1.INTRODUCTION
Current uncertainties in factors related to investment alternatives, such as initial investment and annual return, require manufacturing companies to pay careful attention to methods for evaluating profitability and rigorousness against expected risks.
Methods for evaluating economic performance for a set of multiple investment alternatives, for both cases of stable return over the planning horizon, and of fluctuating return year by year, are the main areas of focus of this paper. This problem has been investigated in the field of engineering economy, and basic procedures have been clarified and modified/extended in previous research (Senju and Fushimi, 1982; Senju et al., 1986, 1994; Nakamura, 1985, 2002). Furthermore, general economic evaluation procedures with consideration of risk have been discussed in previous research (Kono, 2003, 2010; Kono and Ichikizaki, 2015; Kono and Mizumachi, 2004, 2009). The paper extends outcomes of previous research in the field of engineering economy, to propose effective tools in managerial decision making under today’s uncertain economic situation.
In evaluating profitability for investment alternatives, this paper pays attention to the net value of profit, incorporating the burden of capital cost under estimated interest rate, either at the present time (net present profit) or annual mean value (net annual profit). On the other hand, risk in this paper refers to either increase in initial investment, or decrease in annual return. Increase in initial investment results from such as increase in equipment price, or increase in construction labor charge. Decrease in annual return results from decrease in sales price or sales volume in practical situations. Increase in initial investment has direct impact to the net present profit if initial investment is large, and its impact to the net annual profit is evaluated with consideration of the interest rate. On the other hand, for calculating the net present profit, decrease in annual return is discounted by the interest rate and therefore decrease in late years will have less damage to the net present profit. Thus, the profitability and risk must be evaluated taking interest rate into consideration. The purpose of this paper is to present a method to quantitatively evaluate risk on a simple model of investment alternatives.
The paper presents the basic model for analysis in the next section, and then proceeds to the case of stable return in Section 3, to be followed by the case of fluctuating return in Section 4. Then simple numerical examples in Sections 5 are used to examine the effectiveness of the methods proposed in Sections 3 and 4. Since the paper assumes the case of evaluating investment alternatives for a specific product item, it assumes the common period n for all alternatives under consideration.
2.MODEL FORMULATION
The paper assumes the following investment alternative:
Figure 1 represents consistent return type. Figure 2 represents another case of fluctuating return year by year, where R_{j} refer to return for the jth year.
The paper incorporates two variables, α and β, in later sections, for evaluating impact of risk for investment alternatives. α denotes the increase ratio in the value of initial investment. For an investment alternative with positive net profit, the value of profit is decreased as α becomes larger. The breakeven point to realize zero profit will be denoted by α^{*} .
On the other hand, β denotes the decrease ratio in annual return. Along with decrease in the value of β , annual profit is decreased. The breakeven point to result in zero profit will be denoted by β^{*} in this paper.
The paper investigates profitability and safety under uncertainties for the abovementioned cases of stable return and fluctuating return over the planning horizon. It then proposes selection procedure for the both cases, when mutually exclusive multiple investment alternatives are given, from the viewpoint of profitability and safety against expected risks, using above mentioned variables α and β .
where each notation refers to

C: amount of initial investment

R: annual return (increase of cash inflow and/or decrease in cash outflow)

n: period of investment

i: interest rate to be used as hurdle rate in profit calculation
3.THE CASE OF CONSISTENT RETURN
3.1.Representation of Profit on the CR Domain
In this case, the net present value P and annual mean profit M can be obtained by the following equations.
where ${\left[M\to P\right]}_{n}^{i}$ is called “uniform series present worth factor,” which is obtained by $\frac{{\left(1+i\right)}^{n}1}{i{\left(1+i\right)}^{n}}$ and, ${\left[P\to M\right]}_{n}^{i}$ is called “capital recovery factor” defined by $\frac{i{\left(1+i\right)}^{n}}{{\left(1+i\right)}^{n}1}$. For the purpose of representing such profit values as P and M, this paper proposes a domain whose horizontal axis corresponds to the amount of initial investment C, and vertical axis refers to the annual return R. This domain is hereafter referred to as CR domain. Then, an investment alternative with initial investment C and annual return R is represented as a plot as shown in Figure 3. Depicting a line from (0, 0) whose slope corresponds to ${\left[P\to M\right]}_{n}^{i}$ , annual profit M of the investment alternative, which is obtained by statement (2), can be represented as in Figure 3.
