Time value of money

The time value of money is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

The time value of money explains why interest is paid or earned: Interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the time value of money.

It also underlies investment. Investors are willing to forgo spending their money now only if they expect a favorable return on their investment in the future, such that the increased value to be available later is sufficiently high to offset the preference to have money now.

Economics of climate change chapter3 discounting curves
The present value of $1,000, 100 years into the future. Curves represent constant discount rates of 2%, 3%, 5%, and 7%.

History

The Talmud (~500 CE) recognizes the time value of money. In Tractate Makkos page 3a the Talmud discusses a case where witnesses falsely claimed that the term of a loan was 30 days when it was actually 10 years. The false witnesses must pay the difference of the value of the loan "in a situation where he would be required to give the money back (within) thirty days..., and that same sum in a situation where he would be required to give the money back (within) 10 years...The difference is the sum that the testimony of the (false) witnesses sought to have the borrower lose; therefore, it is the sum that they must pay." [1]

The notion was later described by Martín de Azpilcueta (1491–1586) of the School of Salamanca.

Calculations

Time value of money problems involve the net value of cash flows at different points in time.

In a typical case, the variables might be: a balance (the real or nominal value of a debt or a financial asset in terms of monetary units), a periodic rate of interest, the number of periods, and a series of cash flows. (In the case of a debt, cash flows are payments against principal and interest; in the case of a financial asset, these are contributions to or withdrawals from the balance.) More generally, the cash flows may not be periodic but may be specified individually. Any of these variables may be the independent variable (the sought-for answer) in a given problem. For example, one may know that: the interest is 0.5% per period (per month, say); the number of periods is 60 (months); the initial balance (of the debt, in this case) is 25,000 units; and the final balance is 0 units. The unknown variable may be the monthly payment that the borrower must pay.

For example, £100 invested for one year, earning 5% interest, will be worth £105 after one year; therefore, £100 paid now and £105 paid exactly one year later both have the same value to a recipient who expects 5% interest assuming that inflation would be zero percent. That is, £100 invested for one year at 5% interest has a future value of £105 under the assumption that inflation would be zero percent.[2]

This principle allows for the valuation of a likely stream of income in the future, in such a way that annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream; all of the standard calculations for time value of money derive from the most basic algebraic expression for the present value of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, the future value sum to be received in one year is discounted at the rate of interest to give the present value sum :

Some standard calculations based on the time value of money are:

  • Present value: The current worth of a future sum of money or stream of cash flows, given a specified rate of return. Future cash flows are "discounted" at the discount rate; the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations.[3]
  • Present value of an annuity: An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due.[4]
Present value of a perpetuity is an infinite and constant stream of identical cash flows.[5]
  • Future value: The value of an asset or cash at a specified date in the future, based on the value of that asset in the present.[6]
  • Future value of an annuity (FVA): The future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheet. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).[7]

For any of the equations below, the formula may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).

These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital at the end of the bond's maturity—that is, a future payment. The two formulas can be combined to determine the present value of the bond.

An important note is that the interest rate i is the interest rate for the relevant period. For an annuity that makes one payment per year, i will be the annual interest rate. For an income or payment stream with a different payment schedule, the interest rate must be converted into the relevant periodic interest rate. For example, a monthly rate for a mortgage with monthly payments requires that the interest rate be divided by 12 (see the example below). See compound interest for details on converting between different periodic interest rates.

The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.

For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the present value of an annuity due, you can simply multiply the PV of an ordinary annuity by (1 + i).

Formula

The following formula use these common variables:

  • PV is the value at time=0 (present value)
  • FV is the value at time=n (future value)
  • A is the value of the individual payments in each compounding period
  • n is the number of periods (not necessarily an integer)
  • i is the interest rate at which the amount compounds each period
  • g is the growing rate of payments over each time period

Future value of a present sum

The future value (FV) formula is similar and uses the same variables.

Present value of a future sum

The present value formula is the core formula for the time value of money; each of the other formulae is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.

The present value (PV) formula has four variables, each of which can be solved for by numerical methods:

The cumulative present value of future cash flows can be calculated by summing the contributions of FVt, the value of cash flow at time t:

Note that this series can be summed for a given value of n, or when n is ∞.[8] This is a very general formula, which leads to several important special cases given below.

