Package com.imsl.finance

Class Finance

java.lang.Object

com.imsl.finance.Finance

public class Finance
extends Object

Collection of finance functions.

Field Summary

Fields

Modifier and Type	Field	Description
static final int	AT_BEGINNING_OF_PERIOD	Flag used to indicate that payment is made at the beginning of each period.
static final int	AT_END_OF_PERIOD	Flag used to indicate that payment is made at the end of each period.

Method Summary

Modifier and Type	Method	Description
static double	cumipmt(double rate, int nper, double pv, int start, int end, int when)	Returns the cumulative interest paid between two periods.
static double	cumprinc(double rate, int nper, double pv, int start, int end, int when)	Returns the cumulative principal paid between two periods.
static double	db(double cost, double salvage, int life, int period, int month)	Returns the depreciation of an asset using the fixed-declining balance method.
static double	ddb(double cost, double salvage, int life, int period, double factor)	Returns the depreciation of an asset using the double-declining balance method.
static double	dollarde(double fractionalDollar, int fraction)	Converts a fractional price to a decimal price.
static double	dollarfr(double decimalDollar, int fraction)	Converts a decimal price to a fractional price.
static double	effect(double nominalRate, int nper)	Returns the effective annual interest rate.
static double	fv(double rate, int nper, double pmt, double pv, int when)	Returns the future value of an investment.
static double	fvschedule(double principal, double[] schedule)	Returns the future value of an initial principal taking into consideration a schedule of compound interest rates.
static double	ipmt(double rate, int period, int nper, double pv, double fv, int when)	Returns the interest payment for an investment for a given period.
static double	irr(double[] pmt)	Returns the internal rate of return for a schedule of cash flows.
static double	irr(double[] pmt, double guess)	Returns the internal rate of return for a schedule of cash flows.
static double	mirr(double[] value, double financeRate, double reinvestRate)	Returns the modified internal rate of return for a schedule of periodic cash flows.
static double	nominal(double effectiveRate, int nper)	Returns the nominal annual interest rate.
static double	nper(double rate, double pmt, double pv, double fv, int when)	Returns the number of periods for an investment for which periodic, and constant payments are made and the interest rate is constant.
static double	npv(double rate, double[] value)	Returns the net present value of a stream of equal periodic cash flows, which are subject to a given discount rate.
static double	pmt(double rate, int nper, double pv, double fv, int when)	Returns the periodic payment for an investment.
static double	ppmt(double rate, int period, int nper, double pv, double fv, int when)	Returns the payment on the principal for a specified period.
static double	pv(double rate, int nper, double pmt, double fv, int when)	Returns the net present value of a stream of equal periodic cash flows, which are subject to a given discount rate.
static double	rate(int nper, double pmt, double pv, double fv, int when)	Returns the interest rate per period of an annuity.
static double	rate(int nper, double pmt, double pv, double fv, int when, double guess)	Returns the interest rate per period of an annuity with an initial guess.
static double	sln(double cost, double salvage, int life)	Returns the depreciation of an asset using the straight line method.
static double	syd(double cost, double salvage, int life, int per)	Returns the depreciation of an asset using the sum-of-years digits method.
static double	vdb(double cost, double salvage, int life, int start, int end, double factor, boolean no_sl)	Returns the depreciation of an asset for any given period using the variable-declining balance method.
static double	xirr(double[] pmt, Date[] dates)	Returns the internal rate of return for a schedule of cash flows.
static double	xirr(double[] pmt, Date[] dates, double guess)	Returns the internal rate of return for a schedule of cash flows with a user supplied initial guess.
static double	xnpv(double rate, double[] value, Date[] dates)	Returns the present value for a schedule of cash flows.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

AT_END_OF_PERIOD

public static final int AT_END_OF_PERIOD

Flag used to indicate that payment is made at the end of each period.

See Also:

• Constant Field Values

AT_BEGINNING_OF_PERIOD

public static final int AT_BEGINNING_OF_PERIOD

Flag used to indicate that payment is made at the beginning of each period.

See Also:

• Constant Field Values

Method Details

cumipmt

public static double cumipmt(double rate,
 int nper,
 double pv,
 int start,
 int end,
 int when)

Returns the cumulative interest paid between two periods. It is computed using the following:

$$\sum\limits_{i = {\it start}}^{\it end} {\it interest}_i$$

where \({\it interest}_i$$ is computed from ipmt for the \(i\)th period.

