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In
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
and
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

finance
, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has
interest In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

interest
-earning potential, a characteristic referred to as the
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are co ...
, except during times of zero- or negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed funds (the present value) is less than the total amount of money paid to the lender. Present value calculations, and similarly
future value Future value is the value Value or values may refer to: * Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them ** Values (Western philosophy) expands the notion of value beyond that of ethi ...
calculations, are used to value
loans In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money avai ...
,
mortgages A mortgage loan or simply mortgage () is a loan In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations etc. The recipient (i.e., the borrower) incurs a ...
,
annuities An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account, monthly home mortga ...
,
sinking fund A sinking fund is a fund established by an economic entity by setting aside revenue over a period of time to fund a future capital expense, or repayment of a long-term debt Debt is an obligation that requires one party, the debtor, to pay ...
s,
perpetuities A perpetuity is an Annuity (finance theory), annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; the ...
, bonds, and more. These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times, since time and dates must be consistent in order to make comparisons between values. When deciding between projects in which to invest, the choice can be made by comparing respective present values of such projects by means of discounting the expected income streams at the corresponding project interest rate, or rate of return. The project with the highest present value, i.e. that is most valuable today, should be chosen.


Years' purchase

The traditional method of valuing future income streams as a present capital sum is to multiply the average expected annual cash-flow by a multiple, known as "years' purchase". For example, in selling to a third party a property leased to a tenant under a 99-year lease at a rent of $10,000 per annum, a deal might be struck at "20 years' purchase", which would value the lease at 20 * $10,000, i.e. $200,000. This equates to a present value discounted in perpetuity at 5%. For a riskier investment the purchaser would demand to pay a lower number of years' purchase. This was the method used for example by the English crown in setting re-sale prices for manors seized at the
Dissolution of the Monasteries#REDIRECT Dissolution of the monasteries {{Redirect category shell, 1= {{R from other capitalisation ...
in the early 16th century. The standard usage was 20 years' purchase.


Background

If offered a choice between $100 today or $100 in one year, and there is a positive real interest rate throughout the year, ''
ceteris paribus ' or ' () is a Latin phrase meaning "other things equal"; English translations of the phrase include "all other things being equal" or "other things held constant" or "all else unchanged". A prediction or a statement about a ontic, causal, epist ...
'', a rational person will choose $100 today. This is described by economists as
time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good In most contexts, the concept of good denotes the conduct that ...
. Time preference can be measured by auctioning off a risk free security—like a US Treasury bill. If a $100 note with a zero coupon, payable in one year, sells for $80 now, then $80 is the present value of the note that will be worth $100 a year from now. This is because money can be put in a bank account or any other (safe) investment that will return interest in the future. An investor who has some money has two options: to spend it right now or to save it. But the financial compensation for saving it (and not spending it) is that the money value will accrue through the
compound interest Compound interest is the addition of interest Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

compound interest
that he or she will receive from a borrower (the bank account in which he has the money deposited). Therefore, to evaluate the real value of an amount of money today after a given period of time, economic agents compound the amount of money at a given (interest) rate. Most actuarial calculations use the
risk-free interest rate The risk-free interest rate is the rate of return In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted wit ...
which corresponds to the minimum guaranteed rate provided by a bank's saving account for example, assuming no risk of default by the bank to return the money to the account holder on time. To compare the change in purchasing power, the
real interest rate The real interest rate is the rate of interest an investor, saver or lender receives (or expects to receive) after allowing for inflation. It can be described more formally by the Fisher equation In financial mathematicsMathematical finance, also ...
(
nominal interest rateIn finance and economics, the nominal interest rate or nominal rate of interest is either of two distinct things: # the rate of interest In finance Finance is the study of financial institutions, financial markets and how they operate within ...
minus
inflation In economics, inflation refers to a general progressive increase in prices of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a r ...

inflation
rate) should be used. The operation of evaluating a present value into the
future value Future value is the value Value or values may refer to: * Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them ** Values (Western philosophy) expands the notion of value beyond that of ethi ...
is called a capitalization (how much will $100 today be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of money—is called a discounting (how much will $100 received in 5 years—at a lottery for example—be worth today?). It follows that if one has to choose between receiving $100 today and $100 in one year, the rational decision is to choose the $100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least $105 in one year so that the two options are equivalent (either receiving $100 today or receiving $105 in one year). This is because if $100 is deposited in a savings account, the value will be $105 after one year, again assuming no risk of losing the initial amount through bank default.