Present value of an annuity for n payment periods

In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for by numerical methods:

To get the PV of an annuity due, multiply the above equation by (1 + i).

Present value of a growing annuity

In this case each cash flow grows by a factor of (1+g). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of g as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.

Where i ≠ g :

Where i = g :

To get the PV of a growing annuity due, multiply the above equation by (1 + i).

Present value of a perpetuity

A perpetuity is payments of a set amount of money that occur on a routine basis and continue forever. When n → ∞, the PV of a perpetuity (a perpetual annuity) formula becomes a simple division.

Present value of a growing perpetuity

When the perpetual annuity payment grows at a fixed rate (g, with g < i) the value is determined according to the following formula, obtained by setting n to infinity in the earlier formula for a growing perpetuity:

In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.

This is the well known Gordon Growth model used for stock valuation.

Future value of an annuity

The future value (after n periods) of an annuity (FVA) formula has four variables, each of which can be solved for by numerical methods:

To get the FV of an annuity due, multiply the above equation by (1 + i).

Future value of a growing annuity

The future value (after n periods) of a growing annuity (FVA) formula has five variables, each of which can be solved for by numerical methods:

Where i ≠ g :

Where i = g :

Formula table

The following table summarizes the different formulas commonly used in calculating the time value of money.[9] These values are often displayed in tables where the interest rate and time are specified.

Find Given Formula
Future value (F) Present value (P)
Present value (P) Future value (F)
Repeating payment (A) Future value (F)
Repeating payment (A) Present value (P)
Future value (F) Repeating payment (A)
Present value (P) Repeating payment (A)
Future value (F) Initial gradient payment (G)
Present value (P) Initial gradient payment (G)
Fixed payment (A) Initial gradient payment (G)
Future value (F) Initial exponentially increasing payment (D)

Increasing percentage (g)

  (for i ≠ g)

  (for i = g)

Present value (P) Initial exponentially increasing payment (D)

Increasing percentage (g)

  (for i ≠ g)

  (for i = g)

Notes:

  • A is a fixed payment amount, every period
  • G is the initial payment amount of an increasing payment amount, that starts at G and increases by G for each subsequent period.
  • D is the initial payment amount of an exponentially (geometrically) increasing payment amount, that starts at D and increases by a factor of (1+g) each subsequent period.

Derivations

Annuity derivation

The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.

A single payment C at future time m has the following future value at future time n:

Summing over all payments from time 1 to time n, then reversing t

Note that this is a geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series, we get

The present value of the annuity (PVA) is obtained by simply dividing by :

Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:

Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:

Initially, before any payments, the present value of the system is just the endowment principal (). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (). Plugging this back into the equation:

Perpetuity derivation

Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:

can be seen to approach the value of 1 as n grows larger. At infinity, it is equal to 1, leaving as the only term remaining.

Continuous compounding

Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:

This can be generalized to discount rates that vary over time: instead of a constant discount rate r, one uses a function of time r(t). In that case the discount factor, and thus the present value, of a cash flow at time T is given by the integral of the continuously compounded rate r(t):

Indeed, a key reason for using continuous compounding is to simplify the analysis of varying discount rates and to allow one to use the tools of calculus. Further, for interest accrued and capitalized overnight (hence compounded daily), continuous compounding is a close approximation for the actual daily compounding. More sophisticated analysis includes the use of differential equations, as detailed below.

Examples

Using continuous compounding yields the following formulas for various instruments:

Annuity
Perpetuity
Growing annuity
Growing perpetuity
Annuity with continuous payments

These formulas assume that payment A is made in the first payment period and annuity ends at time t.[10]

Differential equations

Ordinary and partial differential equations (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics. While time value of money can be understood without using the framework of differential equations, the added sophistication sheds additional light on time value, and provides a simple introduction before considering more complicated and less familiar situations. This exposition follows (Carr & Flesaker 2006, pp. 6–7).

The fundamental change that the differential equation perspective brings is that, rather than computing a number (the present value now), one computes a function (the present value now or at any point in future). This function may then be analyzed—how does its value change over time—or compared with other functions.