Parameters:

rate - a double, the interest rate

nper - an int, the total number of payment periods

pv - a double, the present value

start - an int, the first period in the caclulation. Periods are numbered starting with one.

end - an int, the last period in the calculation

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the cumulative interest paid between the first period and the last period

See Also:

• Example

cumprinc

public static double cumprinc(double rate,
 int nper,
 double pv,
 int start,
 int end,
 int when)

Returns the cumulative principal paid between two periods. It is computed using the following:

$$\sum\limits_{i = {\it start}}^{\it end} {\it principal}_i$$

where \({\it principal}_i\) is computed from ppmt for the \(i\)th period.

Parameters:

rate - a double, the interest rate

nper - an int, the total number of payment periods

pv - a double, the present value

start - an int, the first period in the calculation. Periods are numbered starting with one.

end - an int, the last period in the calculation

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD.

Returns:

a double, the cumulative principal paid between the first period and the last period

See Also:

• Example

db

public static double db(double cost,
 double salvage,
 int life,
 int period,
 int month)

Returns the depreciation of an asset using the fixed-declining balance method. Method db varies depending on the specified value for the argument period, see table below.

If period = 1, $${\rm{cost}} \times {\rm{rate}} \times {{{\rm{month}}} \over {12}}$$
If period = life, $$\left( {{\rm{cost}} - {\rm{total}}\,{\rm{depreciation}}\,{\rm{from}}\,{\rm{periods}}} \right) \times {\rm{rate}} \times {{12{\rm{ - month}}} \over {12}}$$
If period other than 1 or life, $$\,\left( {{\rm{cost}} - {\rm{total}}\,{\rm{depreciation}}\,{\rm{from}}\, {\rm{prior periods}}} \right) \times rate$$

where

$$\,rate\, = 1 - \left( {{{{\rm{salvage}}} \over {{\rm{cost}}}}} \right)^{\left( {{1 \over {life}}} \right)}$$

NOTE: \(rate\) is rounded to three decimal places.

Parameters:

cost - a double, the initial cost of the asset

salvage - a double, the salvage value of the asset

life - an int, the number of periods over which the asset is being depreciated

period - an int, the period for which the depreciation is to be computed

month - an int, the number of months in the first year

Returns:

a double, the depreciation of an asset for a specified period using the fixed-declining balance method

See Also:

• Example

ddb

public static double ddb(double cost,
 double salvage,
 int life,
 int period,
 double factor)

Returns the depreciation of an asset using the double-declining balance method. It is computed using the following:

$$\left[ {{\it cost} - {\it salvage}\left( {\it {total\,depreciation\,from\,prior\,periods}} \right)} \right]\, {{{\it factor} \over {\it life}}}$$

Parameters:

cost - a double, the initial cost of the asset

salvage - a double, the salvage value of the asset

life - an int, the number of periods over which the asset is being depreciated

period - an int, the period

factor - a double, the rate at which the balance declines

Returns:

a double, the depreciation of an asset for a specified period

See Also:

• Example

dollarde

public static double dollarde(double fractionalDollar,
 int fraction)

Converts a fractional price to a decimal price. It is computed using the following:

$${\it idollar} + \left( {{\it fractionalDollar} - {\it idollar}} \right) \times {{10^{\left( {{\it ifrac} + 1} \right)} } \over {\it fraction}}$$

where \({\it idollar}\) is the integer part of \({\it fractionalDollar}\), and \({\it ifrac}\) is the integer part of \({\rm {log}}({\it fraction})\).

Parameters:

fractionalDollar - a double, a fractional number

fraction - an int, the denominator

Returns:

a double, the dollar price expressed as a decimal number

See Also:

• Example

dollarfr

public static double dollarfr(double decimalDollar,
 int fraction)

Converts a decimal price to a fractional price. It is computed using the following:

$${\it idollar} + {{{\it decimalDollar} - {\it idollar}} \over {10^{\left( {{\it ifrac} + 1} \right)} / {\it fraction}}}$$

where \({\it idollar}\) is the integer part of the \({\it decimalDollar}\), and \({\it ifrac}\) is the integer part of \(log({\it fraction})\).

