Present value

src: i.ytimg.com

In economics and finance, the current value ( PV ), also known as presenting a discount value , is the value of the expected revenue stream specified on assessment. The current value is always less than or equal to the future value because money has interest earning potential, a characteristic that is referred to as the time value of money, except during a negative interest rate, when the present value will be more than the future. value. The time value can be explained by a simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'more value' means greater value. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and generates one-day interest, making the total accumulation to be worth more than a dollar tomorrow. Interest can be compared with rent. Just as the rent is paid to the landowner by the lessee, without the ownership of the transferred asset, the interest is paid to the lender by the borrower who gains access to the money for a certain period of time before paying back. By letting the borrower have access to money, the lender has sacrificed this exchange rate, and is compensated in the form of interest. The initial amount of loan funds (present value) is less than the total amount of money paid to the lender.

The current value calculation, as well as the calculation of future value, is used to assess loans, mortgages, annuities, sinking, lasting, bond, and more. This calculation is used to make comparisons between cash flows that do not occur at the same time, because the time date should be consistent to make comparisons between values. When deciding between projects to invest, a choice can be made by comparing each of the present value of the projects by discounting the expected income stream at the appropriate project interest rate, or rate of return. The project with the highest current value, the most valuable at the moment, should be selected.

Video Present value

Purchase year

The traditional method of assessing future income flows as the current amount of capital is to double the average expected annual cash flow with some, known as "year purchases". For example, in selling to a third-party property leased to a lessee under a 99-year lease with a $ 10,000-a-year lease, the deal may be triggered on a "20-year purchase", which will reward the rent of 20 * $ 10,000, which is $ 200,000. This is equivalent to the current discounted continuous value of 5%. For more risky investments, buyers will demand to pay for purchases in lower number of years. This is the method used for example by the British crown in setting the resale price for manors confiscated in the Dissolution of Monasteries in the early 16th century. The standard usage is the purchase of 20 years.

Maps Present value

​​â € <â €

If a choice is offered between $ 100 today or $ 100 in one year, and there is a positive real interest rate throughout the year, , a rational person will choose $ 100 today. This is described by economists as a time preference. Timing preferences can be measured by risk-free auctioning - such as US Treasury bills. If the $ 100 note with zero coupon, paid in one year, sells for $ 80 now, then $ 80 is the present value of the note that will be worth $ 100 per year from now. This is because money can be put into a bank account or other (secure) investment that will return interest in the future.

An investor who has some money has two options: to spend it now or to save it. But the financial compensation to keep it (and not spend it) is that the value of money will increase through the compound interest that it will receive from a borrower (the bank account in which he deposited the money).

Therefore, to evaluate the real value of a certain amount of money today after a period of time, the economic agent increases the amount of money at a certain interest rate. Most actuarial calculations use the risk-free rate in accordance with the minimum guarantee level provided by the bank savings account, for example, assuming no risk of default by the bank to return the money to the account holder on time. To compare changes in purchasing power, the real interest rate (the nominal interest rate minus the inflation rate) should be used.

The operation evaluating the present value to the future value is called capitalization (how much $ 100 a day will be worth in 5 years?). The reverse operation - evaluating the present value of the amount of money in the future - is called a discount (how much $ 100 received in 5 years - on the lottery for example - is worth today?).

Therefore if one has to choose between receiving $ 100 today and $ 100 in a year, the rational decision is to choose $ 100 today. If the money is to be received within one year and assuming the savings account's interest rate is 5%, the person should be offered at least $ 105 in one year so the two options are equivalent (either receiving $ 100 today or receiving $ 105 in a year). This is because if $ 100 is stored in a savings account, its value will be $ 105 after one year, again assuming no risk of losing the initial amount through the default bank.

src: open.lib.umn.edu

Interest rate

Interest is the amount of extra money earned between the beginning and the end of the time period. Interest represents the time value of money, and can be considered as a necessary lease from the borrower to use the money from the lender. For example, when someone takes out a bank loan, they are charged interest. Or, when someone deposits money into the bank, their money earns interest. In this case, the bank is the borrower of the funds and is responsible for crediting the interest to the account holder. Similarly, when a person invests in a company (through corporate bonds, or through shares), the company borrows funds, and must pay interest to the individual (in the form of coupon payments, dividends, or stock price appreciation). The interest rate is a change, expressed as a percentage, in the amount of money over a period of compounding. The compounding period is the length of time that must occur before the interest is credited, or added to the total. For example, annually aggravated interest is credited once a year, and the compound period is one year. The quarterly compounded interest is credited four times a year, and the merger period is three months. The period of compounding can be over time, but some common periods are annually, semi-annual, quarterly, monthly, daily, and even continuously.

There are several types and terms related to interest rates:

• Compound interest, interest that increases exponentially over the next period,
• Simple flowers, extra interest that does not increase
• Effective, effective equivalent rate compared to several compound interest periods
• The nominal annual interest, the simple annual interest rate of several interest periods
• The discount rate, the interest rate is reversed when doing the calculation in reverse
• Compound interest constantly, the limit of interest rate mathematics with zero period of time.
• The real interest rate, which takes into account inflation.

src: ezto-cf-media.mheducation.com

Calculation

The operation of evaluating the current amount of money in the future is called capitalization (how much will the value of 100 days be worth in 5 years?). The reverse operation - evaluating the present value of the amount of money in the future - is called a discount (how much will 100 be received in 5 years now?).

