Present value

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The present value of a single or multiple future payments (known as cash flows) is the nominal amounts of money to change hands at some future date, discounted to account for the time value of money, and other factors such as investment risk. A given amount of money is always more valuable sooner than later since this enables one to take advantage of investment opportunities. Present values are therefore smaller than corresponding future values.

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.

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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.

The interest rate used is the risk free interest rate (for example the yield on US treasury bonds). 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.

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 in 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 periodic amount receivable indefinitely is called a perpetuity, although few such instruments exist. A perpetuity is an infinite geometric series which reduces to PV = C / i, where C is the periodic cash flow and i the periodic rate of interest.

A cash flow stream with a limited number (n) of periodic payments ("C"), receivable at times 1 through n, is an annuity. The value of this annuity is determined with this formula:

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

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}).\,

Often, the present value formula is written in a simplified formula (for example, in textbooks on finance) as:

\mathrm{PV} = FV \cdot PVIF(r,n)\,

Similarly, the annuity formula is often simplified and written as follows:

PV = PMT \cdot PVIFA(r,n)
where:
n = number of periods
r = interest rate in the period
PV = present value at time 0
FV = future value at time n

This simplified form is easier to present, and well-adapted to using financial tables, financial calculators and computer spreadsheets.

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