Present Worth of Cash Flow Series and Functions

SECTION III Service Systems

8.4 Present Worth of Cash Flow Series and Functions

Present worth is a monetary value that occurs at the beginning of a project or a value that is equivalent to the algebraic sum of all the cash flows (inflows are + and outflows are –) in a project including accrued interest. Since invested capital earns money at the rate of i or an equivalent j percent or more, a cash flow of Fk dollars after k time periods have elapsed will have a present value that is calculated as follows:

P = F_k e^–jk = F_k (1 + i)^–k (5) It may also explain that when i or j is the minimally acceptable (or attractive) rate of return (MARR), the resulting present value is the least current value acceptable for any future value. Thus, if all future values were rescaled at equivalent current value by multiplying the actual cash flow that occurs at k by the factor e^–jk or (1 + i)^–k, the algebraic sum of these equivalent current values represents the total present worth.

The typical convention is to sign all outflows as negative and inflows as positive. Projects with greater positive present worth are more preferred than those with smaller positive present worths, since all cash inflows add to wealth and all outflows detract from wealth, and the accumulation of present worth of inflows minus the worth of outflows represents the surplus current value. If the net present worth is zero, the investment is paid off exactly at the minimum attractive rate of return (MARR). The surplus in present worth tells one how much more the future returns on the investment accumulate beyond the minimum investment coverage.

Discrete Serial Models for Discounted Cash Flows

The future cash flows shown above consist of a single cash flow at the beginning of a project which is equivalent to several future cash flows. In most cases the equivalent present worth was found by finding the present worth equivalent to each separate future cash flow and then accumulating these equivalents algebraically using the sign convention of positive inflows and negative outflows. While that computa- tional practice is correct, it is inefficient. An alternative procedure that is computationally efficient is the use of present worth factors for future time series. There are factors to several cash flow series which convert each member of the series into an equivalent present worth accumulation over the series. These factors provide computational convenience for analysts because the computation of a single formula handles the entire series rather than treating each member of the series individually.

These series also come with different functional forms over time, as shown in Table 8.1. The most elementary is the step function series which consists of a uniform series of C dollars at the end of each time period and going on for k serial flows. The present worth of the step series over k time periods is:

8-6 Occupational Ergonomics: Design and Management of Work Systems

where the final term in the equation above is the sum of the series (1+i)^-1+ (1+i)^-2+…+ (1+i)^-K. Table 8.1 shows the step and four additional functional cash flow series where each successive time in the series is an interest compounding time in which cash flow events occur. The final column in Table 8.1 describes the present worth formula for the step and other series. Actual time series of values and typical kinds of cash flows where the series is often appropriate are shown in the second column. The fourth column in Table 8.1 is simply the cumulative sum of undiscounted cash flows. These cumulative sums are useful in estimating cash flow constants R and C in these series from accounting or other data sources.

In the case of a step series of $100 each time period for 3 years, with a minimum attractive rate of return of 10%, the formula and computation of present worth is as follows:

In effect, this is equivalent to 3 individual cash flows of $100 each at the end of years 1, 2, and 3. Individual computations over the three cash flows are more cumbersome, as the following equation demonstrates:

P(0.1) = 100 (1.1)^–1+100 (1.1)^–2 + 100 (1.1)^–3 = 90.91 + 82.64 + 75.13 = $248.69 Obviously, the computational savings of a series improves with the length of the series, and that is one important advantage of series formulae.

While the process of cash flow generation is really a separate process from interest generation, calculations describing these processes in Table 8.1 are not totally separable. The reason is because the fundamental time period between successive time points in a series must correspond to the interest-generating process, as this TABLE 8.1 Five Functional Series, Serial Magnitudes, Cumulative Quantities over k Periods, and Present Worths.

Functional

Series Magnitude at Time k

Cumulative Cash Flow from Time 0 to k

Present Worth at the Minimum Attractive Rate of Return i or P(i) =

STEP f(k) = C

typical of operating cost UP-RAMP f(k) = Ck

typical of maintenance and deterioration costs DOWN-

RAMP

f(k) = R – Ck

typical of lost sales due to fewer customers (R > Ck)

DECAY f(k) = Ce^–rk startup costs GROWTH f(k) = R – Ce^–rk

maintenance costs of aging equipment

f x Ck

x=1

k ( )=

∑

^Ci

[

1− +( )1 i^−k

]

f x Ck k +1

x=1 k

( )⁼ ( )

∑

2

C

i i C

i k +1 1+ i

2

k k

1− +( )1 1

[

⁻

]

