Editing Present value (section)

==Calculation==
The operation of evaluating a present sum of money some time in the future is called a capitalization (how much will 100 today be worth in five years?). The reverse operation—evaluating the present value of a future amount of money—is called discounting (how much will 100 received in five years be worth today?).<ref name=Ross>{{cite book|last=Ross|first=Stephen|title=Fundamentals of Corporate Finance|year=2010|publisher=McGraw-Hill|location=New York|isbn=9780077246129|pages=145–287|edition=9|author2=Randolph W. Westerfield |author3=Bradford D. Jordan }}</ref>

Spreadsheets commonly offer functions to compute present value. In Microsoft Excel, there are present value functions for single payments - "=NPV(...)", and series of equal, periodic payments - "=PV(...)". Programs will calculate present value flexibly for any cash flow and interest rate, or for a schedule of different interest rates at different times.

===Present value of a lump sum===
The most commonly applied model of present valuation uses [[compound interest]]. The standard formula is:

:<math>PV = \frac{C}{(1+i)^n} \,</math>

Where <math>\,C\,</math> is the future amount of money that must be discounted, <math>\,n\,</math> is the number of compounding periods between the present date and the date where the sum is worth <math>\,C\,</math>, <math>\,i\,</math> is the interest rate for one compounding period (the end of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily). The interest rate, <math>\,i\,</math>, is given as a percentage, but expressed as a decimal in this formula.

Often, <math>v^{n} = \,(1 + i)^{-n}</math> is referred to as the Present Value Factor <ref name=Broverman>{{cite book|last=Broverman|first=Samuel|title=Mathematics of Investment and Credit|year=2010|publisher=ACTEX Publishers|location=Winsted|isbn=9781566987677|pages=4–229}}</ref>

This is also found from the [[Future value#Compound interest|formula for the future value]] with negative time.

For example, if you are to receive $1000 in five years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is

:<math>PV = \frac{\$1000}{(1+0.10)^{5}} = \$620.92 \, </math>

The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $1000 in five years, or $620.92 today.<ref name="Moyer"/>

The [[purchasing power]] in today's money of an amount <math>\,C\,</math> of money, <math>\,n\,</math> years into the future, can be computed with the same formula, where in this case <math>\,i\,</math> is an assumed future [[inflation rate]].

If we are using lower discount rate(''i'' ), then it allows the present values in the discount future to have higher values.

===Net present value of a stream of cash flows===
A cash flow is an amount of money that is either paid out or received, differentiated by a negative or positive sign, at the end of a period. Conventionally, cash flows that are received are denoted with a positive sign (total cash has increased) and cash flows that are paid out are denoted with a negative sign (total cash has decreased). The cash flow for a period represents the net change in money of that period.<ref name=Ross/> Calculating the net present value, <math>\,NPV\,</math>, of a stream of cash flows consists of discounting each cash flow to the present, using the present value factor and the appropriate number of compounding periods, and combining these values.<ref name=Moyer>{{cite book|last=Moyer|first=Charles|title=Contemporary Financial Management|year=2011|publisher=South-Western Publishing Co|location=Winsted|isbn=9780538479172|pages=147–498|edition=12|author2=William Kretlow |author3=James McGuigan }}</ref>

For example, if a stream of cash flows consists of +$100 at the end of period one, -$50 at the end of period two, and +$35 at the end of period three, and the interest rate per compounding period is 5% (0.05) then the present value of these three Cash Flows are:

:<math>PV_{1} = \frac{\$100}{(1.05)^{1}} = \$95.24 \, </math>
:<math>PV_{2} = \frac{-\$50}{(1.05)^{2}} = -\$45.35 \, </math>
:<math>PV_{3} = \frac{\$35}{(1.05)^{3}} = \$30.23 \, </math> respectively

Thus the net present value would be:

:<math>NPV = PV_{1}+PV_{2}+PV_{3} = \frac{100}{(1.05)^{1}} + \frac{-50}{(1.05)^{2}} + \frac{35}{(1.05)^{3}} = 95.24 - 45.35 + 30.23 = 80.12, </math>

There are a few considerations to be made.
* The periods might not be consecutive. If this is the case, the exponents will change to reflect the appropriate number of periods
* The interest rates per period might not be the same. The cash flow must be discounted using the interest rate for the appropriate period: if the interest rate changes, the sum must be discounted to the period where the change occurs using the second interest rate, then discounted back to the present using the first interest rate.<ref name="Broverman" /> For example, if the cash flow for period one is $100, and $200 for period two, and the interest rate for the first period is 5%, and 10% for the second, then the net present value would be:

:<math>NPV = 100\,(1.05)^{-1} + 200\,(1.10)^{-1}\,(1.05)^{-1} = \frac{100}{(1.05)^{1}} + \frac{200}{(1.10)^{1}(1.05)^{1}} = \$95.24 + \$173.16 = \$268.40 </math>
* The interest rate must necessarily coincide with the payment period. If not, either the payment period or the interest rate must be modified. For example, if the interest rate given is the effective annual interest rate, but cash flows are received (and/or paid) quarterly, the interest rate per quarter must be computed. This can be done by converting effective annual interest rate, <math>\, i \, </math>, to nominal annual interest rate compounded quarterly:
:<math> (1+i) = \left(1+\frac{i^{4}}{4}\right)^4 </math><ref name="Broverman"/>

Here, <math> i^{4} </math> is the nominal annual interest rate, compounded quarterly, and the interest rate per quarter is <math>\frac{i^{4}}{4}</math>

====Present value of an annuity====
{{See also|Annuity#Valuation}}
Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules; payments of the same amount at regular time intervals. Such an arrangement is called an [[annuity]]. The expressions for the present value of such payments are [[summation]]s of [[geometric series]].