Interest rates

Interest is the additional amount of money gained between the beginning and the end of a time period. Interest represents the
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are co ...
, and can be thought of as rent that is required of a borrower in order to use money from a lender. For example, when an individual takes out a bank loan, the individual is charged interest. Alternatively, when an individual deposits money into a bank, the money earns interest. In this case, the bank is the borrower of the funds and is responsible for crediting interest to the account holder. Similarly, when an individual invests in a company (through
corporate bond A corporate bond is a bond Bond or bonds may refer to: Common meanings * Bond (finance) In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with ...
s, or through
stock In finance, stock (also capital stock) consists of all of the shares In financial markets A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities i ...

stock
), the company is borrowing funds, and must pay interest to the individual (in the form of coupon payments,
dividend A dividend is a distribution of profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit ...

dividend
s, or stock price appreciation). The interest rate is the change, expressed as a percentage, in the amount of money during one compounding period. A compounding period is the length of time that must transpire before interest is credited, or added to the total. For example, interest that is compounded annually is credited once a year, and the compounding period is one year. Interest that is compounded quarterly is credited four times a year, and the compounding period is three months. A compounding period can be any length of time, but some common periods are annually, semiannually, quarterly, monthly, daily, and even continuously. There are several types and terms associated with
interest In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

interest
rates: *
Compound interest Compound interest is the addition of interest Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

Compound interest
, interest that increases exponentially over subsequent periods, *
Simple interest In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, i ...
, additive interest that does not increase *
Effective interest rate The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the interest rate An interest rate is the amount of interest Interest, in finance and economics, is payment from a debt ...
, the effective equivalent compared to multiple compound interest periods *
Nominal annual interestIn finance and economics, the nominal interest rate or nominal rate of interest is either of two distinct things: # the rate of interest before adjustment for inflation (in contrast with the real interest rate); or, # for interest rates "as stated" ...
, the simple annual interest rate of multiple interest periods * Discount rate, an inverse interest rate when performing calculations in reverse *
Continuously compounded interest Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then ea ...
, the
mathematical limit In mathematics, a limit is the value that a function (mathematics), function (or sequence) "approaches" as the input (or index) "approaches" some Value (mathematics), value. Limits are essential to calculus and mathematical analysis, and are used ...
of an interest rate with a period of zero time. *
Real interest rate The real interest rate is the rate of interest an investor, saver or lender receives (or expects to receive) after allowing for inflation. It can be described more formally by the Fisher equation In financial mathematicsMathematical finance, also ...
, which accounts for inflation.


Calculation

The operation of evaluating a present sum of money some time in the future is called a capitalization (how much will 100 today be worth in five years?). The reverse operation—evaluating the present value of a future amount of money—is called discounting (how much will 100 received in five years be worth today?). Spreadsheets commonly offer functions to compute present value. In Microsoft Excel, there are present value functions for single payments - "=NPV(...)", and series of equal, periodic payments - "=PV(...)". Programs will calculate present value flexibly for any cash flow and interest rate, or for a schedule of different interest rates at different times.


Present value of a lump sum

The most commonly applied model of present valuation uses
compound interest Compound interest is the addition of interest Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

compound interest
. The standard formula is: :PV = \frac \, Where \,C\, is the future amount of money that must be discounted, \,n\, is the number of compounding periods between the present date and the date where the sum is worth \,C\,, \,i\, is the interest rate for one compounding period (the end of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily). The interest rate, \,i\,, is given as a percentage, but expressed as a decimal in this formula. Often, v^ = \,(1 + i)^ is referred to as the Present Value Factor This is also found from the with negative time. For example, if you are to receive $1000 in five years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is :PV = \frac = \$620.92 \, The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $1000 in five years, or $620.92 today. The
purchasing power Purchasing power is the amount of goods and services that can be purchased with a unit of currency A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" in the most specific sense is money I ...
in today's money of an amount \,C\, of money, \,n\, years into the future, can be computed with the same formula, where in this case \,i\, is an assumed future
inflation rate In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the be ...
. If we are using lower discount rate(''i'' ), then it allows the present values in the discount future to have higher values.


