . . . . . . n |
Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|
V
PV
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
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
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
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
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)
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]
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
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)
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
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
The amortization schedule is computed for six different situations:
The amortization schedule may be computed and displayed in three manners:
At the end of each year a summary is computed and displayed and the total interest paid is displayed at the end.
The total interest paid is displayed at the end.
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
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.
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) <>
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).
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
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
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
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
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
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:
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
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
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
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
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
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"
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
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
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
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 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
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