Financial management notes :-Unit IWhat is the objective of financial management? What do you think should be the objective? What do a finance manager do? Suppose he makes available the required funds at an acceptable cost and those funds are suitably invested and that every thing goes according to plan because of the effective control measures he uses. If the firm is a commercial or profit seeking then the results of good performance are reflected in the profits the firm makes. How are profits utilized? They are partly distributed among the owners as dividends and partly reinvested in to the business. As this process continues over a period If you dont know where you are going, it does not matter how you get there of time the value of the firm increases. If the share of the organization is traded on stock exchange the good performance is reflected through the market price of the share, which shows an upward movement. When the market price is more a shareholder gets more value then what he has originally invested thus his wealth increases. Therefore we can say that the objective of financial management is to increase the value of the firm or wealth maximization. Objective: Maximize the Value of the Firm Brealey & Myers: "Success is usually judged by value: Shareholders are made better off by any decision which increases the value of their stake in the firm... The secret of success in financial management is to increase value." Copeland & Weston: The most important theme is that the objective of the firm is to maximize the wealth of its stockholders." Brigham and Gapenski: Management's primary goal is stockholder wealth maximization, which translates into maximizing the price of the common stock. The Objective in Decision Making In traditional corporate finance, the objective in decision-making is to maximize the value of the firm. A narrower objective is to maximize stockholder wealth. When the stock is traded and markets are viewed to be efficient, the objective is to maximize the stock price. All other goals of the firm are intermediate ones leading to firm value maximization, or operate as constraints on firm value maximization. The Criticism of Firm Value Maximization Maximizing stock price is not incompatible with meeting employee needs/objectives. In particular: - Employees are often stockholders in many firms - Firms that maximize stock price generally are firms that have treated employees well. Maximizing stock price does not mean that customers are not critical to success. In most businesses, keeping customers happy is the route to stock price maximization. Maximizing stock price does not imply that a company has to be a social outlaw. Why traditional corporate financial theory focuses on maximizing stockholder wealth? Stock prices are easily observable and constantly updated (unlike other measures of performance, which may not be as easily observable, and certainly not updated as frequently). If investors are rational, stock prices reflect the wisdom of decisions, short term and long term, instantaneously. As it is, it is believed that market discounts all the information in the form of market price of the share. Why not profit maximization? Profitabilityobjectivemaybestatedintermsof profits, returnoninvestment, or profit to-sales ratios. According to this objective, all actions such as increase income and cut down costs should be undertaken and those that are likely to have adverse impact on profitability of the enterprise should be avoided. Advocates of the profit maximisation objective are of the view that this objective is simple and has the in-built advantage of judging economic performance of the enterprise. Further, it will directtheresourcesinthosechannels that promise maximum return. This, in turn, would help in optimal utilisation of society's economic resources. Since the finance manager is responsible for the efficient utilisation of capital, it is plausible to pursue profitabilitymaximisationas theoperational standardtotest theeffectiveness of financial decisions. However, profit maximisation objective suffers from several drawbacks rendering it an ineffective decisional criterion. These drawbacks are: (a) It is Vague It is not clear in what sense the term profit has been used. It may be total profitbeforetaxoraftertaxorprofitabilityrate. Rateofprofitabilitymay again be in relation to Share capital; owner's funds, total capital employed or sales. Whichofthesevariantsofprofitshouldthemanagementpursueto maximise so as to attain the profit maximisation objective remains vague? Furthermore, the word profit does not speak anything about the short-term and long-term profits. Profits in the short-run may not be the same as those in the long run. A firm can maximise its short-term profit by avoiding current expenditures on maintenance of a machine. But owing to this neglect,the machinebeingput tousemaynolonger becapableof operationafter sometime with the result that the firm will have to defray huge investment outlay to replace the machine. Thus, profit maximisation suffers in the long run for the sake of maximizing short-termprofit. Obviously, long-term consideration of profit cannot be neglected in favor of short-term profit. (b) It Ignores Time Value factor Profit maximisation objective fails to provide any idea regarding timing of expected cash earnings. For instance, if there are two investment projects and suppose one is likely to produce streams of earnings of Rs. 90,000 in sixth year from now and the other is likely to produce annual benefits of Rs. 15,000 in each of the ensuing six years, both the projects cannot be treated as equally useful ones although total benefits of both the projects are identical because of differences in value of benefits received today and those received a year two years after. Choice of more worthy projects lies in the study of time value of future flows of cash earnings. The interest of the firmanditsownersisaffectedbythetimevalueor. Profit maximisation objective does not take cognizance of this vital factor and treats all benefits, irrespective of the timing, as equally valuable. (c) It Ignores Risk Factor Anotherseriousshortcomingoftheprofitmaximisationobjectiveisthatit overlooks risk factor. Future earnings of different projects are related with risksof varyingdegrees. Hence, different projectsmayhavedifferent valueseventhoughtheirearningcapacityisthesame. Aproject with fluctuating earnings is considered more risky than the one with certainty of earnings. Naturally, an investor would provide less value to the former thantothelatter. Riskelementof aproject isalsodependent onthe financingmixof theproject. Projectlargelyfinancedbywayofdebtis generally more risky than the one predominantly financed by means of share capital. Inviewoftheabove, theprofitmaximisationobjectiveisinappropriate and unsuitable an operational objective of the firm. Suitable and operationally feasible objective of the firm should be precise and clear cut and should give weightage to time value and risk factors. All these factors are well taken care of by wealth maximisation objective. That is why we have Wealth Maximisation as an Objective Wealth maximisation objective is a widely recognised criterion with which the performance a business enterprise is evaluated. The word wealth refers to the net present worth of the firm. Therefore, wealth maximisation is also stated as net present worth. Net present worthis difference betweengross present worthandthe amount of capital investment required to achieve the benefits. Gross present worth represents the present valueofexpectedcashbenefitsdiscountedat arate, whichreflectstheir certaintyor uncertainty. Thus, wealth maximisation objective as decisional criterion suggests that any financialaction, whichcreateswealth or which, has a net present value above zerois desirable one and should be accepted and that which does not satisfy this test should be rejected. The wealth maximisation objective when used as decisional criterion serves as a veryuseful guidelineintakinginvestment decisions. Thisisbecausetheconcept of, wealthisveryclear. It representspresent valueofthebenefitsminusthecost ofthe investment. The concept of cash flowis more precise in connotation than that of accountingprofit. Thus, measuringbenefit interms of cashflows generatedavoids ambiguity. Thewealthmaximisationobjectiveconsiders timevalueof money. It recognises that cash benefits emerging from a project in different years are not identical in value. This is why annual cash benefits of a project are discountedat a discount rate to calculate total value of these cash benefits. At the same time, it also gives due weightage to risk factor by making necessary adjustments in the discount rate. Thus, cash benefits of a project with higher risk exposure is discounted at a higher discount rate (cost of capital), while