The underlying theory of the following discussion relates to Portfolio Theory. Assessing a risk-return profile of a portfolio entails not only determining the portfolio expected return and comparing it with the expected portfolio standard deviation of returns, but also assessing the correlation of expected returns between the assets making up the portfolio.
The Portfolio Expected Return and Standard Deviation The portfolio expected return is the weighted average (percentage value) of the expected returns for the portfolio's individual assets. For a two asset portfolio, the Portfolio Expected Return is calculated as follows:- The portfolio standard deviation encompasses not only the variances of the individual assets, but also the covariance between the rates of return for the pair/s of assets in the portfolio. Covariance measures the degree to which two variables move together relative to their individual mean values over time. Its magnitude depends on the variances of the individual return series and also the relationship between the series. Correlation is a statistical measure of the degree of relationship between two variables in terms of a range between -1.00 and +1.00. A portfolio standard deviation (in terms of correlation) is calculated as follows:- In order to evaluate the risk return profile and the diversification effect of a portfolio made up of two assets (Example illustrated in Table 1), we are going to take four Cases with differing correlations. For each Case, we are also going to illustrate three Portfolios with different weightings. Table 1 Case A A -1 correlation depicts a perfect negative relationship and represents the true benefits of diversification for minimizing risk. Had the two asset's individual returns and standard deviations been equal, the standard deviation would have been zero . Amongst the four Cases, this Case yields the lowest value of portfolio standard deviation. Case B Perfectly positive correlation means that the standard deviation for the portfolio is the weighted average of the standard deviations of the individual assets. A value of +1 indicates a perfect positive linear relationship between the two assets - thus exhibiting no diversification benefits and highest portfolio standard deviation value. Diversification benefits would only be observed in cases where there is less than perfect correlation. Case C A zero correlation means that the change in price of one asset has no effect on the price change of the other asset. The returns have no linear relationship, that is, they are statistically uncorrelated. Varying portfolio weightings under such scenario, would not explain anything on the overall portfolio risk. Case D The actual correlation of 0.67% has a positive value closer to 1 indicating a positive linear association between the two assets. Similar to what happened in Cases A and C, when comparing the outcomes for Portfolios 1 and 2, it would transpire that despite of the increasing proportion of the riskier asset (Canon), the overall standard deviation is lower. This is due to the diversification effect and the less than perfect correlation is acting as a 'hedge' to a portfolio composed solely of Toyota. As the percentage held in Canon is increased, it eventually causes the portfolio standard deviation to rise. Plotting the data for the three Portfolios for the four cases would result in four opportunity sets that would reflect potential combinations of efficient frontiers - outlining also, the minimum-variance portfolio . The Portfolio having the lowest correlation would have the bandiest type of curve to the left. Thus, investors could maintain their rate of return while reducing the risk level of their portfolio, by combining portfolios that have a less than perfect positive correlation. A notable result of Cases A, C and D in Table2 is that with a correlation of less than 1, it is possible to derive portfolios that have lower risk than either single asset. A basic principle of correlation is that as correlation falls below 1, the portfolio standard deviation must fall as more diversification effect would be present. This feature is evident in Table2. Toyota and Canon, both Japanese export companies, tend to be heavily effected by the fluctuations in the Japanese Yen . Thus, a priori, one would expect some level of positive correlation (Exhibit 3). Given a correlation of 0.67, a diligent investor could still be careful in choosing the best combination of assets yielding maximum return for a given level of risk. In our example, for Case D (out of the three possible portfolios), a risk averse investor would choose Portfolio B, as it carries the highest return at lowest risk. Despite of effective asset allocation, there may still be systematic risks which would affect even a portfolio made up of totally uncorrelated assets. Nonetheless, diversification is a key tool against random events in the market, especially during periods when the correlation level across various assets is high.
