corresponding net income NPV 1 , NPV 2 , NPV 3 … (these values can be positive or negative numbers).
Then, plot the values (r 1 , NPV 1 ), (r 2 , NPV 2 ), (r 2 , NPV 3 )… on the graph to get a curve. This curve intersects the x-axis at a point, where NPV = 0 and that point is the IRR we are looking for (see figure 6.3).
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i
IRR r
NPV 1
2
1
( r r )
NPV 1 NPV 2
This method requires very precise drawing or the use of appropriate computer software.
• IRR is determined by the interpolation method, which is the method of determining a value to be found between two selected values. According to this method, it is necessary to find 2 discount rates r 1 and r 2 ( r 2 > r 1 ) so that for r 1 we have NPV 1 > 0; for r 2 we have NPV 2 < 0. The IRR to be found (corresponding to NPV = 0) will be between the two discount rates r 1 and r 2 . Interpolating the third value (IRR) between the two discount rates above is done according to the following formula:
i
IRR r
NPV 1
2
1
( r r )
NPV 1 NPV 2
In which: r 2 > r 1 and r 2 – r 1 ≤ 5%
NPV 1 > 0 is close to 0, NPV 2 < 0 is close to 0
The interpolation method to find IRR can be shown in Figure 6.4: From the figure we see: IRR = r 1 + BC
∆ABC is similar to ∆AB'C'
1.3.1.7. Break-Even Point (BEP)
The break-even point is the point at which revenue is just enough to cover the costs incurred. At the break-even point, total revenue equals total costs, so the project is not yet profitable but also not at a loss. Therefore, this indicator indicates the lowest volume of products or revenue (from selling those products) that the project must achieve to ensure that it covers the costs incurred.
The break-even point is expressed by the physical indicator (output at the break-even point). If the output or revenue of the entire project life is greater than the output or revenue at the break-even point, the project is profitable; conversely, if it is lower, the project is at a loss. Therefore, the smaller the break-even point indicator, the better, the higher the safety level of the project, and the shorter the payback period.
There are two methods for determining the break-even point.
• Algebraic method: Apply this method to find the theoretical formula to determine the break-even point, the relationship between the break-even point and related factors. From there, it helps us have measures to influence those factors in the project management process to make the break-even point the smallest.
According to this method, to determine the break-even point, we assume that: X is the number of products produced and sold for the entire project life (if calculating the break-even point for the entire project life or one year (if calculating the break-even point for one year of the project life).
x f
p v
x: Number of products produced and sold at the break-even point. p: Selling price of 1 product.
v: Variable cost or variable cost calculated for a product (variable cost includes cost of raw materials, energy, fuel, direct labor wages, transportation costs, short-term loan interest payments... This cost changes proportionally to the volume of products produced).
f: Total fixed costs for the entire project life (if calculating the break-even point for the entire project life) or fixed costs for 1 year (if calculating the break-even point for 1 year of the project life). Fixed costs or constant costs include indirect machinery, loan interest, depreciation of fixed assets, maintenance costs, insurance, annual fixed taxes, annual real estate rent, etc. These costs do not change according to the products produced.
If calling:
y 1 = xp is the revenue equation
y 2 = xv + f is the cost equation. At break-even point total revenue equals total cost, therefore:
y 1 = y 2 => xp = xv + f
From that we can deduce:
x f
p v
This is the formula for determining the break-even point in physical units.
Break-even output (x) is directly proportional to f and inversely proportional to (p – v). Investors are interested in x→ min. To do so, they must find every way to reduce f, increase p, and reduce v. However, any changes to f, p, and v can only be within the limits acceptable to the market and investors.
• Break-even revenue (break-even point calculated in value units). In the case of producing 1 type of product, we have the following formula:
Oh h P x
p f
p v
f
1 v
p
In there:
x: Number of products produced and sold at the break-even point. P: selling price of a product.
v: variable cost or variable cost of a product
f : Total fixed costs of the entire project life if calculating the break-even point for the entire project life or fixed costs of one year if calculating the break-even point for one year of the entire project life.