Formally, the statement that "value decreases over time" is given by defining the linear differential operator as:

This states that values decreases (−) over time (∂t) at the discount rate (r(t)). Applied to a function it yields:

For an instrument whose payment stream is described by f(t), the value V(t) satisfies the inhomogeneous first-order ODE ("inhomogeneous" is because one has f rather than 0, and "first-order" is because one has first derivatives but no higher derivatives) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow (if you receive a £10 coupon, the remaining value decreases by exactly £10).

The standard technique tool in the analysis of ODEs is Green's functions, from which other solutions can be built. In terms of time value of money, the Green's function (for the time value ODE) is the value of a bond paying £1 at a single point in time u – the value of any other stream of cash flows can then be obtained by taking combinations of this basic cash flow. In mathematical terms, this instantaneous cash flow is modeled as a Dirac delta function

The Green's function for the value at time t of a £1 cash flow at time u is

where H is the Heaviside step function – the notation "" is to emphasize that u is a parameter (fixed in any instance—the time when the cash flow will occur), while t is a variable (time). In other words, future cash flows are exponentially discounted (exp) by the sum (integral, ) of the future discount rates ( for future, r(v) for discount rates), while past cash flows are worth 0 (), because they have already occurred. Note that the value at the moment of a cash flow is not well-defined – there is a discontinuity at that point, and one can use a convention (assume cash flows have already occurred, or not already occurred), or simply not define the value at that point.

In case the discount rate is constant, this simplifies to

where is "time remaining until cash flow".

Thus for a stream of cash flows f(u) ending by time T (which can be set to for no time horizon) the value at time t, is given by combining the values of these individual cash flows:

This formalizes time value of money to future values of cash flows with varying discount rates, and is the basis of many formulas in financial mathematics, such as the Black–Scholes formula with varying interest rates.

See also

Notes

  1. ^ "Makkot 3a William Davidson Talmud online".
  2. ^ Carther, Shauna (3 December 2003). "Understanding the Time Value of Money".
  3. ^ Staff, Investopedia (25 November 2003). "Present Value - PV".
  4. ^ "Present Value of an Annuity".
  5. ^ Staff, Investopedia (24 November 2003). "Perpetuity".
  6. ^ Staff, Investopedia (23 November 2003). "Future Value - FV".
  7. ^ Hovey, M. (2005). Spreadsheet Modelling for Finance. Frenchs Forest, N.S.W.: Pearson Education Australia.
  8. ^ http://mathworld.wolfram.com/GeometricSeries.html Geometric Series
  9. ^ "NCEES FE exam".
  10. ^ "Annuities and perpetuities with continuous compounding".

References

External links

Accounting rate of return

Accounting rate of return, also known as the Average rate of return, or ARR is a financial ratio used in capital budgeting. The ratio does not take into account the concept of time value of money. ARR calculates the return, generated from net income of the proposed capital investment. The ARR is a percentage return. Say, if ARR = 7%, then it means that the project is expected to earn seven cents out of each dollar invested (yearly). If the ARR is equal to or greater than the required rate of return, the project is acceptable. If it is less than the desired rate, it should be rejected. When comparing investments, the higher the ARR, the more attractive the investment. More than half of large firms calculate ARR when appraising projects.The key advantage of ARR is that it is easy to compute and understand. The main disadvantage of ARR is that it disregards the time factor in terms of time value of money or risks for long term investments. The ARR is built on evaluation of profits and it can be easily manipulated with changes in depreciation methods. The ARR can give misleading information when evaluating investments of different size.

Cost–benefit analysis

Cost–benefit analysis (CBA), sometimes called benefit costs analysis (BCA), is a systematic approach to estimating the strengths and weaknesses of alternatives used to determine options which provide the best approach to achieving benefits while preserving savings (for example, in transactions, activities, and functional business requirements). A CBA may be used to compare completed or potential courses of actions, or to estimate (or evaluate) the value against the cost of a decision, project, or policy. It is commonly used in commercial transactions, business or policy decisions (particularly public policy), and project investments.

CBA has two main applications:

To determine if an investment (or decision) is sound, ascertaining if – and by how much – its benefits outweigh its costs.

