Some Views on Quantitative Analysis of Risk Management By Using Net Present Value Model

Introduction
Since a number of corporate and finance scandals including Enron, WorldCom and Tyco International incurred, tremendous losses were suffered by most business participators like investors, debtors, company personnel and public confidence in the markets was serious injured. In order to rebuilt the public confidence and protect the interests of investors, many regulations and standards like the SarbanesOxley Act of 2002 and Enterprise Risk Management – Integrated Framework of 2004 were established, so as to improve and enhance information disclosure, corporate governance and risk management (Committee of Sponsoring Organizations of the Treadway Commission, 2004).
With both development of theories and practices in finance and investment, methodologies and methods applied for investment and risk management by enterprise have got much more enriched. From game theory, portfolio theory, arbitrary pricing theory, utility function model in theoretical aspects to options, futures, hedging, forward contracts in practical aspects, companies are provided indeed lots of options to access their investment and manage their risks (Brealey etc., 2003). However, it is not so easy to evaluate these options and make the most appropriate investment decision. And net present value (NPV), as a widely used method for investment appraisal, has met these requirements by allowing most aspects to be considered in one framework including initial outlays, residuals, timing and time value of cash flows, opportunity distributions and opportunity costs. Risk assessment and management is included in NPV method, too, even though not as obvious as other aspects mentioned above, and this report would try to analysis the risk management by using NPV in quantitative way.

The Understanding of Net Present Value Equation
Many methods and tools are available for helping the investors to make better economic decisions, and these methods can be applied to single independent project or several available projects for determining which should be chosen. Generally, these various methods are categorized into five basic types: payback methods, accounting methods, ratio methods, rate of return methods and net present value methods (Remer etc., 1995), and the most popular one is NPV.
NPV is usually recognized as the bestconceived and most informative method to make investment decisions for it brings time value of money and opportunity cost into consideration. For example, compared with the cash flows generated from an investment project at the present time, the same amount of cash flows in the future is less worthwhile because it both losses the opportunity to be invested to earn profit during the period between the present and the future and confronts the uncertainty to be got. By discounting the future cash flows to present value and comparing the present value with the initial outlays, the difference between them will decide if the investment is acceptable. If the difference is positive or not negative, the investment is acceptable, and the higher the positive difference, the more attractive the investment (Weygandt etc., 2009). The difference is referred to as the net present value, and the method is called NPV method. A simple form of NPV equation is as below:
(1)
Where is the expected life of the investment, is the expected net cash flows at period , is the opportunity cost at period , and is the initial cash payments for the investment.
As it is seen from the equation above, when applying for NPV method there are three primary variables confronted to be estimated through the whole life of the investment: the net cash flows in the future, the opportunity costs and their attached probability distribution. However, the variables are not so easy to estimate as is shown above, and here are the detailed descriptions of the three variables as following:
The net cash flows equal cash receipts minus cash payments, and all cash flows related to the investment should be included. Sometimes the related cash flows are easy to identify. For example, to buy company A’s outstanding shares, the cash payments would probably be the amount to buy in these stocks, and the cash receipts would be the dividends and the amount received from selling out the stocks. But sometimes the related cash flows are not so easy to estimate. For example, product X is produced by both machine A and B in an assembly line, but X is finished by B with the semifinished products produced by A. When X is sold out and money is received, how to separate the money into machine A and B as their respective cash receipts? It is a similar subject for costs allocation in management accounting research, and definitely it is not a simple subject (Fabozzi etc., 2003). In order to simply the process of this report, an assumption of clear identification of related cash flows is applied.
The opportunity cost, deriving from economics for interpreting the relationship between scarcity and choice, is the cost related to the second best alternative which is foregone for the chosen investment. Then, with the help of risk preference conception and utility function, the opportunity cost is further developed to the cost related to the second best alternative whose risk is on the same level or similar with the invested project. For example, if A wants to invest some free money in bank X, the opportunity cost of the money is not the expected return from security markets or company bond markets because in which the risks of return is higher than that in bank X and A would not want to bear the higher risks to take these alternatives as the second best choices. The opportunity cost should be with the similar risk with that in bank X, like the return in bank Y, whose size, reputation and other aspects are all similar to band X’s. What’s more, even if A always puts the money in bank X, the opportunity cost would not stay stable because the return of bank Y would be changeable with macroeconomics, policies, technology development. However, the opportunity costs is aim to measure the risks of the invested projects rather than that of the second best alternative. For the risks of the invested projects in the future is unknown and hard to estimate, it is usually represented by some known and similar estimations. What’s more, risks are variable and various like systematic risks including financial crisis, depressions, wars, and unsystematic risks including credit risk, operating risk, finance risk, so opportunity costs would be both complicated to estimate and less likely to stay unchangeable. In order to simply the process of this report, the opportunity cost is supposed to be able to be estimated precisely like the cash flows assumption made above.
The probability distribution is aim to describe the different cash flows and opportunity costs in different situations at certain period. In reality, things are and will be always changing, and it cannot be certain that at period during the investment that the net cash flows would be and the opportunity cost is. In order to better describe the reality, the probability distribution concept is applied to the estimation of cash flows and opportunity costs. For example, at period , there is probability to realize net cash flows with opportunity cost, and with and . Assumption is made in this report that the probability of each situation can be identify so as to focus main attention on the main purpose of this report.
Another scenario is necessary to be specified here is that the three variables usually covary with each other together, like higher net cash flows may come with higher opportunity costs at lower probability, which means a situation should be described with four variables as a united group: the period , the probability of , the net cash flows of and the opportunity costs of . In this report, the function of is served for the description. For example, represents that at period , there is probability to realize net cash flows with opportunity cost, and is totally matched with the risk of the investment to realize .
By using the function of , the outcomes of the invested project at period can be stated completely as below:
Where , .
The expected present value of cash flows from the invested project at period is:
Where is the expected present value, are the expected value of opportunity costs in period respectively.
The equation (2) is simplified by using the expected value of as the opportunity costs in prior periods. And if the opportunity costs at period is supposed to be stable at with the risks attached on the investment keeping fixed and unchanged, the equation (2) can be rewritten into:
(3)
Where is the expected net cash flows at period , and is the expected opportunity cost at period .
Supposed the risks of the invested project stay stable during the whole life by effective risk management, it is reasonable to imply that the opportunity cost is fixed at during the life of the investment, so the equation can be transferred to:
(4)
To add up all the present values and compare the sum with the initial payments, the difference is net present value:
(5)