Parameters:

decimalDollar - a double, a decimal number

fraction - an int, the denominator

Returns:

a double, a dollar price expressed as a fraction

See Also:

• Example

effect

public static double effect(double nominalRate,
 int nper)

Returns the effective annual interest rate. The nominal interest rate is the periodically-compounded interest rate as stated on the face of a security. The effective annual interest rate is computed using the following:

$$\left( {1 + {{\it nominalRate} \over {\it nper}}} \right)^{\it nper} - 1$$

Parameters:

nominalRate - a double, the nominal interest rate

nper - an int, the number of compounding periods per year

Returns:

a double, the effective annual interest rate

See Also:

• Example

fv

public static double fv(double rate,
 int nper,
 double pmt,
 double pv,
 int when)

Returns the future value of an investment. The future value is the value, at some time in the future, of a current amount and a stream of payments. It can be found by solving the following:

If \(rate = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \(rate \ne 0\), $${\it pv} (1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

rate - a double, the interest rate

nper - an int, the total number of payment periods

pmt - a double, the payment made in each period

pv - a double, the present value

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the future value of an investment

See Also:

• Example

fvschedule

public static double fvschedule(double principal,
 double[] schedule)

Returns the future value of an initial principal taking into consideration a schedule of compound interest rates. It is computed using the following:

$$\sum\limits_{i = 1}^{\it count} {\left( {{\it principal} \times {\it schedule}_i } \right)}$$

where \({\it schedule}_i =\) interest rate at the \(i\)th period, and the count is schedule.length.

Parameters:

principal - a double, the present value

schedule - a double array of interest rates to apply

Returns:

a double, the future value of an initial principal

See Also:

• Example

ipmt

public static double ipmt(double rate,
 int period,
 int nper,
 double pv,
 double fv,
 int when)

Returns the interest payment for an investment for a given period. It is computed using the following:

$$\left\{ {{\it pv}\left( {1 + {\it rate}} \right)^{{\it nper} - 1} + {\it pmt} \left( {1 + {\it rate} \times {\it when}} \right) {{{\left( {1 + {\it rate}} \right)^{{\it nper} - 1} } \over {\it rate}}} } \right\} {\it rate}$$

Parameters:

rate - a double, the interest rate

period - an int, the payment period

nper - an int, the total number of periods

pv - a double, the present value

fv - a double, the future value

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the interest payment for a given period for an investment

See Also:

• Example

irr

public static double irr(double[] pmt)

Returns the internal rate of return for a schedule of cash flows. It is found by solving the following:

$$0 = \sum\limits_{i = 1}^{\it count} {{{\it value}_i } \over {\left( {1 + {\it rate}} \right)^i }}$$

where \({\it value}_i\) = the \(ith\) cash flow, \({\it rate}\) is the internal rate of return, and count is pmt.length.

Parameters:

pmt - a double array which contains cash flow values which occur at regular intervals

Returns:

a double, the internal rate of return

See Also:

• Example

irr

public static double irr(double[] pmt,
 double guess)

Returns the internal rate of return for a schedule of cash flows. It is found by solving the following:

$$0 = \sum\limits_{i = 1}^{\it count} {{{\it value}_i } \over {\left( {1 + {\it rate}} \right)^i }}$$

where \({\it value}_i\) = the \(ith\) cash flow, \({\it rate}\) is the internal rate of return.

Parameters:

pmt - a double array which contains cash flow values which occur at regular intervals

guess - a double value which represents an initial guess at the return value from this function

Returns:

a double, the internal rate of return

mirr

public static double mirr(double[] value,
 double financeRate,
 double reinvestRate)

Returns the modified internal rate of return for a schedule of periodic cash flows. The modified internal rate of return differs from the ordinary internal rate of return in assuming that the cash flows are reinvested at the cost of capital, not at the internal rate of return. It also eliminates the multiple rates of return problem. It is computed using the following:

$${\left\{ {{{ - \left( {\it pnpv} \right)\left( {1 + {\it reinvestRate}} \right)^{\it nper} } \over {\left( {\it nnpv} \right)\left( {1 + {\it financeRate}} \right)}}} \right\}^{{1 \over {{\it nper} - 1}}} } - 1$$

where \({\it pnpv}\) is calculated from npv for positive values in value using reinvestRate, \({\it nnpv}\) is calculated from npv for negative values in value using financeRate, and nper = value.length.

