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Present value, also known as present discount value, is the current value of a future sum of money. Money has interest-earning potential, and interest is referred to as the time value of money. This concept of time value can be described with the phrase, “A dollar today is worth more than a dollar tomorrow”. Here, 'worth more' means that its value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be thought of as rent. Just as rent is paid to a landlord, without the ownership of the asset being transferred, interest is paid by a borrower of money to the lender for the ability to use the money for a time before paying it back. By letting the borrower have authority over the money, the lender has sacrificed the interest-earning potential of the money, and must be compensated for it, in the form of interest.

Present value calculations, and similarly future value calculations, are used to evaluate loans, mortgages, annuities, sinking funds, perpetuities, and more. These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times. The idea is much like algebra, where variable units must be consistent in order to compare or carry out addition and subtraction; time dates must be consistent in order to make comparisons between values or carry out simple calculations. When deciding between projects in which to invest, the choice can be made by comparing respective present values discounted at the same interest rate, or rate of return. The project with the least present value, i.e. that costs the least today, should be chosen.

=Interest Rates= Interest is the added value of a sum of money between the beginning and the end of a period of time. Interest is referred to as the time value of money, and can be thought of as rent that is required of the borrower in order to use money from a lender. For example, when an individual takes out a loan, they are charged interest. Alternatively, when an individual deposits money into a bank, their money earns interest. In this case, the bank is viewed as the borrower of the funds and is responsible for crediting interest to the account holder. Similarly, when an individual invests in a company (through corporate bonds, or through stock), the company is borrowing funds, and must pay interest to the individual (in the form of coupon payments, dividends, or stock price appreciation). The interest rate is the change, expressed as a percentage, in the amount of money during one compounding period. A compounding period is the length of time that must transpire before interest is credited, or added to the total. For example, interest that is compounded annually is credited once a year, and the compounding period is one year. Interest that is compounded quarterly is credited 4 times a year, and the compounding period is three months. A compounding period can be any length of time, but some common periods are annually, semiannually, quarterly, monthly, daily, and even continuously.

There are several types and terms associated with interest rates.

Compound Interest
Compound interest is multiplicative. This means that interest is earned on the interest that has already accrued (credited to the sum) in addition to the principal (initial) value.

The term $$ (1+i)^{t} \,$$ is the factor often used to make various calculations concerning compound interest

Simple Interest
Simple interest is additive. Simple interest is earned only on the principal amount, and there is no interest earned on interest already accrued.

The term $$ (1+it) \,$$ is the factor used to make various calculations when simple interest is applied. Simple interest is often used when calculating interest earned during a fraction of the year. Simple interest earns a higher return during the first year, and compound interest earns a higher return after the first year. At the end of the first year, simple interest and compound interest earn the same return.

Effective Annual Rate of Interest
Effective annual rate of interest is the percentage increase in an amount after a year of accumulating interest, regardless of what happens during the year. It can be calculated by
 * $$ \frac{A(1)-A(0)}{A(0)} \,$$

Where $$A(t)$$ represents the amount of money at time $$t$$.

Nominal Annual Rate of Interest
Nominal annual rate of interest is NOT the same as effective annual rate of interest. Instead, nominal annual interest is compounded more than once a year, say, $$m$$ times. For instance, if the nominal annual interest rate is 8% and interest is compounded quarterly (4 times a year, or every three months), then the interest rate for ¼ of the year is (8/4)%=2%. Because it is compounded more often, the effective annual interest rate is actually more than 8%

The formula to convert between effective annual interest rate and nominal annual interest rate is
 * $$(1+i)=(1+\frac{i^{m}}{m})^{m} \,$$

Where $$i$$ is the effective annual interest rate and $$i^{m}$$ is the nominal annual interest rate, and $$m$$ is the number of times interest is compounded in a year

Rate of Discount
The rate of discount, $$d$$, refers to interest that is payable in advance, before funds have been transferred.
 * $$d = \frac{(A(1)-A(0)}{A(1)} \,$$

There also exists a nominal annual discount rate, $$d^{m}$$, which is analogous to nominal interest rate

The formula to convert between effective annual interest rate, $$i$$ and effective annual discount rate $$d$$ is
 * $$d = \frac{i}{(1+i)} \,$$

Where $$i$$ is the effective annual interest rate, and $$d$$ is the effective annual discount rate

Continuous Compounding
Continuous compounding is nominal interest that is compounded infinitely throughout the year. It is the instantaneous growth rate in the principal amount, i.e. the interest rate can change as a function of time. A force of interest, which describes the accumulated amount of money as a function of time, characterizes continuous compounding:
 * $$\delta_{t}=\frac{A'(t)}{A(t)}\,$$

Where $$\delta_{t}$$ is the force of interest, and $$A(t)$$ is the amount of money as a function of time, $$t$$, and its derivative.

The formula to convert between effective annual interest rate, $$i$$ and the force of interest $$\delta$$ is
 * $$(1+i) = e^{\delta}$$

Real Rate of Interest
Real rate of interest refers to the interest rate that has been adjusted to account for inflation; it is the percent change in buying power due to inflation. The real return refers to the surplus associated with an interest rate compared to an investment rate that matches the inflation rate.
 * $$ i_\text{real} = \frac{(1-r)}{1+r} \,\!$$

Where $$r$$ is the inflation rate.

=Calculations= The operation of evaluating a present sum of money some time in the future called a capitalization (how much will 100 today be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of money—is called discounting (how much will 100 received in 5 years be worth today?).

