Corporate Finance Self-Learning Manual

1-1 UNIT 1 UNDERSTANDING CORPORATE FINANCE 1.0 OBJECTIVE The objective of this unit is to a) Explain the scope of corporate finance b) Examine carious possible objectives of the firm c ) Elaborate on the key decisions of finance d) Elaborate on agency issues e) Explain how three decisions of the firm are interrelated f) Describe various activities of finance function 1.1 INTRODUCTION Finance function in an organisation performs a key role in devising strategies, evaluation of most profitable opportunities and monitoring of selected projects. It encompasses cost control, budgeting and evaluation of reasons for failure and success alike. With increased local and global competiveness in the markets in this modern era, projects are no more a situation of exploiting the opportunities by select individuals and group with financial muscle. With increased access of resources including capital the field has become much wider with many not only ready to capitalise on the slightest of available opportunities but also create them of their own. In this sense finance function in an organisation has assumed greater significance and has started playing increasing role in decision making process of strategic importance. This is in sharp contrast to the role of finance function in the olden days where finance function was mainly confined to routine aspects of regulatory and monitoring of resources. The increased importance of finance function in the corporate world has primarily emerged from the development of capital markets since 1950s. Advanc ements in the capital markets with regard to behaviour of prices, developments of asset pricing models, techniques of risk management, approaches to valuation etc have placed onerous demands on the finance function in terms of increased literacy, intelligence and comprehension. An analytical approach has become almost mandatory in all financial decision making areas. Subjective approach with qualitative reasoning is being replaced by objective approach with quantitative details as far as possible. However, it does not mean to suggest that subjective approach would be completely and effectively be replaced by quantitative figures. 1.2 STAKEHOLDERS IN THE FIRM For any decision making the objective of such decision must be absolutely clear. Clarity of objective provides an unambiguous framework for an organisation to make suitable decisions especially when there are conflicting views amongst those who take decisions. Most decisions in an organisation are collective and conflict of opinions among people is inevitable. 1-2 Many believe that objective of the firm is dependent upon how the organisation is constituted. There are broadly three kinds of firms Proprietorship, Partnership and Corporations. Proprietorship is a firm owned by single individual. Such forms of organisations are small where single individual contributes the entire capital and solely responsible for all decisions in the firm. Here the individual and firm are one and same. When resources required become large, several persons pool together resources and run the enterprise jointly. Such firms are called partnerships. When resources required (not necessarily financial alone) become too large to be beyond the scope of few individuals the firm has to mobilise resources from public at large. Those who contribute capital are called shareholders. In order to provide an exit to the shareholders, such shares are normally listed and traded on stocks exchanges to facilitate the transfer of ownership from those wanting to exit to those wanting to enter. While not much conflict is seen in proprietorship and partnership organisations due to commonality of management and ownership there is a conflicting situation regarding corporations. Broadly speaking the capital can be provided either by way of debt or by way of equity. Debt providers are content with fixed return not linked to fortunes of the firm. Equity providers expect return dependent upon the performance of the firm. People who contribute equity capital have two distinct categories a) those with motive to invest to seek control and management of the enterprise, and b) those with motive to invest to make financial gain and not much concerned with management and control the firm. More over these shareholders are sparsely located with little Figure 1-1: Ownership and Management of a Corporation Corporation Shareholders Managers Need not be shareholders Involved in day-to-day functioning of the firm Mostly work for own professional careers sac rific ing loyalty Supposed to work in the i t t f h h ld Investors Need not be managers Sparsely located Generally no role in the management of enterprise Expect financial gains commensurate with the performance that may not fructify If not satisfied with the management it b lli hi Debt Holders Provide capital for fixed return Fortunes of returns not linked with performance No role in management 1-3 interaction among themselves. The ownership structure of a corporation is depicted in Figure 1-2 highlighting the differences among the providers of capital. 1.3 THE AGENCY PROBLEM The organisation structure of the firm poses a serious question with regard to management and ownership. Shareholders being owners have a right to manage the firm. However, if the motive is to earn a reward on capital it is not necessary that shareholders individually or collectively have the required skills and knowledge to successfully run the firm. In their own interest they need to appoint professionals to manage the enterprise they own. These managers are supposed to act in the interest of shareholders. Relationship between the managers and shareholders is akin to that of principal and agent, with managers acting as agents of the principal; the shareholders. Since management and ownership lies in separate hands the question that arises is that do the agents always act in the interest of shareholders; referred as agency problem. Managers have their own goals to pursue, which may conflict with the interest of the shareholders. Top management living in palatial bungalows in prime locations jetting around the world and riding expensive automobiles, at the expense of the firm they work in and supposedly in the interest of the shareholders, is a common occurrence. The cost of such luxuries is borne by the firm and is actually a personal reward for the mangers. Though certain luxuries may be admissible and actually may enhance the productivity but an overdose of such awards dents the bottom line of the firm and hits the shareholders. It is indeed difficult to draw a line between use and misuse and make a distinction. The conflict of interest between the interest of the shareholders and managers pursuing their personal enrichment at the expense of the firm is referred to as agency problem. A similar conflict may arise between other stakeholders in the firm where interest of one group comes in the way of goals of another group. Besides managers there are other stakeholders of in the firm. The next prominent stakeholder is the debt providers. Though debt providers are only interested in their known and fixed returns, but in order to ensure that these minimum required returns are generated they may put some restrictive covenants while providing debt. These covenants such as restrictions on acquisitions, major expansion plans etc. even though in the long term interest of all stakeholders constrain the freedom and pursuit the goals of shareholders. Similar conflict situations may arise in respect of other stakeholders such as customers, suppliers and government. This is condensed in Table 1-1. 1-4 1.4 OBJECTIVE OF THE FIRM In the context of several stakeholders and conflicting interest with the owners (shareholders) there needs to be an objective that serves the interest of all stakeholders alike. The framework of decision making must eliminate or minimise conflicts. In this perspective there are several objectives that come to mind such as maximisation of profit/ EPS, maximisation of market share, minimisation of cost, maximisation of shareholders wealth etc. At first sight they all seem to be same, but in fact it is not so. There are situations which can cause conflicts. Maximisation of profit or EPS would indirectly imply that the firm sells at higher price (conflict with customers), procures inputs at lesser cost (c onflict with suppliers), save cost of compliances (c onflict with government) etc. While maximisation of profit/ EPS serves the interest of the shareholders it does not mean that interests of other stakeholders are equally well served. Similarly maximisation of market share sounds good for customers but it may be sacrificing the interest of the shareholders. Same holds true for minimisation of cost. The other limitation is that such objective lack quantitative framework. The undisputed objective of the firm is maximisation of shareholders wealth. Shareholders are owners of the residual only i.e. get whatsoever is left only after obligations of all other stakeholders are served in full. Some may argue that maximisation of shareholders wealth is a narrow self-serving goal lacking the holistic view. Ac cording to such belief the organisation must exist for the welfare of the entire society and not one group of the society; i.e. shareholders. However, such belief is incorrect because this seemingly narrowed objective encompasses much broader aspect of social welfare. The interest of the shareholders comes in the last. They own only the residual howsoever small or big. In case the residual does not exist they fund the gap in resources. If firm continues to make losses it ultimately winds up and from that situation no one benefits. Table 1-1: Stakeholders and Shareholders: Conflict of Interest Stakeholder Goals Conflict Managers Personal career growth, increased remuneration and perquisites Increases cost reducing the benefit to the shareholders Debt Providers To ensure promised return is generated and principal is repaid on time and debt is secured by assets Covenants may restrict freedom of decision making and may come in the way of growth of the firm Government To place suitable laws on functioning of the firm and