Net present value of a stream of cash flows

A cash flow is an amount of money that is either paid out or received, differentiated by a negative or positive sign, at the end of a period. Conventionally, cash flows that are received are denoted with a positive sign (total cash has increased) and cash flows that are paid out are denoted with a negative sign (total cash has decreased). The cash flow for a period represents the net change in money of that period. Calculating the net present value, \,NPV\,, of a stream of cash flows consists of discounting each cash flow to the present, using the present value factor and the appropriate number of compounding periods, and combining these values. For example, if a stream of cash flows consists of +$100 at the end of period one, -$50 at the end of period two, and +$35 at the end of period three, and the interest rate per compounding period is 5% (0.05) then the present value of these three Cash Flows are: :PV_ = \frac = \$95.24 \, :PV_ = \frac = -\$45.35 \, :PV_ = \frac = \$30.23 \, respectively Thus the net present value would be: :NPV = PV_+PV_+PV_ = \frac + \frac + \frac = 95.24 - 45.35 + 30.23 = 80.12, There are a few considerations to be made. * The periods might not be consecutive. If this is the case, the exponents will change to reflect the appropriate number of periods * The interest rates per period might not be the same. The cash flow must be discounted using the interest rate for the appropriate period: if the interest rate changes, the sum must be discounted to the period where the change occurs using the second interest rate, then discounted back to the present using the first interest rate. For example, if the cash flow for period one is $100, and $200 for period two, and the interest rate for the first period is 5%, and 10% for the second, then the net present value would be: :NPV = 100\,(1.05)^ + 200\,(1.10)^\,(1.05)^ = \frac + \frac = \$95.24 + \$173.16 = \$268.40 * The interest rate must necessarily coincide with the payment period. If not, either the payment period or the interest rate must be modified. For example, if the interest rate given is the effective annual interest rate, but cash flows are received (and/or paid) quarterly, the interest rate per quarter must be computed. This can be done by converting effective annual interest rate, \, i \, , to nominal annual interest rate compounded quarterly: : (1+i) = \left(1+\frac\right)^4 Here, i^ is the nominal annual interest rate, compounded quarterly, and the interest rate per quarter is \frac


Present value of an annuity

Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules; payments of the same amount at regular time intervals. Such an arrangement is called an
annuity An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account A savings acco ...
. The expressions for the present value of such payments are
summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

summation
s of
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
. There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, \, n \, payments are received (or paid) at the end of each period, at times 1 through \, n \, , while for an annuity due, \, n \, payments are received (or paid) at the beginning of each period, at times 0 through \, n-1 \, . This subtle difference must be accounted for when calculating the present value. An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ by a factor of (1+i): : PV_\text = PV_\text(1+i) \,\! The present value of an annuity immediate is the value at time 0 of the stream of cash flows: :PV = \sum_^ \frac = C\left frac\right \qquad (1) where: :\, n \, = number of periods, :\, C \, = amount of cash flows, :\, i \, = effective periodic interest rate or rate of return.


An approximation for annuity and loan calculations

The above formula (1) for annuity immediate calculations offers little insight for the average user and requires the use of some form of computing machinery. There is an approximation which is less intimidating, easier to compute and offers some insight for the non-specialist. It is given by Swingler, D. N., (2014), "A Rule of Thumb approximation for time value of money calculations", ''Journal of Personal Finance'', Vol. 13,Issue 2, pp.57-61 :: C \approx PV \left( \frac + \frac i \right) Where, as above, C is annuity payment, PV is principal, n is number of payments, starting at end of first period, and i is interest rate per period. Equivalently C is the periodic loan repayment for a loan of PV extending over n periods at interest rate, i. The formula is valid (for positive n, i) for ni≤3. For completeness, for ni≥3 the approximation is C \approx PV i. The formula can, under some circumstances, reduce the calculation to one of mental arithmetic alone. For example, what are the (approximate) loan repayments for a loan of PV = $10,000 repaid annually for n = ten years at 15% interest (i = 0.15)? The applicable approximate formula is C ≈ 10,000*(1/10 + (2/3) 0.15) = 10,000*(0.1+0.1) = 10,000*0.2 = $2000 pa by mental arithmetic alone. The true answer is $1993, very close. The overall approximation is accurate to within ±6% (for all n≥1) for interest rates 0≤i≤0.20 and within ±10% for interest rates 0.20≤i≤0.40. It is, however, intended only for "rough" calculations.


Present value of a perpetuity

A
perpetuity A perpetuity is an annuity An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings ...
refers to periodic payments, receivable indefinitely, although few such instruments exist. The present value of a perpetuity can be calculated by taking the limit of the above formula as ''n'' approaches infinity. :PV\,=\,\frac. \qquad (2) Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly by summing the present value of the payments :PV = \sum_^\infty \frac = \frac, \qquad i > 0, which form a
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
. Again there is a distinction between a perpetuity immediate – when payments received at the end of the period – and a perpetuity due – payment received at the beginning of a period. And similarly to annuity calculations, a perpetuity due and a perpetuity immediate differ by a factor of (1+i) : : PV_\text = PV_\text(1+i) \,\!