lower discount rate applied to discount expected cashbenefitsofalessriskyproject. Inthisway, discountrateusedto determine present value of future streams of cash earning reflects both the time and risk. . In view of the above reasons, wealth maximisation objective is considered superior profit maximisation objective. It may be noted here that value maximisation objective is simply the extension of profit maximisation to real life situations. Where the time period is short and magnitude of uncertainty is not great, value maximisation and profit maximisation amount almost the same thing. Objective redefined :-Although shareholder wealth maximization is the primary goal, in recent years many firms have broadened their focus to include the interests of stakeholdersaswell asshareholders. Stakeholdersaregroupssuchas employees, customers, suppliers, creditors, and owners who have a direct economic link to the firm. Employees are paid for their labor, customers purchase the firm's products or services, suppliers are paid for the materials and services they provide, creditors provide debt financing, and owners provide equity financing. A firm with a stakeholder focus consciously avoids actions that would prove detrimentalto stakeholders bydamagingtheirwealthpositionsthroughthetransferof stakeholder wealth to the firm. The goal is not to maximize stakeholder well being, but to preserveit. The stakeholderview tendstolimitthe firm'sactionsin order to preserve the wealth of stakeholders. Such a viewis often consideredpart of thefirm's "social responsibility." It is expectedto provide long-run benefit to shareholders by maintaining positive stakeholder relationships. Such relationships should minimize stakeholder turnover, conflicts, and litigation. Clearly, the firm can better achieve its goalof shareholder wealth maximization with the cooperation of- rather than conflict with-its other stakeholders. To achieve the objective of financial management there are four major decisions that a manager takes. The Four Major Decisions in Corporate Finance/Financial management The Allocation (Investment) decision Where do you invest the scarce resources of your business? What makes for a good investment? The Financing decision Where do you raise the funds for these investments? Generically, what mix of owners money (equity) or borrowed money (debt) do you use? The Dividend Decision How much of a firms funds should be reinvested in the business and how much should be returned to the owners? The Liquidity decision How much should a firm invest in current assets and what should be the components with their respective proportions? How to manage the working capital? A firm performs finance functions simultaneously and continuously in the normal course of the business. They do not necessarily occur in a sequence. Finance functions call for skilful planning, control and execution of a firms activities. Let us note at the outset hat shareholders are made better off by a financial decision that increases thevalueof their shares, Thus whileperformingthefinancefunction, the financial manager shouldstrive tomaximize the market value of shares. Whatever decisiondoesamangertakesneedtoresultinwealthmaximisationofashareholder. Investment Decision Investment decision or capital budgeting involves the decision of allocation of capital or commitment of funds to long-term assets that would yield benefits in the future. Two important aspects of the investment decision are: (a) the evaluation of the prospective profitability of new investments, and (b) the measurement of a cut-off rate against that the prospective return of new investments could be compared. Future benefits of investments are difficult to measure and cannot be predicted with certainty. Because of the uncertain future, investment decisions involve risk. Investment proposals should, therefore, be evaluated in terms of both expected return and risk. Besides the decision for investment managers do see where to commit funds when an asset becomes less productive or non-profitable. There is a broad agreement that the correct cut-off rate is the required rate of return or the opportunity cost of capital. However, there are problems in computing the opportunity cost of capital in practice from the available data and information. A decision maker should be aware of capital in practice from the available data and information. A decision maker should be aware of these problems. Financing Decision Financing decision is the second important function to be performed by the financial manager. Broadly, her or she must decide when, where and how to acquire funds to meet the firms investment needs. The central issue before him or her is to determine the proportion of equity and debt. The mix of debt and equity is known as the firms capital structure. The financial manager must strive to obtain the best financing mix or the optimum capital structure for his or her firm. The firms capital structure is considered to be optimum when the market value of shares is maximised. The use of debt affects the return and risk of shareholders; it may increase the return on equity funds but it always increases risk. A proper balance will have to be struck between return and risk. When the shareholders return is maximised with minimum risk, the market value per share will be maximised and the firms capital structure would be considered optimum. Once the financial manager is able to determine the best combination of debt and equity, he or she must raise the appropriate amount through the best available sources. In practice, a firm considers many other factors such as control, flexibility loan convenience, legal aspects etc. in deciding its capital structure. Dividend Decision Dividend decision is the third major financial decision. The financial manager must decide whether the firm should distribute all profits, or retain them, or distribute a portion and retain the balance. Like the debt policy, the dividend policy should be determined in terms of its impact on the shareholders value. The optimum dividend policy is one that maximises the market value of the firms shares. Thus if shareholders are not indifferent to the firms dividend policy, the financial manager must determine the optimum dividend payout ratio. The payout ratio is equal to the percentage of dividends to earnings available to shareholders. The financial manager should also consider the questions of dividend stability, bonus shares and cash dividends in practice. Most profitable companies pay cash dividends regularly. Periodically, additional shares, called bonus share (or stock dividend), are also issued to the existing shareholders in addition to the cash dividend. Liquidity Decision Current assets management that affects afirms liquidityisyet another important financesfunction, inadditiontothemanagement of long-term assets. Currentassetsshouldbemanagedefficientlyforsafeguardingthe firm against the dangers of illiquidity and insolvency. Investment in current assets affects thefirms profitability. Liquidityandrisk. Aconflict exists between profitability and liquidity while managing current assets. If the firm does not invest sufficient funds in current assets, it may become illiquid. But itwouldloseprofitability, asidlecurrentassetswouldnotearnanything. Thus, a proper trade-off must be achieved between profitability and liquidity. In order to ensure that neither insufficient nor unnecessary funds are invested incurrent assets, the financial manager should develop sound techniquesof managingcurrent assets. Heor sheshouldestimatefirms needs for current assets and make sure that funds would be made available when needed. It wouldthusbeclear that financial decisionsdirectlyconcernthefirms decision to acquire or dispose off assets and require commitment or recommitment of funds on a continuous basis. It is in this context that finance functions are said to influence production, marketing and other functions of the firm. This, in consequence, finance functions may affect the size, growth, profitabilityandriskof thefirm, andultimately, thevalueof thefirm. To quote Ezra Solomon The function of financialmanagement is to review and controldecisions to commit or recommit fundstonewor ongoinguses. Thus, inadditionto raisingfunds, financial managementisdirectlyconcernedwithproduction, marketing and other functions, within an enterprise whenever decisions are about the acquisition or distribution of assets. Various financial functions are intimately connected with each other. For instance, decision pertaining to the proportion in which fixed assets and current assetsaremixeddeterminestheriskcomplexionof thefirm. Costs of various methods of financing are affected by this risk. Likewise, dividend decisions influence financing decisions and are themselves influenced by investment decisions. In