collect taxes Increases cost for legal and environmental compliances Customers Better quality products at lower price and extended credit period Constrains liquidity, demands investment adversely affecting shareholders cash flow Suppliers Supply inputs meeting bare quality standards at more than deserved price and early/ cash payment Constrains liquidity, demands investment adversely affecting shareholders cash flow 1-5 Further maximisation of wealth must not be construed as maximisation of profit. Wealth is much broader term that not only includes current profit but the future potential. Wealth is long term in nature while profit has limited time horizon. In case the shares are traded the stock price reflects the present value of the wealth. Profit is merely one constituent of determination of stock price and hence the wealth. Therefore stock price provides exact quantification of the wealth the shareholder would own. The changes in wealth could be measured by changes in the stock price. Wealth therefore would include present and future levels of profit, the firm s utility to the society at large, interest of other stakeholders, and the risk associated with the activities the firm undertakes. This is because the shareholders decide the management and future course of actions. Therefore maximisation of shareholders wealth supersedes all other objectives because of a) the long term view, b) shareholders sub-ordination to other stakeholders, c) consideration of risks involved, d) quantification, and e) shareholders control over management. 1.5 DECISIONS OF THE FIRM From the perspective of financial management the firm is required to make several decisions each of which influences the present and future activities. All these decisions must be viewed with an unambiguous and sole objective of maximisation of shareholders wealth. Finance function is required to make three important decisions: 1. Investment decision 2. Financing decision, and 3. Dividend decision 1.5.1 Investment Decision Investment decision relates to the allocation of financial resources to the contemplated activities of the firm. Alternatively it is referred as capital budgeting decision and broadly involves determination of requirement of financial resourc es including working capital. Investment decision is focussed on desirability of investment in expansions, acquisition, divestment and finding if acceptance or rejection of the business idea would add or destroy the value of the firm i.e. shareholders wealth. Most organisations face a constraint of capital and have larger number of projects available. Not all projects are equally rewarding and therefore need to be placed in an order of preferences. Therefore we need to have a decision making framework to decide the preferences. The decision to accept or reject a project is based on what happens to shareholders wealth. A project is accepted only when it is expected to increase the shareholders wealth. In case of deciding the priorities of acceptance the guiding principle is the accept projects that is expected to result in the greater increase of the shareholders wealth. 1-6 The investment decision can further be classified into two: 1) for fixed assets referred as capital budgeting decision, and 2) for current assets referred as working capital decision. Investment in fixed assets relating to acquisition of land, building, plant and machinery is concerned with long term growth and survival of the firm. In most cases the capital outlay is large and the decision irreversible. To that extent it becomes more risky and demands greater caution and exhaustive evaluation. In contrast the investment decision relating to current assets referred to as working capital decision is essentially a short term decision. It is subject to change c ontinuously and is reviewed periodically. Working capital decisions can be corrected periodically. Also the capital outlay is rather small. Characteristically, since working capital decisions are less risky and involve smaller outlays the decision framework could be different than that of capital budgeting decision. 1.5.2 Financing Decision After establishing the desirability of investment the next question is how to fund such investment. There are primarily two sources of funds available equity and debt as depicted in Figure 1-1. Suppliers of equity capital called shareholders provide capital with no fixed and assured reward. Shareholders fully recognise that there may not any residual left for them and therefore assume risk of getting no reward at all. Debt providers are interested in fixed reward quite independent of the performanc e of the firm. Though capital has no colour and does the same function irrespective of its source the mix of the two has significant influence on the investment decision and the objective of the firm i.e. maximisation of shareholders wealth. How much debt and how muc h equity must be mobilised is called financing decision and commonly referred as capital structure decision. One may wonder how such a decision adds or destroys value. The answer lies in understanding the linkages of investment and financing decisions. In theory the capital structure decision can be proved to be immaterial to the value creation (referred as irrelevance of capital structure and discussed later), but in real world such may not be the case. Because of the priority of claims of debt holders over equity holders the capital provided by latter is more risky and therefore costlier. The cost of debt capital is rather less as it face minimal risk in terms of return. Therefore the mix of two types of capital determines the overall cost of capital for the firm; an important input for determining the financial viability of the project. The risk profile of the nature of capital alters the return expectations of its suppliers. A project to be financially viable must meet the return expectations of the investors as reflected in the cost of capital. The investment decision is concerned with how adequately the cash flows of the project satisfy the required return of the suppliers of capital. Changing capital structure implies changing cost of capital, which in turn determines the acc eptance or rejection of investment decision. Therefore investment decision and financing decision are intricately linked. The financing decision attempts to answer the question if there is an optimal capital structure maximising the shareholders wealth. 1-7 1.5.3 Dividend Decision The third important decision of finance function is the dividend decision. Though the entire residual belongs to the shareholder the dividend decision relates to how much of the cash flow should be distributed now, called payout ratio and how much must be retained to fund the future growth of the firm. Again, though in theory the decision can be proved to be irrelevant to the value of the firm (referred as irrelevance of dividend and discussed later) but in practice several factors may make it relevant. How much of the profit should be retained for the future growth would impact the proportion of funds the shareholders would contribute in the investment. Thus dividend decision affects the future capital structure which has bearing on the investment decision. In this perspective dividend decision can be viewed either as independent decision capable of affecting the value of the firm of its own or as a passive residual decision i.e. investment decision and financing decisions precede it. Three dec isions as desc ribed above are c losely int errelat ed. Invest ment decision cannot be made without the capital requirement which can be met from external equity, retained earnings (internal equity) and debt. The other input required for investment decision is the cost of capital. All three sources of c apit al having different expec t at ions of ret urn have differing c ost s, put Figure 1-2: Inter Linkages of the Decisions of the Firm Investment Decision Financing Decision Dividend Decision Retained Cash Cash Disbursed Debt Equity Cost of CapitalCash Generated 1-8 together determine the overall cost of capital. The outcome of the investment dec ision is t he c ash flows generat ed whic h eit her c an be dist ribut ed or retained. Three decisions and their inter linkages are depicted in Figure 1-2. 1.6 FUNCTIONAL VIEW OF ORGANISATION All decisions of the firm require a collective view from various departments that constitute an organisation. Though there can be large number of departments in an organisation depending upon its size, nature and other factors we can clearly see minimum of four important functions of marketing, production, finance and human resources. Each of these functions would have sub-goals that not only may be at cross purposes with each other but also may not be consistent with the sole objective of the firm of maximisation of shareholders wealth. For example marketing department of any organisation would like to pass on the maximum benefit to the customer for increasing market share, gaining customer s favour in the interest of the firm. In their enthusiasm to gain sales the importance of top line they tend to lose the bottom line. Similarly, production department objective of producing standard products only maintaining no inventory of goods would come in direct conflict with the objectives of marketing function. Finance function would like to ensure that capital be deployed most efficiently and hence forc e faster collection. The sub-goals of the departments are made to ensure that personnel remain focussed and these sub-goals are meant to facilitate achievement of the broader objective of the firm. Some of the objectives of the departments are shown in Figure 1-3. However they may end up as hindrance and the top management must ensure congruence of departmental goals with the organisational goals. Figure 1-2: Functional Description of a Firm Board of Directors Marketing Production Finance HR Provide suitable manpower, train and retain them to ensure employee satisfaction Produce only on receipt of orders, and standardised goods to derive home the advantages of economies of scale To generate required resources at least c ost, and put them to most efficient use to optimise use of capital Supply customised