To provide a basis for comparing investments (or decisions), comparing the total expected cost of each option with its total expected benefits.CBA is related to cost-effectiveness analysis. Benefits and costs in CBA are expressed in monetary terms and are adjusted for the time value of money; all flows of benefits and costs over time are expressed on a common basis in terms of their net present value, regardless of whether they are incurred at different times. Other related techniques include cost–utility analysis, risk–benefit analysis, economic impact analysis, fiscal impact analysis, and social return on investment (SROI) analysis.

Cost–benefit analysis is often used by organizations to appraise the desirability of a given policy. It is an analysis of the expected balance of benefits and costs, including an account of any alternatives and the status quo. CBA helps predict whether the benefits of a policy outweigh its costs (and by how much), relative to other alternatives. This allows the ranking of alternative policies in terms of a cost–benefit ratio. Generally, accurate cost–benefit analysis identifies choices which increase welfare from a utilitarian perspective. Assuming an accurate CBA, changing the status quo by implementing the alternative with the lowest cost–benefit ratio can improve Pareto efficiency. Although CBA can offer an informed estimate of the best alternative, a perfect appraisal of all present and future costs and benefits is difficult; perfection, in economic efficiency and social welfare, is not guaranteed.The value of a cost–benefit analysis depends on the accuracy of the individual cost and benefit estimates. Comparative studies indicate that such estimates are often flawed, preventing improvements in Pareto and Kaldor–Hicks efficiency. Interest groups may attempt to include (or exclude) significant costs in an analysis to influence its outcome.

Discounted cash flow

In finance, discounted cash flow (DCF) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. All future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value of the cash flows in question.Using DCF analysis to compute the NPV takes as input cash flows and a discount rate and gives as output a present value. The opposite process takes cash flows and a price (present value) as inputs, and provides as output the discount rate; this is used in bond markets to obtain the yield.

Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management and patent valuation. It was used in industry as early as the 1700s or 1800s, widely discussed in financial economics in the 1960s, and became widely used in U.S. Courts in the 1980s and 1990s.

Discounting

Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The discount, or charge, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

The discount is usually associated with a discount rate, which is also called the discount yield. The discount yield is the proportional share of the initial amount owed (initial liability) that must be paid to delay payment for 1 year.

Since a person can earn a return on money invested over some period of time, most economic and financial models assume the discount yield is the same as the rate of return the person could receive by investing this money elsewhere (in assets of similar risk) over the given period of time covered by the delay in payment. The concept is associated with the opportunity cost of not having use of the money for the period of time covered by the delay in payment. The relationship between the discount yield and the rate of return on other financial assets is usually discussed in economic and financial theories involving the inter-relation between various market prices, and the achievement of Pareto optimality through the operations in the capitalistic price mechanism, as well as in the discussion of the efficient (financial) market hypothesis. The person delaying the payment of the current liability is essentially compensating the person to whom he/she owes money for the lost revenue that could be earned from an investment during the time period covered by the delay in payment. Accordingly, it is the relevant "discount yield" that determines the "discount", and not the other way around.

As indicated, the rate of return is usually calculated in accordance to an annual return on investment. Since an investor earns a return on the original principal amount of the investment as well as on any prior period investment income, investment earnings are "compounded" as time advances. Therefore, considering the fact that the "discount" must match the benefits obtained from a similar investment asset, the "discount yield" must be used within the same compounding mechanism to negotiate an increase in the size of the "discount" whenever the time period of the payment is delayed or extended. The "discount rate" is the rate at which the "discount" must grow as the delay in payment is extended. This fact is directly tied into the time value of money and its calculations.

The "time value of money" indicates there is a difference between the "future value" of a payment and the "present value" of the same payment. The rate of return on investment should be the dominant factor in evaluating the market's assessment of the difference between the future value and the present value of a payment; and it is the market's assessment that counts the most. Therefore, the "discount yield", which is predetermined by a related return on investment that is found in the financial markets, is what is used within the time-value-of-money calculations to determine the "discount" required to delay payment of a financial liability for a given period of time.

Future value

Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is the present value multiplied by the accumulation function.

The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money calculations.

HP-19B

HP-19B, introduced on January 4, 1988, along with the HP-17B, HP-27S and the HP-28S, and replaced by the HP-19BII (F1639A) in January 1990, was a simplified Hewlett Packard business model calculator, like the 17B. It had a clamshell design, like the HP-18C, HP-28C and 28S.Two common issues with the clamshell case were the plastic surrounding the battery door would break under pressure from the batteries; and the ribbon connecting the two keyboards would begin to fail after numerous case openings.