Parameters:

value - a double array of cash flows

financeRate - a double, the interest you pay on the money you borrow

reinvestRate - a double, the interest rate you receive on the cash flows

Returns:

a double, the modified internal rate of return

See Also:

• Example

nominal

public static double nominal(double effectiveRate,
 int nper)

Returns the nominal annual interest rate. The nominal interest rate is the interest rate as stated on the face of a security. It is computed using the following:

$$\left[ {\left( {1 + {\it effectiveRate}} \right)^{^{{1 \over {\it nper}}} } - 1} \right] \times {\it nper}$$

Parameters:

effectiveRate - a double, the effective interest rate

nper - an int, the number of compounding periods per year

Returns:

a double, the nominal annual interest rate

See Also:

• Example

nper

public static double nper(double rate,
 double pmt,
 double pv,
 double fv,
 int when)

Returns the number of periods for an investment for which periodic, and constant payments are made and the interest rate is constant. It can be found by solving the following:

If \(rate = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \(rate \ne 0\), $${\it pv} (1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

rate - a double, the interest rate

pmt - a double, the payment

pv - a double, the present value

fv - a double, the future value

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

an int, the number of periods for an investment

See Also:

• Example

npv

public static double npv(double rate,
 double[] value)

Returns the net present value of a stream of equal periodic cash flows, which are subject to a given discount rate. It is found by solving the following:

$$\sum\limits_{i = 1}^{count} {{{\it value}_i} \over {\left( {1 + {\it rate}} \right)}^i }$$

where \({\it value}_i\) = the \(i\)th cash flow, and count is value.length.

Parameters:

rate - a double, the interest rate per period. It must not be -1.

value - a double array of equally-spaced cash flows

Returns:

a double, the net present value of the investment

See Also:

• Example

pmt

public static double pmt(double rate,
 int nper,
 double pv,
 double fv,
 int when)

Returns the periodic payment for an investment. It can be found by solving the following:

If \(rate = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \({\it rate} \ne 0\), $$ {\it pv}(1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

rate - a double, the interest rate

nper - an int, the total number of periods

pv - a double, the present value

fv - a double, the future value

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the interest payment for a given period for an investment

See Also:

• Example

ppmt

public static double ppmt(double rate,
 int period,
 int nper,
 double pv,
 double fv,
 int when)

Returns the payment on the principal for a specified period. It is computed using the following:

$${\it payment}_i - {\it interest}_i$$

where \({\it payment}_i\) is computed from pmt for the \(i\)th period, \({\it interest}_i\) is calculated from ipmt for the \(i\)th period.

Parameters:

rate - a double, the interest rate

period - an int, the payment period

nper - an int, the total number of periods

pv - a double, the present value

fv - a double, the future value

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the payment on the principal for a given period

See Also:

• Example

pv

public static double pv(double rate,
 int nper,
 double pmt,
 double fv,
 int when)

Returns the net present value of a stream of equal periodic cash flows, which are subject to a given discount rate. It can be found by solving the following:

If \({\it rate} = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \({\it rate} \ne 0\), $$ {\it pv}(1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

rate - a double, the interest rate per period

nper - an int, the number of periods

pmt - a double, the payment made each period

fv - a double, the annuity's value after the last payment

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the present value of the investment

See Also:

• Example

rate

public static double rate(int nper,
 double pmt,
 double pv,
 double fv,
 int when)

Returns the interest rate per period of an annuity. rate is calculated by iteration and can have zero or more solutions. It can be found by solving the following:

If \({\it rate} = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \({\it rate} \ne 0\), $$ {\it pv}(1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

nper - an int, the number of periods

pmt - a double, the payment made each period

pv - a double, the present value

fv - a double, the annuity's value after the last payment

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

Returns:

a double, the interest rate per period of an annuity

See Also:

• Example

rate

public static double rate(int nper,
 double pmt,
 double pv,
 double fv,
 int when,
 double guess)

Returns the interest rate per period of an annuity with an initial guess. rate is calculated by iteration and can have zero or more solutions. It can be found by solving the following:

If \({\it rate} = 0\), $${\it pv} + {\it pmt} \times {\it nper} + {\it fv} = 0$$

If \({\it rate} \ne 0\), $$ {\it pv}(1 + {\it rate})^{\it nper} + {\it pmt} \left[ {1 + {\it rate} \left( {\it when} \right)} \right] {{(1 + {\it rate})^{\it nper} - 1} \over {\it rate}} + {\it fv} = 0$$

Parameters:

nper - an int, the number of periods

pmt - a double, the payment made each period

pv - a double, the present value

fv - a double, the annuity's value after the last payment

when - an int, the time in each period when the payment is made, either AT_END_OF_PERIOD or AT_BEGINNING_OF_PERIOD

guess - a double value which represents an initial guess at the interest rate per period of an annuity

Returns:

a double, the interest rate per period of an annuity

See Also:

• Example

sln

public static double sln(double cost,
 double salvage,
 int life)

Returns the depreciation of an asset using the straight line method. It is computed using the following:

$${{\it cost} - {\it salvage}} / {\it life}$$

Parameters:

cost - a double, the initial cost of the asset

salvage - a double, the salvage value of the asset

life - an int, the number of periods over which the asset is being depreciated

Returns:

a double, the straight line depreciation of an asset for one period

See Also:

