what are the required elements of a stream of cash flow to be called annuity?
A Primer on the Time Value of Money
The notion that a dollar today is preferable to a dollar some time in the future is intuitive enough for most people to grasp without the use of models and mathematics. The principles of present value provide more backing for this statement, however, and enable united states to calculate exactly how much a dollar some time in the future is worth in today�s dollars and to move cash flows across time. Present value is a concept that is intuitively appealing, simple to compute, and has a wide range of applications. Information technology is useful in decision-making ranging from simple personal decisions (buying a firm, saving for a kid�southward educational activity and estimating income in retirement) to more complex corporate financial decisions (picking projects in which to invest also as the correct financing mix for these projects).
Fourth dimension Lines and Annotation
Dealing with cash flows that are at different points in fourth dimension is made easier using a time line that shows both the timing and the amount of each cash flow in a stream. Thus, a cash catamenia stream of $100 at the cease of each of the next 4 years can be depicted on a time line like the one depicted below.
In the effigy, 0 refers to right at present. A cash catamenia that occurs at time 0 is therefore already in present value terms and does not need to be adjusted for time value. A distinction must exist fabricated here between a menstruation of fourth dimension and a point in time. The portion of the time line betwixt 0 and i refers to catamenia 1, which, in this example, is the first year. The cash flow that occurs at the point in time �one� refers to the greenbacks flow that occurs at the end of menstruum one. The disbelieve rate, which is 10% in this example, is specified for each period on the time line and may be different for each catamenia. Note that in present value terms, a greenbacks menstruum that occurs at the terminate of period 1 is the equivalent of a cash flow that occurs at the starting time of period 2.
Cash flows can be either positive or negative; positive cash flows are called cash inflows and negative cash flows are called cash outflows. For notational purposes, the following abbreviations are used:
| Notation | Stands for |
| PV | Present value |
| FV | Future value |
| Cft | Greenbacks flow at the cease of menstruum t |
| A | Annuity: Constant cash flows over several periods |
| k | Discount Charge per unit (likewise oftentimes abbreviated every bit r or i) |
| g | Expected growth rate |
| n | Number of periods over which greenbacks flows are received or paid |
Intuitive Basis for Present Value
There are three reasons why a cash flow in the future is worth less than a like cash flow today.
- Individuals prefer present consumption to hereafter consumption. People would have to be offered more in the futurity to give up present consumption. If the preference for current consumption is stiff, individuals will accept to exist offered much more than in terms of future consumption to give upward current consumption, a trade-off that is captured by a high �real� rate of render or discount rate. Conversely, when the preference for current consumption is weaker, individuals will settle for less in terms of future consumption and, by extension, a low real rate of return or disbelieve charge per unit.
- When in that location is monetary aggrandizement, the value of currency decreases over time. The greater the aggrandizement, the greater the difference in value between a cash catamenia today and the same cash flow in the hereafter.
- A promised cash menstruum might non exist delivered for a number of reasons: the promisor might default on the payment, the promisee might not be effectually to receive payment, or some other contingency might intervene to forestall the promised payment from beingness made or to reduce information technology. Whatever dubiety or risk associated with the cash flow in the futurity reduces the value of the greenbacks flow today.
The process by which future greenbacks flows are adjusted to reverberate these factors is called discounting, and the magnitude of these factors is reflected in the discount rate. The disbelieve rate incorporates all of the above-mentioned factors. In fact, the discount rate tin be viewed as a composite of the expected existent render (reflecting consumption preferences in the amass over the investing population), the expected inflation charge per unit (to capture the deterioration in the purchasing ability of the cash flow), and the incertitude associated with the cash flow.
The Mechanics of Time Value
Greenbacks flows at different points in time cannot exist compared and aggregated. All greenbacks flows have to exist brought to the aforementioned point in fourth dimension before comparisons and aggregations tin can exist fabricated. The process of discounting hereafter greenbacks flows converts them into greenbacks flows in present value terms. Conversely, the process of compounding converts present cash flows into time to come greenbacks flows.