⁺

[

^{−( ) ( )−}

]

f x Rk C k k +1

x=1

k ( )= − ( )

∑

2

Ri C

i i C

i k +1 1+ i

2

k k

−

[

¹− +( )^{1 −}

]

⁺

[

^{1−( ) ( )−}

]

f x C e e e

r x=1

k rk

( )^{= ⎡} r₋⁻

⎣⎢ ⎤

⎦⎥

− −

∑

− 1¹

C

1+ i e e e

r i

r k+1 r

− − k

( )+

⎡

⎣⎢

⎢

⎤

⎦⎥

− − −( ) ⎥

1

R k C e e e

r rk

− r −

−

⎡

⎣⎢ ⎤

⎦⎥

− −

−

1 1

R

i i C

1+ i e e e i

k

r r

k+1 r

1 1 k

− +( ) 1

[ ]

⁻ ₋ ^{− ⎡}_⎣^⎢_⎢^{− −}_{( )}⁻₊^{( ) ⎤}_⎦^⎥_⎥

P i C 1+ i C1 i

i

k k=1

K k

( )

=

∑ ( )

⁻ = − +

(

¹

)

⁻

P i C

i i ^k

( )

⁼

[

^{1− +}

( )

^{1 −}

]

⁼$0 1¹⁰⁰

[

1^{− +}

⁽

1 0 1

⁾

⁻³

]

⁼ 248 69

. . $ .

from the gradient, although it bears similarities. The down-ramp series is nothing but a step function of R dollars per period minus an up-ramp of $C, so long as Rk > Ck. A common example of a down- ramp series is reduced revenues expected as the result of deteriorating machinery or seasonal decreases in revenues from the sales of off-season products. Two other serial functions, the decay and growth series, are shown in Table 8.1. Startup or learning costs can often be captured by decay series, and maintenance costs frequently follow a growth series. Here again, present worth formulae are presented in Table 8.1 for these series as well.

In addition to the present worth formulae, Table 8.1 describes the magnitude of the series at specific points in time (in column 2) and the cumulative sums over time from t = 1 to t = k (in column 3). As noted above, before the economics of the situation can be analyzed, the benefits and costs must be identified and estimated. These other formulae in Table 8.1 are useful in this estimation once the different benefit and cost streams are identified. In fact, part of the identification involves a recognition of the general nature of the costs and benefits over time. For example, a maintenance cost would be expected to increase more with time and so an up-ramp series or a growth series would be likely candidates for describing this situation. Then it is merely a case of finding which functional series best describes these costs. If past records indicate increasing values at a diminishing rate over time, the growth series would be a better choice than the up-ramp. By fitting the function f(nt) for the magnitude of that series to previously recorded data on costs, the following series of equations will result:

f(1t) = R – Ce^–r and f(2t) = R – Ce^–2r and f(3t) = R – Ce^–3r

The values of R, C, and r that best satisfy those estimates are the most reasonable values. More data facilitate a statistically better fit. Sometimes available data (e.g., accounting records) show cumulative expenditures, rather than serial expenditures. In such a case, the cumulative values can be used as follows:

Any combination of the series magnitudes and cumulative serial amounts can be used.

As stated above, the use of a series carries the assumption that the time period between successive cash flows is constant and can be designated as the interest-generating time period. If one chooses to use a quarterly time period, the interest rate charged per time period should be the nominal MARR divided by four. It also follows that the series magnitudes are the costs or benefits over that quarter-of-the-year time interval or whatever period the analyst chooses. Although interest and cash-flow generating processes are theoretically separate, the algebra describing a series makes it difficult to separate them. One could use monthly series with an interest rate per month as one-third of a quarterly interest rate. That computation would contain only a very slight error and one far less than the expected errors of estimation, but highly frequent series computations become cumbersome.

2See Au and Au (1992), Newnan (1996), Fabrycky, Thuesen, and Verma (1998).

f t R Ce f t R2 Ce e

e f t R3 Ce e

n=1 e

K

r n=1

2

r r r

n=1 3

r r

( )

= −

( )

= − − r

−

⎡

⎣⎢ ⎤

⎦⎥

( )