Here, it is clear that the values of ${\left[P\to M\right]}_{n}^{i}$ and ${\left[M\to P\right]}_{n}^{i}$ are mutually inverse and the next statement holds:(3)
Therefore, on the CR domain, the line with the value of ${\left[M\to P\right]}_{n}^{i}$ can be represented as shown in Figure 4.
This implies that the net present value P, defined by statement (1), and net annual profit, defined by statement (2), can be represented as the horizontal and virtual length on the CR domain as in Figure 5.
3.2.Representation of IRR and Payback Period on the CR Domain
Along with the increase in interest rate, the slope of the line of ${\left[P\to M\right]}_{n}^{i}$ becomes steeper. Since IRR (Internal Rate of Return) r is defined to be the interest rate which makes the value of net profit M zero, the following statement is satisfied.(4)
It follows,(5)
Therefore,
Then, the value of IRR is depicted on the CR domain as shown in Figure 6.
On the other hand, as the period n becomes smaller, the value of ${\left[P\to M\right]}_{n}^{i}$ becomes larger. The value of payback period N is given by the period where the net profit is zero, and thus described by the next statement.(7)
Therefore, it follows,
Then the value of N can be described on the CR domain as in Figure 6.
The analysis conducted in Figure 6 implies that, among multiple alternatives, one with higher IRR always achieves shorter payback period. Therefore if we evaluate alternatives based on IRR, naturally ones with shorter payback period are selected. It should be noted, however, that selection by IRR is identical with selection by payback period, which leads to selection of alternatives of low risk, not necessarily guaranteeing high economic profit.
3.3.Evaluation of Risk
The paper then analyzes the risk of investment alternatives. In this paper, risks encompass those relating to increase in interest rate, increase in initial investment, and decrease in annual return.
Along with the increase in interest rate, the value of ${\left[P\to M\right]}_{n}^{i}$ increases. Therefore, the slope of the line of ${\left[P\to M\right]}_{n}^{i}$ on the CR domain becomes steeper. If the value of C is increased toαC(α>1), the present profit and the annual profit are decreased to(9)(10)
This statement implies that the decrease in net annual profit can be evaluated in the same context as the case of increase in interest rate. The breakeven point (BEP) α^{*}, at which net profit becomes zero, is given by(11)
Applying statement (8),
This implies that an alternative is safer if payback period N is small because the denominator of statement (12) is also small.
On the other hand, the decrease in annual profit from original value R to β R (β <1) can be evaluated by the next statement.(13)
For all the profit on the line connecting (0, 0) and plot (C, R), the decrease in annual return at the ratio β decreases the profit equally. Therefore, it is clear that the risk against decrease in annual return can be evaluated according to the same logic as the former two cases. BEP β^{*}, is given by the next statement.(14)
Then, applying statement (6),(15)
This implies that an alternative with large internal rate of return is safer against decrease in annual return.
From the above discussion, risk against expected changes can be evaluated simultaneously on the CR domain, based on the slope of the line connecting the point (0, 0) and each plot (C, R). The alternative with larger value of R/C (namely, steeper slope) is more rigorous in terms of risk. But it requires attention that the alternative with higher risk aversion level does not guarantee higher economic profitability.
3.4.Evaluation of Multiple Alternatives
The paper then examines the case of selecting the best one out of mutually exclusive alternatives, which can be delineated in Figure 7. The two alternatives can be represented on the CR domain as shown in Figure 8.
When line segments connecting (0,0), (C_{A}, R_{A}) and (C_{B}, R_{B}) create convex, it should be remembered that alternative A with smaller initial investment always guarantees larger IRR and shorter payback period, although the value of net annual profit for B may be larger than A. From the viewpoint of risk of increase in interest rate, increase by the same ratio in initial investment, and/or same ratio of decrease in annual return, alternative A is more rigorous than alternative B.
What requires attention is that, when comparing more than three alternatives, there might be cases where line segments connecting alternatives form concave as in Figure 9. In such a case, even if the interest rate fluctuates and the slope of ${\left[P\to M\right]}_{n}^{i}$ changes, the profit for the alternative B is always smaller than the alternatives A or C. Thus, the alternative B becomes disqualified in terms of profitability. It should be noted that when alternatives A, B and C create concave line segment on the CR domain, the next statement is satisfied.