There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, <math>\, n \, </math> payments are received (or paid) at the end of each period, at times 1 through <math>\, n \, </math>, while for an annuity due, <math>\, n \, </math> payments are received (or paid) at the beginning of each period, at times 0 through <math>\, n-1 \, </math>.<ref name="Ross"/> This subtle difference must be accounted for when calculating the present value.

An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ by a factor of <math>(1+i)</math>:

:<math> PV_\text{annuity due} = PV_\text{annuity immediate}(1+i) \,\!</math><ref name="Broverman"/>

The present value of an annuity immediate is the value at time 0 of the stream of cash flows:

:<math>PV = \sum_{k=1}^{n} \frac{C}{(1+i)^{k}} = C\left[\frac{1-(1+i)^{-n}}{i}\right], \qquad (1) </math>

where:

:<math>\, n \, </math> = number of periods,

:<math>\, C \, </math> = amount of cash flows,

:<math>\, i \, </math> = effective periodic interest rate or rate of return.

====An approximation for annuity and loan calculations====
The above formula (1) for annuity immediate calculations offers little insight for the average user and requires the use of some form of computing machinery. There is an approximation which is less intimidating, easier to compute and offers some insight for the non-specialist. It is given by <ref>Swingler, D. N., (2014), "A Rule of Thumb approximation for  time value of money calculations", ''Journal of Personal Finance'', Vol. 13, Issue 2, pp.57-61</ref>
:: <math>C \approx  PV \left( \frac {1}{n} + \frac {2}{3} i \right) </math>

Where, as above, C is annuity payment, PV is principal, n is number of payments, starting at end of first period, and i is interest rate per period. Equivalently C is the periodic loan repayment for a loan of PV extending over n periods at interest rate, i. The formula is valid (for positive n, i) for ni≤3. For completeness, for ni≥3 the approximation is <math> C \approx PV  i</math>.

The formula can, under some circumstances, reduce the calculation to one of mental arithmetic alone. For example, what are the (approximate) loan repayments for a loan of PV = $10,000 repaid annually for n = ten years at 15% interest (i = 0.15)? The applicable approximate formula is C ≈ 10,000*(1/10 + (2/3) 0.15) = 10,000*(0.1+0.1) = 10,000*0.2 = $2000 pa by mental arithmetic alone. The true answer is $1993, very close.

The overall approximation is accurate to within ±6% (for all n≥1) for interest rates 0≤i≤0.20 and within ±10% for interest rates 0.20≤i≤0.40. It is, however, intended only for "rough" calculations.

====Present value of a perpetuity====
A [[perpetuity]] refers to periodic payments, receivable indefinitely, 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.

:<math>PV\,=\,\frac{C}{i}. \qquad (2)</math>

Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly by summing the present value of the payments

:<math>PV = \sum_{k=1}^\infty \frac{C}{(1+i)^{k}} = \frac{C}{i}, \qquad i > 0,</math>

which form a [[geometric series]].

Again there is a distinction between a perpetuity immediate – when payments received at the end of the period – and a perpetuity due – payment received at the beginning of a period. And similarly to annuity calculations, a perpetuity due and a perpetuity immediate differ by a factor of <math>(1+i) </math>:

:<math> PV_\text{perpetuity due} = PV_\text{perpetuity immediate}(1+i) \,\!</math><ref name="Broverman"/>

====PV of a bond====
:''See: [[Bond valuation#Present value approach]]''
A corporation issues a [[Bond (finance)|bond]], an interest earning debt security, to an investor to raise funds.<ref name=Ross/> The bond has a face value, <math> F </math>, coupon rate, <math> r </math>, and maturity date which in turn yields the number of periods until the debt matures and must be repaid. A bondholder will receive coupon payments semiannually (unless otherwise specified) in the amount of <math> Fr </math>, until the bond matures, at which point the bondholder will receive the final coupon payment and the face value of a bond, <math> F(1+r) </math>.

The present value of a bond is the purchase price.<ref name="Broverman"/>
The purchase price can be computed as:
:<math>PV = \left[\sum_{k=1}^{n} Fr(1+i)^{-k}\right]</math> <math> + F(1+i)^{-n} </math>

The purchase price is equal to the bond's face value if the coupon rate is equal to the current interest rate of the market, and in this case, the bond is said to be sold 'at par'. If the coupon rate is less than the market interest rate, the purchase price will be less than the bond's face value, and the bond is said to have been sold 'at a discount', or below par. Finally, if the coupon rate is greater than the market interest rate, the purchase price will be greater than the bond's face value, and the bond is said to have been sold 'at a premium', or above par.<ref name="Ross"/>

=====Technical details=====
Present value is [[Additive inverse|additive]]. The present value of a bundle of [[cash flow]]s is the sum of each one's present value. See [[time value of money]] for further discussion.
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; see [[Inflation-indexed bond]].
* That the likelihood of receiving the payments is high — or, alternatively, that the [[default risk]] is incorporated into the interest rate; see [[Corporate bond#Risk analysis]].
(In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the [[Laplace transform]] of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate. The full Laplace transform is the curve of all present values, plotted as a function of 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 [[Compound interest#Continuous compounding|mathematics of continuous functions]] can be used as an approximation.)

===Variants/approaches===
There are mainly two flavors of Present Value. 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. With Present Value under uncertainty, future dividends are replaced by their conditional expectation.
* '''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]] 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]]. 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.