So far, we have assumed that minimum rate of return is fixed over the life of the project. But there are situations where other opportunities for investment (that determine the minimum rate of return) can make different rate of returns in different time. Thus, minimum rate of return can change over time. For example, other opportunities for investment of capital can give i*=12% now; and three years from now, we might expect a project that has a return on investment of i*=15%.
For analyses with minimum rate of return that change with time, NPV and PVR are recommended as the best methods. ROR is not a reliable approach for such analyses.
Example 45:
Cash flows for two mutually exclusive investment projects A and B are given as:
C=$40  I=$20  I=$20  I=$20  I=$20  I=$20  L=$40  
A) 


0  1  2  3  ...  10 
C=$50  I=$25  I=$25  I=$25  I=$25  I=$25  L=$50  
B) 


0  1  2  3  ...  10 
C: Cost, I:Income, L:Salvage
Analyze these alternatives, assuming the minimum rate of return for the first and second years is 25% and for third to tenth year it is 15%.
$$\begin{array}{l}NP{V}_{A}=40+20\left(P/{A}_{25\%,2}\right)+20\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+40\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)=\$54.61\text{}\hfill \\ NP{V}_{B}=50+25\left(P/{A}_{25\%,2}\right)+25\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+50\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)=\$68.26\text{}\hfill \end{array}$$Results indicate that project B is a better economic investment.
Note:
After year 2, minimum rate of return changes from 25% to 15%. In order to calculate the NPV of the cash flow, we have to separate the payments that happened at and before year 2 from payments that occurred after year 2.
Payments at year 2 and before that are not going to be affected by the change:
PV of payments from year 0 to year 2:
Project A: Present value of year 0 to year 2 payments $=40+20\left(P/{A}_{25\%,2}\right)$
Project B: Present value of year 0 to year 2 payments $=50+25\left(P/{A}_{25\%,2}\right)$
But payments after year 2 will be affected by the change.
To calculate the NPV of those payments and apply the change in i, first, we need to discount all the payments occurred after year 2 to this year (we set the year 2 as the base year) by i* = 15% and we calculate value of all payments at year 2:
Project A: Value of year 3 to year 10 payments at year 2 $=20\left(P/{A}_{15\%,8}\right)+40\left(P/{F}_{15\%,8}\right)$
Project B: Value of year 3 to year 10 payments at year 2 $=25\left(P/{A}_{15\%,8}\right)+50\left(P/{F}_{15\%,8}\right)$
Second, we discount the year 2 values for 2 years by i* = 25%to get the present value (value at year 0) of the payments:
Project A: Present Value of year 3 to year 10 payments $=20\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+40\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)$
Project B: Present Value of year 3 to year 10 payments $=25\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+50\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)$
In the end, we add all the values together:
$$\begin{array}{l}NP{V}_{A}=40+20\left(P/{A}_{25\%,2}\right)+20\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+40\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)=\$54.61\\ NP{V}_{B}=50+25\left(P/{A}_{25\%,2}\right)+25\left(P/{A}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)+50\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25\%,2}\right)=\$68.26\end{array}$$Another Method:
You can also treat each payment separately. This method is especially helpful when payments are not equal or when you are using spreadsheet to calculate the NPV.
We separate the payments that happened at and before year 2 from payments after year 2. Payments at and before year 2 will be discounted just by 25%:
PV of payments from year 0 to year 2:
Project A: PV year 0 to year 2 $=40+20\left(P/{F}_{25\%,1}\right)+20\left(P/{F}_{25\%,2}\right)$
Project B: PV year 0 to year 2 $=50+25\left(P/{F}_{25\%,1}\right)+25\left(P/{F}_{25\%,2}\right)$
For payments after year 2, first we calculate their value at year 2:
Project A: Value of year 3 to year 10 payments at year 2
$$\begin{array}{l}=20\left(P/{F}_{15\%,}1\right)+20\left(P/{F}_{15\%,}2\right)+20\left(P/{F}_{15\%,3}\right)+20\left(P/{F}_{15\%,4}\right)+20(P/{F}_{15\%,5})\text{}\\ +20\left(P/{F}_{15\%,6}\right)+20\left(P/{F}_{15\%,7}\right)+20\left(P/{F}_{15\%,8}\right)+40\left(P/{F}_{15\%,8}\right)\end{array}$$
Project B: Value of year 3 to year 10 payments at year 2
$$\begin{array}{l}=25\left(P/{F}_{15\%,}1\right)+25\left(P/{F}_{15\%,}2\right)+25\left(P/{F}_{15\%,}3\right)+25\left(P/{F}_{15\%,}4\right)+25\left(P/{F}_{15\%,}5\right)\\ +25\left(P/{F}_{15\%,}6\right)+25\left(P/{F}_{15\%,}7\right)+25\left(P/{F}_{15\%,}8\right)+\text{}50\left(P/{F}_{15\%,8}\right)\end{array}$$
Second step, we discount the year 2 values for 2 years by i* = 25% to get the present value (value at year 0) of the payments:
Project A: Present Value of year 3 to year 10 payments
$$\begin{array}{l}=20\left(P/{F}_{15\%,1}\right)\left(P/{F}_{25\%,2}\right)+20\left(P/{F}_{15\%,2}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,3}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +20\left(P/{F}_{15\%,4}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,5}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,6}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +20\left(P/{F}_{15\%,7}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)+40\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)\end{array}$$
Project B: Present Value of year 3 to year 10 payments
$$\begin{array}{l}=25\left(P/{F}_{15\%,1}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,2}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,3}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +25\left(P/{F}_{15\%,4}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,5}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,6}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +25\left(P/{F}_{15\%,7}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)+50\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)\end{array}$$
In the end we add all the values together:
$$\begin{array}{l}NP{V}_{A}=40+20\left(P/{F}_{25\%,1}\right)+20\left(P/{F}_{25\%,2}\right)+20\left(P/{F}_{15\%,1}\right)\left(P/{F}_{25\%,2}\right)\\ +20\left(P/{F}_{15\%,2}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,3}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,4}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +20\left(P/{F}_{15\%,5}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,6}\right)\left(P/{F}_{25}{}_{\%,2}\right)+20\left(P/{F}_{15\%,7}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +20\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)+40\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)=\$54.61\end{array}$$ $$\begin{array}{l}NP{V}_{B}=50+25\left(P/{A}_{25\%,1}\right)+25\left(P/{F}_{25\%,2}\right)+25\left(P/{F}_{15\%,1}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +25\left(P/{F}_{15\%,2}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,3}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,4}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +25\left(P/{F}_{15\%,5}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,6}\right)\left(P/{F}_{25}{}_{\%,2}\right)+25\left(P/{F}_{15\%,7}\right)\left(P/{F}_{25}{}_{\%,2}\right)\\ +25\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)+\text{}50\left(P/{F}_{15\%,8}\right)\left(P/{F}_{25}{}_{\%,2}\right)=\$68.26\end{array}$$Microsoft Excel or Spreadsheet
If you are using Microsoft Excel or another spreadsheet to calculate the Net Present Value for the cash flow that has different discount rates over the life of project, be careful! You can not use the NPV function. However, you can calculate the Net Present Value by making a summation over calculated discounted cash flow. Figure 43 displays how Net Present Value for Project A cash flow with a changing minimum rate of return can be calculated. Note the formula in the cell D3 to D12.