Is a series of regular payments received made at the end of each period for a fixed period of time?

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 us 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. It is useful in decision-making ranging from simple personal decisions (buying a house, saving for a child�s education and estimating income in retirement) to more complex corporate financial decisions (picking projects in which to invest as well as the right financing mix for these projects).

Time Lines and Notation

Dealing with cash flows that are at different points in time 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 flow stream of $100 at the end of each of the next 4 years can be depicted on a time line like the one depicted below.

In the figure, 0 refers to right now. A cash flow 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 be made here between a period of time and a point in time. The portion of the time line between 0 and 1 refers to period 1, which, in this example, is the first year. The cash flow that occurs at the point in time �1� refers to the cash flow that occurs at the end of period 1. The discount rate, which is 10% in this example, is specified for each period on the time line and may be different for each period. Note that in present value terms, a cash flow that occurs at the end of period 1 is the equivalent of a cash flow that occurs at the beginning 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:

Intuitive Basis for Present Value

There are three reasons why a cash flow in the future is worth less than a similar cash flow today.

1. Individuals prefer present consumption to future consumption. People would have to be offered more in the future to give up present consumption. If the preference for current consumption is strong, individuals will have to be offered much more in terms of future consumption to give up current consumption, a trade-off that is captured by a high �real� rate of return 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 discount rate.
2. When there is monetary inflation, the value of currency decreases over time. The greater the inflation, the greater the difference in value between a cash flow today and the same cash flow in the future.
3. A promised cash flow might not be delivered for a number of reasons: the promisor might default on the payment, the promisee might not be around to receive payment, or some other contingency might intervene to prevent the promised payment from being made or to reduce it. Any uncertainty or risk associated with the cash flow in the future reduces the value of the cash flow today.

The process by which future cash flows are adjusted to reflect these factors is called discounting, and the magnitude of these factors is reflected in the discount rate. The discount rate incorporates all of the above-mentioned factors. In fact, the discount rate can be viewed as a composite of the expected real return (reflecting consumption preferences in the aggregate over the investing population), the expected inflation rate (to capture the deterioration in the purchasing power of the cash flow), and the uncertainty associated with the cash flow.

The Mechanics of Time Value

Cash flows at different points in time cannot be compared and aggregated. All cash flows have to be brought to the same point in time before comparisons and aggregations can be made. The process of discounting future cash flows converts them into cash flows in present value terms. Conversely, the process of compounding converts present cash flows into future cash flows.

There are five types of cash flows: simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.

A. Simple Cash Flows

A simple cash flow is a single cash flow (CF) in a specified future time period t, usually depicted as CFt. This cash flow can be discounted back to the present using a discount rate that reflects the uncertainty of the cash flow. Concurrently, cash flows in the present can be compounded to arrive at an expected value of the future cash flow.

I. Discounting a Simple Cash Flow

Discounting a cash flow converts it into present value dollars and enables the user to do several things. First, once cash flows are converted into present value dollars, they can be aggregated and compared. Second, if present values are estimated correctly, the user should be indifferent between the future cash flow and the present value of that cash flow. The present value of a cash flow can be written as follows:

where CFt equals the cash flow at the end of time period t, and where k equals the discount rate.

Other things remaining equal, the present value of a cash flow will decrease as the discount rate increases and continue to decrease the further into the future the cash flow occurs.

Illustration: Discounting a Cash Flow

Assume you expect to receive a lump-sum payment of $500,000 ten years from now and that an appropriate discount rate for this cash flow is 10%. The present value of this cash flow can then be estimated as:

Note the present value is a decreasing function of the discount rate 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 be moved to the future by compounding the cash flow at the appropriate discount rate.

where CF0 equals the cash flow now, and where k equals the discount rate.

This compounding effect increases with both the discount rate and time; the larger the discount rate or the longer the time period, the larger the future value.

III. The Frequency of Discounting and Compounding

The frequency of compounding affects both the future and present values of cash flows. In simple examples, cash flows are often assumed to be discounted and compounded annually, i.e., interest payments and income are computed at the end of each year, based on the balance at the beginning of the year. In some cases, however, the interest may be computed more frequently, 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 effective or true interest rate. The effective interest rate can be computed as follows:

where n equals the number of compounding periods during the year (2=semiannual; 12=monthly), and k equals the stated annual interest rate.

As compounding becomes continuous, the effective interest rate can be computed as follows:

where e equals the exponential function and k equals the stated annual interest rate.

The table below provides the effective rates as a function of the compounding frequency for a nominal interest rate of 10 percent.

B. Annuities

An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. An annuity can occur at the end of each period or at the beginning of each period.