Introduction This part of the essay shall outline commonly-used Capital Budgeting measures. A discussion will follow detailing the benefits of two popular measures, which would at the same time, shed light about the short-comings of competing measures. Capital Budgeting is a cost-benefit exercise, entailing the process that companies use for making corporate investing and financing decisions on capital projects. Projects to be undertaken would be accepted or rejected based on the criterion being used. Capital Budgeting Measures (CBM) Net Present Value (NPV) NPV represents the difference between the present value of all future after-tax cash-flows (discounted to their present value by the required return 'r' - the opportunity cost of capital). The NPV is then compared with the capital outlay required by the investment. An investment would be worth undertaking if its return is at least equal to the return available in the financial markets. Since the NPV represents the amount by which the investor's wealth increases as a result of the investment, the decision rule for the NPV is to accept a project if NPV is greater than zero. Conversely, a project would be rejected if NPV is less than zero. Internal Rate of Return (IRR) IRR is calculated by discounting the net cash-flows at the IRR, until NPV is equal to zero. The discount rate in this measure is the IRR instead of r and does not depend on the interest rate prevailing in the capital market. Therefore, the number is intrinsic to the project and depending only upon the project's cash-flows. The decision rule for the IRR is to invest if the IRR exceeds the required rate of return for a project. Payback Period (PP) This method focuses on payback rather than profitability and it measures the number of years required for a project to recover its initial investment. PP considers that the shorter the payback period, the more attractive the investment is. Discounted Payback Period (DPP) This method follows the same principle followed by the PP method but also takes into consideration the time value of money. Accounting Rate of Return (ARR) ARR is the average project earnings after taxes and depreciation, divided by the average book value of the investment during its life. Profitability Index (PI) PI is the present value of a project's future cash-flows divided by the initial investment. This measure is closely related to NPV, in the sense that the PI is the ratio of the present value of future cash-flows, while the NPV is the difference between the present value of future cash-flows and the initial investment. A basic principle of this measure is that whenever the NPV is positive, the PI will be greater than one. In such case, one should accept the project and when PI is less than one, the project should be rejected.
Limitations of CBM Graham and Harvey (2001) presented a Survey regarding the most comprehensive measures used for capital budgeting. The Survey delves into the usefulness of these tools and reports the frequency of the use of commonly used measures used by U.S. and European corporations. From the Survey it transpires that besides NPV and IRR, other measures are also heavily used. The key attributes of NPV include the fact that it uses cash-flows rather than earnings and that the discounting factor caters for the time value of money. IRR is also widely used and occasionally preferred by certain companies, in view of its underlying benefits. For independent projects, using NPV and IRR often yields the same findings. Yet, there are cases involving two mutually exclusive projects, where the two measures would be in conflict. In such cases, using IRR would not be as effective as when using NPV, thereby transpiring some of the IRR's deficiencies. A limitation of IRR relates to the problem when a project has a mixture of positive and negative cash-flows - possibly resulting in multiple IRR values. In this regard, NPV would be able to handle multiple discount rates as each cash flow would be discounted separately. Furthermore, in the case of mutually exclusive projects, differing cash flow patterns can cause two projects to rank differently with NPV and IRR. Another circumstance that may cause differing results between the two measures, relates to the size of projects. In view of the more realistic discount rate that the NPV uses, when conflicts exist between the two measures, NPV criterion is deemed as more appropriate. The results of the Survey depict IRR as the most popular approach. This is presumably not surprising, in view of the several benefits of IRR which nearly equal those of NPV. Many firms may perceive NPV as complex and that it requires assumptions at each stage, whereas IRR simplifies a project to a single number that decision makers may utilise to determine a project's viability. Further to the above, albeit being the strongest measure, NPV has also other pitfalls, such as the sensitivity to discount rates, which when varied, may yield very different NPV results. Moreover, the possibility that an investment would not have equal risk throughout its entire time horizon may require the decision maker to make use of more than one discount rates and hence resulting in a more cumbersome process. Payback method ignores the time value of money and the risk of the project. Moreover, it also overlooks cash-flows after the payback period is reached. Yet, the Survey outlines that it is still heavily used in view of its simplicity and its usefulness as a measure of liquidity. By choosing this method, the risk of loss related with changes in economic conditions and obsolesces, may also be lessened. DPP does account for the time value of money and risk, but it ignores cash-flows after the discounting payback period is reached. Thus, it may not be deemed a good measure of profitability since it ignores cash-flows. ARR is based on accounting numbers rather than cash-flows and it does not account for the time value of money. There is no conceptually sound cut-off for the ARR that distinguishes between un/profitable investments. PI indicates the value one should receive in exchange for each unit invested, considers all cash-flows and time value of money. It is also relatively easy to communicate and considered a useful guide for capital rationing and when available investment funds are limited. Potential flaws of this method include the drawback of having to calculate the cost of capital and the possibility of not giving the correct decision when comparing mutually exclusive investments. When choosing an appropriate measure, besides the underlying limitations, other factors are usually taken in consideration, such as, the firm's size and the type of industry . The valuation principles used in capital budgeting are similar to the valuation principles used in security analysis and portfolio management. In fact, many of the methods used in this area are based on these Capital Budgeting measures.
