PART
6.2 ONE-DIMENSIONAL AND 2-DIMENSIONAL SENSITIVITY ANALYSIS
In this section we will deal with the creation of 1-dimensional and 2-dimensional tables in Excel as a means of sensitivity analysis. We have already considered the concept of sensitivity analysis in Chapter 4. You may recall that, based on SteelCo’s financial model, we built a control panel where we were able to change dynamically certain input parameters and see the changing outcomes of the model. You may also recall that in Section 4.1 we concluded that if the covenants were not to be breached in 2014, a much higher gross margin should be achieved than the one originally planned. This particular kind of uncertainty handling is called What if analysis and is often used to compare different scenarios and their potential outcomes based on changing conditions.
Now consider the following example. A company plans to take on a project that is fore-cast to give the cash flows in Exhibit 6.1 (cells C5:G5). The initial cost of the project is esti-mated at €10,000. It is important to calculate an estimate of how profitable the project will be since the company has an internal policy of accepting projects with an Internal Rate of Return (IRR) higher than 10%. Internal rate of return is used for investment appraisal and is the discount rate at which the Net Present Value (NPV) of an investment becomes zero. In other words, IRR is the discount rate which equates the present value of the future cash flows of a project with the initial outflow. You will recall from Chapter 5 that the NPV formula is used to determine the present value of the project by adding the discounted cash flows received from the project. The formula for the discounted sum of all cash flows is written as:
NPV Cf cf
r
i i i
T
= − +
= +
∑
0
1 (1 )
where Cf0 is the initial cash outflow of the project and r the discount rate.
Considering that since the money going out is subtracted from the discounted sum of the incoming cash flows, the net present value of a project always needs to be positive in order to be considered a viable project.
The above model has 2 inputs: the growth rate of cash flows from the second year onwards and the discount rate. The project will bring in €2,400 in the first year of implementation and then it is assumed that each following year the cash inflow will be that of the previous year increased at a constant growth rate (in our case Cell B2 = 5%). Thus year 2 cash inflow will be:
Year2 Cash Inflow = Year1 Cash Inflow × (1+5%) = €2,400 × 1.05 = €2,520
and so on. Moreover the company’s cost of capital, which is roughly 10% (see Cell B3), is used as the discount rate. Instead of calculating discount factors manually as we did in Chapter 5 and then multiplying them with the future cash flows, we will use Excel’s NPV function to calculate the net present value of the project immediately (cell B6). Finally we will use another Excel function (IRR) to calculate the Internal Rate of Return of the project (cell B7). Please note that while we included cells B5 to G5 in the IRR function (that is we included the initial cash outflow), we excluded cell B5 from the NPV function for the calcu-lation of net present value. We just added cell B5 to the result of the NPV function instead (see cell B6 of Exhibit 6.1). The reason is that the cash outflow is considered to take place at the beginning of year 1 whereas the first cash inflow is considered to take place at the end of year 1. The first input value in the NPV function is discounted by the discount rate and that is why we exclude cash outflow at year 0. This should not be discounted at all. The function of cell B6 {NPV(B3;C5:G5)+B5} results in the discounted sum of all cash flows from years 1 to 5 described above.
We see from Exhibit 6.1 that the NPV of the project, under the specific assumptions of growth and discount rate, is negative. That is, the project is “No Go” as we say in finance jargon. Moreover the IRR is below 10%. Again the project is No Go. The project manager wants to know under what circumstances he can improve the project outcome. In order to help him, we could start changing the input assumption manually. Or instead we can use Excel’s Data Table command to perform sensitivity analysis for a range of values for the growth rate at first and then for both the growth rate and the discount rate. To start building a data table for the IRR, as a function of the growth rate, we need a set of possible values for the growth rate.
Let us say that the possible growth rate’s values range from 3% to 8%. That is, we surround the growth base assumption with higher and lower estimates as shown in Exhibit 6.2. Then, to build the 1-dimensional sensitivity analysis table we just need to copy the formula calculating the IRR in the adjacent cell 1 row below to the left of the range values (cell C11) as shown in Exhibit 6.2 and select the cell range of both rows (cells C10:I11).
EXHIBIT 6.1 Simple investment appraisal model presenting selective cell functions
Next, in the Excel ribbon we choose Data, then the What If Analysis tab, and finally the Data Table option. A Data Table dialog box like the one in Exhibit 6.3 appears. Since we want to construct a 1-dimensional data table in order to see how the IRR of the project varies with respect to changes in growth rate values, we need to fill in only the row input cell with cell B2. Finally we click the OK button. The various IRRs according to the different growth rates appear on the worksheet (cells D11:I11) as shown in Exhibit 6.4. Instead of a row, we could use a range of values in a column. In that case we would use the Column input cell of Exhibit 6.3.