product ready delivery at least cost with extended credit period to gain increased market share 1-9 1.7 ACTIVITIES OF FINANCE FUNCTION The activities of finance function can broadly be classified under following two heads: 1. Raising the required financial resources for the survival and growth of the firm 2. Monitoring and controlling the end-use of the financial resources There are many ac tivities that are required for mobilisation of resources. Management of public issues, raising debt from banks and financial institutions, coordination with rating agencies, Investment of temporary surplus in marketable securities etc are some of the activities. These are referred as treasury functions. In larger organisation coordination with the stock exchanges and maintaining of investor relations would also form the part of the treasury function. Under the second function normally referred as controller function the various activities are preparation of budgets and MIS, costing, accounting and audits, taxation. Development of systems and procedures for preventing misuse of funds also forms the part of this activity. In many of the activities the finance function performs the role of conflict resolution. As we have seen that sub-goals of various departments come in conflict with each other. In suc h situations finance function has to resolve the conflicts in the interest of the overall objective of maximisation of shareholders wealth. It is in this context finance function becomes more relevant and crucial to the functioning of the organisation. KEY TERMS Agency Problem Do the managers who are not owners act as agents of owners, the shareholders is referred as agency problem. Investment Decision The decision with respect to acc eptance and rejection of a project is called investment decision Financing Decision How much of debt and how muc h of equity is optimal for the objective of maximisation of shareholders wealth is called financing decision Dividend Decision What proportion if the earnings must be distributed and what proportion retained for future use is referred as dividend decision. SUMMARY The increased importance of finance function in the corporate world has primarily emerged from the development of capital markets since 1950s. Advanc ements in the capital markets with regard to behaviour of prices, developments of asset pricing models, techniques of risk management, 1-10 approaches to valuation etc have placed onerous demands on the finance function in terms of increased literacy, intelligence and comprehension. To understand the role of the firm one need to understand who the stakeholders in the firm are and what their objectives are. All the stakeholders such as owners, lenders, employees, customers, suppliers and government have different expectations from a firm. These different expectations make the objective of the organisation hazy. This needs to be replaced by an unambiguous objective. Large firms typically are managed by professionals who are non-owners and owned by vast number of individuals who are non-managers. Owners appoint the managers who must act as agents to their principals. The interests of managers may not be in the interest of owners and in case of conflict managers may take a decision favouring them. Such happenings are referred as agency problem. In order to resolve the conflicts among the stakeholders there must be an objective that is capable of resolving all conflicts. The undisputed objective of the firm is maximisation of shareholders wealth. The objective of maximisation of shareholders wealth supersedes all other objectives because of a) the long term view, b) shareholders sub-ordination to other stakeholders, c ) consideration of risks involved, d) quantification, and e) shareholders control over management. There are three important decisions a firm has to make the investment decision, the financing decision and the dividend decision. Investment decision relates to the allocation of financial resources to the contemplated activities of the firm. A project is accepted only when it is expected to increase the shareholders wealth. Of the total resources required, how much debt and how much equity is called financing decision and commonly referred as capital structure decision. Though the entire residual belongs to the shareholder the dividend decision relates to how much of the cash flow should be distributed now, called payout ratio and how much must be retained to fund the future growth of the firm. Three decisions of investment, financing and dividend are closely interrelated. Investment decision cannot be made without the capital requirement which can be met from external equity, retained earnings (internal equity) and debt. The other input required for investment decision is the cost of capital. The activities of finance function can broadly be classified under two heads: a) Raising the required financial resources for the survival and growth of the firm, and b) Monitoring and controlling the end-use of the financial resources. SELF ASSESSSMENT QUESTIONS 1. What is universally acceptable objective of the firm? 2. What are three important decisions of the firm? 1-11 3. How do investment decision and financing decision effect eac h other? Explain. 4. How investment decision and dividend decision are interrelated? 5. What do you understand by agency problem? 6. What are various functions of finance function? FURTHER READINGS 1. Srivastava & Misra (2008), Financial Management, Oxford University Press, Chapter 1 2. Prasanna Chandra (2009), Financial Management: Theory and Practice, Tata McGraw Hill, Chapter 1 2-1 UNIT 2 TIME VALUE OF MONEY 2.0 OBJECTIVE The objective of this unit is to a) Explaining what is meant by time value of money b) How to compute the time value of money c ) What is meant by annuity d) How to calculate the future value or the present value of money e) What is the present value and future value of annuities f) What is meant by Equated Monthly Instalments (EMIs) and how to compute them g) How to segregate the components of interest and principal from the EMIs. 2.1 INTRODUCTION One of the central themes of finance is the time value of money. It forms the backbone of most financial analysis starting from simple computation of interest one can earn from deposit in banks to the complex situations of determining net present values, valuation of securities, valuation of bonds in the debt markets, capital budgeting and mergers and acquisition. It is also widely used in personal finance to determine instalments of loans in leasing and hire-purchase transactions. More complex applications of time value of money can be found in mathematical proposition of finding the value of derivatives. The applic ations are too numerous to be listed down. Possibly no financial analysis would ever be complete without using the concept of time value of money. In fact one can be certain that analysis is incomplete and most probably wrong if it does not use the concept of time value of money. 2.2 MEANING OF TIME VALUE The time value of money recognises the fact that value of money changes with time. Most commodities lose value with time. With passage of time food gets deteriorated, water is evaporated, metals get rusted, land loses productivity, etc . However, such is not the case with money. Money grows with time. Even if one does not put it to use and instead lends it to someone who puts it to use and derive the benefits. The user would pay a price for this opportunity. Because capital is scarce its value enhances with time. The concept of time value of money recognises that value of the money is different at different points of time. Since money can be put to productive use, irrespective of whether it is actually done, the value of money is different depending upon when it is received or paid. In simpler terms that need no explanation that the money today is more valuable than the money tomorrow. 2-2 It is not because of the uncertainty involved with time but purely on account of timing. The difference in the value of money today and tomorrow is referred as time value of money. 2.2.1 Importance of Time Value Time value is important because we need to compare the money available at different points of time. Generally the evaluation and analysis is carried out on the basis of money generated or spent at different points of time. For meaningful comparison we need to compare apples with apples and oranges with oranges. We cannot mix the two. Likewise we need to compare the value of money at same point of time for effective and appropriate comparison. To make a judicious comparison we need to compares the likes and not dislikes. The application of time value of money is an inevitable tool in financial analysis. The scope of application of this concept is really vast. Since all financial issues almost invariably deal with the analysis of cash flows that occur at different points of time, it makes them non-comparable. For consistency of approach we need to aggregate these cash flows and they simply can t be added unless they are assume to occur at the same point of time. To bring these cash flows at the same point one needs the concept of time value of money. 2.2.2 Sources of Time Value Why should the value of money be different at different points of time is because of several reasons. These reasons are a) presence of inflation, b) preference of individuals for current consumption over future consumption, and c ) investment opportunities that make the money grow with time possibly without taking risk. Generally speaking money today is more valuable as compared to tomorrow is because of inflation. By inflation it is meant that the goods would be more expensive in future than what they are today. As we go deeper into the future the prices of goods would keep increasing. Hence what can be bought today with Rs 100, lesser quantity of the same goods can be bought in future. This applies in general and some products like computers, durables may defy the trend because technology absorbs the increased cost. In general the presence of inflation makes money more valuable today than tomorrow. The second reason having its origin in inflation is the preference of individuals to consume now rather than later. If we sacrifice today s consumption it may be done in the hope of having more in future. Possibly one would defer consumption if more of product were available tomorrow than what is available today. But due to inflation that is not likely to happen. Therefore, consumers deserve a reward for postponing the current consumption to future. This reward is likely to be increased amount of money in future. Finally, why the money has time value is because capital can be put to productive use, if not consumed now. Capital is scarce and there are people who can always put it to produc tive use. Banks perform the activity of passing on the capital from those with surplus to those with deficit. 