I. Present Value of an End-of-the-Period Annuity

The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present and then adding up the present values. Alternatively, a formula can be used in the calculation. In the case of annuities that occur at the end of each period, this formula can be written as

where A equals the annuity (periodic cash flow), k equals the discount rate, and n 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 year for 5 years for the same copier. If the opportunity cost is 12%, which would you rather do?

The present value of the installment payments exceeds the cash-down price; therefore, you would want to pay the $10,000 in cash now.

Alternatively, the present value could have been estimated by 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 you trying to estimate the present value of a firm�s expected pension obligations, which amount in nominal terms to the following:

����������� Years�������� Annual Cash Flow
����������� 1 � 5��������� $200 million
����������� 6 � 10������� $300 million
����������� 11 � 20����� $400 million

If the discount rate is 10%, the present value of these three annuities can be estimated as follows:

Present Value of the first annuity = $200 million x PV(A,10%,5) = $758 million
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,10%,10) / 1.1010 = $948 million

The present values of the second and third annuities can be estimated in two steps. First, the standard present value of the annuity is computed over the period that the annuity is received. Second, that present value is brought back to the present. Thus, for the second annuity, the present value of $300 million each year for 5 years is computed to be $1,137 million; this present value is really as of the end of the fifth year so it must be discounted back 5 more years to arrive at today�s present value of $706 million. Similarly, the value of the third annuity ($2,458 million) is discounted back 10 more years to a present value of $948.

The sum of the cumulated present values = $758 + $706 + $948 = $2,412 million.

II. Amortization Factors - Annuities Given Present Values

In some cases, the present value of the cash flows is known and the annuity needs to be estimated. This is often the case with home and automobile loans, for example, 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 House Loan

Suppose you are trying to borrow $200,000 to buy a house on a conventional 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%. The monthly payments on this loan can be estimated using the annuity due formula:

Note: Monthly interest rate on the loan = APR / 12 = 0.08 / 12 = 0.0067

III. Future Value Of End-Of-The-Period Annuities

In some cases, an individual may plan to set aside a fixed annuity each period for a number of periods and will want to know how much he or she will have at the end of the period. The future value of an end-of-the-period annuity can be calculated as follows:

Illustration: Individual Retirement Accounts (IRA)

Individual retirement accounts (IRAs) allow taxpayers to make tax-free investments into a designated retirement account. Assume that an individual invests $2,000 at the end of every year, starting at age 25, for an expected retirement at age 65, and expects to make 8% a year on the investments. The expected value of the account on the retirement date can be estimated as follows:

IV. Annuity Given Future Value

Individuals or businesses who have a fixed obligation or a target to meet (in terms of savings) some time in the future need to know how much they should set aside each period to reach this target. If you are given the future value and are looking for an annuity, the following equation can be used:

Illustration: Sinking Fund Provision

In any balloon payment loan, only interest payments are made during the life of the loan, while the principal is paid at the end of the period. Companies that borrow money using balloon payment loans or conventional bonds (which share the same 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 principal on the loan or the face value of the bonds. Thus, a company with debt having a face value of $100 million coming due in 10 years would need to set aside the following amount each year (assuming an interest rate of 8%):

The company would need to set aside $6.9 million at the end of each year to ensure that there are enough funds ($100 million) to retire the debt at maturity.

V. Effect Of Annuities At The Beginning Of Each Year

The annuities considered thus far have had end-of-the-period cash flows. Both the present and future values will be affected if the cash flows occur at the beginning of each period instead of the end. To illustrate this effect, consider an annuity of $100 at the end of each year for the next 4 years, with a discount rate of 10%, as contrasted with an annuity with payments made at the beginning of each year.

Since the first of these annuities occurs right now, and the remaining cash flows take the form of an end-of-the-period annuity over 3 years, the present value of this annuity can be written as follows:

In general, the present value of a beginning-of-the-period annuity over n years can be written as follows:

or as:

Note that this present value will be higher than the present value of an equivalent annuity at the end of each period.

The future value of a beginning-of-the-period annuity typically can be estimated by allowing for one additional period of compounding for each cash flow:

This future value will be higher than the future value of an equivalent annuity at the end of each period.

Illustration: IRA - Saving At The Beginning Of Each Period Instead Of The End

Consider again the example of an individual who sets aside $2,000 at the end of each year for the next 40 years in an IRA account earning 8%. The future value of these deposits amounted to $518,113 at the end of the 40th year. If the deposits had been made at the beginning of each year instead of the end, the future value would have been higher:

As you can see, the gains from making payments at the beginning of each period can be substantial.

VI. Effect of Cash Flows Occurring Evenly Throughout the Year

Cash flows do not always occur at either the beginning or at the end of each period. Many of a firm�s operating cash flows (e.g., sales, expenses) tend to take place more or less evenly throughout the year. One way to handle this situation is to calculate present or future 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 beginning. In this situation, one may calculate a present or future value as if the cash flows occur at the end of each period, and then multiply the result by the square root of (1+k).