Risk Management and NPV Method
Risk management should be considered all the time no matter the investment has been made or will be, and as an important investment appraisal method, NPV does put a considerable weight on the assessment of the effectiveness of risk management. The ways of risk management are usually categorized into four basic types: risk retention, risk transfer, risk control and risk avoidance, and their respective features and effectiveness would be detailed by using NPV method as below.
In order to make the viewpoints in this report much clearer, an example would be used and each type of risk management would be specified in the same example so as to improvement the comparability between them.
The example: At period 0, person A invests money into company X by buying its common stocks, and the stocks are going to be hold for periods. At period , A is supposed to get dividends with expected value of . The estimation of discount factor would be measured by using CAPM model. According to CAPM model, company A faces two types of risks: systematic risks and unsystematic risks. Systematic risks are mainly from macroeconomics like financial crisis, depressions and wars, and unsystematic risks are mainly connected with business transactions in company A, like credit risk, operating risk, financial risk and reputation risk. The equation of discount factor is:
(6)
and (7)
Where measures the discount factor at period , is the riskfree rate, is the expected return of the market, is the market premium, measures the systematic risks of company A, measures the unsystematic risks of company A, and is the expected value of .

Risk Retention involves accepting the losses caused by risks. Supposed in the investment of company X, person A mainly faces the unsystematic risks, which means in the equation (6) , all the variables except remain unchangeable, and satisfies the random walk hypothesis with , then . So the net present value of the investment in this situation is:
(8)
If person A wants to eliminate all the unsystematic risks completely, he/she should increase his/her initial payments, so the net present value of the investment would be:
(9)
Compared the value of with , and if, the reasonable reaction that person A should take is to accept the risk, which is called risk retention. Risk retention is usually used in the situation that the level of risk is low, and the losses caused is acceptable, but to control or eliminate the risk would cost a lot.
As it is shown above, by taking the risk management of retention, the cash flows person A gets are mainly from dividends which are mainly affected by unsystematic risks, and timely assessment about the risk is necessary so as to ensure the risk is at the acceptable level. At the same time, person A need to take some supplementary preparation like to keep more free money to resist the effects from unsystematic risks.

