Similar to what we had in previous sections (such as Example 26), there are problems that require you to calculate present value (as an unknown variable) for payments occurring in the future as revenue, with interest rate or rate of return (as known variable). These types of calculations are called breakeven and enable you to determine the initial investment cost that can breakeven the future payments considering a specified interest rate. It gives you the equivalent amount of money that needs to be invested at present time for receiving the given payments in the future with the desired interest rate.
As explained in Lesson 1, time value of money affects present value calculations. Consequently, the size of the payments, interest rate, and also payment schedule are influential factors in determining present value and breakeven calculations.
Example 28:
Assume two investments of A and B with the payment schedule as shown in Figure 23. Calculate the present value of these investments considering minimum rates of return of 10% and 20%. The calculation will give the initial cost that can be invested to breakeven with 10% and 20% rate of return.
Please notice that cumulative payments for investment A and B are equal and the difference between two investments is in the payment schedule.
Investment A P=? 
A=100  A=200  A=300  A=400  


0  1  2  3  4 
Investment B P=? 
A=400  A=300  A=200  A=100  


0  1  2  3  4 
Figure 23a: In investment A, the payment (revenue) schedule will be 100, 200, 300, and 400 dollars at the end of the first, second, third and fourth year. In investment B, the payment (revenue) schedule will be 400, 300, 200, and 100 dollars at the end of the first, second, third and fourth year.
Assuming rate of return 10%:
$$\begin{array}{l}{P}_{A}=100*\left(P/{F}_{10\%,1}\right)+\text{}200*\left(P/{F}_{10\%,2}\right)\text{}+\text{}300*\left(P/{F}_{10\%,3}\right)\text{}+\text{}400*\left(P/{F}_{10\%,4}\right)\\ {P}_{A}=100*0.9091+200*0.8264+300*0.7513+400*0.6830\text{}=\text{\$}754.80\end{array}$$ $$\begin{array}{l}{P}_{B}=400*\left(P/{F}_{10\%,1}\right)\text{}+\text{}300*\left(P/{F}_{10\%,2}\right)+\text{}200*\left(P/{F}_{10\%,3}\right)\text{}+\text{}100*\left(P/{F}_{10\%,4}\right)\\ {P}_{B}=400*0.9091\text{}+\text{}300*0.8264\text{}+\text{}200*0.7513\text{}+\text{}100*0.6830\text{}=\text{\$}830.13\end{array}$$Assuming rate of return 20%:
$$\begin{array}{l}{P}_{A}=100*\left(P/{F}_{20\%,1}\right)+\text{}200*\left(P/{F}_{20\%,2}\right)\text{}+\text{}300*\left(P/{F}_{20\%,3}\right)\text{}+\text{}400*\left(P/{F}_{20\%,4}\right)\\ {P}_{A}=100*0.8333\text{}+\text{}200*0.6944\text{}+\text{}300*0.5787\text{}+\text{}400*0.4823\text{}=\text{\$}588.73\end{array}$$ $$\begin{array}{l}{P}_{B}=400*\left(P/{F}_{20\%,1}\right)+\text{}300*\left(P/{F}_{20\%,2}\right)+\text{}200*\left(P/{F}_{20\%,3}\right)+\text{}100*\left(P/{F}_{20\%,4}\right)\\ {P}_{B}=400*0.8333\text{}+\text{}300*0.6944\text{}+\text{}200*0.5787\text{}+\text{}100*0.4823\text{}=\text{\$}705.63\end{array}$$This example shows the effect of time on future payments.Cumulative payments for investment A and B are equal, and the difference between two investments is in the payment schedule. In investment B, the investor receives a larger amount of revenue in the closer future, which amortizes the investor’s principal more rapidly than “A."
Example 29:
Investing on an asset is expected to yield 2,000 dollars per year in income after all expenses for each of the next ten years. It is also expected to have a resale value of $25,000 in ten years. How much can be paid for this asset now if a 12% annual compound interest rate of return before taxes is desired? Note that the wording of this example can be changed to describe a mineral reserve, petroleum, chemical plant, pipeline, or other general investment, and the solution will be identical.
C=?  I=2000  I=2000  I=2000  ...  I=2000  L=$25,000  


0  1  2  3  ...  10 
Figure 23b: Cash flow: 2,000 dollars per year in income after all expenses for 10 years and resale value of $25,000 in the tenth year.
C: Cost
I: Income
L: Salvage Value
Present Value Equation:
Let’s equate costs and income at the present time.
Present value of all costs =present value of all incomes plus present value of salvage
The result will be similar, if costs and revenue plus salvage is equated in any time.
Future Value Equation
If we equate costs and income by the end of the 10^{th}year, then:
future value of cost = future value of income + future value of salvage
Annual Value Equation
Let’s equate the annual value costs and incomes,
annual value of cost = annual value of income + annual value of return
Please note that an equation can be written to equate costs and incomes at any point in time and the same breakeven initial cost of $19,350 can be obtained.