ReturnIn this loan, each total payment is different, with each succeeding payment +less than the preceeding payment. Each payment is the total of the constant +ammount to the principal plus the interest for the period. The constant payment +to the principal is computed as: + +
+ C = -PV / N + ++Where PV is the loan amount to be repaid in N payments (periods). Thus the +principal after the first payment is: + +
+ PV[1] = PV[0] + C = PV + C ++ +after the second payment, the principal is: + +
+ PV[2] = PV[1] + C = PV[0] + 2C ++ +In general, the remaining principal after n payments is: + +
+ PV[n] = PV[0] + nC = PV + nC ++ +If the effective interest per payment period is i, then the interest for the +first payment is: + +
+ I[1] = -i*PV[0] ++ +and for the second: + +
+ I[2] = -i * PV[1] ++ +and in general, for the n'th payment the interest is: + +
+ I[n] = -i * PV[n-1] + = -i * (PV + (n - 1)C) ++ +The total payment for any period, n, is: + +
+ P[n] = C + I[n] + = C - i * (PV + (n - 1)C) + = C(1 + i) - i * (PV + nC) ++ +The total interest paid to period n is: + +
+ T[n] = I[1] + I[2] + I[3] + ... + I[n] + T[n] = sum(j = 1 to n: I[j]) + T[n] = sum(j = 1 to n: -i * (PV + (j-1)C)) + T[n] = sum(j=1 to n: -i*PV) + sum(j=1 to n: iC) + sum(j=1 to n: -iCj) + T[n] = -i*n*PV + i*n*C - i*C*sum(j=1 to n:j) + sum(j=1 to n:j) = n(n+1)/2 + T[n] = -i*n*(PV + C) - i*C*n(n+1)/2 + T[n] = -i*n*(PV + (C*(n - 1)/2)) ++ +Note: substituing for C = -PV/N, in the equations for PV[n], I[n], P[n], and T[n] +would give the following equations: + +
+ PV[n] = PV*(1 - n/N) + I[n] = -i*PV*(1 + N - n)/N + P[n] = -i*PV*(2 + N - n)/N + T[n] = -i*n*PV*(2*N - n + 1)/(2*N) ++ +Using thses equations for the calculations would eliminate the dependence +on C, but only if C is always defined as above and would eliminate the +possibility of another value for C. If the value of C was less than -PV/N +then a balloon payment would be due at the final payment and this is a possible +alternative for some people. diff --git a/src/doc/finderv.html b/src/doc/finderv.html new file mode 100644 index 0000000000..d850168f87 --- /dev/null +++ b/src/doc/finderv.html @@ -0,0 +1,337 @@ + +
+ +Financial Equations Documentation + + + +Return +
+
+
The financial equation is derived in the following manner: + +
Start with the basic equation to find the balance or Present Value, PV[1], after +one payment period. Note PV[1] is the Present Value after one payment and PV[0] +is the initial Present Value. PV[0] will be shortened to just PV. + +
The interest due at the end of the first payment period is the original present value, +PV, times the interest rate for the payment period plus the periodic payment times the +interest rate for beginning of period payments: + +
ID[1] = PV * i + X * PMT * i = (PV + X * PMT) * i + +
The Present Value after one payment is the original Present Value with the periodic +payment, PMT, and interest due, ID[1], added: + +
+ PV[1] = PV + (PMT + ID[1]) + PV[1] = PV + (PMT + (PV + X * PMT) * i) + PV[1] = PV * (1 + i) + PMT * (1 + Xi) ++ +
This equation works for all of the cash flow diagrams shown previously. The Present Value, +money received or paid, is modified by a payment made at the beginning of a payment +period and multiplied by the effective interest rate to compute the interest +due during the payment period. The interest due is then added to the payment +to obtain the amount to be added to the Present Value to compute the new Present Value. + +
For diagram 1): PV < 0, PMT == 0, PV[1] < 0
+
For diagram 2): PV == 0, PMT < 0, PV[1] < 0
+
For Diagram 3): PV > 0, PMT < 0, PV[1] >= 0 or PV[1] <= 0
+
For Diagram 4): PV < 0, PMT > 0, PV[1] <= 0 or PV[1] >= 0
+
+
X may be 0 or 1 for any diagram. + +