As we can see from Exhibit 6.4, the project can succeed with an IRR greater than 10%
assuming growth rates starting from 6% onwards.
At this point, one might ask the obvious question: at which exact growth rate is the IRR 10%? To answer this question we will use another feature of Excel, that of Goal Seek. Goal Seek helps you find what inputs you need to assign to a variable of a function in order to get a certain output. To do that, we choose Data, then the What If Analysis tab, and finally the Goal Seek option (Exhibit 6.5a). The Goal Seek window has 3 input options (Exhibit 6.5b):
the cell that we want to change (Set cell), the value that we want this cell to take (To Value), and last the cell that we will change in order to achieve the requested value of the Set cell (By changing cell). In other words Goal Seek helps us set a cell at a certain value by changing
EXHIBIT 6.2 Constructing a 1-dimensional data table
Output area
= B7
EXHIBIT 6.3 Data Table dialog box. We input the cell we want to vary with a range of values.
another cell. So what we want is to set cell B7 (the IRR) at a value of 10% by changing cell B2 (the growth rate). Once we press OK the Goal Seek function will find the closest value to achieve the goal and will display it in the cell denoted in the Set cell input option. This value is 5.2% and the growth rate is adjusted automatically to 5.2% once we push the OK button.
What we have done so far is to change a single variable in our simple model and moni-tor how the output of the model reacts to this change. Now let us try to change 2 variables at the same time and see how the output of the model responds. The real value of sensitivity EXHIBIT 6.4 One-dimensional data table presenting various IRR values according to various
growth rates
EXHIBIT 6.5 Using the Goal Seek function to find the required input of growth rate that gives an IRR of 10%
(a)
(b)
analysis resides in the 2-dimensional data tables. The reason for this is simple. If we were to calculate manually the outcome of the 1-dimensional data table of Exhibit 6.4, we would need to change the growth rate cell 6 times. In a 2-dimensional 6 × 6 data table we would need to make 36 changes! Let us illustrate the point with another example of how we can automate this. Let us construct the table in Exhibit 6.6 in order to see which combinations of growth rate and discount rate give rise to positive project net present values. Note that, again, the first row of the table contains the possible input values for the growth rate whereas the first column of the table contains the possible input values for the discount rate. We have copied the formula calculating the NPV at the top left corner of the table (cell C16 of Exhibit 6.6). We set cell B2 as the row input cell in the Data Table dialog box (the growth rate input variable) and as the column input cell we set cell B3 (the discount rate input variable). Then by clicking OK the values of the table appearing at the bottom of Exhibit 6.6 fill the table range D17:I22.
We see a range of possible NPV values according to the particular growth and discount rates respectively. We can easily identify the pairs of growth and discount rates that result in a positive NPV. Moreover, by using the so-called conditional formatting feature of Excel we can easily visualize the positive and the negative NPV values. Conditional Formatting is an Excel tool that enables us to distinguish patterns and trends that exist in raw data. With condi-tional formatting we can apply formatting to 1 or more cells (in our case the whole sensitivity table ranging from cell D17 to I22) based on the value of the cell. We simply have to select the cells that we want to add the formatting to and then, in the Excel ribbon, in the Home tab to click the Conditional Formatting command. A drop-down menu will appear. The Con-ditional Formatting menu gives us various options (see Exhibit 6.7). We select the Highlight Cells Rules and then the Greater Than option where, for example, we indicate that we want NPVs higher than €1,000 to be presented in a dark colour. In the Less Than option we choose negative NPVs to be presented as gray text.
EXHIBIT 6.6 Two-dimensional sensitivity analysis table of a project NPV
EXHIBIT 6.7 Conditional formatting: entering a value and formatting style
Make sure that you always use conditional formatting, when working with 2-dimensional sensitivity analysis tables, to visualize the desired pairs of variables that give the requested output.
The model described above had only 2 input variables: growth rate and discount rate.
Therefore we did not have much choice as to which variable to choose in the sensitivity analysis. Recall that SteelCo’s “Assumptions” worksheet in Chapter 3 included more than 20 variables. The question which may be asked is how we know which variables to choose in the sensitivity analysis. The next section deals with this question.