= − −

−

⎡

⎣⎢ ⎤

⎦⎥

∑

^{− ,}

∑

^{− −−2} 1^{1 ,}

∑

^{− −−3}

1 1

8-8 Occupational Ergonomics: Design and Management of Work Systems

Continuous Models of Discounted Cash Flow

An alternative to modeling cash flows as series is to model cash flows as continuous flow functions over time. For example, a continuous step function accumulates $C over a year and the accumulation each month is 1/12th of C. One of the distinct differences between series modeling and continuous cash flow modeling is that series parameters correspond to the individual serial flows, but continuous model param- eters pertain to the flow over a year. For example, a continuous up-ramp that has a parameter of $C per year means that the flow magnitude each year increases the flow density by $C more each year. At the beginning of the first year, the flow was $0 per year and, by the end of the year, cash is flowing at $C per year. During that first year, the accumulated cash flow is ($0 + $C)/2 or $C/2 because the flow going from

$0 to $C can be represented as a triangle with a base of 1 year and an area that contains $ C/2. That same up-ramp reaches a magnitude of $2C over two years and the accumulation is $4 C/2 or $2C. It is often helpful to think of continuous cash flows over time as analogous to putting a pail below a water faucet. The magnitude of the cash flow rate at any point in time is analogous to the amount the faucet value is open, but the amount of flow into the pail depends upon how long the faucet is set at a given opening. Step flows describe the case in which the faucet is set at a constant opening over time. An up-ramp flow is analogous to putting the pail below the faucet and then uniformly increasing the faucet opening. The reverse is true for a down-ramp flow. When that down-ramp represents benefits, the stream of benefits gets thinner and thinner, supporting fewer and fewer business operations. Continuous cash flows accord with the notion of stream flow and this form of modeling has some mathematical and conceptual advantages. For one thing, a continuous flow can be stopped at any point without concern about the next serial value, and whether or not it should be included in the calculation. Continuous flows carry continuous compounding so that compounding time periods are not a problem; the continuous interest rate must be set to be equivalent with the nominal rate or with a natural monthly, quarterly, or semiannual rate as shown above. Moreover, some of the formulae for continuous cash flows are a bit simpler than the discrete series, yet they yield the advantage of describing flows over various time periods.

Table 8.2 shows formulae for computing the present worth of continuous cash flow functions that start immediately, accumulate up to time k, and then stop. In all of those formulae, it is assumed that both the pattern of changing cash flow and the start of cash flow occur at time zero relative to the project start. As with the Table 8.1 formulae, Table 8.2 shows cash flow densities at any future point in time as well as cumulative functions of cash flow. Those density magnitude and cumulative flow formulae are particularly useful in setting the C and r parameters in these continuous flow models.

Note that when k is a very long time, the value of e^–jk in the formula in Table 8.2 approaches zero.

Similar to the series formulae, this situation greatly simplifies the formula in Table 8.2. In the case of a step function, the present worth calculated for a very long time period approaches C/j. With an up-ramp function, the present worthapproaches C/j².

The formulae for continuous cash flows in Table 8.2 correspond closely to those for discrete cash flow series in Table 8.1. For example, in the step function with a constant flow per year over k years at j percent interest, if j is set at 9.53%, the equivalent continuous interest rate for 10%, the present worth over three years, is calculated as follows:

The reader may recall that a similar calculation using the Table 8.1 formula and 10% annual discrete interest yielded only $248.69 rather than $260.93. Since the exponential term in the fraction part of the equation above is equal to the numerator in the discrete series step function, the only cause of a difference is the j in the denominator, in contrast to the i in the denominator for a series step formula. In other words, $260.93 times 0.0953/0.1 is $248.67. This example demonstrates that present worths of continuous flows and discrete serial flows are different, even when the interest rates are termed “equivalent.” The amount of this difference, however, is typically small.

P j C1 e

j ^jk year e

( )

^{= − − =$100 1−}_{0 0953.}^{−0 0953 3. ( ) =$260 93.}

Estimating Parameter r in Serial and Continuous Models

Sometimes analysts have difficulty estimating the exponential parameter r in decay and growth models.

One aid to making these parameter estimates is to find the expected cumulative cash flows over one and two time periods. Theoretically, it does not make any difference what time period is selected, but in practice a period long enough to reveal changes in the flow is most desired. Thus, if the changes were relatively rapid, monthly time periods would be adequate. However, quarterly, semiannual, or annual periods are preferable for flows that change less frequently over time. Note in Tables 8.1 and 8.2 that the cumulative cash flow of a decay cash flow over k time periods is as follows:

Now consider the ratio S(1)/S(2) describing the occurrence of a two-period flow during the first period:

(6)

In a similar fashion, the growth model has the following ratio:

(7)