view of this, finance manager is expected to call upon the expertise of other functional managers of the firm particularly in regard to investment of funds. Decisions pertaining to kinds of fixed assets to be acquired for the firm, level of inventories to be kept in hand, type of customers to be granted credit facilities, terms of credit should be made after consulting production and marketing executives. However, in the management of income finance manager has to act on hisown. Thedeterminationof dividendpoliciesisalmostexclusivelya finance function. Afinance manager has afinal say indecisions on dividends than in asset management decisions. Financial management is looked on as cutting across functional even disciplinary boundaries. It is in such an environment that finance manager works as a part of total management. In principle, a finance manager is held responsible to handle all such problem: that involve money matters. But in actual practice, as noted above, he has to call on the expertise of those in other functional areas to discharge his responsibilities effectively. You have studied separate legal entity concept in financial accounting the following paragraph is extension of the same. Separation of Ownership and Management In large businesses separation of ownership and management is a practical necessity. Major corporations may have hundreds of thousands of shareholders. There is no way for all of them to be actively involved in management: Authority has to be delegated to managers. The separation of ownership and management has clear advantages. It allows share ownership to change without interfering with the operation of the business. It allows the firmto hire professional managers. But it also brings problems if the man-agers' and owners' objectives differ. You can see the danger: Rather than attending to the wishes of shareholders, managers may seek a more leisurely or luxurious working lifestyle; they may shun unpopular decisions, or they may attempt to build an empire with their shareholders' money. Such conflicts between shareholders and managers' objectives create principal agent problems.The shareholders are the principals; the managers are their agents. Shareholders want management to increase the value of the firm, but managers may have their own axes to grind or nests to feather. Agency costs are incurred when (1) managers do not attempt to maximize firm value and (2) shareholders incur costs to monitor the managers and influence their actions. Of course, there are no costs when the shareholders arealso the managers.That is one of the advantages of a sole proprietorship. Owner-managers have no conflicts of interest. Conflicts between shareholders and managers are not the only principal-agent problemsthat thefinancial manager islikelytoencounter. For example, just as shareholders need to encourage managers to work for the shareholders' interests, so senior management needs to think about howto motivate everyone else in the company. In this case senior management are the principals and junior management and other employees are their agents. Thinkofthecompany'soverall valueasapiethatisdividedamonga number of claimants. These include the management and the shareholders, aswell asthecompany'sworkforceandthebanksand investorswhohavebought thecompany'sdebt. Thegovernment isa claimant too, since it gets to tax corporate profits. All these claimants are bound together in a complex web of contracts and un-derstandings. For example, when banks lend money to the firm, they insist onaformal contract statingtherateof interest andrepayment dates, perhaps placing restrictions on dividends or additionalborrowing. But you can't devise written rules to cover every possible future event. So written contracts are incomplete and need to be supplemented by understandings and by arrangements that help to align the interests of the various parties. Principal-agent problems would be easier to resolve if everyone had the same information. That is rarely the case in finance. Managers, shareholders, andlendersmayall havedifferentinformationaboutthe value of a real or financial asset, and it may be many years before all the information is revealed. Financial managers need to recognize these information asymmetriesand find ways to reassure investors that there are no nasty surprises on the way. The Agency Issue The control of the modern corporation is frequently placed in the hands of professional non-owner managers. Wehaveseenthat thegoal of the financial manager should be to maximize the wealth of the owners of the firmand given themdecision-making authority to manage the firm. Technically, any manager who owns less than 100 percent of the firm is to some degree anagent of theother owners.Intheory, most financial managerswouldagreewiththegoal of ownerwealthmaximization. In practice, however, managers are also concerned with their personal wealth, job security, and fringe benefits, such as country club memberships, limousines, and posh offices, all provided at company expense. Suchconcernsmaymakemanagersreluctant orunwillingto take more that, moderate risk if they perceive that too much risk might result in a loss of job and damage to personal wealth. The result is a less-than-maximum return and a potential loss of wealth for the owners. How do we resolve the agency problem? From this conflict of owners and managers arises what has been called theagencyproblem-thelikelihoodthat managers mayplacepersonal goalsaheadof corporategoals. Twofactors-market forcesandagency costs-act to prevent or minimize agency problems. Market Forces One market force is major shareholders, particularly large institutional investors, suchasmutual funds, lifeinsurancecompanies, andpensionfunds. Theseholdersof largeblockofafirm'sstockhave beguninrecent years toexert pressureonmanagement toperform. When necessary they exercise their voting rights as stockholders to replace under performing management. Anothermarket forceisthethreatoftakeoverbyanother firmthat believes that it can enhance the firm's value by restructuring its management, operations, and financing. The constant threat of takeover tendstomotivate management to act in the best interest of the firm's owners by attempting to maximize share price. Agency CostsTo minimize agency problems and contribute to the maximization of owners' wealth, stockholders incur agency costs. These are the costs of monitoring management behavior, ensuring against dishonest acts of management, and giving managers the financial incentive to maximize share price. The most popular, powerful, and expensive approach is tostructure management compensationto correspond with share price maximization. The objective is to compensate managers for acting in the best interests of the owners. This is frequently accomplishedbygrantingstockoptionstomanagement. Theseoptions allowmanagerstopurchasestockatasetmarketprice;ifthemarket price rises, the higher future stock price would result in greater management compensation. In addition, well-structured compensation packagesallowfirmstohirethebestmanagersavailable. Todaymore firmsaretyingmanagement compensationtothefirm'sperformance. Thisincentiveappearstomotivatemanagers tooperateinamanner reasonably consistent with stock price maximization. Social Responsibility Maximizing shareholder wealth does not mean that management should ignore social responsibility, such as protecting the consumer, paying fair wages to employees, maintainingfairhiring practicesandsafeworking conditions, supporting education, and becoming involved in such environmental issues as clean air and water .It is appropriate for management to consider the interests of stakeholders other than shareholders. These stakeholders include creditors, employees, customers, suppliers, communities inwhichacompanyoperates, and others. Onlythroughattentiontothelegitimateconcernsof thefirms various stakeholders can the firm attain its ultimate goalof maximizing shareholder wealth. Is stock price maximization the same as profit maximization? No, despite a generally high correlation amongst stock price, EPS, and cash flow. Current stock price relies upon current earnings, as well as future earnings and cash flow. Someactionsmaycauseanincreaseinearnings, yet cause the stock price to decrease (and vice versa).Questions 1. Contrast theobjectiveof maximizingearnings withthat of maximizing wealth. 2. What is financial management all about? 3. In large corporations, ownership and management are separated. What are the main implications of this separation? 