2-3 2.3 RISK AND TIME VALUE OF MONEY Time value of money should not be construed as return on investment. They are distinctly different. When one invests the expectations of returns comprise of two factors 1. a reward for the risk undertaken in the investment and 2. a reward for waiting the investment to return. The reward for the risk must be commensurate with the risk. Risk can broadly be defined as uncertainty of the returns. When we talk of time value of money we refer to the second component of the return that is completely associated with the element of time and not risk. For the reasons stated in the preceding sub-section the value of money differs with time we are concerned only with the factor of deferment of consumption. Time value should not account for the risk associated with the investment. 2.4 PRESENT VALUE AND FUTURE VALUE Analysis of cash flow must be done at the same point of time. Usually analysis involves cash flows that occur at different points of time and they need to be brought to the same instant of time. Consider an example. Let us consider investment of Rs 100 in two alternative projects A and B that yield the following cash flows in 2 years: Year 0 1 2 Project A - 100.00 80.00 70.00 Project B - 100.00 90.00 60.00 Projects A and B seem equivalent as they return a cash flow of Rs 150 over two years for investment of Rs 100 toady. But are they really equal? Though initial investment is same and the aggregate cash flows are also equal but they occur at different points of time. That makes these projects unequal. To make the right assessment as to which of the two projects A and B is better and preferable, we need to compare the cash flows at same point of time. To do this we bring cash flows occurring at different times at the same point of time. We have following two choices: 1. To bring the cash flows of the projects to time t = 0, i.e. calculate the present value of all the cash flows, or 2. To bring the cash flows of the projects to time t = 2 i.e. find the future value of the cash flows at time t = 2. Present value: Present value answers the question how much is the present worth of the cash flow that occurs in future. Future value: Future value answers the question how much worth is the worth of the cash flow if it has to occur at future date. Whether we use present value or the future value we would reach the same decision even though the numbers would be different. 2-4 2.5 COMPOUNDING AND DISCOUNTING How do we get the present and future values? To know the present value or future value we need at what rate the value changes with time. The rate of change in the value of money due to time is normally spec ified in terms of percent per period normally a year. For example if the rate of change is 10% per annum then the value of cash flow of Rs 100 today is equal to Rs 110 after one year. Similarly a cash flow of Rs 100 a year later would have the present value of Rs 100/ 1.10 = Rs 90.91. As time elapses the present cash flows grows and the process of converting the present cash flow to the future value is referred as compounding. Similarly, as we go back in time the value of money decreases. The process of finding the present value (value at time t = 0) for the cash flow occurring some time in future is called discounting. Figure 2-1 and Figure 2-2 depict the process of compounding and discounting for the value of money with respect to time. 2.6 FINDING FUTURE VALUE Let us find the future value of the cash flows of the projects A and B. We assume that the rate of change of value of money with time is 10% per annum. It implies that if you are given Rs 100 today and decide not to use it now and instead deposit in a bank for a year your money grows by 10% to Rs 110. This is the opportunity cost. At the rate of 10% the value after one year becomes: Figure 2-1: Compounding Increasing Value with Time T= 0 T= 1 T= 2 T= 4 Figure 2-2: Discounting Decreasing Value with Time T= 0 T= 1 T= 2 T= 4 2-5 Rs 100 + 10% of Rs 100 = 100 + 10 = Rs 110 Or P x (1+ r) = 100 x (1 + 0.1) = 100 x 1.1 = 110; where P is principal amount invested and r is the rate of change per annum represented in decimal form. Further for the second year the amount becomes 110 x (1.1) = Rs 121.00, with Rs. 110 becoming principal amount for second period. The interest for the first period becomes entitled for interest during the second period. The earning of interest over interest is called compounding. The future value of an amount P at rate of change of r after n periods is given by Equation 2-1 Future Value F = P x (1+r)n Equation 2-1 Using Equation 2-1 we find the future values of all the cash flows of Project A and B at time T = 2 as shown below: Year 0 1 2 Project A - 100.00 80.00 70.00 70.00 88.00 - 121.00 Future value of Project A 37.00 Project B - 100.00 90.00 60.00 60.00 99.00 -121.00 Future value of Project B 38.00 The future value at time T = 2 implies that if all the cash flows that occur at time t = 0, 1, and 2 are combined they are equivalent to Rs 37 for Project A and Rs 38 for Project B. Clearly, despite the fact that the net cash flows for Project A and Project B were at Rs 50 (Rs 150 Rs 100), Project B is preferable than Project A. 2.7 FINDING PRESENTVALUE As we find the future value of a cash flow that occurs today, in similar fashion we can find the present value of the cash flow that occur in future. The proc ess used to find present value is called discounting. The present value for the future cash flow is given by Equation 2-2 (an alternative form of Equation 2-1) as below: n) r + 1 (F= P ..Equation 2-2 While computing the future value we transformed the cash flows at different periods to time T = 2, in computation of present value we would convert the cash flows of different periods to time T = 0. The underlying concept is the value all the cash flows at same instant of time. The value at T = 0 could be found by disc ounting the cash flows using Equation 2-2. The present values of the cash flows are computed below: 2-6 Year 0 1 2 Project A - 100.00 80.00 70.00 - 100.00 72.73 57.85 Present value 30.58 of Project A Project B - 100.00 90.00 60.00 - 100.00 81.82 49.59 Present Value 31.41 of Project B Again it is clear that Project B is preferable to Project A since the present value of B at Rs 31.41 is higher than that of A at Rs 30.58. 2.7.1 Reconciling Present and Future Values The above analysis of present and future values Future Value at T= 2 Present Value at T= 0 Project A Rs 37.00 Rs 30.58 Project B Rs 38.00 Rs 31.41 The future values and present values as stated above are different. But are they equivalent? Certainly yes. Let us examine if it is so. We may do so by converting either i) the future values to the present values, or ii) present values to future values. The value of cash flows at T=2 is equal to present value of 37.00/ 1.102 = Rs 30.58, and 38/ 1.102 = 31.41 Similarly, future value of the cash flows at T = 0 are 30.58 x 1.102 = Rs 37.00, and 31.41 x 1.102 = Rs 38.00 Since the present values and future values are equivalent we can make our judgement about desirability of project on either basis and yet we would reac h the same conclusion. 2.7.2 Future Value and Present Value Tables The future values and present values are so frequently used that tables providing these values become very handy. These tales are provided at the end of the study material as Appendices 1 and 2 for different rates from 1% to 20% and from periods of one year to 30 years. 2.7.3 Impact of Compounding and Discounting: Compounding increases value of money with time. There would be increase in value as we move one period ahead from Period 1 to Period 2. What about 2-7 increase in value in one period from Period 11 to Period 12. The increase in latter case would be much higher. The effects of compounding become larger and larger as time progresses. It is because of the reason that growth applies not only to the original sum at T = 0 but also to the money earned over last 10 year on it. 1. The money grows at increasing rate as time elapses. 2. The impact of compounding is more and more pronounced as the rate of interest increases. Like compounding, the effect of discounting is more pronounced if the cash flows are more distant in future. Discounting decreases value. 1. The decrease in value would be at increasing rate for more distant future cash flows. 2. Discounting is more pronounced as the discount rate increases. 2.8 ANNUITY Sometimes same cash flows occur at evenly spaced intervals. We also make investment on a periodic basis with same amount of investment in each period, like a recurring deposit in a bank, premium paid for life insurance, or investing in provident fund every month. The tenure of investment is fixed and known at the time of making the investment. We either receive or pay a fixed some in each period for specified number of periods. A stream of equal cash flows on a recurring basis at uniform intervals of time is referred as annuity. Though the term annuity refers to annual periods but it can be used for any interval of time other than a year but they must be equally spaced. 