• Example

syd

public static double syd(double cost,
 double salvage,
 int life,
 int per)

Returns the depreciation of an asset using the sum-of-years digits method. It is computed using the following:

$$({\it cost} - {\it salvage})({\it per})\,\, {{({\it life} + 1)\left( {\it life} \right)} \over 2}$$

Parameters:

cost - a double, the initial cost of the asset

salvage - a double, the salvage value of the asset

life - an int, the number of periods over which the asset is being depreciated

per - an int, the period

Returns:

a double, the sum-of-years digits depreciation of an asset

See Also:

• Example

vdb

public static double vdb(double cost,
 double salvage,
 int life,
 int start,
 int end,
 double factor,
 boolean no_sl)

Returns the depreciation of an asset for any given period using the variable-declining balance method. It is computed using the following:

If \(no\_sl = 0\), $$ \sum\limits_{i = {\it start} + 1}^{\it end} {{\it ddb}_i }$$

If \(no\_sl \ne 0\), $$A + \sum\limits_{i = k}^{\it end} {{{{{\it cost}} - A - {\it salvage}} \over {{\it end} - k + 1}}}$$

where \({\it ddb}_i\) is computed from ddb for the \(i\)th period. \(k\) = the first period where straight line depreciation is greater than the depreciation using the double-declining balance method.

$$A = \sum\limits_{i = {\it start} + 1}^{k - 1} {{\it ddb}_i}$$

Parameters:

cost - a double, the initial cost of the asset

salvage - a double, the salvage value of the asset

life - an int, the number of periods over which the asset is being depreciated

start - an int, the initial period for the calculation

end - an int, the final period for the calculation

factor - a double, the rate at which the balance declines

no_sl - a boolean flag. If true, do not switch to straight-line depreciation even when the depreciation is greater than the declining balance calculation.

Returns:

a double, the depreciation of the asset

See Also:

• Example

xirr

public static double xirr(double[] pmt,
 Date[] dates)

Returns the internal rate of return for a schedule of cash flows. It is not necessary that the cash flows be periodic. It can be found by solving the following:

$$0 = \sum\limits_{i = 1}^{\it count} {{{{\it value}_i } \over {\left( {1 + {\it rate}} \right)^{{{d_i - d_1 } \over {365}}} }}}$$

In the equation above, \(d_i\) represents the \(i\)th payment date. \(d_1\) represents the 1st payment date. \({\it value}\) represents the \(i\)th cash flow. \({\it rate}\) is the internal rate of return, and count is pmt.length.

Parameters:

pmt - a double array which contains cash flow values which correspond to a schedule of payments in dates

dates - a Date array which contains a schedule of payment dates

Returns:

a double, the internal rate of return

See Also:

• Example

xirr

public static double xirr(double[] pmt,
 Date[] dates,
 double guess)

Returns the internal rate of return for a schedule of cash flows with a user supplied initial guess. It is not necessary that the cash flows be periodic. It can be found by solving the following:

$$0 = \sum\limits_{i = 1}^{\it count} {{{{\it value}_i } \over {\left( {1 + {\it rate}} \right)^{{{d_i - d_1 } \over {365}}} }}}$$

In the equation above, \(d_i\) represents the \(i\)th payment date. \(d_1\) represents the 1st payment date. \({\it value}\) represents the \(i\)th cash flow. \({\it rate}\) is the internal rate of return. Count is pmt.length.

Parameters:

pmt - a double array which contains cash flow values which correspond to a schedule of payments in dates

dates - a Date array which contains a schedule of payment dates

guess - a double value which represents an initial guess at the return value from this function

Returns:

a double, the internal rate of return

xnpv

public static double xnpv(double rate,
 double[] value,
 Date[] dates)

Returns the present value for a schedule of cash flows. It is not necessary that the cash flows be periodic. It is computed using the following:

$$\sum\limits_{i = 1}^{\it count} {{{{\it value}_i } \over {\left( {1 + {\it rate}} \right)^{\left( {d_i - d_1 } \right)/365}}}}$$

In the equation above, \(d_i\) represents the \(i\)th payment date, \(d_1\) represents the first payment date, \({\it value}_i\) represents the \(i\)th cash flow. and count is value.length

Parameters:

rate - a double, the interest rate

value - a double array containing the cash flows

dates - a Date array which contains a schedule of payment dates

Returns:

a double, the present value

See Also:

• Example