4. What are agency costs & what causes them? Multiple Choice Questions 1. __________ is concerned with the acquisition, financing, and management of assets with some overall goal in mind. a) Financial management b) Profit maximization c) Agency theory d) Social responsibility 2. __________ is concerned with the maximization of a firm's earnings after taxes. a) Shareholder wealth maximization b) Profit maximization c) Stakeholder maximization d) EPS maximization 3. What is the most appropriate goal of the firm? a) Shareholder wealth maximization b) Profit maximization c) Stakeholder maximization d) EPS maximization. 4. Which of the following statements is correct regarding profit maximization as the primary goal of the firm? a) Profit maximization considers the firm's risk level. b) Profit maximization will not lead to increasing short-term profits at the expense of lowering expected future profits. c) Profit maximization does consider the impact on individual shareholder's EPS. d) Profit maximization is concerned more with maximizing net income than the stock price. 5. __________ is concerned with the branch of economics relating the behavior of principals and their agents. a) Financial management b) Profit maximization c) Agency theory d) Social responsibility 6. A concept that implies that the firm should consider issues such as protecting the consumer, paying fair wages, maintaining fair hiring practices, supporting education, and considering environmental issues. a) Financial management b) Profit maximization c) Agency theoryd) Social responsibility 7. The __________ decision involves determining the appropriate make-up of the right-hand side of the balance sheet. a) Asset management b) Financing c) Investment d) Capital budgeting 8. You need to understand financial management even if you have no intention of becoming a financial manager. One reason is that the successful manager of the not-too-distant future will need to be much more of a __________ who has the knowledge and ability to move not just vertically within an organization but horizontally as well. Developing __________ will be the rule, not the exception. a) Specialist; specialties b) Generalist; general business skills c) Technician; quantitative skills d) Team player; cross-functional capabilities 9. The __________ decision involves a determination of the total amount of assets needed, the composition of the assets, and whether any assets need to be reduced, eliminated, or replaced. a) Asset management. b) Financing c) Investment d) Accounting 10.How are earnings per share calculated? a) Use the income statement to determine earnings after taxes (net income) and divide by the previous period's earnings after taxes. Then subtract 1 from the previously calculated value. b) Use the income statement to determine earnings after taxes (net income) and divide by the number of common shares outstanding. c) Use the income statement to determine earnings after taxes (net income) and divide by the number of common and preferred shares outstanding. d) Use the income statement to determine earnings after taxes (net income) and divide by the forecasted period's earnings after taxes. Then subtract 1 from the previously calculated value. 11. What is the most important of the three financial management decisions? a) Asset management decision b) Financing decision c) Investment decision d) Accounting decision 12. The __________ decision involves efficiently managing the assets on the balance sheet on a day-to-day basis, especially current assets. a) Asset management b) Financing c) Investment d) Accounting 13. Which of the following is not a perquisite (perk)? a) Company-provided automobile b) Expensive office c) Salary d) Country club membership 14. All constituencies with a stake in the fortunes of the company are known as __________. a) Shareholders b) Stakeholders c) Creditors d) Customers 15. Which of the following statements is not correct regarding earnings per share (EPS) maximization as the primary goal of the firm? a) EPS maximization ignores the firm's risk level. b) EPS maximization does not specify the timing or duration of expected EPS. c) EPS maximization naturally requires all earnings to be retained. d) EPS maximization is concerned with maximizing net income. 16. __________ is concerned with the maximization of a firm's stock price. a) Shareholder wealth maximization b) Profit maximization c) Stakeholder welfare maximization d) EPS maximization Answers to above 1. Financial management 2. Profit maximization 3. Shareholder wealth maximization 4. Profit maximization is concerned more with maximizing net income than the stock price. 5. Agency theory 6. Social responsibility 7. Financing 8. Team player; cross-functional capabilities 9. Investment 10. Use the income statement to determine earnings after taxes (net income) and divide by the number of common shares outstanding. 11. Investment decision 12. Asset management 13. Asset management 14. Stakeholders 15. EPS maximization is concerned with maximizing net income. 16. Shareholder wealth maximization The time value of moneyYou all instinctively know that money loses its value with time. Why does this happen? What doesaFinancial Managerhavetodotoaccommodatethislossinthevalueof money with time? In this section, we will take a look at this very interesting issue. Why should financial managers be familiar with the time value of money? The time value of money shows mathematically how the timing of cash flows, combined with the opportunity costs of capital, affect financial asset values. A thorough understandingoftheseconceptsgivesafinancial managerpowerfultool tomaximize wealth. What is the time value of money? Thetimevalueofmoneyservesasthefoundationforall othernotionsinfinance. It impacts business finance, consumer financeandgovernment finance. Timevalueof money results from the concept of interest. This overview covers an introduction to simple interest and compound interest, illustratestheuseof timevalueof moneytables, showsa approach to solving time value of money problems and introduces the concepts of intra year compounding, annuities due, and perpetuities. A simpleintroductiontoworkingtimevalueof moneyproblemsona financial calculator is included as well as additional resources to help understand time value of money. Time value of money The universal preference for a rupee today over a rupee at some future time is because of the following reasons: - Alternative uses/ Opportunity cost Inflation Uncertainty The manner in which these three determinants combine to determine the rate of interest can be represented symbolically as Nominal or market rate of interest rate = Real rate of interest + Expected rate of Inflation + Risk of premiums to compensate uncertainty Basics Evaluating financial transactions requires valuing uncertain future cash flows. Translating a value to the present is referred to as discounting. Translating a value to the future is referred to as compounding .The principal is the amount borrowed. Interest is the compensation for the opportunity cost of funds and the uncertainty of repayment of the amount borrowed; that is, it represents both the price of time and the price of risk. The price of time is compensation for the opportunity cost of funds and the price of risk is compensation for bearing risk. Interest is compound interest if interest is paid on both the principal and any accumulated interest.Most financial transactions involve compound interest, though there are a few consumer transactions that use simple interest (that is, interest paid only on the principal or amount borrowed).Under the method of compounding, we find the future values (FV)ofall thecashflowsattheendofthetimehorizonataparticularrateof interest.Therefore, inthiscasewewill becomparingthefuturevalueoftheinitial outflow of Rs. 1,000 as at the end of year 4 with the sum of the future values of the yearly cashinflowsat theendof year 4. Thisprocesscanbeschematicallyrepresentedas follows: PROCESS OF DISCOUNTING Under the method of discounting,we reckon the time value of money now, i.e. at time 0 on the time line. So, we will be comparing the initial outflow with the sum of the present values (PV) of the future inflows at a given rate of interest. Translating a value back in time -- referred to as discounting -- requires determining what a future amount or cash flow is worth today. Discounting is used in valuation because we often want to determine the value today of future value or cash flows. The equation for the present value is: Present value = PV = FV / (1 + i) n Where: PV = present value (today's value), FV = future value (a value or cash flow sometime in the future), i = interest rate per period, and n = number of compounding periods And [(1 + i) n] is the compound factor. We can also represent the equation a number of different, yet equivalent ways: Where PVIFi,n is the present value interest factor, or discount factor. In other words future value is the sum of the present value and interest: Future value = Present value + interest From the formula for the present value you can see that as the number of discount periods, n, becomes