For the standard loan, PV is the money borrowed, PMT is the periodic payment to repay +the loan, i is the effective interest rate agreed upon and FV is the residual loan amount +after the agreed upon number of periodic payment periods. If the loan is fully paid off +by the periodic payments, FV is zero, 0. If the loan is not completely paid off after the +agreed upon number of payments, a balloon payment is necessary to completely pay off the loan. +FV is then the amount of the needed balloon payment. For a loan in which the borrower pays +only enough to repay the interest due during a payment period, interest only loan, the +balloon payment is equal to the negative of PV. + +
To calculate the Present Value after the second payment period, the above calculation +is applied iteratively to PV[1] to obtain PV[2]. In fact to calculate the Present Value +after any payment period, PV[n], the above equation is applied iteratively to PV[n-1] +as shown below. + +
+ PV[2] = PV[1] + (PMT + (PV[1] + X * PMT) * i) + = PV[1] * (1 + i) + PMT * (1 + iX) + = (PV * (1 + i) + PMT * (1 + iX)) * (1 + i) + PMT * (1 + iX) + = PV * (1 + i)^2 + PMT * (1 + iX) * (1 + i) + + PMT * (1 + iX) ++ +
Similarly, PV[3] is computed from PV[2] as: + +
+ PV[3] = PV[2] + (PMT + (PV[2] + X * PMT) * i) + = PV[2] * (1 + i) + PMT * (1 + iX) + = (PV * (1 + i)^2 + PMT * (1 + iX) * (1 + i) + + PMT * (1+ iX)) * (1 + i) + + PMT * (1+ iX) + = PV * (1 + i)^3 + PMT * (1 + iX) * (1 + i)^2 + + PMT * (1 + iX) * (1 + i) + + PMT * (1 + iX) ++ +
And for the n'th payment, PV[n] is computed from PV[n-1] as: + +
+ PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i) + PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * (1 + i)^(n-1) + + PMT * (1 + iX) * (1 + i)^(n-2) + + . + . + . + + PMT * (1 + iX) * (1 + i) + + PMT * (1 + iX) + PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^(n-1) + ... + (1 + i) + 1] ++ +
The formula for PV[n] can be proven using mathematical induction.
+
+
+Basic Financial Equation
+
As shown above, the basic financial transaction equation is simply: + +
+ PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i) + = PV[n-1] * (1 + i) + PMT * (1 + iX) + for: n >= 1 ++ +
relating the Present Value after n payments, PV[n] to the previous Present Value, PV[n-1]. + + + +
The sum of the finite series: + +
1 + k + (k^2) + (k^3) + ... + (k^n) = (1-k^(n+1))/(1-k) + +
as can be seen by the following. Let S(n) be the series sum. Then + +
S(n) - k * S(n) = 1 - k^(n+1) + +
and solving for S(n): + +
S(n) = (1-k^(n+1))/(1-k) = 1 + k + (k^2) + (k^3) + ... + (k^n) + + +
Using this in the equation above for PV[n], we have: + +
+ PV[n] = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^(n-1) + ... + (1 + i) + 1] + = PV * (1 + i)^n + PMT * (1 + iX) * [1 - (1 + i)^n]/[1 - (1 + i)] + = PV * (1 + i)^n + PMT * (1 + iX) * [1 - (1 + i)^n]/[-i] + = PV * (1 + i)^n + PMT * (1 + iX) * [(1 + i)^n - 1]/i ++ +
or: + +
+ PV * (1 + i)^n + PMT * [(1 + i)^n - 1]/i - PV[n] = 0 ++ +
If after n payments, the remaining balance is repaid as a lump sum, the lump sum +is known as the Future Value, FV[n]. Since FV[n] is negative if paid and positive +if received, FV[n] is the negative of PV[n]. + +
Setting: FV[n] = -PV[n] + +
Since n is assumed to be the last payment, FV[n] will be shortened to simply +FV for the last payment period. + +
+ PV*(1 + i)^n + PMT*(1 + iX)*[(1 + i)^n - 1]/i + FV = 0 ++ +
Up to this point, we have said nothing about the value of PMT. PMT can be any value mutually +agreed upon by the lender and the borrower. From the equation for PV[1]: + +
+ PV[1] = PV + (PMT + (PV + X * PMT) * i), ++ +
Several things can be said about PMT. + +
The payment is exactly equal to the interest due and PV[1] = PV. In this case, the borrower +must make larger future payments to reduce the balance due, or make a single payment, after +some agreed upon number of payments, with PMT = -PV to completely pay off the loan. This is +an interest only payment with a balloon payment at the end. + +
The payment is insufficient to cover even the interest charged and the balance due grows + +