2.9 FUTURE VALUE OF ANNUITY Future value often referred as terminal value can be found using the principle of compounding. Compounding is used for every cash flow to find the value at maturity. The first cash flow would last till maturity while the second cash flow would last one period less. The last cash flow would be invested only for one period. For example consider a deposit of Rs 100 in a bank account every year for a period of 5 years that pays 10% per annum. The maturity value or terminal value (MV or TV) can be found by treating the each year s stream as a separate investment with first year payment being invested for five years, second year payment invested for 4 years and so on with last payment at 5th year invested for one year. What would be the value of such savings at the end of 5 years? Figure 2-3 depicts the outcome of such investment where each contribution is treated independently. The maturity value of such a recurring investment for an interest rate of r is found mathematically as = 100(1+r)5 +100(1+r)4 + 100(1+r)3 + 100 (1+r)2 + 100 (1+r) = 100(1.1)5 +100(1.1)4 + 100(1.1)3 + 100 (1.1)2 + 100 (1.1) = 100 x 1.6105 +100 x 1.4641 + 100 x 1.3310 + 100 x 1.2100 + 100 x 1.1 = Rs 771.56 2-8 The terminal value as computed above is stated in terms of a formula through Equation 2-3. n1nr) (1 n) (r, FVA 1, Rs of Annuity of Value Future Equation 2-3 The value in Equation 2-3 is referred as Future Value Annuity Factor at interest rate of r for n periods. Alternative form of Equation 2-3 is given below as Equation 2-41. r1 - r) + (1= n) FVA(r, years, n for r% at Factor Annuity Value Futuren .Equation 2-4 2.10 PRESENT VALUE OF ANNUITY We considered the issue of making investment on a periodic basis and determined the value of the investment at a future date in the earlier section by compounding the each cash flow. How do we find the value of the uniform stream of cash flow that occurs at regular intervals of time in future? To do so we reverse the process i.e. compounding is replaced by discounting. In the earlier section the issue was considered by making a recurring deposit in the bank for 5 years and found the terminal value. We now try to find the present value of the cash flow of Rs 100 that is available at the end of each year for 5 years. 1 Equation 2-3 is a geometric progression with subsequent value increased by a factor of (1+r). The sum of geometric progression Sn is derived as follows: r1 - r) + (1= S or,1 - r) + (1 = S x r: (2) from (1) g Subtractin) 2 .......( ) r + 1 ( + r) + (1 + .. .......... r) + (1 + r) + (1 + r) + (1 + r) + (1 = S ) r + 1 (get we r) + (1 by sides both g Multiplyin) 1 ........( r) + (1 + .. .......... r) + (1 + r) + (1 + r) + (1 + r) + (1 + 1 = Snnnnn 1 - n 4 3 2n1 - n 4 3 2n Figure 2-3: Future Value of Annuity Figures in Rs Time 1 2 3 4 5 End of 5 Amount 100.00 100.00 100.00 100.00 100.00 110.00 121.00 133.10 146.41 161.05 Terminal value of the cash flows after 5 years 771.56 2-9 Figure 2-4 gives the present value of Rs 100 occurring at the beginning of each period by using Equation 2-2 with rate of change at 10% per annum. The present value of recurring cash flows for an interest rate of r is found mathematically as = 100/ (1+r)5 +100/ (1+r)4 +100/ (1+r)3 + 100/ (1+r)2 + 100/ (1+r)1 = 100/ (1.1)5 + 100/ (1.1)4 +100/ (1.1)3 + 100/ (1.1)2 + 100 / (1.1)1 = 100 x 0.6209 + 100 x 0.6830 +100 x 0.7513 + 100 x 0.8264 + 100 x 0.9091 = Rs 379.07 Mathematically, the present value of an amount of Rs 1 received for n years at interest rate of r is represented as Equation 2-5. n1nr) (11 n) (r, PVA 1, Rs. of Annuity of Value Present Equation 2-5 The value is in Equation 2-5 is referred as Present Value Annuity Factor. These values are derived from following formula2: nnr) r(11 - r) + (1= n) PVA(r, years, n for r% at Factor Annuity Value Present...Equation 2-6 2.11 PERIODICITY OF COMPOUNDING AND DISCOUNTING 2 Equation 2-6 is a geometric progression with subsequent value decreased by a factor of 1/ (1+r). The sum of geometric progression Sn is derived as follows: nnnn -n1) - (n - 3 - 2 - 1 -n-n -4 -3 -2 -1nr) + r(11 - r) + (1= S or,r) + (1 - 1 = S x r: (2) from (1) g Subtractin) 2 .......( r) + (1 + .. .......... r) + (1 + r) + (1 + r) + (1 + r) + (1 = S ) r + 1 (get we r) + (1 by sides both g Multiplyin) 1 ........( r) + (1 + .. .......... r) + (1 + r) + (1 + r) + (1 + r) + (1 = S Figure 2-4: Present Value of Annuity Figures in Rs Time 0 1 2 3 4 5 Amount 100.00 100.00 100.00 100.00 100.00 90.91 82.64 75.13 68.30 62.09 379.07 Present value of annuity of 5 years2-10 How do we compute the present and future values if the periodicity of rate of growth is changed from annual to some other period? As a simple exposition consider Rs 100 growing at the rate of 10% per annum but the compounding period is reduced to 6 months. It means that the money would grow at the rate of 5% every six months. After 6 months the sum would be Rs 105. For the second six months Rs 105 becomes the principal and it would become 105 x 1.05 = Rs 110.25. In case of annual periodicity of compounding the sum would be Rs 110. The extra amount of Rs 0.25 comes from interest over interest in the second period of 6 months. If the periodicity is increased to 3 months the computation of amount would be as follows: Time Amount Interest % Interest Closing balance 1st quarter 100.0000 2.5% 2.5000 102.5000 2nd quarter 102.5000 2.5% 2.5625 105.0625 3rd quarter 106.0625 2.5% 2.6265 107.6890 4th quarter 107.6890 2.5% 2.6922 110.3812 We may generalise the above in the following Equation 2-7. Future Value F = P x (1+r/ m)mxn .Equation 2-7 Where r= annual rate of interest m = total number of periods and n = number of years Similarly we may generalise the present value of the future cash flows by Equation 2-8 by re-arranging the Equation 2-7. mxnr/ m) + (1F= P Value, Present .. Equation 2-8 Continuous Compounding & Discounting The maturity value of the deposit keeps increasing as periodicity increases from quarterly to monthly to daily. How far can we reduce the interval of compounding what ultimate value can be achieved? In the ultimate case the compounding becomes continuous. In case of continuous compounding the future value and discounted value may be given by following Equation 2-9 and 2-10 respectively. 10 - 2 Equation . .......... .......... e x F =e1 x F = P Value, Present9 - 2 Equation ...... .......... .......... .......... . e x P = F Value, Futurert -rtrt For a two year deposit of Rs. 100 the maximum maturity value (with continuous compounding) at 10% is 100 x e0.10x2 = Rs 122.1403 as against Rs 121.00 over a two year period with annual compounding. 2.12 EQUATED MONTHLY INSTALMENTS In personal fianc for housing, car loans etc the concept of time value of money is extensively used. Time value of money forms the basis of fixing the periodic repayment. In most cases these repayments are in regular equally 2-11 spaced in time and equal instalments payable at specific intervals usually monthly called Equated Monthly Instalments (EMIs). The EMIs are regular periodic payments whose present value, discounted at specified rate of interest, adds to the loan value. Through EMIs the lender recovers the original amount as well as the desired interest on the loan. As an example consider a loan of Rs 1,00,000 repayable in 5 yearly instalments with interest rate of 10%. Referring to Appendix A-4 for 10% for 5 periods we find PVA(10%,5) at 3.7908 implying that present value of Rs 1 received each year for 5 years is Rs 3.7908 at 10%. Therefore for a loan of Rs 1,00,000 the annual instalment of Rs 26,379.75 (1,00,000/ 3.7908) may be fixed. 2.12.1 Finding out EMIs with EXCEL The EMIs are fixed in such a manner that the cash flows of the bank yield the desired return of 10% p.a. This can easily be done with the help of EXCEL using PMT function, as displayed below: Principal Amount (Rs.) 20000 Interest rate (%) 10% Period (years) 3 EMI (Rs. Per month) =-PMT(10%/ 12,3*12,20,000,0,1) The syntax for PMT function uses five fields; 1) interest rate for the period 2) number of instalments (periods), 3) value of the loan (the loan amount), 4) residual value at the end of the period of loan (taken as 0 if all loan has to be repaid), and finally 5) a field describing the nature of payment, 1 for payment in advance and 0 for payment in arrears. This is shown in Figure 2-5. Figure 2-5: Calculating Equated Monthly Instalments (Payable in Advance) 2-12 2.12.2 Segregating the EMIs into Principal and Interest Each EMI paid consists of two parts i) interest on the outstanding loan and ii) repayment towards principal. It would be wrong to assume that the interest and principal repayment would be same in each EMI. Since in the beginning the loan is at maximum and therefore the interest component in the EMI is the largest. As we pay instalment 1. The amount of interest decreases, and 2. The amount of principal repayment increases However, though the two components vary in each instalment the total of the two remains same for entire duration of the loan. The procedure to segregate the amount into interest and principal repayment would be as follows; a) For the first instalment the entire loan amount would carry interest at 10%, b) Calculate the interest on principal (for the first EMI entire loan would be outstanding) at the given rate, c) Subtract the interest in (b) from the EMI, to get the repayment, d) Reduce the principal outstanding by the amount in (c). This would be the principal outstanding for next period, e) Repeat steps (b) to (d) till the last EMI. Using the above procedure the EMI of Rs 26,379.75 for the loan of Rs 1,00,000 for 5 years at 10% EMI is shown in Table 2-1. Table 2-1: Segregating EMIs in Interest & Principal Figures in Rs Year Principal outstanding at beginning Instalment paid Interest Principal repayment 1 1,00,000.00 26,379.75 10,000.00 16,379.75 2 83,620.25 26,379.75 8,362.03 18,017.73 3 65,602.53 26,379.75 6,560.25 19,819.50 4 45,783.03 26,379.75 4,578.30 21,801.45 5 23,981.58 26,379.75 2,398.16 23,981.59 2-13 2.13 FINDING PRESENTAND FUTURE VALUES OF ANNUITY USING EXCEL Future value of an annuity can be found using EXCEL. We need three inputs the rate of interest per period, the number of periods for which the amount is received/ paid, and the amount in each period. Then we may go to Insert- Function FV and find the value as shown in Figure 4-5. Figure 4-5: Finding Future Value of an annuity using EXCEL Like future value of an annuity we can also find present value of an annuity using EXCEL. We need three inputs the rate of interest per period, the number of periods for which the amount is received/ paid, and the amount in each period. Then we may go to Insert- Function PV and find the value as shown in Figure 4-7. Figure 4-7: Finding Present Value of an annuity using EXCEL 2-14 SOLVED PROBLEMS Example 2-1: Compounding and Future Value You have Rs 50,000 available today for investment. A bank has offered 12% interest payable annually. 