The calculator included functions for solving financial calculations like time value of money, amortizing, interest rate conversion and cash flow. Business functionalities included percentage change, markup, currency exchange and unit conversions. It also had math capabilities such as trigonometry and graphing. Upscale functionality, at the time of release, included the ability to design your own problem solving equations and storing text directly in the calculator using the letter keyboard on the left side. The calculator could also be connected to a printer using a special cable; which allowed you to print out the generated graphs.

HP-22

The HP-22 was a finance-oriented pocket calculator produced by Hewlett-Packard between 1975 and 1978. It was designed as a replacement for the short-lived HP-70, and was one of a set of three calculators, the others being the HP-21 and HP-25, which were similarly built but aimed at different markets.As with most HP calculators then and now, the HP-25 used RPN entry logic, with a four-level stack. It also had ten user-accessible memory registers. As was normal at the time, memory was not preserved on power-down. Its principal functions were (1) time value of money (TVM) calculations, where the user could enter any three of the variables and the fourth would be calculated, and (2) statistics calculations, including linear regression. Basic logarithmic and exponential functions were also provided. For TVM calculations, a physical slider switch labelled "begin" and "end" could be used to specify whether payments would be applied at the beginning or end of periods. It had a 12-digit LED display. A shift key provided access to functions whose legends were printed on the faceplate above the corresponding keys.

Its HP development codename was Turnip, and it was a member of the Woodstock series. Its US price was $165 in 1975, $125 in 1978.

HP-27S

The HP-27S was a pocket calculator produced by Hewlett-Packard, introduced in 1988, and discontinued between 1990 and 1993 (sources vary). It was the first HP scientific calculator to use algebraic entry instead of RPN, and though it was labelled scientific, it also included features associated with specialised business calculators.

The device featured standard scientific functions, including statistics and probability. Equations could be stored in memory, and solved and integrated for specified variables. Binary, octal, and hexadecimal number bases could be used. Business features included a real-time clock and calendar, as well as functions such as time value of money calculations.The calculator had 7k bytes of usable memory, shared among variables and formulas.

Its hardware features included a dot-matrix display of two rows of 22 characters. Depending on context, either the top row was used to display the current expression or a message, or the bottom row was used to show menu options, which could be selected with the corresponding keys. An infra-red transmitter was also included, allowing the machine to be used with a compatible printer, such as the HP 82240B. A beeper could be used to sound date/time alarms.

The 27S was not programmable in the conventional way, but it included an advanced formula-storage system with programming features. Within stored formulas, sub-formulas could be defined and later referred to by name. Loops and conditional execution could also be embedded within formulas.

Martín de Azpilcueta

Martín de Azpilcueta (Azpilikueta in Basque) (13 December 1491 – 1 June 1586), or Doctor Navarrus, was an important Spanish canonist and theologian in his time, and an early economist, the first to develop monetarist theory.

Net present value

In finance, 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 rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

Time value of money dictates that time affects the value of cash flows. For example, a lender may offer 99 cents for the promise of receiving $1.00 a month from now, but the promise to receive that same dollar 20 years in the future would be worth much less today to that same person (lender), even if the payback in both cases was equally certain. This decrease in the current value of future cash flows is based on a chosen rate of return (or discount rate). If for example there exists a time series of identical cash flows, the cash flow in the present is the most valuable, with each future cash flow becoming less valuable than the previous cash flow. A cash flow today is more valuable than an identical cash flow in the future because a present flow can be invested immediately and begin earning returns, while a future flow cannot.

Net present value (NPV) is determined by calculating the costs (negative cash flows) and benefits (positive cash flows) for each period of an investment. The period is typically one year, but could be measured in quarter-years, half-years or months. After the cash flow for each period is calculated, the present value (PV) of each one is achieved by discounting its future value (see Formula) at a periodic rate of return (the rate of return dictated by the market). NPV is the sum of all the discounted future cash flows. Because of its simplicity, NPV is a useful tool to determine whether a project or investment will result in a net profit or a loss. A positive NPV results in profit, while a negative NPV results in a loss. The NPV measures the excess or shortfall of cash flows, in present value terms, above the cost of funds. In a theoretical situation of unlimited capital budgeting a company should pursue every investment with a positive NPV. However, in practical terms a company's capital constraints limit investments to projects with the highest NPV whose cost cash flows, or initial cash investment, do not exceed the company's capital. NPV is a central tool in discounted cash flow (DCF) analysis and is a standard method for using the time value of money to appraise long-term projects. It is widely used throughout economics, finance, and accounting.

