# Present value

Present value, also known as present discounted value, is a future amount of money that has been discounted to reflect its current value, as if it existed today. The present value is always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money.[1] Time value can be described with the simplified 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 compared to rent.[2] Just as rent is paid to a landlord by a tenant, without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed their authority over the money, and is compensated for it in the form of interest. The initial amount of the borrowed funds (the present value) is less than the total amount of money paid to the lender.

Present value calculations, and similarly future value calculations, are used to value loans, mortgages, annuities, sinking funds, perpetuities, bonds, and more. These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times.[1] 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.

## Years' Purchase

The traditional method of valuing future income streams as a present capital sum is to multiply the average expected annual cash-flow by a multiple, known as Years' purchase. Thus in selling to a third party a property leased to a tenant under a 99 year lease at a rent of $10,000 per annum, a deal might be struck at "20 years' purchase", which would value the lease at 20 *$10,000, i.e. $200,000. This equates to a present value discounted in perpetuity at 5%. For a riskier investment the purchaser would demand to pay a lower number of years' purchase. This was the method used for example by the English crown in setting re-sale prices for manors seized at the Dissolution of the Monasteries in the early 16th century. The standard usage was 20 years' purchase.[3] ## Background If offered a choice between 100 today or 100 in one year and there is a positive real interest rate throughout the year ceteris paribus, a rational person will choose 100 today. This is described by economists as time preference. Time preference can be measured by auctioning off a risk free security—like a US Treasury bill. If a 100 note, payable in one year, sells for 80 now, then 80 is the present value of the note that will be worth 100 a year from now. This is because money can be put in a bank account or any other (safe) investment that will return interest in the future. An investor who has some money has two options: to spend it right now or to save it. But the financial compensation for saving it (and not spending it) is that the money value will accrue through the compound interest that he will receive from a borrower (the bank account on which he has the money deposited). Therefore, to evaluate the real value of an amount of money today after a given period of time, economic agents compound the amount of money at a given (interest) rate. Most actuarial calculations use the risk-free interest rate which corresponds to the minimum guaranteed rate provided by a bank's saving account for example. To compare the change in purchasing power, the real interest rate (nominal interest rate minus inflation rate) should be used. The operation of evaluating a present value into the future value is 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 a discounting (how much will 100 received in 5 years—at a lottery for example—be worth today?). It follows that if one has to choose between receiving 100 today and 100 in one year, the rational decision is to choose the 100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least 105 in one year so that the two options are equivalent (either receiving 100 today or receiving 105 in one year). This is because if 100 is deposited in a savings account, the value will be 105 after one year. ## Interest Rates Interest is the additional amount of money gained between the beginning and the end of a time period. Interest represents the time value of money, and can be thought of as rent that is required of a borrower in order to use money from a lender.[2][4] For example, when an individual takes out a bank loan, they are charged interest. Alternatively, when an individual deposits money into a bank, their money earns interest. In this case, the bank is 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).[1] 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.[2] 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. Interest is earned on the interest that has already accrued (credited) in addition to the principal (initial amount).[1] The term $(1+i)^{{t}}\,$ is the factor often used to make various calculations concerning compound interest[2] ### 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.[1] 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.[2] ### Effective Annual Rate of Interest Effective annual rate of interest is the percentage increase in an amount of money after a year of accumulating interest. This does not depend on 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 $t$.[2] ### 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%[1] The formula to convert between effective annual interest rate and nominal annual interest rate is $(1+i)=\left(1+{\frac {i^{{m}}}{m}}\right)^{{m}}\,$ 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[2] ### 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[2] ### 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 constant force of interest, $\delta$, is $(1+i)=e^{{\delta }}$ [2] ### 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.[1] The real return refers to the surplus associated with an interest rate compared to an investment rate that matches the inflation rate. $i_{{\text{real}}}={\frac {(i-r)}{1+r}}\,\!$ Where $r$ is the inflation rate.[2] ## Calculation 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?).[4] 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: $PV={\frac {C}{(1+i)^{n}}}\,$ 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 (the end of 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 [2] This is also found from the formula for the future value with negative time. 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.[1]

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

### 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.[4] 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.[1]

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 $PV_{{1}}={\frac {\100}{(1.05)^{{1}}}}=\95.24\,$ $PV_{{2}}={\frac {-\50}{(1.05)^{{2}}}}=-\45.35\,$ $PV_{{3}}={\frac {\35}{(1.05)^{{3}}}}=\30.23\,$ respectively Thus the net present value would be $NPV=PV_{{1}}+PV_{{2}}+PV_{{3}}={\frac {100}{(1.05)^{{1}}}}+{\frac {-50}{(1.05)^{{2}}}}+{\frac {35}{(1.05)^{{3}}}}=95.24-45.35+30.23=80.12,$ 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.[2] 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:
$NPV=100\,(1.05)^{{-1}}+200\,(1.10)^{{-1}}\,(1.05)^{{-1}}={\frac {100}{(1.05)^{{1}}}}+{\frac {200}{(1.10)^{{1}}(1.05)^{{1}}}}=\95.24+\173.16=\268.40$
• 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, $\,i\,$, to nominal annual interest rate compounded quarterly:
$(1+i)=\left(1+{\frac {i^{{4}}}{4}}\right)^{4}$[2]

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; payments 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 $\,n-1\,$.[4] 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 $(1+i)$:

$PV_{{\text{annuity due}}}=PV_{{\text{annuity immediate}}}(1+i)\,\!$[2]

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 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.

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

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.

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 $(1+i)$:

$PV_{{\text{perpetuity due}}}=PV_{{\text{perpetuity immediate}}}(1+i)\,\!$[2]

#### PV of a Bond

A corporation issues a bond, an interest earning debt security, to an investor to raise funds.[4] The bond has a face value, $F$, coupon rate, $r$, 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 $Fr$, 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.[2] 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.[4] The purchase price can be computed as:

$PV=\left[\sum _{{k=1}}^{{n}}Fr(1+i)^{{-k}}\right]$ $+F(1+i)^{{-n}}$

#### Technical details

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 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 mathematics of continuous functions can be used as an approximation.

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.

### Variants/Approaches

There are mainly two flavors of PV. 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.

• 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.

## 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.[1] 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.[2]