2. Uncertainty vs. Measurement Error

Example: Suppose we want to open a rooftop bar on the Willis Tower (formerly Sears Tower) in Chicago. When assessing the financial value of the project, we realize that one of the main drivers of revenues of a rooftop bar is the outside temperature. To model the monthly revenues of the bar, we therefore collect weather data (f. ex. from www.wunderground.com). For example, these data show that, over the last 15 years, Chicago's average November temperature was 5°C. Based on this information, your business plan assumes an average November temperature of 5°C for the years to come. You proceed accordingly for all other months to model the temperature curve that ultimately drives revenues. 

Now how do uncertainty and measure error come into play here? Let us look at these two dimensions separately:

Uncertainty:

  • Based on the historical weather data, we know that not each year had the same average November temperature of 5°C. There were relatively warm years such as 2001 with an average temperature of 9°C, and there were comparatively cooler years such as 2013 with an average November temperature of only 3°C.
  • Looking forward, we know that the average November temperature will not be exactly 5°C each year. We know that some future years will be warmer whereas some other years will be cooler. This is uncertainty. The value driver is not a data point but a random variable with an expected mean and some dispersion around that mean.
  • Uncertainty per se is not a problem for valuation. We capture uncertainty by using risk-adjusted discount rates. All else the same, the larger the uncertainty the higher is the risk-adjusted discount rate that we apply to the firm's expected cash flow.
  • Absent measurement error, discounting expected cash flows at a risk-adjusted rate yields a point estimate of firm value.


Measurement error:

  • The problem is that our measurement of the distribution parameters of the value drivers is often erroneous. In our example, it could be that we systematically over- or underestimate the expected average November temperature of 5°C as well as its dispersion. This is the problem of measurement error.
  • In our example, there are many potential sources of measurement error. For example:
    • The temperature data we use are from the Chicago O'Hare airport and not from the specific location of the Willis Tower. The Willis Tower is closer to the Lake Michigan, which could affect the average temperature and its variation.
    • The temperature data we use are measured at ground level. The Willis Tower, however, is more than 400 meters high. We know that temperature drops with altitude.
    • We ignore the fact that the future climate could be systematically different from the past climate. Global warming, for example, would imply that the historical average temperature underestimates the expected average temperature as well as its variation.
  • Using risk-adjusted discount rates does not cure measurement error. To assess the influence of measurement error, we have to go back to our valuation model and investigate how firm value changes if the expected average values of the key value drivers are biased.
  • Practitioners often use a combination of scenario analysis and sensitivity analysis to answer these questions. More sophisticated users rely on simulations. In fact, there are several easy-to-use Excel add-ins that allow us to conduct powerful simulations on any computer.


For valuation purposes, we are primarily interested in the influence of measurement error. The reason is that, as we have already mentioned, general uncertainty is captured by the risk-adjusted discount rate. This means that we want to know how changes in the expected average value affect firm value. Going back to our example of the rooftop bar in Chicago, we want to know the value implications if the long-term expected average November temperature is not 5°C but, for example, 4°C or 6°C. In this context, potential extreme individual outcomes (f.ex., a very hot November with an average temperature of 10°C or a very cold November with an average temperature of 1°C) are irrelevant.

Extreme individual outcomes are typically assessed in the context of risk management or project management. There, we want to know the consequences of a particularly good (f.ex. a very hot November) or a particularly bad outcome (f.ex. a very cold November). Risk and project management is often conducted with scenario analysis as well. However, because it focuses more on extreme outcomes, the value intervals used are typically much wider than the ones we use to assess the implications of measurement error for valuation.


To illustrate the three techniques--scenario analysis, sensitivity analysis, and simulation--we go back to the hypothetical company that we have discussed throughout the various modules of this course. Let's assume that we have conducted the following steps for this firm:

  • Projections of the cash flows for the following 2 years,
  • Estimation of the WACC,
  • Estimation of continuing value.

Let's also assume that the result of these steps is an estimated firm value of 57'031 (point estimate). The details of the analysis is summarized in the attached Excel file.


In what follows, we assess this valuation for potential measurement errors. In reality, we would have to go through the individual assumptions step by step and question their validity, for example by conducting expert interviews or the analysis of comparable companies. Here, the focus is again on the more technical aspects. For simplicity, we assume that there are four potential sources of measurement error:


Affected variable

Estimated average values

Potential average values

Cost of sales (ex D&A)

44%

Between 42% and 46%

Equity beta

1.2

Between 1.0 and 1.4

Credit  spread

3%

Between 2% and 4%

EBIT-margin in steady state

22%

Between 21% and 23%