At that place are five types of cash flows: unproblematic cash flows, annuities, growing annuities, perpetuities and growing perpetuities.
A. Simple Cash Flows
A elementary cash flow is a single greenbacks flow (CF) in a specified futurity time menses t, usually depicted as CFt . This cash menses tin can be discounted dorsum to the present using a discount rate that reflects the uncertainty of the cash menstruation. Concurrently, cash flows in the nowadays can be compounded to arrive at an expected value of the future cash flow.
I. Discounting a Elementary Greenbacks Flow
Discounting a cash catamenia converts information technology into present value dollars and enables the user to do several things. Commencement, in one case greenbacks flows are converted into nowadays value dollars, they tin can be aggregated and compared. Second, if present values are estimated correctly, the user should be indifferent betwixt the hereafter cash flow and the present value of that cash period. The nowadays value of a cash period tin be written as follows:
where CFt equals the greenbacks menses at the end of fourth dimension period t, and where m equals the discount rate.
Other things remaining equal, the present value of a cash flow will decrease as the discount rate increases and proceed to decrease the farther into the future the greenbacks flow occurs.
Illustration: Discounting a Cash Flow
Presume you expect to receive a lump-sum payment of $500,000 ten years from now and that an advisable discount charge per unit for this cash catamenia is 10%. The present value of this cash menses tin and then be estimated as:
Notation the present value is a decreasing function of the discount charge per unit and time; the larger the discount rate or the longer the time period, the smaller the present value.
II. Compounding a Cash Flow
Current cash flows can exist moved to the future past compounding the greenbacks flow at the appropriate discount rate.
where CF0 equals the cash flow now, and where grand equals the discount rate.
This compounding upshot increases with both the discount rate and time; the larger the discount charge per unit or the longer the fourth dimension period, the larger the future value.
III. The Frequency of Discounting and Compounding
The frequency of compounding affects both the time to come and present values of cash flows. In uncomplicated examples, cash flows are often assumed to be discounted and compounded annually, i.eastward., interest payments and income are computed at the cease of each twelvemonth, based on the residuum at the beginning of the yr. In some cases, however, the interest may exist computed more oft, such as on a monthly or a semi-annual basis. In these cases, the present and future values may be very different from those computed on an annual basis; the stated interest rate, on an annual basis, can deviate significantly from the constructive or truthful involvement rate. The constructive involvement rate tin can be computed as follows:
where northward equals the number of compounding periods during the yr (2=semiannual; 12=monthly), and k equals the stated annual involvement rate.
Every bit compounding becomes continuous, the constructive interest rate can exist computed as follows:
where e equals the exponential role and 1000 equals the stated almanac interest rate.
The table below provides the effective rates every bit a function of the compounding frequency for a nominal interest rate of 10 percent.
B. Annuities
An annuity is a constant cash period that occurs at regular intervals for a fixed period of time. An annuity can occur at the cease of each menses or at the beginning of each period.
I. Present Value of an Stop-of-the-Period Annuity
The present value of an annuity can be calculated by taking each cash flow and discounting it dorsum to the nowadays and and so calculation upwardly the present values. Alternatively, a formula can exist used in the calculation. In the case of annuities that occur at the end of each catamenia, this formula tin can be written as
where A equals the annuity (periodic greenbacks flow), k equals the discount rate, and north equals the number of periods (years).
Illustration: Estimating the Present Value of Annuities
Assume that you have a choice of buying a copier for $10,000 cash down or paying $3,000 a twelvemonth for 5 years for the same copier. If the opportunity cost is 12%, which would you rather exercise?
The present value of the installment payments exceeds the cash-downward price; therefore, you would want to pay the $10,000 in cash now.