Illustration: IRA - Investing Throughout the Year Instead of the End

Consider again the example of an individual who sets aside $2,000 at the end of each year for the next 40 years in an IRA account earning 8%. The future value of these deposits amounted to $518,113 at the end of the 40th year; with payments made at the beginning of each year the future value were $559,562. If the deposits had instead been made evenly throughout each year, the future value would have been $538,439 as calculated below:

C. Growing Annuities

A growing annuity is a cash flow that grows at a constant rate for a specified period of time. Note that to qualify as a growing annuity, the growth rate in each period has to be the same as the growth rate in the prior period.

In most cases, the present value of a growing annuity can be estimated by using the following formula:

where g equals the constant growth rate of the annuity.

Note also that this formulation works even when the growth rate is greater than the discount rate. The present value of a growing annuity can be estimated in all cases but one�where the growth rate is equal to the discount rate. In that case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect (i.e., n x A).

Alternatively, the present value of a growing annuity can be found using the standard Present Value of an Annuity formula, but using an adjusted discount rate that factors in the growth rate so that k would equal: [(1 + k) / (1 + g)] - 1.

Illustration: The Value Of A Gold Mine

Suppose you have the rights to a gold mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The current price per ounce is $300, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold to be extracted from this mine can be estimated as follows:

The standard Present Value of an Annuity calculation (with k = (1.10 / 1.03) - 1 = 6.8%) would give the same results (other than small rounding errors).

D. Perpetuities

A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity can be written as:

where A equals the perpetual cash flow and k 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. Assume that you have $100 par value preferred stock paying a 6% dividend ($6.00 per year). The value of this stock, if the discount rate is 9%, is as follows:

The value of preferred stock will be equal to its face (par) value only if the dividend rate is equal to the discount rate.

E. 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 flow next year, g equals the constant growth rate, and k equals the discount 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.

Illustration: 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 year and are expected to grow at the same rate in the long term. The rate of return required by investors on stocks of equivalent risk was 12.23%.

As an interesting aside, assume that the stock was actually trading at $70 per share. This price could be justified by using a higher expected growth rate. The growth rate would have to be approximately 8% to justify a price of $70. This growth rate is often referred to as an implied growth rate.

F. Combinations and Uneven Cash Flows

In the real world, a number of different types of cash flows may exist simultaneously, including annuities, simple cash flows, and sometimes perpetuities: Some examples are discussed below.

I. Bond Valuation

A conventional bond pays a fixed coupon every period 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 cash flow that has to be discounted separately. The value of a straight bond can then be written as follows:

Illustration: The Value of a Straight Bond

Say you are trying to value a straight bond with a 15-year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%. The value of the bond is found as the sum of the present value of an annuity (the coupon payments) and a single amount (the par value).

If the bond paid interest on a semiannual basis, we would need to adjust the valuation equation accordingly:

II. Common Stock Valuation

Take the case of the stock of a company that expects high growth in the near future and lower and more stable growth forever after that. The expected dividends over the high growth period represent a growing annuity, while the dividends after that satisfy the conditions of a growing perpetuity. The value of the stock can thus be written as the sum of the two present values.

where P0 equals the present value of expected dividends, g equals the extraordinary growth rate for the first n years (n = high growth period), gn equals the growth rate forever after year n, D0 equals the current dividends per share, Dt equals the dividends per share in year t, and k equals the required rate of return (discount rate).

Illustration: The Value of a High Growth Stock

Assume Eli Lilly had earnings per share of $4.50 and paid dividends per share of $2.00. Analysts expected both to grow 9.81% a year for the next 5 years. After the fifth year, the growth rate was expected to drop to 6% a year forever, while the payout ratio was expected to increase to 67.44%. The required return on Eli Lilly stock is 12.78%.

The price at the end of the high growth period (P5) can be estimated using the growing perpetuity formula:

The present value of dividends and the terminal price can then be calculated as follows:

The value of Eli Lilly stock, based on the expected growth rates and discount rate, is $52.74.

There are some cases where one annuity follows another. In this case, the present value will be the sum of the present values of the two (or more) annuities. The present value of these two annuities can be calculated separately and cumulated to arrive at the total present value. The present value of the second annuity has to be discounted back to the present.

Conclusion

Present value remains one of the simplest and most powerful techniques in finance, providing a wide range of applications in both personal and business decisions. Cash flow can be moved back to present value terms by discounting and moved forward by compounding. The discount rate at which the discounting and compounding are done reflects three factors: (1) the preference for current consumption, (2) expected inflation and (3) the uncertainty associated with the cash flows being discounted.

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This page last updated on Tuesday, January 17, 2006.

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