PV of a bond

:''See: Bond valuation #Present value approach'' A corporation issues a
bond Bond or bonds may refer to: Common meanings * Bond (finance) In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of ...
, an interest earning debt security, to an investor to raise funds. The bond has a face value, F , coupon rate, r , and maturity date which in turn yields the number of periods until the debt matures and must be repaid. A bondholder will receive coupon payments semiannually (unless otherwise specified) in the amount of Fr , until the bond matures, at which point the bondholder will receive the final coupon payment and the face value of a bond, F(1+r) . The present value of a bond is the purchase price. The purchase price can be computed as: :PV = \left sum_^ Fr(1+i)^\right/math> + F(1+i)^ The purchase price is equal to the bond's face value if the coupon rate is equal to the current interest rate of the market, and in this case, the bond is said to be sold 'at par'. If the coupon rate is less than the market interest rate, the purchase price will be less than the bond's face value, and the bond is said to have been sold 'at a discount', or below par. Finally, if the coupon rate is greater than the market interest rate, the purchase price will be greater than the bond's face value, and the bond is said to have been sold 'at a premium', or above par.


=Technical details

= Present value is
additive Additive may refer to: Mathematics * Additive function In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer ''n'' such that whenever ''a'' and ''b'' are coprime, the function of the product is the ...
. The present value of a bundle of
cash flow A cash flow is a real or virtual movement of money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic contex ...
s is the sum of each one's present value. See
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are co ...
for further discussion. These calculations must be applied carefully, as there are underlying assumptions: * That it is not necessary to account for price
inflation In economics, inflation refers to a general progressive increase in prices of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a r ...

inflation
, or alternatively, that the cost of inflation is incorporated into the interest rate; see
Inflation-indexed bond Daily inflation-indexed bonds (also known as inflation-linked bonds or colloquially as linkers) are bonds where the principal is indexed to inflation In economics, inflation refers to a general progressive increase in prices of goods and s ...
. * That the likelihood of receiving the payments is high — or, alternatively, that the default risk is incorporated into the interest rate; see Corporate bond #Risk analysis. (In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the
Laplace transform In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate. The full Laplace transform is the curve of all present values, plotted as a function of interest rate. For discrete time, where payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on an almost continual basis, the mathematics of continuous functions can be used as an approximation.)


Variants/approaches

There are mainly two flavors of Present Value. Whenever there will be uncertainties in both timing and amount of the cash flows, the expected present value approach will often be the appropriate technique. With Present Value under uncertainty, future dividends are replaced by their conditional expectation. * Traditional Present Value Approach – in this approach a single set of estimated cash flows and a single interest rate (commensurate with the risk, typically a weighted average of cost components) will be used to estimate the fair value. * Expected Present Value Approach – in this approach multiple cash flows scenarios with different/expected probabilities and a credit-adjusted risk free rate are used to estimate the fair value.


Choice of interest rate

The interest rate used is the
risk-free interest rate The risk-free interest rate is the rate of return In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted wit ...
if there are no risks involved in the project. The rate of return from the project must equal or exceed this rate of return or it would be better to invest the capital in these risk free assets. If there are risks involved in an investment this can be reflected through the use of a
risk premium Overview A risk premium is a measure of excess return that is required by an individual to compensate them for being subjected to an increased level of risk. It is used widely in finance and economics with the general definition being the expec ...
. The risk premium required can be found by comparing the project with the rate of return required from other projects with similar risks. Thus it is possible for investors to take account of any uncertainty involved in various investments.


Present value method of valuation

An investor, the lender of money, must decide the financial project in which to invest their money, and present value offers one method of deciding. A financial project requires an initial outlay of money, such as the price of stock or the price of a corporate bond. The project claims to return the initial outlay, as well as some surplus (for example, interest, or future cash flows). An investor can decide which project to invest in by calculating each projects’ present value (using the same interest rate for each calculation) and then comparing them. The project with the smallest present value – the least initial outlay – will be chosen because it offers the same return as the other projects for the least amount of money.


See also

*
Capital budgeting Capital most commonly refers to: * Capital letter Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minusc ...
*
Derivative (finance) In finance, a derivative is a contract A contract is a legally binding agreement that defines and governs the rights and duties between or among its parties Image:'Hip, Hip, Hurrah! Artist Festival at Skagen', by Peder Severin Krøyer (1 ...
*
Lifetime value In marketing, customer lifetime value (CLV or often CLTV), lifetime customer value (LCV), or life-time value (LTV) is a prognostication of the net profit contributed to the whole future relationship with a customer. The prediction model can have ...
*
Liquidation Liquidation is the process in accounting by which a company A company, abbreviated as co., is a Legal personality, legal entity representing an association of people, whether Natural person, natural, Legal person, legal or a mixture of ...
*
Net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ra ...


References


Further reading

* {{DEFAULTSORT:Present Value Income Mathematical finance fr:Valeur actuelle