In case of manufacturing multiple products:
Oh h m
f
vxp
p
(1 i )ii
m
i 1
m
i x i p i i 1
In which: m: number of product types
p i : selling price of one unit of product ix i : number of product i
i = 1, 2, 3, …, m
From the break-even point x, determine the project's break-even activity level (denoted M).
M x *100%
(%)X
Or:
In there:
M (%)
f O V
*100%
X: Number of products produced and sold during the entire project life. O: Revenue from sales of manufactured products during the entire project life.
(O = Xp)
V: Total variable costs over the life of the project (V = Xv).
Thus, the smaller the break-even level, the more profitable the project is, so investors are interested in making the break-even level the smallest. From the break-even level, it is possible to determine the safety for producing the product (symbol L) as follows:
L 100% x *100%
(%)X
Or:
L (%)
100%
x O V
*100%
The break-even point can be determined when the break-even level of activity is known.
Then the output at the break-even point is determined: x = M . X
Revenue at break-even point:
Oh = M . O
+ Graphing method: Create a coordinate axis, the horizontal axis represents the number of products, the vertical axis represents the cost or revenue from selling products. Because the break-even point is the point where revenue equals cost, the break-even point is the intersection of the revenue curve and the cost curve.
The revenue equation and the cost equation intersect at point M(x M , y M ). M is the break-even point.
X M : break-even output.
Y M : break-even revenue.
x MX : safety margin for product manufacturing.
If we take a point in the range OX M . For example x 1 , we can calculate the amount of capital not yet recovered at x 1 (corresponding to the number of products produced x 1 ).
If we take a point in the range x M X (profit range), for example x 2, we can calculate the profit at time x 2 (corresponding to the production quantity x 2 ).
Thus, using the graphical method, it is possible to visually see the break-even point, determine the profit and loss levels corresponding to the specific level of the number of products produced, and the break-even level of the project.
In reality, costs and revenues are not always proportional to the quantity of products. For small projects, when put into operation, the impact on input and output prices is often insignificant. For large projects, when put into operation, it has an impact on input and output prices.
The break-even point can also be calculated for each year depending on the goal to be analyzed. The following types of break-even points can be calculated:
• Theoretical break-even point (profit-loss break-even point) is determined by the same formula as presented above, but the only difference is that the total fixed cost is only calculated for one year of the project life.
The formula for calculating the theoretical break-even point for one year of the project is as follows:
- Output at theoretical break-even point:
x f
p v
In which: f is the fixed cost calculated for the considered year of the project life.
- Revenue at theoretical break-even point:
Oh = xp or
Oh h
f
1 v
p
• The monetary break-even point (cash break-even point) is the point at which the project begins to have money to pay off the loan, even with depreciation.
Formula for determining the monetary break-even point for one year of the project life
as follows:
- Output at the monetary break-even point:
x f D
1 p v
In there:
D: depreciation of the year under consideration.
f : fixed cost calculation for the considered year of the project life.
- Revenue at the monetary break-even point of the year under consideration is calculated:
O XP
or
O f D
ht t
ht 1 v p
• The debt service break-even point is the point at which the project has enough money to repay the loan and pay income taxes.
The formula for determining the debt repayment break-even point for one year of the project life is as follows:
- Output at the debt repayment break-even point:
x n
f D N T p v
- Revenue at the debt repayment break-even point is determined:
Oh hn X n . p
Or:
Oh
f D N T
1 v
p
In there:
N: principal payable in year T: corporate income tax.
1.3.2. Assessment of financial safety of investment projects
The financial safety of the project is a content that needs to be considered in the process of analyzing and appraising the financial investment project. It is an important basis for assessing the financial feasibility of the project.
The financial feasibility of a project is assessed not only through indicators reflecting the project's financial efficiency such as: IRR, NPV... but also on the following aspects:
• Capital security.
• Safety in the ability to pay short-term financial obligations and debt repayment capacity.
• High safety for calculated efficiency indicators (certainty of expected efficiency indicators of the project). This analysis is performed through sensitivity analysis of the project.
1.3.2.1. Capital safety
To consider the safety of project capital, it is necessary to pay attention to the following issues: