Editing Present value (section)

===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.)