$\Delta AB<\Delta AC\hspace{0.17em}\text{and}\hspace{0.17em}\Delta BC>\Delta AC,$
where(16)
If the initial investment for the three alternatives is increased by the ratio of α(α > 1) , then the slope of line segments, $\overline{{A}^{\prime}{B}^{\prime}},\hspace{0.17em}\overline{{A}^{\prime}{C}^{\prime}},\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\overline{{B}^{\prime}{C}^{\prime}}$ under Figure 10 are respectively represented by,(17)
Therefore, the relationship of(18)
holds. Thus, the line segments connecting plots A, B, and C on the CR domain remain concave and alternative B stays disqualified. In the same manner, alternative B stays disqualified, when annual return is decreased at the same ratio for the three alternatives.
It follows that the set of qualified alternatives on the CR domain creates convex line segments as in Figure 11.
4.THE CASE OF INCONSISTENT RETURN
4.1.Representation of Profit on the CR Domain
First, the cash flow pattern under investigation is described in Figure 12.
In this case, the net present profit P can be calculated by the next statement, whereas net annual profit is dependent on the return pattern and cannot be obtained directly.(19)
On the CR domain, where the vertical axis is con verted to the sum of annual return $\sum _{j=1}^{n}{R}_{j}$, depicting the line with slope 1 from (0, 0), the value of P can be represented as shown in Figure 13.
It should be noted that the values of IRR and payback period are dependent on the pattern of annual return, and therefore cannot be represented on the CR domain for the case of inconsistent return over the horizon.
4.2.Evaluation of Multiple Alternatives
This section analyzes the comparison of multiple alternatives, as illustrated in Figure 14.
The net present profit for alternatives A and B can be calculated by the following statements, where subscripts and superscripts refer to the name of respective alternatives.(20)(21)
Then, these values can be represented on the CR domain as in Figure 15. Profitability can be evaluated by the length of P_{A} and P_{B} as in Figure 15.
For the purpose of evaluating rigidity under uncertainties, this section considers the case of increase in the same ratio in initial investment (C → αC, α > 1) and decrease in the same ratio in annual return $\left({R}_{j}\to \beta {R}_{j},\hspace{0.17em}\beta <1,\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}\cdots ,\hspace{0.17em}n\right)$
In the case where the value of initial investment increases from C to αC, as shown in Figure 16, when the plot A reaches plot A′ the net profit becomes zero. Thus, the BEP α^{*} , can be given by the next statement.
On the other hand, if the value of $\sum}_{j=1}^{n}\frac{{R}_{j}}{{\left(1+i\right)}^{j}$ is decreased, the plot A on the CR domain moves downward. And if it reaches A′′ (refer to Figure 17), then the net present profit becomes zero. Therefore, BEP β^{*} for the case of annual return decrease can be given by the next statement (23).
Statements (22) and (23) show that α^{*} and β^{*} are mutually inverse, satisfying the next equation.(24)
4.3.Elimination of Disqualified Alternatives
This section examines the case of multiple alternatives represented on the CR domain, where line segments connecting adjacent plots may be concave, as illustrated in Figure 18.
As regards the increase or decrease in value of C for the same ratio among candidate alternatives, its impact can be shown as change in the slope of the line starting from (0, 0), upward with investment increase, and downward with investment decrease. In both cases, alternative B cannot attain maximum value of P, being less than either A or C.
In the same context, decrease or increase in annual return for the same ratio over each year among candidate alternatives, can be evaluated by the slope of the line connecting (0, 0) and each plot. Any plot on the same line gets the same impact from the change in annual return.
Therefore, the decrease (or increase) in annual return can be evaluated by the shift of the line with slope 1 starting (0, 0) upward (for return decrease) or downward (for return increase). In any case, alternative B in Figure 18 cannot achieve the maximum profit, and can therefore be disqualified in selecting the most profitable alternative.
Above discussion leads to a conclusion that a set of qualified alternatives, not from the viewpoint of profit but from that of risk aversion under uncertainty in initial investment and annual return, can create a set of convex line segments as shown in Figure 19.