Risk Transfer means to shift the risks from one party to another. Supposed that person A thinks the unsystematic risks which company X faces is beyond his/her acceptance, and he/she wants to sell out these stock and to buy some more conservative securities like government bonds, then the net present value of the investment is:
(10)
Where is the income from selling out the stocks.
Compared the value of with the expected net present value of when the stocks were not sold out, and if , to sell out these stocks is reasonable activities, which is called risk transfer. Risk transfer management is usually in the situation that the operation policy or strategy of the invested company is shifted away from the previous one, and the new one will bring more deviation to the value of the company even though it may create more value for shareholders. According to preference theory, an investment decision is made based on both returns and risks, and if risks’ level is beyond the intended level which one is willing to take, the investment would be unacceptable even though the return is beyond one’s expectation. By taking the risk management of transfer, the cash flows person A would get is the incomes from selling out the stocks.

Risk Avoidance involves taking steps to remove risks or engage into another available investment. In the example of this report, if person A want to quit the investment for higher risk level, he/she could sell out the stock to avoid the risks, just like what he/she does in risk transfer management. The difference between risk transfer and risk avoidance is that by taking risk transfer person A can transfer partial risks to others until to his/her acceptable level, while by taking risk avoidance person A would undertake none of risks from the investment in company X at all, and he/she would quit the investment totally and put the money into other alternative options, like stocks of company Y, or government bonds. The cash flows person A could get from risk avoidance management is similar with that from risk transfer management, and the discount factor would depend on his/her alternative options.

Risk Control involves reducing the probability of the loss from occurring or the severity of the loss. Supposed company is with higher unsystematic risk, person A doesn’t want to take these risks but either to miss the potential higher profitability of company X, so person A decides to take a portfolio which includes the stocks of company X. Given that the portfolio is so large that the unsystematic risks from each company can be cancelled out, person A could get the expected market return with taking average systematic risks, so the net present value for person A is:
(11)
Where is the cash flows from the portfolio at period , is the expected return of the market, and is the initial payments for the portfolio.
Compared the net present value of the portfolio with , if , then the portfolio should be taken. Risk control is the most widely used method by companies to manage their risks, for risks are both widely existing and some are hard or costly to avoid or transfer, and there are many available methods can be applied by companies to control their risks, like options, futures, hedging and forward contracts.

Conclusion
From theoretical aspect, NPV can be recognized as the bestconceived method to make investment decisions for it involves the most elements of an investment, like timing and time value of money, initial payments, risks and opportunity cost. However, in the realistic world, there are so many strong assumptions are required in the estimation of variables that it reduces the accuracy and precision of the results from NPV method. In order to make better investment decisions, some other methods used as the supplement of NPV is necessary.
Reference
Committee of Sponsoring Organizations of the Treadway Commission. (2004). Enterprise Risk Management – Integrated Framework.
Fama, Eugene F. & Miller, Merton H. (1971). The Theory of Finance. Chicago: Dryden Press
Marcus, Bodie Kane (2003). Investment. 5th edition. London: the McGraw – Hill Companies.
Brealey, Richard A. & Myers, Stewart C. (2003). Principles of Corporate Finance. 7th edition. London: the McGraw – Hill Companies.
Remer, Donald S. & Nieto, Armando P. (1995). A Compendium and Comparison of 25 Project Evaluation Techniques. Part 1: Net present value and rate of return methods. International Journal of Production Economics, 42: 7996
Robison, L.J. & Barry, P.J. (1996). Present Value Models and Investment Analysis. The Academic Page, Northport AL.
Weygandt, Jerry J., Kimmel, Paul D., Kieso, Donald E. & Aly, Ibrahim M. (2009). Managerial Accounting: Tools for Business DecisionMaking. 2nd Canadian edition. Ontario: John Wiley & Sons Canada, Ltd.
Fabozzi, Frank J. & Peterson, Pamela P. (2003). Financial Management and Analysis. 2nd edition. Ontario: John Wiley & Sons Canada, Ltd.
Ang, Andew & Liu, JUN (2004). How to Discount Cash flows with TimeVarying Expected Returns. The Journal of Finance, 6.
Markiw, N. Gregory (2008). The Principles of Microeconomics. 5th edition. Mason: SouthWestern Cengage Learning.