Alternatively, the nowadays value could have been estimated past discounting each of the cash flows back to the present and aggregating the present values as illustrated below:
Illustration: Present Value of Multiple Annuities
Suppose yous trying to estimate the present value of a firm�due south expected alimony obligations, which amount in nominal terms to the following:
����������� Years�������� Almanac Greenbacks Flow
����������� 1 � 5��������� $200 meg
����������� 6 � 10������� $300 million
����������� eleven � twenty����� $400 meg
If the discount rate is ten%, the present value of these three annuities can be estimated as follows:
Present Value of the offset annuity = $200 1000000 x PV(A,10%,five) = $758 one thousand thousand
Present Value of the second annuity = $300 million x PV(A,10%,5) / 1.105 = $706 million
Present Value of the third annuity = $400 million x PV(A,ten%,10) / 1.xx = $948 million
The present values of the second and 3rd annuities can be estimated in 2 steps. First, the standard present value of the annuity is computed over the menstruation that the annuity is received. Second, that nowadays value is brought back to the nowadays. Thus, for the second annuity, the present value of $300 million each yr for 5 years is computed to be $1,137 one thousand thousand; this present value is actually every bit of the end of the fifth yr so it must be discounted back 5 more years to arrive at today�s nowadays value of $706 meg. Similarly, the value of the third annuity ($ii,458 1000000) is discounted back 10 more years to a nowadays value of $948.
The sum of the cumulated present values = $758 + $706 + $948 = $two,412 million.
2. Amortization Factors - Annuities Given Nowadays Values
In some cases, the present value of the greenbacks flows is known and the annuity needs to be estimated. This is often the case with domicile and car loans, for case, where the borrower receives the loan today and pays it back in equal monthly installments over an extended period of time. This process of finding an annuity when the present value is known is examined below:
Illustration: Calculating The Monthly Payment On A Business firm Loan
Suppose yous are trying to infringe $200,000 to buy a house on a conventional 30-year mortgage with monthly payments. The annual percentage rate on the loan is eight%. The monthly payments on this loan tin can be estimated using the annuity due formula:
Note: Monthly interest charge per unit on the loan = April / 12 = 0.08 / 12 = 0.0067
III. Future Value Of End-Of-The-Period Annuities
In some cases, an private may plan to prepare bated a stock-still annuity each period for a number of periods and volition want to know how much he or she will have at the cease of the period. The future value of an end-of-the-period annuity can exist calculated as follows:
Illustration: Private Retirement Accounts (IRA)
Private retirement accounts (IRAs) allow taxpayers to brand tax-free investments into a designated retirement account. Assume that an individual invests $2,000 at the end of every twelvemonth, starting at age 25, for an expected retirement at age 65, and expects to make viii% a year on the investments. The expected value of the account on the retirement date can be estimated as follows:
Four. Annuity Given Futurity Value
Individuals or businesses who have a fixed obligation or a target to meet (in terms of savings) some time in the hereafter need to know how much they should set aside each menstruum to accomplish this target. If you lot are given the future value and are looking for an annuity, the post-obit equation can be used:
Illustration: Sinking Fund Provision
In whatever airship payment loan, only involvement payments are made during the life of the loan, while the chief is paid at the end of the period. Companies that borrow money using balloon payment loans or conventional bonds (which share the aforementioned features) often set aside money in sinking funds during the life of the loan to ensure that they have enough at maturity to pay the primary on the loan or the face value of the bonds. Thus, a company with debt having a confront value of $100 1000000 coming due in 10 years would demand to prepare bated the following corporeality each year (assuming an interest rate of 8%):
The company would demand to prepare aside $6.9 one thousand thousand at the end of each year to ensure that there are plenty funds ($100 1000000) to retire the debt at maturity.
V. Effect Of Annuities At The Kickoff Of Each Year
The annuities considered thus far accept had end-of-the-period cash flows. Both the present and time to come values volition be affected if the greenbacks flows occur at the commencement of each menses instead of the end. To illustrate this effect, consider an annuity of $100 at the stop of each year for the side by side 4 years, with a discount charge per unit of 10%, every bit assorted with an annuity with payments made at the showtime of each year.