Spreadsheets generally offer a function to calculate the present value. In Microsoft Excel, there is a function of the present value for a single payment - "= NPV (...)", and the same set of payments, periodically - "= PV (...)". The program will calculate the present value flexibly for any cash flows and interest rates, or for different interest rate schedules at different times.

The current value of lump sum

The most common appraisal model is now used using compound interest. The standard formula is:

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

Where ${\ displaystyle \, C \,}$ is the amount of money in the future to be discounted, ${\ displaystyle \, n \,}$ is the number of combined periods between the current date and the date at which the amount is ${\ displaystyle \, C \,}$ , ${\ displaystyle \, i \,}$ is the interest rate for one compound period (the end of the merger period is when the interest is applied, for example, every year, semiannual, quarterly, monthly, daily). Interest rate, ${\ displaystyle \, i \,}$ , given as a percentage, but expressed as decimal in this formula.

Often, ${\ displaystyle v ^ {n} = \, (1 i) ^ {- n}}$ is called the Current Rate Factor

This is also found from the formula for future value with negative time.

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

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

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

The purchasing power in today's money amounts to ${\ displaystyle \, C \,}$ money, ${\ displaystyle \, n \,}$ ${\ displaystyle \, i \,}$ is the future inflation rate assumed.

Net present value of the cash flow stream

Cash flow is the amount of money paid or received, distinguished by a negative or positive sign, at the end of a period. Conventionally, the cash flow received is denoted by a positive sign (total cash increases) and the cash flow paid is marked with a negative sign (total cash has decreased). The cash flows for a period represent a net change in the money for that period. Calculating the current net value, ${\ displaystyle \, NPV \,}$ , from a cash flow stream consisting of discounting each cash flow to date, using present value factors and the number of appropriate merge periods, and combining these values.

For example, if the cash flow flow consists of $ 100 at the end of the first period, - $ 50 at the end of the second period, and $ 35 at the end of the three period, and the interest rate per period of merger is 5% (0.05) then the present value of the third The Cash Flow is

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â $ Â Â Â Â Â Â Â Â Â Â Â Â 100 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â $ Â Â Â Â Â Â Â 95.24 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle PV_ {1} = {\ frac {\ $ 100} {(1.05) ^ {1}}} = \ $ 95.24 \,} Â Â}

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂ ÂÂÂÂ Â Â Â Â Â Â Â Â Â Â Â Â Â $ Â Â Â ÂÂÂÂÂÂ ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <50% Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â - Â Â Â Â Â Â Â $ Â Â Â Â Â Â Â 45,35 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle PV_ {2} = {\ frac {- \ $ 50} {(1.05) ^ {2}}} = - \ $ 45.35 \, } Â Â}

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 3 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â $ Â Â ÂÂÂÂÂÂÂ ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> <Â> Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 3 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â $ Â Â Â Â Â Â Â 30.23 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle PV_ {3} = {\ frac {\ $ 35} {(1.05) ^ {3}}} = \ $ 30.23 \,} Â Â} each

Thus the net present value will be

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂNÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â\; ÂVÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â P Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â P Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 3 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 100 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂ ÂÂÂÂ Â Â Â ÂÂÂÂÂÂ ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <50% Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ( Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ <Â> Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... Â Â Â Â Â Â Â Â Â Â Â Â Â ) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 3 Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â 95.24 Â Â Â Â Â Â Â Â - Â Â Â Â Â Â Â 45,35 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 30.23 Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â 80.12 Â Â Â Â Â Â Â Â , Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle 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,} Â Â}

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

The current value of the annuity

Many financial arrangements (including bonds, other loans, leases, salaries, membership fees, annual allowances including direct annuity and annuity straight-line depreciation) establish a structured payment schedule; payment of the same amount at regular time intervals. Such arrangements are called annuities. The expression for the present value of the payout is a summary of the geometry series.

There are two types of annuities: immediate annuities and maturity annuities. For immediate annuities, ${\ displaystyle \, n \,}$ payment received (or paid) at the end of each period, at time 1 to ${\ displaystyle \, n \,}$ , whereas for the maturity annuity, ${\ displaystyle \, n \,}$ payment received (or paid) at the beginning of each period, at time 0 to ${\ displaystyle \, n-1 \,}$ . This subtle difference must be taken into account when calculating the present value.

The maturity annuity is an immediate annuity with one more period of interest. Thus, the two current values ​​are different from the factor ${\ displaystyle (1 i)}$ :

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{         V                        annuity because                          =          P                 V                        immediate annuity                         (         1                   me        )                           Annotation encoding = "application/x-tex"> {\ displaystyle PV _ {annuity due}} = PV _ {\ text {annuity immediate}} (1 i) \, \!}  Â}

The present value of the immediate annuity is the value at the time of 0 cash flow flow:

${\ displaystyle PV = {_ = C \ left [{\ frac {1- (1 i) ^ {- n}} {i}} \ right], \ qquad (1)}$

Where:

${\ displaystyle \, n \,}$ = number of periods,

${\ displaystyle \, C \,}$ = total cash flow,

${\ displaystyle \, i \,}$ = effective interest rate or periodic rate of return.

