# Present value

Present 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 and other factors such as investment risk. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.

Calculation

The most commonly applied model of the time value of money is compound interest. To someone who can lend or borrow for $,t,$ years at an interest rate $,i,$ per year (where interest of "5 percent" is expressed fully as 0.05), the present value of the receiving $,C,$ monetary units $,t,$ years in the future is:

:$C_t = C\left(1 + i\right)^\left\{-t\right\}, = frac\left\{C\right\}\left\{\left(1+i\right)^t\right\} ,$

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

The purchasing power in today's money of an amount "C" of money, "t" years into the future, can be computed with the same formula, where in this case "i" is an assumed future inflation rate.

The expression $,\left(1 + i\right)^\left\{-t\right\}$ 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 $,i,$ may be included; an investment over a two year period would then have PV of:

:$mathrm\left\{PV\right\} = frac\left\{C\right\}\left\{\left(1+i_1\right)\left(1+i_2\right)\right\} ,$

Technical details

Present value is additive. The present value of a bundle of cash flows 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.

Choice of interest rate

The interest rate used is the risk-free interest rate. If there are no risks involved in the project, the expected 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. 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, other loans, leases, salaries, membership dues, annuities, straight-line depreciation 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 summations 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. Future payments are discounted by the periodic rate of interest ("i"). The present value of this ordinary annuity is determined with this formula: [cite web |url=http://www.college-cram.com/study/finance/presentations/1116|title=Annuities: Present and Future Value|accessdate=2008-07-10 ]

:$PV ,=,frac\left\{C\right\}\left\{i\right\}cdot \left[1-frac\left\{1\right\}\left\{left\left(1+i ight\right)^n\right\}\right]$

A periodic amount receivable indefinitely is called a perpetuity, 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:

:$PV,=,frac\left\{C\right\}\left\{i\right\}$

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, 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 for further discussion.

References

ee also

* Capital budgeting
* Net present value
* Future value
* Time value of money

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