Present Value of a Lump Sum
The most commonly applied model of present valuation uses compound interest. The standard formula is:


 * $$PV = \frac{C}{(1+i)^n} \,$$

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

Often, $$v^{n} = \,(1 + i)^{-n}$$ is referred to as the Present Value Factor

For example if you are to receive $1000 in 5 years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is


 * $$PV = \frac{$1000}{(1+0.10)^{5}} = $620.92 \, $$

The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $1000 in 5 years, or $620.92 today.

Net Present Value of a stream of Cash Flows
A cash flow is a sum of money that is either paid out or received, differentiated by a positive or negative sign, at the end of a period. Conventionally, cash flows that are received are denoted with a positive sign (money coming in) and cash flows that are paid out are denoted with a negative sign (money is leaving). The cash flow for a period represents the net change in money of that period. Calculating the net present value, $$\,NPV\,$$, 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.

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


 * $$PV = \frac{$100}{(1.05)^{1}} = $95.24 \, $$
 * $$PV = \frac{-$50}{(1.05)^{2}} = -$45.35 \, $$
 * $$PV = \frac{$35}{(1.05)^{3}} = $30.23 \, $$   respectively

Thus the net present value would be


 * $$NPV = \frac{100}{(1.05)^{1}} + \frac{-50}{(1.05)^{2}} + \frac{35}{(1.05)^{3}} = 95.24 - 45.35 + 30.23 = 80.12, $$

There are a few considerations to be made.
 * The periods might not be consecutive. If this is 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 . 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:


 * $$NPV = 100\,(1.05)^{-1} + 200\,(1.10)^{-1}\,(1.05)^{-1} = \frac{100}{(1.05)^{1}} + \frac{200}{(1.10)^{1}(1.05)^{1}} = $95.24 + $173.16 = $268.40 $$


 * It is necessary to modify the interest rate to coincide with the payment period, or vice versa. 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, $$\, i \, $$, to nominal annual interest rate compounded quarterly:
 * $$ (1+i) = (1+\frac{i^{4}}{4})^4 $$

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

Present Value of an Annuity
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, 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.

There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, $$\, n \, $$ payments are received (or paid) at the end of each period, at times 1 through $$\, n \, $$, while for an annuity due, $$\, n \, $$ payments are received (or paid) at the beginning of each period, at times 0 through $$\, n-1 \, $$. This subtle difference must be accounted for when calculating the present values.

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


 * $$ PV_\text{annuity due} = PV_\text{annuity immediate}(1+i) \,\!$$

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


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

where:


 * $$\, n \, $$ = number of periods,


 * $$\, C \, $$ = amount of cash flows,


 * $$\, i \, $$ = effective periodic interest rate or rate of return.

Present Value of a Perpetuity
A perpetuity is a periodic payment 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.


 * $$PV\,=\,\frac{C}{i}. \qquad (2)$$

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


 * $$PV = \sum_{k=1}^\infty C(1+i)^{-k} = C \sum_{k=1}^\infty \frac{1}{(1+i)^{k}} = \frac{C}{i}, \qquad i > 0,$$

which form a geometric series.

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.

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 $$(1+i) $$:


 * $$ PV_\text{perpetuity due} = PV_\text{perpetuity immediate}(1+i) \,\!$$

PV of a Bond
A bond is an interest earning debt security issued by a corporation to an investor (or lender). The bond has a face value, $$ F $$, coupon rate, $$ r $$, and maturity date which denotes 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 $$ Fr $$, until the bond matures, at which point the bondholder will receive the final coupon payment and the face value of a bond, $$ F(1+r) $$. The present value of a bond is the purchase price. The purchase price is equal to the bond's face value if the coupon rate is equal to the market interest rate, 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. The purchase price can be computed as:


 * $$PV = [\sum_{k=1}^{n} Fr(1+i)^{k}]$$ $$ + F $$

The present value of a bond can also be computed after coupon payments have already begun. This happens if a bondholder wishes to sell the bond after having received a number of coupons, but the bond still hasn't matured. In this case, let $$p$$ denote the number of periods that have passed (or the number of coupon payments that have been received). Then the PV of the bond will be


 * $$PV = [\sum_{k=p+1}^{n} Fr(1+i)^{k}]$$   $$ + F $$

=Present Value Method of Valuation= An investor, the lender of money, must decide the financial project in which to invest their money, and present value offers one method of deciding. A financial project requires an initial outlay of money, such as the price of stock or the price of a corporate bond. The project claims to return the initial outlay, as well as some surplus (for example, interest, or future cash flows). An investor can decide which project to invest in by calculating each projects’ present value (using the same interest rate for each calculation) and then comparing them; the project with the smallest present value – the least initial outlay – will be chosen because it offers the same return as the other projects for the least amount of money.

Annuity Valuation

 * Time Value of Money can be summarized by the phrase "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that it's value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, and will then accumulate to more than a dollar by tomorrow
 * Present Value factor is v^n = (1+i)^(-n) where i is the interest rate for one compounding period (a compounding period is when interest is applied, for example, annually, semianually, quarterly, monthly, daily), and n is the number of compounding periods
 * Present Value is the value amount associated with a future amount of money, that has been discounted to the present with the appropriate present value factor
 * Future Value factor is (1+i)n where i is the interest rate for one compounding period and n is the number of compounding periods
 * Future Value is the value amount associated with a present amount of money, that has been