larger, the discount factor becomes smaller and the present value becomes less, and as the interest rate per period, i, becomes larger, the discount factor becomes smaller and the present value becomes less. Therefore, the present value is influenced by both the interest rate (i.e., the discount rate) and the numbers of discount periods. Example Suppose you invest 1,000 in an account that pays 6% interest, compounded annually. How much will you have in the account at the end of 5 years if you make no withdrawals? After 10 years? Solution FV5 = Rs 1,000 (1 + 0.06) 5 = Rs 1,000 (1.3382) = Rs 1,338.23 FV10 = Rs 1,000 (1 + 0.06) 10 = Rs 1,000 (1.7908) = Rs 1,790.85 What if interest was not compounded interest? Then we would have a lower balance in the account: FV5 = Rs 1,000 + [Rs 1,000(0.06) (5)] = Rs 1,300 FV10 = Rs 1,000 + [Rs 1,000 (0.06)(10)] = Rs 1,600 Simple interest is the product of the principal, the time in years, and the annual interest rate. In compound interest the principal is more than once during the time of the investment.Compound interest is another matter. It's good to receive compound interest, but not so good to pay compound interest. With compound interest, interest is calculated not only on the beginning interest, but also on any interest accumulated in the meantime. I hope you have understood the concept of simple interest and compound interest. It is explained with the help of a graph, which is self-explanatory. Now let us solve a problem for Compound Interest vs. Simple Interest Example Suppose you are faced with a choice between two accounts, Account A and Account B. Account A provides 5% interest, compounded annually and Account B provides 5.25% simple interest. Consider a deposit of Rs 10,000 today. Which account provides the highest balance at the end of 4 years? Solution Account A: FV4 = Rs 10,000 (1 + 0.05) 4 = Rs 12,155.06 Account B: FV4 = Rs 10,000 + (Rs 10,000 (0.0525)(4)] = Rs 12,100.00 Account A provides the greater future value. Present value is simply the reciprocal of compound interest. Another way to think of present value is to adopt a stance out on the time line in the future and look back toward time 0 to see what was the beginning amount. Present Value = P0 = Fn / (1+I) n Table A-3 shows present value factors: Note that they are all less than one. Therefore, whenmultiplyingafuturevaluebythesefactors, thefuturevalueis discounted down to present value. The table is used in much the same way as the other time value of money tables. To find the present value of a future amount, locate the appropriate number of years and the appropriate interest rate, take the resulting factor and multiply it times the future value. How much would you have to deposit now to have Rs 15,000 in 8 years if interest is 7%? = 15000 X .582 = 8730 Rs Consider a case in which you want to determine the value today of $ 1,000 to be received five years from now. If the interest rate (i.e., discount rate) is 4%, Problem Suppose that you wish to have Rs 20,000 saved by the end of five years. And suppose you deposit funds today in account that pays 4% interest, compounded annually. How much must you deposit today to meet your goal? Solution Given: FV = Rs 20,000; n = 5; i = 4% PV = Rs 20,000/(1 + 0.04) 5 = Rs 20,000/1.21665 PV = Rs 16,438.54 Q. If you want to have Rs 10,000 in 3 years and you can earn 8%, how much would you have to deposit today? Rs 7938.00 Rs 25,771 Rs 12,597 Using Tables to Solve Future Value Problems A-1 for future value at the end of n yrs A-3 for present value at the beginning of the year Compound Interest tables have been calculated by figuring out the (1+I) n values for various time periods and interest rates. Look at Time Value of Money Future Value Factors. This table summarizes the factors for various interest rates for various years. To use the table, simply go down the left-hand column to locate the appropriate number of years. Then go out along the top row until the appropriate interest rate is located. For instance, to find the future value of Rs100 at 5% compound interest, look up five years on the table, and then go out to 5% interest. At the intersection of these two values, a factor of 1.2763 appears. Multiplying this factor times the beginning value of Rs100.00 results in Rs127.63, exactly what was calculated using the Compound Interest Formula. Note, however, that there may be slight differences between using the formula and tables due to rounding errors. An example shows how simple it is to use the tables to calculate future amounts. You deposit Rs2000 today at 6% interest. How much will you have in 5 years? =2000*1.338=2676 The following exercise should aid in using tables to solve future value problems. Please answer the questions below by using tables 1. You invest Rs 5,000 today. You will earn 8% interest. How much will you have in 4 years? (Pick the closest answer) Rs 6,802.50 Rs 6,843.00 Rs 3,675 2.You have Rs 450,000 to invest. If you think you can earn 7%, how much could you accumulate in 10 years? ? (Pick the closest answer) Rs 25,415 Rs 722,610 Rs 722,610 3.If a commodity costs Rs500 now and inflation is expected to go up at the rate of 10% per year, how much will the commodity cost in 5 years? Rs 805.25 Rs 3,052.55 Cannot tell from this information Now we will talk about the cases when the interest is given semi annually, quarterly, monthly. The interest rate per compounding period is found by taking the annual rate and dividing it by the number of times per year the cash flows are compounded. The total number of compounding periods is found by multiplying the number of years by the number of times per year cash flows is compounded. The formula for this shorter compounding period is = PV0 (1+i/m)n*m Consider the following example. You deposited Rs 1000 for 5 yrs in a bank that offers 10% interest p.a. compounded semiannually, what will be the future value. =1000 (1+. 10/2) 5*2 For instance, suppose someone were to invest Rs 5,000 at 8% interest, compounded semiannually, and hold it for five years. The interest rate per compounding period would be 4%, (8% / 2) The number of compounding periods would be 10 (5 x 2) To solve, find the future value of a single sum looking up 4% and 10 periods in the Future Value table. FV = PV (FVIF) FV = Rs 5,000(1.480) FV = Rs 7,400 Now let us solve a problem for Frequency of Compounding FVn Problem Suppose you invest Rs 20,000 in an account that pays 12% interest, compounded monthly. How much do you have in the account at the end of 5 years? Solution FV = Rs 20,000 (1 + 0.01) 60 = Rs 20,000 (1.8167) = Rs 36,333.93 In what period of time money will be doubled? Investor most of the times wants to know that in what period of time his money will be doubled. For this the rule of 72 is used. Suppose the rate of interest is 12%, the doubling period will be 72/12=6 yrs. Apart from this rule we do use another rule, which gives better results, is the rule of 69 = .35 + 69 int rate = .35 + 69 12 = .35 + 5.75 = 6.1 yrs Practice Problems What is the balance in an account at the end of 10 years if Rs 2,500 is deposited today and the account earns 4% interest, compounded annually? Quarterly? If you deposit Rs10 in an account that pays 5% interest, compounded annually, how much will you have at the end of 10 years? 50 years? 100 years? How much will be in an account at the end of five years the amount deposited today is Rs 10,000 and interest is 8% per year, compounded semi-annually? Answers 1.Annual compounding: FV = Rs 2,500 (1 + 0.04) 10 = Rs 2,500 (1.4802) = Rs 3,700.61 Quarterly compounding: FV = Rs 2,500 (1 + 0.01) 40 = Rs 2,500 (1.4889) = Rs3,722.16 2. 