The payment is sufficient to cover the interest charged with a residual amount to be +applied to reduce the balance due. The larger the residual amount, the faster the loan is +repaid. For most mortgages or other loans made today, the lender and borrower agree upon +a certain number of repayment periods and the interest to be charged per payment period. +The interest may be multiplied by 12 and stated as an annual interest rate. Then the +lender and borrower want to compute a periodic payment, PMT, which will reduce the balance +due to zero after the agreed upon number of payments have been made. If N is the agreed +upon number of periodic payments, then we want to use: + +
+ PV * (1 + i)^N + PMT*(1 +iX)*[(1 + i)^N - 1]/i + FV = 0 ++ +
with FV = 0 to compute PMT: + +
+ PMT = -[PV * i * (1 + i)^(N - X)]/[(1 + i)^N - 1] ++ +
The value of PMT computed will reduce the balance due to zero after N periodic payments. +Note that this is strictly true only if PMT is not rounded to the nearest cent as is the +usual case since it is hard to pay fractional cents. Rounding PMT to the nearest cent has +an effect on the FV after N payments. If PMT is rounded up, then the final Nth payment +will be smaller than PMT since the periodic PMTs have paid down the principal faster than +the exact solution. If PMT is rounded down, then the final Nth payment will be larger than +the periodic PMTs since the periodic PMTs have paid down the principal slower than the +exact solution. +
With a simple alegebraic re-arrangement, The financial Equation becomes: + +
+ 2) [PV + PMT*(1 + iX)/i][(1 + i)^n - 1] + PV + FV = 0 ++ +
or + +
+ 3) (PV + C)*A + PV + FV = 0 ++ +
where: +
+ 4) A = (1 + i)^n - 1 + + 5) B = (1 + iX)/i + + 6) C = PMT*B ++ +
The form of equation 3) simplifies the calculation procedure for all five +variables, which are readily solved as follows: + +
+ 7) n = ln[(C - FV)/(C + PV)]/ln((1 + i) + + 8) PV = -[FV + A*C]/(A + 1) + + 9) PMT = -[FV + PV*(A + 1)]/[A*B] + + 10) FV = -[PV + A*(PV + C)] ++ +
Equations 4), 5) and 6) are computed by the functions in the "fin.exp" utility:
+
+
_A
+
_B
+
_C
+
+
respectively. Equations 7), 8), 9) and 10) are computed by functions:
+
+
_N
+
_PV
+
_PMT
+
_FV
+
+
respectively. + +
The solution for interest is broken into two cases: + +
Equation 3) can be solved exactly for i: + +
+ i = [FV/PV]^(1/n) - 1 ++ +
Since equation 3) cannot be solved explicitly for i in this case, an +iterative technique must be employed. Newton's method, using exact +expressions for the function of i and its derivative, are employed. The +expressions are: + +
+ 12) i[k+1] = i[k] - f(i[k])/f'(i[k]) + where: i[k+1] == (k+1)st iteration of i + i[k] == kth iteration of i + and: + + 13) f(i) = A*(PV+C) + PV + FV + + 14) f'(i) = n*D*(PV+C) - (A*C)/i + + 15) D = (1 + i)^(n-1) = (A+1)/(1+i) ++ +
To start the iterative solution for i, an initial guess must be made +for the value of i. The closer this guess is to the actual value, +the fewer iterations will have to be made, and the greater the +probability that the required solution will be obtained. The initial +guess for i is obtained as follows: + +
+ | n*PMT + PV + FV | + 16) i[0] = | ----------------| + | n*PV | + + = abs[(n*PMT + PV + FV)/(n*PV)] ++ +
+ | FV - n*PMT | + 17) i[0] = |---------------------------| + | 3*[PMT*(n-1)^2 + PV - FV] | + + = abs[(FV-n*PMT)/(3*(PMT*(n-1)^2+PV-FV))] ++ + +
+ | FV + n*PMT | + 18) i[0] = |---------------------------| + | 3*[PMT*(n-1)^2 + PV - FV] | + + = abs[(FV+n*PMT)/(3*(PMT*(n-1)^2+PV-FV))] ++ +
Return
+
diff --git a/src/doc/finutil.html b/src/doc/finutil.html
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+
+
+
+
+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 interst. 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
+amoung 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 occuring 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 positve, 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. Somtimes
+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 calender. 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 diplayed at the end. + +
The total interest paid is diplayed 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 calender 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 addtional (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 ++ + +