UP-RAMP f(k) = C k

typical of maintenance, and deterioration costs DOWN-RAMP f(k) = R – Ck

typical of lost sales due to fewer customers (R > Ck)

step less up-ramp R > Ck

DECAY f(k) = Ce^–rk startup costs GROWTH f(k) = R – Ce^–rk

maintenance

costs of aging equipment

step less decay C x dx = Ck

0

k 2

∫

^{2 C 1 e}j

ke j

jk 2

− − jk

⎛

⎝⎜

⎞

⎠⎟

− −

R dx C x dx R C k 2 k

0 k

0

∫

⁻

∫

k ^{=⎛⎝ − ⎞⎠}

C e dx C 1 e r

r x 0

k r k

− −

∫

^{= ⎡}_⎣^{⎢ − ⎤}_⎦^{⎥ j + rC}

[

^{1−e−( )j+r k}

]

C 1 e dx = C rk e r

r x

k rk

(

−

)

^⎡_⎣⎢ ^{− +} ⎤

⎦⎥

− −

∫

^{0 1}

S k C

r e ^kr

( )

=

−

[

⁻ −¹

]

S 1 S 2

e e

r r

( ) ( )

⁼

[

⁻

]

[

−

]

−

−

1

2 1

S 1 S 2

e e

r 2r

( ) ( )

^{= −−}

1 1

8-10 Occupational Ergonomics: Design and Management of Work Systems

It follows with the step function that the S(1)/S(2) ratio is 0.5, and with an up-ramp function, it is 0.25.

A growth function with a very small r value increases in time almost uniformly, and so that ratio should be similar to the up-ramp when r is small. The r values of growth functions thus lie between 0.25 (with a small r) and 0.5 (with a large r). In a similar manner, decay functions with a small r behave similarly to step functions. Hence, the S(1)/S(2) ratios of decay functions vary from about 0.50 with a very small r and increase up to 1.00 as r increases. Table 8.3 verifies these observations by describing associated S(1)/S(2) ratios for growth and decay functions at various r values from 0.01 to 2.70.

Delayed Cash Flow Streams

All of the present worth functions in Tables 8.1 and 8.2 are assumed to start immediately and continue for k years. If the stated pattern of cash flow is delayed b time units before the pattern starts but the pattern is otherwise exactly the same after starting i, then the present worth without delay (P′) can be computed exactly as if it had started immediately. To find the correct present worth for the pattern delay of b time units, the following formula may be used:

(8) For instance, suppose that a 3-year-long step function of $100 each year were to experience a 2-year delay but remain otherwise unchanged. The present worth could then be figured out as $260.93, as shown earlier, and the delay of 2 years is computed as follows:

Note that once a cash flow pattern is recognized and fitted with parameter values to reflect actual cash flows, those patterns and parameters can be directly used along with the equivalent continuous interest rate to compute present worths. In this use, continuous interest rates are considered to be the minimum acceptable rate of return on company-invested capital.

Repeated Cash Flows Over Time

A typical situation in modeling the cash flows of a project finds that costs or returns associated with maintenance, production, and such other processes repeat themselves over time. Such situations are similar to that of a homeowner who must perform maintenance on the heating and air-conditioning system twice a year, year in and year out. In that situation the present worth of the first event in the

TABLE 8.3 Ratios of S(1)/S(2) for the Growth and Decay Continuous Cash Flow Functions Corresponding to Selected Values of Parameter r

parameter r = 0.01 0.03 0.05 0.07 0.09 0.10 0.20 0.30

growth S(1)/S(2) .2508 .2525 .2542 .2558 .2574 .2567 .2660 .2742

decay S(1)/S(2) = .5050 .5086 .5026 .5176 .5228 .5251 .5499 .5745

parameter r = 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

growth S(1)/S(2) = .2820 .2895 .2969 .3041 .3109 .3176 .3141 .3303

decay S(1)/S(2) = .5987 .6225 .6456 .6682 .6900 .7109 .7310 .7502

parameter r = 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90

growth S(1)/S(2) = .3362 .3419 .3475 .3528 .3579 .3623 .3674 .3719

decay S(1)/S(2) = .7685 .7859 .8022 .8176 .8320 .8455 .8581 .8699

parameter r = 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70

growth S(1)/S(2) = .3761 .3802 .3841 .3879 .3914 .3949 .3981 .4012

decay S(1)/S(2) = .8808 .8909 .9003 .9089 .9168 .9241 .9309 .9370

P = e^−jbP′

P = e^{−0 11333 2. ( )}260 93. =$208 01.

(9)

Accordingly, it is useful to recognize repeated cash flow patterns over time and use the relationship above to simplify calculations.