Introduction This section of this essay examines how market risk models may be utilised for calculating a firm's cost of capital.
Market Risk Methods When using these methods, the discount rate to be used in evaluating a capital project is the rate of return required on the project by the diversified investor. The discount rate should thus be a risk-adjusted discount rate, which includes a premium to compensate investors for risk. This risk premium should reflect factors that are priced or valued in the marketplace. The Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) capture this risk premium and are two approaches used for calculating cost of equity and finding market based risk adjusted rates. The CAPM approach stems from basic relationship pertaining to CAPM theory:- In this regard, the selection of RF would be guided by the duration of projected cash-flows . Regarding the equity risk premium, one would typically estimate beta relative to an equity market index to estimate the equity risk premium. CAPM captures both systematic and unsystematic risk and assigns a single market risk premium for each security, whereas APT develops a set of risk premia. For an all-equity firm and for a project whose beta risk equals to that of a firm, a project's discount rate would thus be equal to the CAPM's/ APT's estimate of the expected return. APT accommodates the risks that may not be covered by the market portfolio. Infact, it allows for an explanatory model of asset returns and captures the sensitivity of the underlying asset to economic factors. Like CAPM, APT assumes that discount rates are based on the systematic risk exposure of the security. In general: Once the required rate of return is determined, this can be used either to find the NPV and hence to determine whether a project is acceptable or not. It may also be used as a rate to compare with the project's IRR. The required rate of return is therefore a hurdle rate and is specific to the project's risk. If we had to remove the assumption that a project has equal risk to that of the firm, the project would then need to be discounted at the rate equal to its own beta and an adjustment would be made to the project's beta. When a firm's cost of capital includes both equity and debt, the cost of capital would be the weighted average of each. Hence, after determining cost of equity one also estimate the cost of debt in order to be able to obtain the weighted average cost of capital (WACC) - which becomes the applicable discount rate to utilise for the NPV's evaluation. The cost of debt is the firm's borrowing rate - usually obtained by looking at the yield to maturity on the firm's debt. In general, the WACC is:- A mixed-capital structured firm that is considering a project having a different risk to that of the firm, would need to adjust the WACC calculation for the risk of the project. Various studies have shown evidence that CAPM is the most popular method for estimating the cost of equity capital and that it is used less by smaller, private firms. This result is understandable considering the difficulties related when estimating systematic risk where the firm's equity is not publicly traded.
References and Websites Bodie Z., Kane A., Marcus A. J. (2007) Part 3 in: 'Investments' 6th Edition (McGrawHill International Edition); Chartered Financial Analyst Program Curriculum; Level I (2008) Volume 4 and Level II (2009) Volume 3 (Pearson Custom Publishing); Hilier D., Ross S., Westerfiled R., Jaffe J., Jordan. B (2010) 'Corporate Finance' Chapters 6, 10, 12 (McGrraw-Hill Higher Education); Research Papers Graham J.R and Harvey C. R (2001) 'The Theory and Practice of Corporate Finance: Evidence from the field' (Journal of Financial Economics 61 (2001) - Accessed from the Online Library; Akalu M.M and Turner R (2001) 'The Practice of Investment Appraisal: An empirical Enquiry?' - Available from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=370935; Web pages
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