Since the starting time of these annuities occurs right now, and the remaining greenbacks flows accept the form of an end-of-the-catamenia annuity over 3 years, the present value of this annuity tin be written as follows:
In general, the present value of a beginning-of-the-catamenia annuity over n years tin be written as follows:
or as:
Note that this nowadays value will be higher than the present value of an equivalent annuity at the end of each period.
The future value of a outset-of-the-period annuity typically can be estimated by allowing for one additional period of compounding for each cash catamenia:
This future value will be college than the hereafter value of an equivalent annuity at the terminate of each catamenia.
Illustration: IRA - Saving At The First Of Each Menses Instead Of The End
Consider again the example of an private who sets bated $2,000 at the stop of each year for the side by side forty years in an IRA account earning 8%. The future value of these deposits amounted to $518,113 at the terminate of the fortythursday year. If the deposits had been made at the get-go of each year instead of the end, the futurity value would have been college:
Every bit you tin see, the gains from making payments at the get-go of each period can be substantial.
Six. Effect of Greenbacks Flows Occurring Evenly Throughout the Year
Cash flows practice not always occur at either the beginning or at the terminate of each flow. Many of a house�south operating cash flows (e.g., sales, expenses) tend to take identify more or less evenly throughout the year. Ane manner to handle this situation is to calculate present or hereafter values based on the estimated daily cash flows and to use daily compounding, as described above. A simpler (albeit slightly less accurate) method is to assume that the cash flows occur in the middle of each year, rather than the end or the kickoff. In this situation, one may calculate a nowadays or future value as if the cash flows occur at the finish of each period, and and then multiply the result by the square root of (1+m).
Illustration: IRA - Investing Throughout the Yr Instead of the Finish
Consider once more the example of an individual who sets bated $2,000 at the end of each yr for the next 40 years in an IRA account earning 8%. The futurity value of these deposits amounted to $518,113 at the end of the 40thursday year; with payments made at the commencement of each year the time to come value were $559,562. If the deposits had instead been fabricated evenly throughout each year, the future value would take been $538,439 every bit calculated below:
C. Growing Annuities
A growing annuity is a cash flow that grows at a constant rate for a specified period of time. Annotation that to qualify as a growing annuity, the growth charge per unit in each period has to be the aforementioned equally the growth rate in the prior period.
In most cases, the nowadays value of a growing annuity can be estimated by using the following formula:
where yard equals the constant growth charge per unit of the annuity.
Note as well that this formulation works even when the growth charge per unit is greater than the disbelieve rate. The nowadays value of a growing annuity can be estimated in all cases simply 1�where the growth charge per unit is equal to the discount rate. In that example, the nowadays value is equal to the nominal sums of the annuities over the flow, without the growth effect (i.east., n 10 A).
Alternatively, the nowadays value of a growing annuity can be plant using the standard Present Value of an Annuity formula, only using an adjusted discount rate that factors in the growth rate and then that thou would equal: [(1 + k) / (1 + m)] - 1.
Illustration: The Value Of A Gilded Mine
Suppose you lot accept the rights to a gold mine for the adjacent xx years, over which period you plan to extract v,000 ounces of gold every yr. The current price per ounce is $300, but information technology is expected to increment 3% a year. The appropriate discount rate is 10%. The nowadays value of the golden to exist extracted from this mine can be estimated every bit follows:
The standard Present Value of an Annuity adding (with k = (1.10 / 1.03) - 1 = half-dozen.8%) would give the same results (other than small rounding errors).
D. Perpetuities
A perpetuity is a constant greenbacks flow at regular intervals forever. The present value of a perpetuity can be written as:
where A equals the perpetual cash menstruum and yard equals the discount rate. The future value of a perpetuity is infinite.
Illustration: Valuing Preferred Stock
Regular preferred stock has no maturity, and it normally pays a fixed dividend forever. Presume that you have $100 par value preferred stock paying a six% dividend ($6.00 per yr). The value of this stock, if the disbelieve rate is ix%, is as follows:
The value of preferred stock will be equal to its face up (par) value only if the dividend rate is equal to the disbelieve charge per unit.