Approach for annuity and loan calculation

The above formula (1) for the immediate calculation of annuities offers little insight for the average user and requires the use of some form of computing machine. There is a less intimidating approach, easier to calculate and offer some insight for non-specialists. It was given by

Where, as above, C is the annuity payment, PV is the principal, n is the amount of payment, beginning at the end of the first period, and i is the interest rate per period. Equivalent C is the repayment of a periodic loan for a PV loan extended during n periods at the rate of interest, i. This formula is valid (for positive n, i) for ni <= 3. For completeness, for ni> = 3 the approximation is ${\ displaystyle C \ approx PVi}$ .

The formula can, in some circumstances, reduce the calculation to one of mental arithmetic alone. For example, what is the loan payment (forecast) for PV loan = $ 10,000 paid annually for n = 10 years at 15% interest (i = 0.15)? The approximate formula is C? 10,000 * (1/10 (2/3) 0,15) = 10,000 * (0,1 0,1) = 10,000 * 0,2 = $ 2000 per year by mental arithmetic only. The correct answer is $ 1993, very close.

The overall approach is accurate in  ± 6% (for all n> = 1) for the interest rate 0 <= i <= 0.20 and in Ã,  ± 10% for the interest rate 0.20 <= i <= 0.40. However, it is intended only for "rough" calculations.

Current rating of length

A continuity refers to periodic payments, unlimited receivables, although only a handful of such instruments. The present value of immortality can be calculated by taking the limit of the above formula as n closer to infinity.

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

Formula (2) can also be found by subtracting from (1) the present value of a period n pending perpetually, or directly by summing up the present value of the payment

${\ displaystyle PV = \ sum _ {k = 1} ^ {\ infty} {\ frac {C} {(1 i) ^ {k}} } = {\ frac {C} {i}}, \ qquad i & gt; 0,}$

which form a geometry series.

Again there is a difference between the length of time - when the payment is received at the end of the period - and the duration due to the payment received at the beginning of a period. And as with the annuity calculation, continuity and sustainability are immediately different from the factor ${\ displaystyle (1 i)}$ :

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{         V                       immutability because                          =          P                 V                       perpetuity immediate                         (         1                   me        )                           {\ Displaystyle PV _ {\ text {perpetuity due}} = PV _ {\ text {perpetuity immediate}} (1 i) \, \!}  Â}

PV from a bond

Corporations issue bonds, interest earning debt, to investors to raise funds. The bond has a nominal value, ${\ displaystyle F}$ , coupon rate, ${\ displaystyle r}$ , and the due date which in turn generates the number of periods until the debt matures and must be repaid. The bondholder will receive a coupon payment every semester (unless otherwise specified) in the amount of ${\ displaystyle Fr}$ , until the bond matures, at which point the bondholder will receive the last coupon payment and the bond's nominal value, ${\ displaystyle F (1 r)}$ . The present value of the bond is the purchase price. The purchase price is equal to the nominal value of the bond if the coupon rate is equal to the current market interest rate, and in this case, the bond is said to be sold parallel. If the coupon rate is less than the market interest rate, the purchase price will be less than the nominal value of the bond, 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 rate, the purchase price will be greater than the nominal value of the bond, and the bond is said to have been sold 'at a premium', or above par. The purchase price can be calculated as:

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

Technical details

The present value is additional. The present value of a bundle of cash flows is the sum of each of the current values.

In fact, the present value of cash flows at a constant mathematical interest rate is a point in the Laplace transform of the cash flows, evaluated by the transformation variable (usually denoted "s") equal to the interest rate. The full Laplace transform is the curve of all current values, plotted as a function of the interest rate. For discrete time, where payments are separated by large periods of time, the transformation is reduced to sum, but when payments are ongoing almost continuously, the mathematical continuous function can be used as an estimate.

This calculation should be applied with caution, because there is an underlying assumption:

• That it is not necessary to take into account price inflation, or alternatively, that inflation costs are included in the interest rate.
• That the likelihood of receiving a high payment - or, alternatively, the default risk is included in the interest rate.

See the time value of money for further discussions.

Variants/Approach

There are mainly two sense values ​​now. Whenever there is uncertainty in time and amount of cash flow, the current expected value approach will often be the right technique.

• Traditional Present Value Approach - in this approach a set of estimated cash flows and a single interest rate (equivalent to risk, usually the weighted average cost component) will be used to estimate fair value.
• The Expected Present Value Approach - in this approach several cash flow scenarios with different/expected probabilities and the risk-free rate adjusted for credit are used to estimate fair value.

Interest rate selection

The interest rate used is a risk-free interest rate if there are no risks involved in the project. The returns from th

Source of the article : Wikipedia