Four Subcomponents of Cash Flow

The first part of the definition, “Cash flows produced by that asset, over its useful life,” includes four subcomponents: timing, duration, magnitude, and growth, as shown in Figure 1.2.

The first subcomponent, timing, addresses the question, “When will we get the cash?” Will we get the cash flow next year or in five years? While the amount of the cash flow is the same in Figures 1.3A and B, the cash flow is received sooner in A than in B. All else being equal, getting cash sooner is better than getting it later.

The second subcomponent, duration, addresses the question, “How long will the cash flows last?” Duration¹ can be thought of as an asset’s estimated useful life. For example, an annuity that pays each year for eight years is more valuable than one that pays for only four years, as shown in Figure 1.4. A longer duration of cash flows is better than a shorter one.

The third subcomponent, magnitude, addresses the question, “How much cash will we get?” Figure 1.5 shows a stream of $4 payments versus a similar stream of $2 payments. It should be obvious that larger cash flows are better than smaller ones.

The fourth subcomponent, growth, addresses the questions, “Will the cash flows grow over time?” or “How fast will the cash flow grow over time?” A growing stream of cash flow is preferable to one that is not growing (A versus B), as shown in Figure 1.6.

When starting at the same level, a stream of cash flows with a faster growth rate is preferable to one with a slower growth rate (A versus B), as shown in Figure 1.7.

A stable cash flow is preferable to one with a negative growth rate or losses, as shown in Figure 1.8. (Note: Arrows pointing down in B represent cash outflows reflecting losses.)

It is important to note that it is necessary to make estimates for all four subcomponents—timing, duration, magnitude, and growth—to calculate the asset’s future cash flow.

Uncertainty

The second part of the definition states that the cash flows need to be “discounted for . . . the uncertainty of receiving those cash flows.” There was no uncertainty to the cash flows up to this point in the discussion as we assumed that they were known and guaranteed (similar to the coupon payments from a U.S. Treasury Bond). However, an asset’s cash flows will be dependent on events that will happen in the future, and because the future is inherently uncertain, we must take uncertainty into account when addressing the question, “How certain are the future cash flows?”

It is important to note that uncertainty will have an impact on all four cash flow subcomponents and, in turn, affect the asset’s value, as shown in Figure 1.9.

Even the most predictable cash flows have some degree of uncertainty to them, however. Therefore, we need to think of any cash flow we calculate as an estimate of the expected cash flow rather than a guaranteed amount.

Before we proceed further we need to alter slightly the definition of an asset’s value to reflect this observation:

Rather than thinking about future cash flows as single-point estimates, as shown in Figure 1.10, it is more appropriate to think about a range of cash flow estimates, around a single-point estimate.

For instance, while we expect the cash flow to equal $4 in this example, we need to recognize that there is a possibility that the actual cash flow will be greater or less than $4, within a forecasted range of $2 to $6, as shown in Figure 1.11.

Another way of depicting the full range of possible estimates is to think of it as a distribution of potential cash flow estimates spreading out, in both directions, around a single-point estimate, as shown in Figure 1.12. The graph shows the range of possible outcomes, with values closest to the single-point estimate representing the outcomes with higher probabilities of being the true cash flow received, while cash flow estimates in the tails of the distribution represent outcomes that have lower probabilities of occurring.

It is hard to predict the future. The charts in Figure 1.13 show how the distributions of possible outcomes widen and the expected point estimates become less predictable as uncertainty increases the further we forecast cash flows into the future. We use the increasingly blurry $100 bill to provide a visual representation of this reality.

(Note: We rotated the diagram to create a 3D image so that the probability distribution of the estimate can be seen on the z-axis, while time remains on the x-axis and estimated cash flows on the y-axis.)

Figure 1.14 combines the individual cash flow distributions from Figure 1.13 into a single chart, showing how the uncertainty of estimating cash flow increases as we look further into the future.

The Time Value of Money

The final component of the definition of an asset’s value states that its cash flows must be “discounted for the time value of money.” Discounting for time is a straightforward concept: A dollar today is worth more than a dollar in the future, or thought of another way, “A bird in the hand is worth two in the bush.”² The value of cash flows today is referred to as their present value.

