Present value

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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.

1 Calculation
2 Choice of interest rate
3 Annuities, perpetuities and other common forms
4 Present value formula
5 See also
6 External links

[edit] Calculation

The most commonly applied model of the time value of money is compound interest. To someone who has the opportunity to invest an amount of money $C$ for $t$ years at a rate of interest of $i$ % (where interest of "5 percent" is expressed fully as 0.05) compounded annually, the present value of the receipt of $C$ , $t$ years in the future, is:

$C_t = C(1 + i)^{-t}\, = C / {(1+i)^ t} \$

The expression (1 + i)^−t 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 (Present Value) of:

$\mathrm{PV} = C(1+i_1)^{-1}\cdot(1+i_2)^{-1} \,$

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.

[edit] 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.

[edit] 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 annuity is determined with this formula:

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

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:

$PV\,=\,\frac{C}{i}$

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.

[edit] Present value formula

One hundred units 1 year from now at 5% interest rate is today worth:

${\rm Present\ value}=\frac{\rm future\ amount}{(1+{\rm interest\ rate})^{\rm term}}=\frac{100}{(1+.05)^1}=\ 95.23.$

So the present value of 100 units 1 year from now at 5% is 95.23 units.

The above is in regard to a single lump sum amount. There is a separate formula to calculate PV of annuities. For present value of annuities, use this formula:

$\mbox{PV annuity} = \frac{1-(1+r)^{-n}}{r}\cdot(\mbox{payment amount}).\,$