1. What would be the maturity value of the investment after 5 years? 2. What would be the value of investment if compounding is done a) every 6 months, b) every 3 months. Solution: 1. The maturity value of the investment after 5 year at 12% is given by: 50,000 x (1 + 0.12)5 = 50,000 x 1.7623 = Rs 88,117.08 2. If interest is payable 6 monthly the rate would be 6% and number of periods would be 5 x 2 = 10. For quarterly compounding the interest rate is 3% with number of periods at 5 x 4 = 20. Therefore the maturity value would be For 6 monthly compounding: 50,000 x (1 + 0.06)10 = 50,000 x 1.7908 = Rs 89,542.38 For 3 monthly compounding: 50,000 x (1 + 0.03)20 = 50,000 x 1.8061 = Rs 90,305.56 Example 2-2: Finding Annuity Values If you were to receive Rs 1,00,000 every year for next 10 years what worth would it be today if current rate of return is 8%? Solution: The present value of the sum of Rs 1 lac for 10 years at 8% is given by: = Rs 1 lac x PV of Annuity (8%, 10 yrs) = 1.0 x 6.7101 = Rs 6.7101 lacs Example 2-3: Finding EMI AB Ltd. is borrowing Rs 1.50 lacs for a period of 5 years at interest rate of 11% repayable in 5 equal annual instalments at the end of each year. Find out the instalment amount, the interest paid each year and the total interest paid for the loan. Solution: We may find the amount of instalment using EXCEL function PMT(11%, 5,150000, 0, 0) or use the annuity table. The value of annuity at 8% for 5 years is 3.6959. Therefore the instalments would be 1,50,000/ 3.6959 = Rs 40,585.55. The break-up of each instalment into interest and principal is given below: Year Principal outstanding at beginning Instalment paid Interest Principal repayment 1 1,50,000.00 40,585.55 16,500.00 24,085.552 1,25,914.45 40,585.55 13,850.59 26,734.963 99,179.50 40,585.55 10,909.74 29,675.804 69,503.70 40,585.55 7,645.41 32,940.145 36,563.56 40,585.55 4,021.99 36,563.56TotalInterest 52,927.73 2-15 KEY TERMS Future Value The value of money at a future date with the given interest rate Present value The worth of the money today that is receivable or payable at a future date. Compounding The process of application of interest over interest period after period over a given sum at specified rate for specified time to know the worth of the money at a future date. Discounting The process of removal of interest over interest period after period on the given money at a future date to find out its worth in today s date. Annuity A fixed and equal amount of money receivable or payable at periodic intervals evenly spaced over time, usually a year. Equated Monthly Instalments An equal amount of money payable or receivable at periodic intervals of time usually a month that is equal to the amount of loan principal and the interest thereon at a given rate. SUMMARY Time value of money is the most important concepts in finance that forms the basis of decision making in almost all areas of finance. The applications range from personal finance areas to corporate finance like capital budgeting and valuation and derivatives and risk management. The time value of money means that besides the amount of money it is important when is it received. The reason for the time value of money is that it has capacity to increase in value even when it is not put to any use. The value of money increases with time due to application of interest. It grows at a higher rate when interest is applied on the interest. Application of interest over interest is known as compounding. Similarly the value of the money received or paid later is less than what it is today by the amount of interest for the time. The process of reduction in value eliminating the interest that could have accrued is known as discounting. Annuities refer to the equal amounts of cash flows spaced uniformly over time, normally a year. The value of equal amount of receivable or payable at evenly spaced intervals of time at a given rate of interest is called future value of annuity. Similarly for a sum receivable or payable at a future date can be equated with the equal amounts evenly spaced over time at a known rate of interest. 2-16 Present values and future values of a single cash flows or recurring equal cash flows are normally available in the Tables for computational use because of very frequent applications of these values in finance. One prominent application of the time value of money that is very widely used in the field of finance is the determination of equated monthly instalments for recovery of loans in specified time period and at a given rate of interest. SELF ASSESSSMENT QUESTIONS 1. What is meant by time value of money? 2. What are the possible reasons that the money has time value? 3. Would you consider the reward for the risk undertaken in an investment while calculating the time value of money? Explain. 4. What do you understand by a) future value and b) present value of money c) annuities? 5. What principle is used in determining the Equated Monthly Instalment? 6. For how long should one invest to get double the amount a) at 10% b) at 8%? 7. Suppose you have Rs. 12,000 today and need to preserve it for next 8 years when you are required to pay fee for your son estimated to be Rs. 40,000. a) At what rate should you invest this money so as to have the required sum at the end of 8 years? b) If you needed Rs 40,000 after 8 years and could invest at 12% only 8. AB Ltd. is borrowing Rs 50 lacs for a period of 4 years at interest rate of 15% repayable in equal instalments at the end of each year. Find out the instalment amount, the interest paid each year and the total interest paid for the loan. FURTHER READINGS 1. Srivastava & Misra (2008), Financial Management, Oxford University Press, Chapter 4 2. Prasanna Chandra (2009), Investment Analysis and Portfolio Management: Theory and Practice, Tata McGraw Hill, Chapter 5 3-1 UNIT 3 INTRODUCTION TO RISK AND RETURN 3.0 OBJECTIVE The objective of this unit is to a) Explain the meaning of return and risk b) Discuss various measures of return c) Draw distinction between arithmetic mean, geometric mean, IRR and expected value d) Discuss various measures of risk e) Demonstrates the superiority of standard deviation as measure o frisk f) Explain the use of historical financial data to find expected value and standard deviation 3.1 INTRODUCTION Risk and returns are the two sides of the same coin. They are inseparable and are so intricately linked that understanding of one without the other becomes difficult. It is common to hear phrases like the returns are attractive but the risk attached with it is too high or it seems a safe investment but the returns are indeed poor. All such statements are subjective and biased by personal preferences and attitudes towards risk and return. What could be risky for one may be too dangerous for another. Similarly, a 10% return on bank deposit would attract old and retired people but may not induce a young executive of a multinational firm. Risk and returns go hand in hand and need to be weighed together. Understanding of risk and return is vital for all. For an individual they become important for making investment decisions for the purpose of investing the savings and planning for retirement, while for firms they are critical dimensions along which all growth oriented projects and decisions of capital budgeting are taken. Irrespective of the motives for investment we need to find a way to measure risk and return. The purpose of this Unit is to render objectivity to the assessment of risk and return while refraining from making subjective interpretation of it being bad or good, adequate or inadequate, acceptable or unacceptable, high or low etc. We shall be dealing with the quantification of return and risk by attaching a number to them so as to enable comparison of different investment opportunities available. It is of significance because each individual or firm has a menu of investments to choose from. Even if all the investments may be acceptable they need to be ranked in order for making preferences along the parameters of risk and return amongst the alternatives available. 3.2 SOURCE OF DATA ON RETURN AND RISK For measurement of returns one needs a) the amount of investment b) its maturity value and c) period of investment. Since by convention returns are specified in annual terms the period of investment is deemed to be one year. Also initial investment if deemed to be Rs 100 the returns become annualised 3-2 percentage. This leaves only maturity value of investment to be determined for an initial investment of Rs 100. Similarly, the risk associated with the investment would be reflected in the returns offered. Here we assume a common sense preposition that increased risk demands increased returns. Risk and return would move in the same direction. To project the future value of investment one usually resorts to financial markets. The prices of the financial assets as reflected in the stock markets, currency markets and derivatives markets are normally used as efficient substitutes and best proxy for the investment of any kind. The returns offered by the financial assets must reflect true returns for several reasons. The reasons include i) the easy availability of price information of financial assets, ii) great reliability of prices as financial assets are frequently traded, iii) financial markets and prices, there reflect collective wisdom of the market independent of individual biases, and iv) the prices are freely determined in a competitive market, and even when markets are not competitive that is the closest to free markets we have. Based on the premise that returns offered by the financial assets would incorporate the risk associated in owning the financial asset the same information can serve as guide to assess and measure the risk as well. All we need to do to assess return and risk of an investment is to find a comparable financial asset that is traded in the financial markets. If we succeed in doing so our job of measuring returns and risk is greatly simplified and half done. Further with ever maturing financial markets world over the identification of comparable financial assets should pose no great threats to the most situations that we are likely to face in the real world. 