In the case when all future cash flows are positive, or incoming (such as the principal and coupon payment of a bond) the only outflow of cash is the purchase price, the NPV is simply the PV of future cash flows minus the purchase price (which is its own PV). NPV can be described as the "difference amount" between the sums of discounted cash inflows and cash outflows. It compares the present value of money today to the present value of money in the future, taking inflation and returns into account.

The NPV of a sequence of cash flows takes as input the cash flows and a discount rate or discount curve and outputs a present value, which is the current fair price. The converse process in discounted cash flow (DCF) analysis takes a sequence of cash flows and a price as input and as output the discount rate, or internal rate of return (IRR) which would yield the given price as NPV. This rate, called the yield, is widely used in bond trading.

Many computer-based spreadsheet programs have built-in formulae for PV and NPV.

Outline of finance

The following outline is provided as an overview of and topical guide to finance:

Finance – addresses the ways in which individuals and organizations raise and allocate monetary resources over time, taking into account the risks entailed in their projects.

Payback period

Payback period in capital budgeting refers to the period of time required to recoup the funds expended in an investment, or to reach the break-even point. For example, a $1000 investment made at the start of year 1 which returned $500 at the end of year 1 and year 2 respectively would have a two-year payback period. Payback period is usually expressed in years. Starting from investment year by calculating Net Cash Flow for each year: Net Cash Flow Year 1 = Cash Inflow Year 1 - Cash Outflow Year 1. Then Cumulative Cash Flow = (Net Cash Flow Year 1 + Net Cash Flow Year 2 + Net Cash Flow Year 3, etc.) Accumulate by year until Cumulative Cash Flow is a positive number: that year is the payback year.

The time value of money is not taken into account. Payback period intuitively measures how long something takes to "pay for itself." All else being equal, shorter payback periods are preferable to longer payback periods. Payback period is popular due to its ease of use despite the recognized limitations described below.

The term is also widely used in other types of investment areas, often with respect to energy efficiency technologies, maintenance, upgrades, or other changes. For example, a compact fluorescent light bulb may be described as having a payback period of a certain number of years or operating hours, assuming certain costs. Here, the return to the investment consists of reduced operating costs. Although primarily a financial term, the concept of a payback period is occasionally extended to other uses, such as energy payback period (the period of time over which the energy savings of a project equal the amount of energy expended since project inception); these other terms may not be standardized or widely used.

Perpetuity

A perpetuity is an 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; these were known as consols and were all finally redeemed in 2015. Real estate and preferred stock are among some types of investments that effect the results of a perpetuity, and prices can be established using techniques for valuing a perpetuity. Perpetuities are but one of the time value of money methods for valuing financial assets. Perpetuities are a form of ordinary annuities.

The concept is closely linked to terminal value and terminal growth rate in valuation.

Present value

In economics and 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 always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be 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. 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 calculations, are used to value loans, mortgages, annuities, sinking funds, perpetuities, bonds, and more. These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times, since time 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.

Rate of return

In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from the investment, such as interest payments or dividends. It may be measured either in absolute terms (e.g., dollars) or as a percentage of the amount invested. The latter is also called the holding period return.

A loss instead of a profit is described as a negative return, assuming the amount invested is greater than zero.

The rate of return is a profit on an investment over a period of time, expressed as a proportion of the original investment. The time period is typically a year, in which case the rate of return is referred to as the annual return.

To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into an annualised return. This conversion process is called annualisation, described below.

The return on investment (ROI) is return per dollar invested. It is a measure of investment performance, as opposed to size (c.f. return on equity, return on assets, return on capital employed).

Return of capital

See also return on capital.Return of capital (ROC) refers to principal payments back to "capital owners" (shareholders, partners, unitholders) that exceed the growth (net income/taxable income) of a business or investment. It should not be confused with Rate of Return (ROR), which measures a gain or loss on an investment. Basically, it is a return of some or all of the initial investment, which reduces the basis on that investment.

