lazarus-ccr/components/fpspreadsheet/examples/other/financemath.pas

185 lines
5.7 KiB
ObjectPascal

unit financemath;
{$mode objfpc}{$H+}
interface
uses
Classes, SysUtils;
{ Cash flow equation:
FV + PV * q^n + PMT (q^n - 1) / (q - 1) = 0 (1)
with
q = 1 + interest rate (RATE)
PV = present value of an investment
FV = future value of an investment
PMT = regular payment (per period)
n = NPER = number of payment periods
This is valid for payments occuring at the end of each period. If payments
occur at the start of each period the payments are multiplied by a factor q.
This case is indicated by means of the parameter PaymentTime below.
The interest rate is considered as "per period" - whatever that is.
If the period is 1 year then we use the "usual" interest rate.
If the period is 1 month then we use 1/12 of the yearly interest rate.
Sign rules:
- Money that I receive is to a positive number
- Money that I pay is to a negative number.
Example 1: Saving account
- A saving account has an initial balance of 1000 $ (PV).
I paid this money to the bank --> negative number
- I deposit 100$ regularly to this account (PMT): I pay this money --> negative number.
- At the end of the payments (NPER periods) I get the money back --> positive number.
This is the FV.
Example 2: Loan
- I borrow 1000$ from the bank: I get money --> positive PV
- I pay 100$ back to the bank in regular intervals --> negative PMT
- At the end, all debts are paid --> FV = 0.
The cash flow equation (1) contains 5 parameters (Rate, PV, FV, PMT, NPER).
The functions below solve this equation always for one of these parameters.
References:
- http://en.wikipedia.org/wiki/Time_value_of_money
- https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_Derivation_of_Financial_Formulas
}
type
TPaymentTime = (ptEndOfPeriod, ptStartOfPeriod);
function FutureValue(ARate: Extended; NPeriods: Integer;
APayment, APresentValue: Extended; APaymentTime: TPaymentTime): Extended;
function InterestRate(NPeriods: Integer; APayment, APresentValue, AFutureValue: Extended;
APaymentTime: TPaymentTime): Extended;
function NumberOfPeriods(ARate, APayment, APresentValue, AFutureValue: Extended;
APaymentTime: TPaymentTime): Extended;
function Payment(ARate: Extended; NPeriods: Integer;
APresentValue, AFutureValue: Extended; APaymentTime: TPaymentTime): Extended;
function PresentValue(ARate: Extended; NPeriods: Integer;
APayment, AFutureValue: Extended; APaymentTime: TPaymentTime): Extended;
implementation
uses
math;
function FutureValue(ARate: Extended; NPeriods: Integer;
APayment, APresentValue: Extended; APaymentTime: TPaymentTime): Extended;
var
q, qn, factor: Extended;
begin
if ARate = 0 then
Result := -APresentValue - APayment * NPeriods
else begin
q := 1.0 + ARate;
qn := power(q, NPeriods);
factor := (qn - 1) / (q - 1);
if APaymentTime = ptStartOfPeriod then
factor := factor * q;
Result := -(APresentValue * qn + APayment*factor);
end;
end;
function InterestRate(NPeriods: Integer; APayment, APresentValue, AFutureValue: Extended;
APaymentTime: TPaymentTime): Extended;
{ The interest rate cannot be calculated analytically. We solve the equation
numerically by means of the Newton method:
- guess value for the interest reate
- calculate at which interest rate the tangent of the curve fv(rate)
(straight line!) has the requested future vale.
- use this rate for the next iteration. }
const
DELTA = 0.001;
EPS = 1E-9; // required precision of interest rate (after typ. 6 iterations)
MAXIT = 20; // max iteration count to protect agains non-convergence
var
r1, r2, dr: Extended;
fv1, fv2: Extended;
iteration: Integer;
begin
iteration := 0;
r1 := 0.05; // inital guess
repeat
r2 := r1 + DELTA;
fv1 := FutureValue(r1, NPeriods, APayment, APresentValue, APaymentTime);
fv2 := FutureValue(r2, NPeriods, APayment, APresentValue, APaymentTime);
dr := (AFutureValue - fv1) / (fv2 - fv1) * delta; // tangent at fv(r)
r1 := r1 + dr; // next guess
inc(iteration);
until (abs(dr) < EPS) or (iteration >= MAXIT);
Result := r1;
end;
function NumberOfPeriods(ARate, APayment, APresentValue, AFutureValue: extended;
APaymentTime: TPaymentTime): extended;
{ Solve the cash flow equation (1) for q^n and take the logarithm }
var
q, x1, x2: Extended;
begin
if ARate = 0 then
Result := -(APresentValue + AFutureValue) / APayment
else begin
q := 1.0 + ARate;
if APaymentTime = ptStartOfPeriod then
APayment := APayment * q;
x1 := APayment - AFutureValue * ARate;
x2 := APayment + APresentValue * ARate;
if (x2 = 0) // we have to divide by x2
or (sign(x1) * sign(x2) < 0) // the argument of the log is negative
then
Result := Infinity
else begin
Result := ln(x1/x2) / ln(q);
end;
end;
end;
function Payment(ARate: Extended; NPeriods: Integer;
APresentValue, AFutureValue: Extended; APaymentTime: TPaymentTime): Extended;
var
q, qn, factor: Extended;
begin
if ARate = 0 then
Result := -(AFutureValue + APresentValue) / NPeriods
else begin
q := 1.0 + ARate;
qn := power(q, NPeriods);
factor := (qn - 1) / (q - 1);
if APaymentTime = ptStartOfPeriod then
factor := factor * q;
Result := -(AFutureValue + APresentValue * qn) / factor;
end;
end;
function PresentValue(ARate: Extended; NPeriods: Integer;
APayment, AFutureValue: Extended; APaymentTime: TPaymentTime): Extended;
var
q, qn, factor: Extended;
begin
if ARate = 0.0 then
Result := -AFutureValue - APayment * NPeriods
else begin
q := 1.0 + ARate;
qn := power(q, NPeriods);
factor := (qn - 1) / (q - 1);
if APaymentTime = ptStartOfPeriod then
factor := factor * q;
Result := -(AFutureValue + APayment*factor) / qn;
end;
end;
end.