3.3 RETURN Admitting that financial assets serve as ideal proxy for measurement of returns lets us assume that we wish to examine the returns from investment in telecom business. For such business the returns offered by investment in Bharti Telecom should serve as an ideal proxy for measurement of returns in such business. Similarly an investment in petro-chemical we can approximate the return to the returns on the stocks of a similar firm listed on the stock exchanges. Having appreciated the suitability of data of financial assets to serve as appropriate measure of returns let us focus on methods of measurement. For convenience assume that the investment horizon is one year. Assume that one makes an investment at t = 0 at in a firm whose price is Rs 100 (P0). After a year the stock price becomes Rs 110 (P1). During this period the firm also gives a dividend of Rs 5. Ignoring the time value of money, the % return on the can also be stated as % 5 1 x100100100 - 115x100Invested AmountInvested Amount - end at ValueReturn %= == This return can be split in two components of dividend yield and capital gains. The total return after a period of one year is Rs 15 on an investment of Rs 100. Of this an amount of Rs 5 is earned as dividend and Rs 110 is the profit upon selling of the asset after investment horizon of one year. It may be noted that whether 3-3 or not the divestment is done is immaterial to the computation of return. If actually divested the investor would realise the capital gain, and if not divested the capital gain would remain unrealised. Therefore, Total Return = Dividend Earned + Capital Gain = 5 + 10 = Rs 15 The percentage return can be expressed in mathematical terms assuming P0 as the initial price, D1 is the dividend received in the period 1, and P1 is the price at the end of period 1, as Equation 3.1 as follows: 1 - 3 ion .....Equat .......... ..........PP PPDP) P (P DInvestment InitialGain Capital DividendReturn Percentage Total00 10100 1 1--+ =+=+= In the above analysis we have ignored the time value of money, made no adjustment for inflation and the returns calculated are in nominal terms, and iii) not provided for taxes that may be payable on dividend and capital gains that are taxed at different rates. 3.4 EXPECTED RETURN The computation shown in the preceding section assumed that prices at the time of investment as well as divestment are known. Normally only the initial value of investment is known and the value of divestment remains unknown. In such a case the returns need to be worked out on expected value of asset in future. The returns so determined would be expected return. There are several ways one can estimate the expected return. These can be By Direct Questionnaire: One way of finding the expected return is to address a direct question to sufficiently large investors as to what price or return is expected by them, and use statistical methods to arrive at a consensus view about the return expectations of the investing community as a whole. Such a method is time consuming, cumbersome and costly. Develop a Valuation Model: Another approach to assess the expected return is to develop valuation model that determines the value of the asset at the end of investment horizon. Such an approach requires thorough understanding of the business and understanding of the critical value determinants. For example internet companies may be valued on the basis of number of hits or a real estate firm on the basis of land area owned etc. Such method of valuation with specific characteristics is adopted by equity research firms, investment advisors etc as deep understanding and intelligence is required for such an approach. Use Standard Model: Those who are unable to comprehend the business intricacies normally adopt more conventional models. These models are popular with large number users and include models such as CAPM (Capital 3-4 Asset Pricing Model), APT (Arbitrage Pricing Model), PE Ratio (Price Earnings Ratio) etc. The approach is easy to implement as data requirements are nominal and publicly available. One simply needs to plug in the publicly available data in to the model to arrive at expected return. Use Historical Data: Yet another effective and inexpensive way is to use the past returns as reflector of the future returns. It is believed that for conventional businesses the changes in the expected return changes too gradually. As such the returns offered in the recent past must give sufficient indication of the returns expected in near future. It is a time tested approach that requires minimal calculations. With the advent of information technology the process of computing past returns can be completed in matter of minutes. 3.5 ARITHMETIC MEAN (AVERAGE) From the historical price data of the financial assets it is easy to compute the average over the past. Assume that we are finding the average return on the stock whose prices have moved from Rs 100 to Rs 130 in 6 years. During these 5 years the firm has also paid dividend of Rs 12 every year in preceding 3 years and Rs 10 every year prior to that. The computation of the average return can be earnings by way of dividend of Rs 36 and Rs 20 in 5 years and Rs 30 as capital gain at the end of 5th year. Thus the average annual return is 86/5 = 17.2%. However, this is not the way returns are computed. In order to have better estimate we need to have returns on annual basis for all the five years. For this we need to have price of the stock at the end of each year. All the data is presented in Table 3-1. Table 3-1: Historical Data of Stock Price Figures in Rs YEAR Share Price Pt Dividend during the year, Dt Capital Gains (Pt Pt1)/Pt1 Dividend Yield Dt/Pt1 Rate of Return (%) 0 100 - - - - 1 110 10 10/100 10/100 20.00 2 120 10 10/110 10/110 18.18 3 100 12 20/120 12/120 6.66 4 90 12 10/100 12/100 2.00 5 130 12 40/90 12/90 57.78 The average rate of return is the simple arithmetic mean of the returns. The arithmetic average Rmathematically is represented by Equation 3-2, 2 3 Equation . .......... Rn1) R .. .......... R R (Rn1R Return; Arithmeticn1n n 3 2 1 = + + + = For the data in Table 3-1 the arithmetic mean of returns works out to 18.26% as follows: % 26 . 1853 . 9157.78) 2.00 6.66 - 18.18 (20.0051R Return; Arithmetic = = + + + = Arithmetic mean represents that on an average the stock yielded the return of 18.26% based on the past price and dividend performance. 3-5 Note that while computing the average return with annual data we obtained a different figure (18.26% vs 17.2%). It is due to the fact that average returns for the period used the price data at the beginning and end of each year, and therefore returns for the year were based on the price at the beginning of the period. Instead if the returns each year were computed on the base price of Rs 100 i.e. the price in the beginning, the annual return would be 17.2%. Also note that we ignored the time value of money in computing the arithmetic mean. 3.6 GEOMETRIC MEAN For the moment let us assume that there is no dividend on the stock. We can recomputed the return based on arithmetic mean as 7.37% (Readers are advised to verify the same as an exercise). If there were to be no dividend the investor would earn Rs 30 as capital gain only from his investment spanning 5 years. If the annual return is assumed to be r then it may be computed using following equation: 100 x (1 + r)5 = 130 gives r = 0.05387 or 5.387% The return computed above is based on geometric mean. The general expression specifying geometric mean return is given by Equation 3-3: Initial value x (1 + r)t = Final value .. Equation 3-3 where t = period of investment in years. We can also calculate the return based on the geometric mean from annual returns. The relationship between the annual return for the period n, Rn and the geometric return, Rg for investment lasting n periods is given by Equation 3-4. 1 - ) R + ..(1 )......... R + )(1 R + )(1 R + )(1 R + (1 = R) R 1 ..( )......... R 1 )( R 1 )( R 1 )( R 1 ( ) R 1 (nn 4 3 2 1 gn 4 3 2 1ng + + + + + = + .Equation 3-4 3.6.1 Arithmetic vs Geometric Mean We discussed the returns based on arithmetic and geometric mean. Which one of them is right is a big question. The answer is dependent upon the objective of finding return. In general when the performance of the firm is to be evaluated as compared to others, or where one wants to take a prospective decision to invest or not it is appropriate to use return based on arithmetic mean. Arithmetic mean keeps the investment period constant (usually a year) so as to facilitate comparison period after period or among the several equally spaced investment horizons. For all future decisions returns based on arithmetic average provide true guidance of what can be expected in future. On the other hand geometric mean provides the returns for the holding period. These are the returns actually earned over the investment horizon. Obviously these returns depend upon the prices at the time of entry and exit. Geometric mean must be used only for computing the realised returns, and cannot be taken as performance measure. 