Present value

# Present value

Discussion

Encyclopedia
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money
Time value of money
The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....

and other factors such as investment risk
Financial risk
Financial risk an umbrella term for multiple types of risk associated with financing, including financial transactions that include company loans in risk of default. Risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss...

. Present value calculations are widely used in business and economics to provide a means to compare cash flow
Cash flow
Cash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, finite period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.Cash flow...

s at different times on a meaningful "like to like" basis.

## 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
Ceteris paribus
or is a Latin phrase, literally translated as "with other things the same," or "all other things being equal or held constant." It is an example of an ablative absolute and is commonly rendered in English as "all other things being equal." A prediction, or a statement about causal or logical...

, a rational person will choose \$100 today. This is described by economists as time preference
Time preference
In economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....

. Time preference can be measured by auctioning off a risk free security - like a US Treasury bill. If a \$100 note, payable in one year, sells for \$80, then the present value of \$100 one year in the future is \$80. This is because you can invest your money today in a bank account or any other (safe) investment that will return you interest.

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 interest that he or she will receive from a borrower (the bank account on 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
Risk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

which corresponds the minimum guaranteed rate provided by your bank's saving account for example. If you want to compare your change in purchasing power, then you should use the real interest rate
Real interest rate
The "real interest rate" is the rate of interest an investor expects to receive after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate...

(nominal interest rate
Nominal interest rate
In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...

minus inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...

rate).

The operation of evaluating a present value into the future value 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 that I receive 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 two options are equivalent (either receiving \$100 today or receiving \$105 in one year). This is because if you cash \$100 today and deposit in your savings account, you will have \$105 in one year.

## Calculation

The most commonly applied model of the time value of money is compound interest
Compound interest
Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...

. To someone who can lend or borrow for years at an interest rate per year (where interest of "5 percent" is expressed fully as 0.05), the present value of the receiving monetary units years in the future is:

This is also found from the formula for the future value with negative time.

Purchasing power is the number of goods/services that can be purchased with a unit of currency. For example, if you had taken one dollar to a store in the 1950s, you would have been able to buy a greater number of items than you would today, indicating that you would have had a greater purchasing...

in today's money of an amount of money, years into the future, can be computed with the same formula, where in this case is an assumed future inflation rate
Inflation rate
In economics, the inflation rate is a measure of inflation, the rate of increase of a price index . It is the percentage rate of change in price level over time. The rate of decrease in the purchasing power of money is approximately equal.The inflation rate is used to calculate the real interest...

.

The expression enters almost all calculations of present value. Where the interest rate is expected to be different over the term of the investment, different values for may be included; an investment over a two year period would then have PV of:

### Technical details

In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

. The present value of a bundle of cash flow
Cash flow
Cash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, finite period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.Cash flow...

s is the sum of each one's present value.

In fact, the present value of a cashflow at a constant interest rate is mathematically the same as the Laplace transform of that cashflow evaluated with the transform variable (usually denoted "s") equal to the 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 PV. 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.
• 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
Risk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

. 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
A risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...

. 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.

## Annuities, perpetuities and other common forms

Many financial arrangements (including bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...

, other loan
Loan
A loan is a type of debt. Like all debt instruments, a loan entails the redistribution of financial assets over time, between the lender and the borrower....

s, lease
Lease
A lease is a contractual arrangement calling for the lessee to pay the lessor for use of an asset. A rental agreement is a lease in which the asset is tangible property...

s, salaries, membership dues, annuities, straight-line depreciation
Depreciation
Depreciation refers to two very different but related concepts:# the decrease in value of assets , and# the allocation of the cost of assets to periods in which the assets are used ....

charges) stipulate structured payment schedules, which is to say payment of the same amount at regular time intervals. The term "annuity" is often used to refer to any such arrangement when discussing calculation of present value. The expressions for the present value of such payments are summation
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...

s of geometric series.

A cash flow stream with a limited number (n) of periodic payments (C), receivable at times 1 through n, is an annuity
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...

. Future payments are discounted by the periodic rate of interest (i). The present value of this ordinary annuity
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...

is determined with this formula:

where:

= number of years

= Amount of cash flows

This formula is usable when the cash flows are spread over the different but in equal intervals and also the amount of these flows is same say \$100 at the end of each year from year one to ten. the is defined as interest rate / required rate of return

A periodic amount receivable indefinitely is called a perpetuity
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...

, 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. The bracketed term reduces to one leaving:

The first formula is found from subtracting from the latter result the present value of a perpetuity delayed n periods.

These calculations must be applied carefully, as there are underlying assumptions:
• That it is not necessary to account for price inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...

, or alternatively, that the cost of inflation is incorporated into the interest rate.
• That the likelihood of receiving the payments is high — or, alternatively, that the default risk is incorporated into the interest rate.

See time value of money
Time value of money
The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....

for further discussion.

• Capital budgeting
Capital budgeting
Capital budgeting is the planning process used to determine whether an organization's long term investments such as new machinery, replacement machinery, new plants, new products, and research development projects are worth pursuing...

• Net present value
Net present value
In finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...

• Future value
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...

• Time value of money
Time value of money
The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....

• Liquidation
Liquidation
In law, liquidation is the process by which a company is brought to an end, and the assets and property of the company redistributed. Liquidation is also sometimes referred to as winding-up or dissolution, although dissolution technically refers to the last stage of liquidation...

• Annuity (finance theory)
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...