10 years: FV = Rs10 (1+0.05) 10 = Rs10 (1.6289) = Rs16.29 50 years:FV = Rs10 (1 + 0.05) 50 = Rs10 (11.4674) = Rs114.67 100 years: FV = Rs10 (1 + 0.05) 100 = Rs10 (131.50) = Rs 1,315.01 3. FV = Rs 10,000 (1+0.04) 10 = Rs10,000 (1.4802) = Rs14,802.44 For example, assume you deposit Rs. 10,000 in a bank, which offers 10% interest per annum compounded semi-annually which means that interest is paid every six months. Now, amount in the beginning = Rs. 10,000 Rs. Interest @ 10% p.a. for first six = 500 Months 10000 x 21.0 =10500 Interest for second 6 months = 10500 x 21.0 = 525 Amount at the end of the year = 11,025 Instead, if the compounding is done annually, the amount at the end of the year will be 10,000 (1 + 0.1) = Rs, 11000. This difference of Rs. 25 is because under semi-annualcompounding,the interestfor first6 moths earns interest in the second 6 months. The generalized formula for these shorter compounding periods is The generalized formula for these shorter compounding periods is FVn = PV(1+k/m) mxnWhere FVn = future value after n years PV = cash flow today K = Nominal Interest rate per annum M = Number of times compounding is done during a year N = Number of years for which compounding is done. Example Under the Vijaya Cash Certificate scheme of Vijaya Bank, deposits can be madefor periodsrangingfrom6monthsto10years. Everyquarter, interest will be added on to the principal. The rate of interest applied is 9% p.a. for periods form 12 to 13 months and 10% p.a. for periods form 24 to 120 months. An amount of Rs. 1,000 invested for 2 years will grow to Where m = frequency of compounding during a year = 1000 (1.025)8 = 1000 x 1.2184 = Rs. 1218Effective vs. Nominal Rate of interest We have seen above that the accumulation under the semi-annual compounding scheme exceeds the accumulation under the annual compounding scheme compounding scheme, the nominal rate of interest is 10% per annum, under the scheme where compounding is done semi annually, theprincipal amount growsat therateof 10.25percent per annum. This 1025 percent is called the effective rate of interest which is the rate of interest per annum under annual compounding that produces the same effect as that produced by an interest rate of 10 percent under semi annual compounding. The general relationship between the effective an nominal rates of interest is as follows:where r = effective rate of interest k = nominal rate of interest m = frequency of compounding per year. Example Find out the effective rate of interest, if the nominal rate of interest is 12% and is quarterly compounded? Effective rate of interest = (1 + mk)m 1 = (+ 412.0)4 1 = (1 + 0.03)4 -1 = 1.126 -1 = 0.126 = 12.6% p.a. compounded quarterly A-1 The Compound Sum of one rupee FVIF A-3 The Present Value of one rupee PVIF IMPORTANT The inverse of FVIF is PVIF i.e. inverse of FVIF is PVIF.Types AnnuitiesTypes of AnnuitiesOrdinary AnnuityOrdinary Annuity: Payments or receipts occur at the endof each period.Annuity DueAnnuity Due: Payments or receipts occur at the beginningof each periodANNUITY Till now we talked about the future value of single payment made at the time zero (PV0). Now we will speak about annuities. An annuity is an equal annual series of cash flows. Annuities may be equal annual deposits, equal annual withdrawals, equal annual payments, or equal annual receipts. The key is equal, annual cash flows. Note that the cash flows occur at the end of the year. This makes the cash flow an ordinary annuity. If the cash flows were at the beginning of the year, they would be an annuity due. Annuity = Equal Annual Series of Cash Flows Assume annual deposits of Rs 100 deposited at end of year earning 5% interest for three years Year 1: Rs100 deposited at end of year = Rs100.00 Year 2: Rs100 x .05 = Rs5.00 + Rs100 + Rs100 = Rs205.00 Year 3: Rs205 x .05 = Rs10.25 + Rs205 + Rs100 = Rs315.25 Translating a series of cash flows into a present value is similar to translating a single amount to the present; we discount each cash flow to the present using the appropriate discount rate and number of discount periods. Translating a series of cash flows into a future value is also similar to translating a single sum: simply add up the future values of each cash flow. Again, there are tables for working with annuities. Future Value of Annuity Factors is the table to be used in calculating annuities due. Basically, this table works the same way as Table 1. Just look up the appropriate number of periods, locate the appropriate interest, take the factor found and multiply it by the amount of the annuity. We use table A-2 for FVIFA For instance, on the three-year, 5% interest annuity of Rs100 per year. Going down three years, out to 5%, the factor of 3.152 is found. Multiply that by the annuity of Rs100 yields a future value of Rs315.20. another example of calculating the future value of an annuity is illustrated. You deposit Rs 300 each year for 15 years at 6%. How much will you have at the end of that time? = 300 X 23.276 = 6982.8 The following exercise should aid in using tables to solve annuity problems. Use table A-2. FVIFA 1.You deposit Rs 2,000 in recurring account each year for 5 years. If interest on this recurring account is 4%, how much will you have at the end of those 5 years? Rs 10,000 Rs 10,832.60 Rs 8,903.60 2.If you deposit Rs 4,500 each year into an account paying 8% interest, how much will you have at the end of 3 years? Rs 13,500 Rs 14,608.80 Rs 11,596.95 To find the present value of an annuity, use Table A-4. Find the appropriate factor and multiply it times the amount of the annuity to find the present value of the annuity. For instance Find the present value of a 4-year, Rs 3,000 per year annuity at 6%. Using the present value of annuity table, going down the left column for 4 yrs and to 6% the corresponding factor is 3.465 =3000 X 3.465 = 10395 Rs FUTURE VALUE OF ANNUITYAnnuityasdiscussedjust nowisthetermusedtodescribeaseriesof periodic flows of equal amounts. These flows can be either receipts or payments. For example, if you are required to pay Rs. 200 per annum as life insurance premium for the next 20 years, you can classify this stream of payments as an annuity. If the equal amounts of cash flow occur at the end of each period over the specified time horizon, then thisstreamofcashflowsisdefinedasaregularannuityordeferred annuity. Whencashflowsoccurat thebeginningofeachperiodthe annuity is known as an annuity due. Which reduces to FVAn = A -+KKn1)1( WhereA=amountdeposited/ investedattheendofeveryyearforn years. K = rate of interest (expressed in decimals) N = time horizon FVAn = accumulation at the end of nThe future value of a regular annuity for a period of n years at a rate of interest k is given by the formula: FVAn = A (1 +K)n-1 + A ( 1+ K)n-2 + A( 1 + k)n-3 + ..+ A Interest factor for Annuity (FVIFA, hereafter) andit represents the accumulation of Re. 1 invested or paid at the end of every year for a period of n years at the rate of interest k.As in the case of the future value of a single flow, this expression has also been evaluated for different combinations of k and n and tabulated in table A.2 at the end of this book. So, given the annuity payment, we have to just multiply it with the appropriate FVIFA value and determine the accumulation. Example UndertherecurringdepositschemeoftheVijayaBank, afixedsumis depositedeverymonthonorbeforetheduedateoptedfor12to120 monthsaccordingtotheconvenienceandneedsof theinvestor. The period of deposit however should be in multiples of 3 months only. The rate of interest applied is 9% p.a. for periods from 12 to 24 months and 10%p.a. for periods form24to120months andis compoundedat quarterly intervals. Based on the above information the maturity value of a monthly installment of Rs. 12 months can be calculated as below: Amount of deposit = Rs. 5 per month Rate of interest = 9% p.a. compounded quarterly Effective rate of interest per annum = 0931.01409.14=-0+ Rate of interest per month = 120931.0 =n 0.78% Alternative method Rate of interest per month = (r + 1 )1/m - 1 = (1 + 0.0931)12 - 1 = 1.0074 - 1 = .0074 = .74% Maturity value cab be calculated using the formula FVAn = A %l2+kkn1)1( = 5 %2+0078.01)0078.1(12l0 = 5 x 12.53 = Rs. 62.65 If the payments are made at the beginning of every year, then the value of such an annuity called annuity due is found by modifying the formula for annuity regular as follows. FVAn (due) = A (1 + k) FVIFAK,n Example Under the Jeevan Mitra plan offered by Life insurance Corporation of India, if a person is insured for Rs. 10,000 and if he survives the full term, then thematuritybenefitswill bethebasicsumofRs. 10,000assuredplus bonus which accrues on the basic sumassured. The minimumand maximum age to propose for a policy is 18 and 50 years respectively. Let us take two examples, one of a person aged 20 and another a person who is 40 years old to illustrate this scheme. The person aged 20, enters the plan for a policy of Rs. 10,000. The term of policyis25 yearsand theannual premiumisRs.41.65.Theperson aged 40, also proposes for the policy of Rs. 10,000 and for 25 years and the annual premium he has to pay comes to Rs. 57. What is the rate of returns enjoyed by these two persons? Solution: Rate of Return enjoyed by the person of 20 years of age Premium = Rs. 41.65 per annum Term of policy = 25 years Maturity value = Rs. 1,000 + bonus which can be neglected as it is a Fixed amount and does not vary with the term of policy. We know that the premium amount when multiplied by FVIFA factor will give us the value at maturity. i.e., P X (1 X k)* FVIFA (k,n) = MV Where P = Annual premium n = Term of policy in years k = Rate of return MV = Maturity Value Therefore 41.65 x (1 + k) FVIFA (k, 25) = 10,000 (1 + k) FVIFA (k, 25) =240.01 From the table A.2 at the end of the book, we can find that (1 + 0.14) FVIFA (14,25) = 207.33 i.e., (1.14) FVIFA (15,25) = 1.15 X 212.793 = 244.71 By Interpolation K = 14% + (15% - 14%) x 33.20771.24433.20701.240 = 14% + 1% X 38.3768.32 = 14% + 0.87% = 14.87% Rate of return enjoyed by the person aged 40 Premium = Rs. 