Eastward. Growing Perpetuities
A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity can be written as:
where CF1 equals the expected cash catamenia next year, 1000 equals the constant growth charge per unit, and thou equals the disbelieve rate.
While a growing perpetuity and a growing annuity share several features, the fact that a growing perpetuity lasts forever puts constraints on the growth rate. It has to be less than the discount rate for this formula to work.
Analogy: Valuing a Stock with Stable Growth in Dividends
BellSouth has just paid annual dividends (CF0) of $2.73 per share. Its earnings and dividends have been growing at 6% a twelvemonth and are expected to grow at the aforementioned charge per unit in the long term. The charge per unit of return required by investors on stocks of equivalent adventure was 12.23%.
Every bit an interesting aside, assume that the stock was really trading at $70 per share. This toll could be justified by using a higher expected growth charge per unit. The growth charge per unit would take to be approximately viii% to justify a price of $70. This growth charge per unit is frequently referred to as an implied growth rate.
F. Combinations and Uneven Cash Flows
In the real world, a number of dissimilar types of cash flows may exist simultaneously, including annuities, simple greenbacks flows, and sometimes perpetuities: Some examples are discussed below.
I. Bail Valuation
A conventional bond pays a fixed coupon every menses for the lifetime of the bond, and the face value of the bond at maturity. Since coupons are fixed and paid at regular intervals, they represent an annuity, while the face value of the bond is a single greenbacks menstruum that has to be discounted separately. The value of a directly bail can so exist written as follows:
Illustration: The Value of a Straight Bond
Say you are trying to value a straight bail with a 15-year maturity and a 10.75% coupon rate. The current involvement rate on bonds of this gamble level is 8.5%. The value of the bond is found as the sum of the nowadays value of an annuity (the coupon payments) and a unmarried amount (the par value).
If the bond paid involvement on a semiannual basis, nosotros would need to adjust the valuation equation accordingly:
II. Common Stock Valuation
Have the case of the stock of a company that expects high growth in the near time to come and lower and more stable growth forever after that. The expected dividends over the high growth period stand for a growing annuity, while the dividends after that satisfy the conditions of a growing perpetuity. The value of the stock tin can thus exist written as the sum of the two present values.
where P0 equals the present value of expected dividends, one thousand equals the boggling growth rate for the first n years (n = loftier growth period), gdue north equals the growth charge per unit forever later on year northward, D0 equals the current dividends per share, Dt equals the dividends per share in year t, and k equals the required rate of render (discount rate).
Illustration: The Value of a High Growth Stock
Assume Eli Lilly had earnings per share of $4.fifty and paid dividends per share of $2.00. Analysts expected both to grow nine.81% a year for the side by side 5 years. Later the fifth year, the growth charge per unit was expected to drib to half dozen% a year forever, while the payout ratio was expected to increase to 67.44%. The required render on Eli Lilly stock is 12.78%.
The cost at the end of the high growth period (Pfive) can exist estimated using the growing perpetuity formula:
The present value of dividends and the concluding price can so be calculated every bit follows:
The value of Eli Lilly stock, based on the expected growth rates and discount charge per unit, is $52.74.
There are some cases where one annuity follows another. In this case, the present value will exist the sum of the present values of the ii (or more) annuities. The present value of these two annuities can be calculated separately and cumulated to arrive at the full present value. The present value of the second annuity has to be discounted back to the present.
Decision
Nowadays value remains one of the simplest and nearly powerful techniques in finance, providing a wide range of applications in both personal and concern decisions. Cash menstruum tin be moved back to nowadays value terms by discounting and moved frontwards by compounding. The discount rate at which the discounting and compounding are washed reflects iii factors: (ane) the preference for current consumption, (two) expected inflation and (three) the uncertainty associated with the greenbacks flows existence discounted.
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This page last updated on Tuesday, January 17, 2006 .
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