5.NUMERICAL EXAMPLES
5.1.The Case of Consistent Return
This section considers the following three alternatives, with the interest rate i = 10%. Annual mean profit M for each alternative can be obtained as follows:Figure 20(25)(26)(27)
Therefore, alternative C is most profitable. But this calculation cannot evaluate robustness against uncertainties. Then each alternative should be plotted on the CR domain, which is shown in Figure 21.
It is clear from this figure that line segments connecting plots A, B, and C are concave, and therefore, alternative B is disqualified. Figure 21 also shows that the slope connecting plots A and C is 0.3. It follows that if the interest rate is increased to satisfy ${\left[P\to M\right]}_{5}^{i}>0.3$, that is, i > 16%, alternative A becomes more profitable than alternative C.
It can also be confirmed that, when initial investment is increased in the ratio α for all alternatives, A and C become equally profitable when α reaches α_{AC} which holds.(28)
Then, α_{AC} obtained is 1.137. This implies that if the initial investment is increased over 13.7% from the expected values, alternative A becomes more profitable.
In the same context, if annual return is decreased in the same ratio β for all alternatives, β_{AC}, in which alternatives A and C are equally profitable, is given by ${\beta}_{AC}=\frac{1}{{\alpha}_{AC}}=0.879$. This implies that if annual return for both alternatives A and C are decreased to 87.9% from the expected values, then alternatives A and C are equally profitable. Thus, BEP α_{AC} and βAC can work as indicators for economic decision making under uncertainties in relevant features.
5.2.The Case of Inconsistent Return
This section assumes the following numerical example (i = 10%).
The net present value of profit is given by:Figure 22(29)(30)
Two alternatives can be represented on the CR domain as in Figure 23. It can be confirmed that P_{A} is larger than P_{B} in this Figure 23.
In the case of decrease in initial investment, BEP for alternatives A and B to attain equal profit, denoted by , AB α is given by the next statement (refer to Figure 24).(31)
where ${R}_{A}={\displaystyle \sum _{j=1}^{n}\frac{{R}_{j}^{A}}{{\left(1+i\right)}^{j}}}$ and ${R}_{B}={\displaystyle \sum _{j=1}^{n}\frac{{R}_{j}^{B}}{{\left(1+i\right)}^{j}}}$.
Therefore, αAB is given by(32)
On the other hand, in the case where annual return decreases, the BEP, βAB necessary to attain equal profit for alternatives A and B is given by(33)
On the CR domain, it can be confirmed that the slope of the line segment connecting plots A and B is 0.938. It follows that if initial investment is decreased for both alternatives to 93.8% from the current estimation, profit for both alternatives becomes equal at the value 642.6. In the same context, if the annual return for each of both alternatives increases up to 1/0.938 = 1.066 from the current estimation, both alternatives become breakeven. If the expected change risk is lower, then it is suggested to select alternative A after consideration of risk under study. This implies that we can select the best investment alternative, not only on a basis of profitability, but also quantitatively evaluating risk factors in practice. Thus, the proposed analysis procedure on the CR domain helps economic evaluation and selection of alternatives under uncertainties.
6.CONCLUDING REMARKS
This paper investigated a problem of evaluating profitability and risk of investment alternatives, for both cases of consistent return and fluctuating return over the planning horizon. In the case of consistent return over the horizon, values of net annual profit and net present profit are represented on the CR domain, and risk evaluation procedure on the CR domain is clarified. In the case of inconsistent return over the horizon, net present profit is represented on the CR domain, and risk evaluation procedure on the CR domain is clarified. Further, breakeven point for the increase in initial investment and decrease in annual return is clarified as a tool for evaluating risk under uncertainties. Thus, major outcome of this paper is the procedure of visually evaluating profitability and risk on the CR domain. Especially, risk evaluation on the CR domain helps practical decision making under uncertain situations in the values of initial investment and annual return. In this context, this paper has practical purpose in addition to theoretical validity of analysis.
The paper simply denoted annual return by R (or R_{j} for the jth year). However, it actually comprises increase of income, such as sales increase, or decrease of production cost, including material cost and processing cost. Therefore, return R can be divided into such factors as sales volume, production volume, unit sales price, and unit variable cost. Then, previous research outcomes in the field of engineering economy applying totalcost unitcost domain, and/or capacity surplus and shortage distinction, can be combined to the analysis on the CR domain. Combining these results of analysis to help practical decision making is left as a topic for future research.