57 per annum Term of policy = 25 years Maturity value = Rs. 10,000 Therefore 57 X ( 1 + k) FVIFA (k,25) = 10,000 (1 + k) FVIFA (k, 25) = 175.87 i.e., (1.13) (155.62) = 175.87 i.e., k. = 13% (appr.) Here we find that the rate of return enjoyed by the 20-year-old person is greater than that of the 40-year-old person by about 2% in spite of the latter paying a higher amount of annual premium for the same period of 25 years and for the same maturity value of Rs. 10,000. This is due to the coverage for the greater risk in the case of the 40 year old person. Now that we are familiar with the computation of future value, we will get into the mechanics of computation of present value. SINKING FUND FACTOR Here is the equation FVA = A -+kKn1)1( We can rewrite it as A = FVA -+1)1(nKK The expression-+1)1(nKKis called the sinking Fund factor. It represents the amount that has to be invested at the end of every year for a period of n years at the rate of interest k, in order to accumulate Re. 1 at the end of the period. PRESENT VALUE OF AN ANNUITY The present value of an annuity A receivable at the end of every year for a period of n years at a rate of interest K is equal to PVAn = )1(...)1()1()1(32KAKAKAKA+++++++ Which reduces to PVAn =A X ()()-++nnkkK1(11 The expression ()() ++ - nnkkK1(11 is called the PVIFA (Present Value Interest Factor Annuity ) and it represents the present value of regular annuity of Rs. 1 for the given values of k and n. The values of PVIFA (k, n) for different combinations of k and n are given in Appendix A.4 given at the end of the book. It must benotedthat thesevaluescanbeusedinanypresent value problem only if the following conditions are satisfied: (a) The cash flows are equal; and (b) The cash flows occur at the end of every year. Example The Swarna Kailash Yojana at rural and semi-urban branches of SBI is a scheme open to all individuals /firms. A lump sum deposit is remitted and the principal is received with interest at the rate of 12% p.a. in 12 or 24 monthly installments. The interest is compounded at quarterly intervals. The amount of initial deposit to receive a monthly installment of Rs. 100 for 12 months can be calculated as below: Firstly, the effective rate of interest per annum has to be calculated. 11""+=mmkr %55.12114=""+=mk Aftercalculatingtheeffectiverateof interestperannum, theeffective rate of interest per month has to be calculated which is nothing but ALWAYS REMEMBER It must also be noted that PVIFA (k, n) is not the inverse of FVIFA (k, n,)although PVIF (k, n) is the inverse of FVIF (k, n). 01046.0121255.0= The initial deposit can now be calculated as below: ()()-++=nnkkkAPVAn111 -++=1212)01046.01(01046.01)01046.01(100 -=01185.0133.0100 1122.22.11100Rsx== Example The annuity deposit scheme of SBI provides for fixed monthly income for suitableperiodsof thedepositorschoice. Aninitial deposit hastobe madefora minimum period of 36months.Afterthefirstmonthofthe deposit, thedepositor receivesmonthlyinstallmentsdependingonthe number of months he has chosen as annuity period. The rate of interest is 11% p.a., which is compounded at quarterly intervals. If aninitial deposit of Rs. 4,549ismadefor anannuityperiodof 60 months, the value of the monthly annuity can be calculated as below: Firstly, the effective rate of interest per annum has to be calculated. 11""+=mmkr = %46.111411.14=-+0 Aftercalculatingtheeffectiverateof interestperannum, theeffective rate of interest per month has to be calculated which is nothing but 00955.0121146.0= The monthly annuity can now be calculated as PVAn = A -++nnkkK)1(1)1( 4549 = A -+6060)00955.1(00955.01)00955.01( 4549 = A X 0169.07688.0 A = Rs. 100 Capital Recovery Factor Manipulating the relationship between PVAn, A, K & n we get an equation: A = PVAn -++1)1()1(nnkkk -++1)1()1(nnkkk is known as the capital recovery factor. Example A loan of Rs. 1,00,000 is to be repaid in five equal annual installments. If the loan carries arate of interest of 14%p.a. the amount of each installment can be calculated as below: If R is defined as the equated annual installment, we are given that R X PVIFA (14.5) = Rs. 1,00,000 There fore R = )5.14(000,00,1.PVIFARs = 433,3000,00,1.Rs = Rs. 29,129 Notes: We have introduced in this example the application of the inverse of the PVIFA factor, which is called the capital recovery factor. The application of the capital recovery factor helps in answering questions like: KEEP IN MIND *Inverse of FVIFA factor is Sinking Fund Factor *Inverse of PVIFA factor is Capital Recovery Factor What should be the amount that must be paid annually to liquidate a loan over a specified period at a given rate of interest? How much can be withdrawn periodically for a certain length of time, if a given amount is invested today? 2. In this example, the amount of Rs. 29,129 represents the sum of the principal and interest components. To get an idea of the break-up of each installment betweentheprincipal andinterest components, theloan- repayment schedule is given below. Year (A) Equated annual installment (Rs.) (B) Interest content of (b) (Rs.) (C) Capital content of (B) (Rs.) [(D) =(B C)] Loan outstanding after payment (Rs.) (E) 0 1 2 3 4 5 - 29,12929,12929,12929,12929,129- 14,00011,8829,4676,7153,577- 15,12917,24719,66222,41425,5521,00,00084,87167,62447,96225,548-- Time Value of Money Numerical:-State whether the following statements are True (T) or false (F) i. Financial analysis requires an explicit consideration of the time value of money because many financial problems involve cash flows occurring at different points of time. ii. A regular annuity in a series of periodic cash flows of equal amounts occurring at the beginning of each period. iv. The inverse of the FVIFA factor is" equal to the PVIFA factor. v. The inverse of the PVIFA factor is called the capital recovery- factor. vi. A bank that pays 10.5 percent interests compounded annually provides a higher effective rate of interest than a bank that pays 10percent compounded semi-annually. vii. The sinking fund factor is used to determine the amount that must be depositedperiodicallytoaccumulateaspecifiedsumat theendof a given period at a given rate of interest. viii.Thenominalrateofinterestisequal to the effectiverate of interest when interest is compounded annually. ix.When debtis amortizedin equal periodicinstallments, the total debt-servicing burden (consisting of interest payment and principal repayment) declines over time. x. The present value interest factor for annuity is equal to the product of the futurevalueinterestfactorforannuityandthepresentvalueinterest factor. xi. The present value of an uneven cash flow stream can be calculated using the PV1FA tables. xii. The rule of 72 is useful in determining the future value of an annuity for 6 years at an interest rate of 12% p.a. xiii. One of the reasons for attributing time value to money is that individuals prefer future consumption to current consumption.Ans:-3.1 T 3.2 F 3.3 F 3.4 T 3.5 T 3.6 T 3.7 T 3.8 F 3.9 T 3.10 F 3.11 F 3.12 F Section (B) Choose the right answer from the' alternatives given. i. Money has time value because a. Money in hand today is more certain than money to be got tomorrow. b. The value of money -gets discounted as time goes by. c. The value of money gets compounded as time goes by. d. Both (a) and (b) above. e. Both (a) and (c) above. ii. Givenaninvestment of Rs1,000tobeinvestedfor 9monthsand interest is credited annually a. It is better to -invest in a scheme, which earns compound interest at 12%. b. It is better to invest in a scheme, which earns simple interest at 12%. c. It is better to invest in a scheme, which earns simple interest at 15%. d. It is better to invest in a scheme, which earns compound interest at 14%. e. The interest rate does not matter. iii. In order to find the value in 1995 of a sum of Rs 100 invested in 1993 at X% interest a. The FVIFA table should be used. b. The PVIFA table should be used. c. The FVIF table should be used. d. The PVIF table should be used. e. Both FVIFA and FVIF tables can be used. iv. The real rate of interest or return is nothing but a) Nominal or market interest rate b) Market interest rate to which expected rate of inflation and risk premium for uncertainty has been added c) Market interest rate, which has been adjusted for inflation d)Nominal interestratefromwhichexpectedrateof inflationandrisk premium for uncertainty has been deducted. e) None of the above. v.The relationship between effective rate of interest (r) and nominal rate of interest (i) is best represented by a) i = (1 + 1)mmr b) r = (1 + 1)nnr c) r = (1 + 1)mmr d) Both (a) and (c) above f) None of the above. vi. Which of the following statement is /are true? a) Present value interest factor Annuity (PVIFA) is the productof future valueinterest factor Annuity(FVIFA) andpresent valueinterest factor (PVIF) b) PVIFA ( i,n ) = nniin)1(1)1(++ c) PVIFA ( i,n) = iin)1(1+ d) PVIFA is the inverse of FVIFA e) None of above. vii. If P= initial amount, i = interest rate, m = frequency of compounding per year, n= number of years and S = accumulation at the end of year n, which of the following expressions are correct. a) S = P mnni)1+( b) P = S mnmi)1+( c) S = [P (1+ nmmi]) d) S= P (mnmi)1+( e) None of the above. viii. The Rule of 72 a) Is used to find the doubling period b) Makes use of the PVIFA tables c) Applies the formula rateerestint72 d) Both (b) and (c) above e) Both (a) and (c) above Solution to Section (B) 3.1 d 3.2 c 3.3 c 3.4 d 3.5 c 3.6 a 3.7 d 3.8 e Section (C) 1. If you invest Rs. 10,000 today for a period of 5 years, what will be the maturity value if the interest rate is? (a) 8% (b) 10% (c) 12% (d) 15% 2. How many years will it take for Rs. 5,000 invested today at 12% rate of interest to grow to Rs. 1,60,000? Use the rule of 72. 