It is often easier for most people to understand the concept of present value after a discussion of future value, which is based on compounding. For example, with a 6% annual rate of return, $100 today increases in value to $106.00 in one year and $112.36 in two years, as shown in Figure 1.15.

The value of $100 in one year is calculated by compounding the initial value by 6%, as shown in the following computation:

The value at the end of the second year is calculated the same way, by compounding the value at the end of the first year by the same 6% for the second year:

Alternatively, the value at the end of year 2 can be calculated by compounding the initial cash flow by two periods of 6% interest, as shown in the following calculation:

The following formulas can be used to calculate the future value of any amount of money:

Or, alternatively:

Which is the equivalent of (using simpler notation):

Where:

• $₀ equals the value of cash today
• $₁ equals the value of cash in one year
• $₂ equals the value of cash in two years
• i represents the return on investment

To calculate the present value of a future $100 payment, we need to discount it to find its value in today’s dollars, which is essentially compounding in reverse. For simplicity, we use the same 6% rate and show that a cash payment of $100 one year from now discounted at 6% is worth $94.34 today, as shown in Figure 1.16.

The formula for discounting future payments is similar to the one used to calculate the future value of cash, with the components in the formula rearranged:

Therefore, the present value of $100 at a 6% discount rate is calculated as follows:

It should be straightforward to show that a cash payment of $100 two years from now discounted at 6% is worth $89.00 today, as shown in Figure 1.17.

Using the same formula from above:

Alternatively, if we assume that there is a $100 payment in year 2, but no payment in year 1, then we can use the slightly more complicated formula to show that the present value of the two payments at a 6% discount rate also equals $89.00, as the following calculation shows:

To calculate the present value of a stream of future cash payments, we need to discount each expected payment separately, as we did with the two-period stream of payments. We use a stream of four annual payments as an example in Figure 1.18.

The calculation is the same, only with more years represented:

We use future cash payments throughout the compounding and discounting examples to simplify the discussion. It should be easy to see that the future payments can be replaced with the asset’s future cash flows and that the present value formula for four years of cash flow is the following:

Where:

• CF₁ equals cash flow in year 1
• CF₂ equals cash flow in year 2
• CF₃ equals cash flow in year 3
• CF₄ equals cash flow in year 4
• i represents the discount return

Some people argue that the timing of cash flow and the time value of money are the same thing. Although we agree, we believe there is significant value in discussing the concepts separately to show the different roles they play when evaluating future cash flows.³ Timing is when you get the money, for example, next year or in five years. The time value of money is the rate used to discount the cash flows in the future back to today’s value, which is their present value.

Now that we have established the framework for valuing an asset, we increase the complexity in the next chapter, and use these tools to value a simple business: the proverbial lemonade stand.

Zoe will not operate her lemonade stand alone, as she will receive help from legendary investor Bill Ackman, who is an expert on the finances of lemonade stands.⁴

Gems:

• Valuing an asset is a relatively simple concept: The value of an asset is the sum of cash flows produced by that asset, over its useful life, discounted for the time value of money and the uncertainty of receiving those cash flows.
• Regarding cash flows:
  • Getting cash sooner is better than getting it later.
  • A longer duration of cash flows is better than a shorter one.
  • Larger cash flows are better than smaller ones.
  • A stream of cash flows with a faster growth rate is preferable to one with a slower growth rate.
  • A growing stream of cash flows is preferable to one that is not growing.
  • A stable stream of cash flows is preferable to one with a negative growth rate or losses.
• Uncertainty will have an impact on all four cash flow subcomponents and, in turn, affect the asset’s value.
• The definition of the value of an asset must be altered to account for the inherent uncertainty of future cash flows: The estimated value of an asset is the sum of the cash flows expected to be produced by that asset, over its useful life, discounted for the time value of money and the uncertainty of receiving those cash flows.
  • It is hard to predict the future as uncertainty increases the further we forecast cash flows into the future.
  • An asset’s cash flows must be “discounted for the time value of money.” Discounting for time is a straightforward concept: A dollar today is worth more than a dollar in the future, or thought of another way, “A bird in the hand is worth two in the bush.” The value of cash flows today is referred to as their present value.