The ROC effectively shrinks the firm's equity in the same way that all distributions do. It is a transfer of value from the company to the owner. In an efficient market, the stock's price will fall by an amount equal to the distribution. Most public companies pay out only a percentage of their income as dividends. In some industries it is common to pay ROC.

Real Estate Investment Trusts (REITs) commonly make distributions equal to the sum of their income and the depreciation (capital cost allowance) allowed for in the calculation of that income. The business has the cash to make the distribution because depreciation is a non-cash charge.

Oil and gas royalty trusts also make distributions that include ROC equal to the drawdown in the quantity of their reserves. Again, the cash to find the O&G was spent previously, and current operations are generating excess cash.

Private business can distribute any amount of equity that the owners need personally.

Structured Products (closed ended investment funds) frequently use high distributions, that include returns of capital, as a promotional tool. The retail investors these funds are sold to rarely have the technical knowledge to distinguish income from ROC.

Public business may return capital as a means to increase the debt/equity ratio and increase their leverage (risk profile). When the value of real estate holdings (for example) have increased, the owners may realize some of the increased value immediately by taking a ROC and increasing debt. This may be considered analogous to cash out refinancing of a residential property.

When companies spin off divisions and issue shares of a new, stand-alone business, this distribution is a return of capital.

Texas Instruments Business Analyst

The Texas Instruments Business Analyst series is a product line of financial calculators introduced in 1976. BA calculators provide time value of money functions and are widely used in accounting and other financial applications. Though originally designed specifically for financial use, current models also include basic scientific calculator and statistics functions. The BA series competes directly with other mid- to high-end financial calculators, particularly the HP-12C and other models from TI competitor Hewlett-Packard. As of November 2015, TI makes two models, the BA II Plus (originally introduced in 1991) and the BA II Plus Professional (introduced in 2004).

Vendor finance

Vendor finance is a form of lending in which a company lends money to be used by the borrower to buy the vendor's products or property. Vendor finance is usually in the form of deferred loans from, or shares subscribed by, the vendor. The vendor often takes shares in the borrowing company. This category of finance is generally used where the vendor's expectation of the value of the business is higher than that of the borrower's bankers, and usually at a higher interest rate than would be offered elsewhere.it is a cheaper option than going to banks.

Vendor finance bridges the valuation gap due to the time value of money. If the buyer of a business doesn't have to repay the vendor for the vendor loan for a few years, then the value of that portion of the purchase price is worthless. In some cases there is an interest charge on vendor loan, but in other cases it is simply a deferred payment. Vendor finance is different from an Earnout because it is not contingent on performance. Since there is no contingency, vendor finance is more risky for the buyer than an earn-out.

Vendor finance can also be used when the buyer does not have the funds to purchase the entire business. In this case the vendor creates a loan with an interest charge to help the buyer complete the purchase and help the seller complete the sale, usually on better terms for the seller.

Zero-coupon bond

A zero-coupon bond (also discount bond or deep discount bond) is a bond where the face value is repaid at the time of maturity. Note that this definition assumes a positive time value of money. It does not make periodic interest payments, or have so-called coupons, hence the term zero-coupon bond. When the bond reaches maturity, its investor receives its par (or face) value. Examples of zero-coupon bonds include U.S. Treasury bills, U.S. savings bonds, long-term zero-coupon bonds, and any type of coupon bond that has been stripped of its coupons.

In contrast, an investor who has a regular bond receives income from coupon payments, which are made semi-annually or annually. The investor also receives the principal or face value of the investment when the bond matures.

Some zero coupon bonds are inflation indexed, so the amount of money that will be paid to the bond holder is calculated to have a set amount of purchasing power rather than a set amount of money, but the majority of zero coupon bonds pay a set amount of money known as the face value of the bond.

Zero coupon bonds may be long or short term investments. Long-term zero coupon maturity dates typically start at ten to fifteen years. The bonds can be held until maturity or sold on secondary bond markets. Short-term zero coupon bonds generally have maturities of less than one year and are called bills. The U.S. Treasury bill market is the most active and liquid debt market in the world.

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