3-6 3.6.2 Reasons of Difference in Arithmetic and Geometric Mean The difference in arithmetic and geometric mean may be visualised rather easily with an exaggerated example. Consider the price of the stock at Rs 100. A year later price falls by 50% to Rs 50. However, in the second year the price rises back to Rs 100 providing the gain of 100%. Assuming that there were no dividends during these two years the returns based on arithmetic mean would be 25% i.e. the average of -50% and 100%. However, realised returns would be none since the values of investment at entry and exit are same. Using Equation 3-3 the returns would be zero as can be seen below: 100 x (1 + r)2 = 100 gives 1 + r = 1 or r = 0 The reason for the differences in the arithmetic and geometric mean is due to the differing investment value from period to period. Geometric mean assumes that the investment is compounded from period to period i.e. gains or loss of one period are re-invested in subsequent periods, and hence are included in the effective return one earns. In contrast arithmetic mean assumes that investment in each period remains constant. One either withdraws or invests more at the end of each period so as to keep the investment constant over time. Therefore the investment is adjusted. By doing so, we eliminate the compounding effect. Lets us consider the same price information as above. Assume that an investor buys one share at Rs 100 at t = 0. At the end of first year the value of investment falls to Rs 50. In order to bring investment back to Rs 100, one needs to buy one more share. The investor invests for one more share at Rs 50 to keep the investment constant for second year. At the end of second year with two shares in hand the end value of the investment is 2 x 100 = Rs 200. Deducting the value of investment of Rs 150 the gain is Rs 50 in two periods over constant investment of Rs 100. It is equivalent to earning an annual return of 25%, precisely equal to the returns given by arithmetic mean. 3.6.3 Relationship Between Arithmetic and Geometric Mean Geometric mean is always less than the arithmetic mean. Since the geometric mean takes into account the compounding while the arithmetic mean keeps the investment constant the difference in the two means would be dependent upon the variability of returns from period to period. The variability of the return is measured by standard deviation (discussed in the remainder chapter). Without getting into the mathematical proof the relationship between the geometric mean and arithmetic mean is approximated by Equation 3-5 where Rg is geometric mean and Ra is the arithmetic mean, and is the standard deviation of the returns. 221a gR R o ~ .Equation 3-5 The above relationship of the geometric mean and arithmetic mean is exact when returns follow normal distribution. 3-7 3.7 INTERNAL RATE OF RETURN (IRR) While computing returns based on arithmetic or geometric mean we ignored the time value of money. All the inflows and outflows were aggregated to find the return irrespective of the time of occurrence of the cash flows. Under the circumstances where cash flows are spread over several periods and vary in each period, the best way to find the realised returns is to use the discounted cash flow approach. The procedure is quite simple. We need to feed the successive cash flows in successive rows (or columns) and compute internal rate of return in the last row (or column) using the formula IRR (range of cash flows). Using EXCEL for the data in Table 3-1 the computation of IRR is shown in Figure 3-1. Figure 3-1: Computation of IRR using EXCEL The computation shows that the holding period return i.e. return over 5 years on the investment is 15.44% after taking into account the timings of the dividends as also the disinvestment. Note that price data for the stock in the interim is not required as we compute the holding period return. Again the question arises as to which of the two, i.e., IRR and Average Rate of Return, provides the correct answer. The difference between the two again lies in the constancy of amount of the investment. While computing IRR the use of actual cash flows implies that the investment in each period gets moderated with the value. While computing arithmetic mean we assume constancy of investment in each period. Therefore while making decisions about investing in future use of neither the geometric mean nor IRR is appropriate. It would be appropriate to make investment decisions on the basis of arithmetic mean as it provides a judicious basis. The geometric mean will distort the opinion depending upon the timings of investment and disinvestment, and IRR is subject to change if the investment amount is changing in each period. 3.8 RISK Besides return the other important dimension to investment decision is risk. Broadly, the risk can be defined as the divergence of the actual outcome from the expected outcome. Since we live in an uncertain world the expectations seldom come out to be true. For investment decisions one expects a return of say 20%. Even though this expectation of return is very realistic the actual return in most cases would be somewhat different than 20%. This variability of return is termed as risk. Hence we say that by making an investment while expecting 3-8 some return one has also assumed some risk. Therefore risk is inherent in investment decisions. The risk can be measured in several ways. Here we shall deal with the ways in which the risk can be measured, their advantages and disadvantages. 3.8.1 The Expected Value: Probability Distribution Before we discuss the ways of measuring risk, lets us understand what is meant by expected return. In the preceding section we stated that arithmetic mean provides a judicious basis of expected return. We defined the average return as the sum of observations divided by the number of observations. Arithmetic mean as basis for expected return 1. We implicitly assumed that past performance would be repeated in future too, 2. We also presupposed that the reliable past data about the return is available, and 3. We assumed that all observations were equally likely while it served as benchmark for future. All the three assumptions though seem reasonable yet may not represent true expectations of future returns. Past performance may or may not be repeated, and there could be several reasons for being so. Enough, reliable and authentic past data may not be readily available. And lastly all the returns on the assets are not equally likely. Some values of returns occur more often, while some values occur fewer times when some extraordinary events happen. In such situations the returns are not evenly distributed. There is a greater chance that some values will occur more often than others. This estimation of how likely each return is can be arrived in many ways including the past data. A graphical plot of the values of the return on the horizontal axis and the frequency of occurrence on the vertical axis is referred as frequency distribution. An example would illustrate the point. Consider the data of returns for 300 observations of returns as given in Table 3-2. Table 3-2: Frequency Distribution of Return Frequency of occurrence 15 25 30 45 60 50 35 25 15 Returns (%) 5.0 8.0 10.0 12.0 15.0 18.0 20.0 25.0 30.0 Probability x Return 0.25 0.67 1.00 1.80 3.00 3.00 2.33 2.08 1.50 Expected Return, % 15.63 In a scenario as depicted in Table 3-2 where the frequency of occurrence is not same, the average will not represent the expected value because arithmetic mean considers all values equally likely. One has to take into account the likelihood of occurrence of each value. Instead the expected value can be calculated as: Expected Value = Sum of Product of Probabilities and Values 3-9 These values are shown in the last row of Table 3-2 and the expected value is the sum of all the values and comes to 15.63%. The expected value can be represented as Equation 3-6. outcomes possibe of nos. noutcome i the for Return Routcome i of occurrence of y Probabilit pWhere6 - 3 Equation .. .......... .......... .......... .......... .......... R p = E(R)thithin1 = ii i=== The frequency distribution of returns is plotted in Figure 3-2 for the data in Table 3-2. The visual presentation leads to better appreciation of the likely returns. Figure 3-2: Frequency Distribution of Returns 3.9 MEASURES OF RISK Different people have different connotations about risk. Some people associate risk with the maximum loss that one can incur in an investment. Some others would talk of the chances of not making the desired gains. Yet another set of people would associate risk with the extreme values the investment may take. Hence different people perceive risk differently. Also we must appreciate the fact that what is termed as risky by someone may be perceived safe by another. Therefore risk becomes a matter of attitude one has. Without challenging the different perceptions of risk of being right or wrong our attempt in this section is to find a suitable measure for assessing risk. We shall be refraining from subjective interpretations such as being safe or risky, acceptable or unacceptable, high or low. These interpretations are dependent upon the profile of the investor and reflect the state of mind. In order to be so there is a need to have a measurement of risk that is objective and unambiguously states the fact but leaves the judgement of it being acceptable or otherwise to the individuals. We now discuss several measures of risk. 3-10 3.9.1 Range Range is one measure that can give an idea about the magnitude of risk. The difference between the maximum and minimum values of the return may be defined as a range. Mathematically it can be stated as: Range = Maximum Value Minimum Value ...Equation 3-7 The measurement of risk by range emphasises the possible extreme values that an investment may take. For example, consider investment A at a price of Rs 200. Best estimates of future price reveals that it can go as high as Rs 400 if good conditions prevail and while under depressed conditions the price fall to as low as Rs 50. For another investment B, the estimated maximum and minimum values are Rs 300 and Rs 75 respectively. On the basis of the difference in the extreme values investment A may be regarded as more risky having a larger possible variation of Rs 350 (400 50) as compared to Rs 225 (300 75) for investment B. While range seems a simple measure of risk it ignores the likelihood of such extreme events happening. If in the above case the likelihood of price of investment A falling to Rs 50 is 10%, while for investment B the probability of price falling to Rs 75 is only 50% the perception of risk would change. To have a clear view of risk one has to not only consider the range of values that the price/return can take but also the probabilities of the different values. Hence, the range cannot be regarded as an appropriate measure of risk. Even if the probabilities are same for extreme values the merit of judging risk on such remote values is doubtful. 3.9.2 Average Deviation Rather than evaluating ris