3.Amount invested today = Rs. 1,000; maturity value = Rs. 8,000; time period = 12 years. Use rule of 69 to calculate the implied interest rate. 4. If you invest Rs. 3,000 a year for 3 years and Rs. 5,000 a year for 7 years therefore at a rate of 12%, what will be the maturity value at the end of 10 years? 5.Sunitaexpectsanexpenditureof Rs. 2,00,000afteraperiodof 10 years. How much should she save annually to have the required sum after 10 years, if she invests her savings at a rate of 12%. 6. Annual payment = Rs. 1,500; maturity value = Rs. 12,500, period = 5 years. Find out the implied interest rate. 7.You invest Rs. 3,000 today and get Rs. 10,000 after 6 years. What is the implied interest rate? 8. What will be the present value of Rs. 12,000 receivable after 10 years if the rate of discount is (i) 10% (ii) 12% (iii) 15% 9. What is the present value of an 5 year annuity of Rs. 3,000 at 12% 10. Mr. Srinivas is going to retire after 6 months. He has a choice between (a) an annual pension of Rs. 8,000 as long as he lives, and (b) A lump sum amount of Rs. Amount of Rs. 50,000. If he expects to live for 20 years and the interest rate is 10% which option would you suggest him to go for? 11. Sunil has deposited Rs. 2,00,000 in a bank, which pays interest @8%. How much can he withdraw every year for a period of 25 years, so that there is no balance left at the end? 12. You invest Rs. 1,500 at the end of year one, Rs. 2,000 at the end of the second year, and Rs. 5,000 each year form the third year to the tenth. Calculate the present value of the stream if the discount rate is 10%. 13. You receive Rs. 1,000 a year for the first 8 years, and Rs. 4,000 a year forever therefore. Calculate the present value if the discount rate is 12%. 14. Suman is due to retire 20 years form now. She wants to invest a lump sum now so as table to withdraw Rs. 10,000 every year, beginning from the end of the 20th year. How much should she invest now if the deposit earns a return of 12%? 15. A company is offering to pay Rs. 10,000 annually for a period of 10 years if you deposit Rs. 50,000 now. What is the implied interest rate? 16.Usingadiscount rate of 10%, calculate thepresent valueof the following cash flow streams. End of yearStream AStream BStream C 1 2 3 4 5 1,0002,0003,0004,0005,00010,0009,0008,0007,0006,0005,0005,0005,0005,0005,000MULTIPLE CHOICE QUESTIONS 1. Compare the interest earned by Rs750 for 8 years at 6% simple interest with that earned by the same amount for 8 years at 6% compounded annually. (A) Simple Interest: I = Rs350; Compound Interest: I = Rs529.48 (B) Simple Interest: I = Rs360; Compound Interest: I = Rs445.39 (C) Simple Interest: I = Rs400; Compound Interest: I = Rs579.46 (D) Simple Interest: I = Rs370; Compound Interest: I = Rs469.25 2. You are considering investing Rs 1,500 at an interest rate of 5% compounded annually for 2 years or investing the Rs1,500 at 7% per year simple interest rate for 2 years. Which option is better? (A) Simple Interest by Rs56.25 (B) Compound Interest by Rs114.05 (C) Compound Interest by Rs52.75 (D) Simple Interest by Rs75.19 3. Suppose you have the alternative of receiving either Rs4,000 at the end of 2 years or P dollars today. Having no current need for the money, you would deposit the P dollars in a bank that pays 7% interest compounded annually. What value of P would make you indifferent in your choice between P dollars today and the promise of Rs 4,000 at the end of 2 years? (A) P = Rs 3,397.48 (B) P = Rs 3,200.39 (C) P = Rs 3,518.86 (D) P = Rs 3,493.75 4. Suppose that you are obtaining a personal loan from your uncle in the amount of Rs 6,000 for three years to cover your college expenses. If your uncle always earns 10% interest (compounded annually) on his money invested in various sources, what minimum lump-sum payment three years from now would make your uncle happy? (A) F = Rs 8,520 (B) F = Rs 7,395 (C) F = Rs 7,784 (D) F = Rs 7,986 5. What will be the amount accumulated by Rs 9,000 in 9 years if it is compounded at a rate of 9% per year? (A) F = Rs 18,229.30 (B) F = Rs 19,547.04 (C) F = Rs 20,978.22 (D) F = Rs 19,055 6. In 7 years, we will have accumulated Rs 17,000. What is the present worth of Rs 17,000 if it is compounded annually at 11%? (A) P = Rs 8,188.19 (B) P = Rs 8,563.05 (C) P = Rs 7,892.46 (D) P = Rs 250.29 7. For an interest rate of 7% compounded annually, find how much can be loaned now if Rs 4,000 will be repaid at the end of 4 years? (A) P = Rs 2,896.22(B) P = Rs 3,190.55 (C) P = Rs 3,051.58 (D) P = Rs 3,789.22 8. For an interest rate of 7% compounded annually, find how much will be required in 3 years to repay Rs 3,000 loan now? (A) F = Rs 3,780.56 (B) F = Rs 3,675.13 (C) F = Rs 4,005.67 (D) F = Rs 3,600.13 9. If you desire to withdraw the following amounts over the next 5 years from a savings account that earns a 9% interest compounded annually, how much do you need to deposit now? n Amount 2 Rs 1,000 3 Rs 1,500 4 Rs 3,000 5 Rs 5,000 (A) P = Rs 6,982.30 (B) P = Rs 7,074.89 (C) P = Rs 7,958.22 (D) P = Rs 7,374.89 10. If Rs300 is invested now, Rs500 two years from now, and Rs700 four years from now at an interest rate of 3% compounded annually, what will be the total amount in 10 years? (A) F = Rs 1,872.40 (B) F = Rs 1,540.27 (C) F = Rs 1,975.11 (D) F = Rs 1,801.36 11. How much invested now at 7% compounded annually would be just sufficient toprovide three withdrawals with the first withdrawal in the amount of Rs1500 occurring three years hence, Rs3000 six years hence, and Rs5000 eight years hence? (A) P = Rs 4606.13 (B) P = Rs 5392.17 (C) P = Rs 6027.51 (D) P = Rs 6133.35 12. What is the future worth at t=7 of a series of equal year-end deposits of Rs750 for 7 years in a savings account that earns 8% compound annual interest? (A) F = Rs 6655.23 (B) F = Rs 6692.10 (C) F = Rs 7582.13 (D) F = Rs 6529.05 13. What equal annual series of payments beginning at t=1 must be paid into a sinking fund to accumulate Rs 13,000 in 20 years at 8% compounded annually? (A) A = Rs419.29 (B) A = Rs485.35 (C) A = Rs284.70 (D) A = Rs387.28 14. An individual deposits an annual bonus into a savings account that pays 5% interest compounded annually. The size of the bonus increases by Rs200 each year and the initial bonus amount at t=1 was Rs250. Determine how much will be in the account immediately after the fifth deposit. (A) F = Rs3019.59 (B) F = Rs3483.89 (C) F = Rs2953.94 (D) F = Rs2752.95 15. Five annual deposits in the amounts beginning at t=1 of (Rs800, Rs700, Rs600, Rs500, and Rs400) are made into a fund that pays interest at a rate of 10% compounded annually. Determine the amount in the fund immediately after the fifth deposit. (A) F = Rs2969.52 (B) F = Rs1127.15 (C) F = Rs3778.99 (D) F = Rs2752.95 16. What is the equal-payment series for 10 years that is equivalent to a payment series of Rs 15,000 at the end of the first year (t=1) decreasing by Rs300 each year over 10 years? Interest is 9% compounded annually. (A) A = Rs 7120.85 (B) A = Rs 10,118.72 (C) A = Rs 12,929.01 (D) A = Rs 13,860.66 17. Suppose that an oil well is expected to produce 10,000 barrels of oil during its first production year. However, its subsequent production (yield) is expected to decrease by 10% over the previous year's production. The oil well has a proven reserve of 100,000 barrels. Suppose that the price of oil is expected to be Rs30 per barrel for the next several years. What would be the present worth of the anticipated revenue stream at an interest rate of 15% compounded annually over the next 7 years? (A) P = Rs 948,629.78 (B) P = Rs 955,013.95 (C) P = Rs 984,228.58 (D) P = Rs 875,629.00 18. If a bank pays you 8% compound annual interest on your balance, how much do you have to deposit now so that you will be able to withdraw Rs75 at the end of the first year, Rs75 at the end of the second year, Rs100 at the end of the third and fourth years, and Rs200 at the end of the fifth and final year. (A) P = Rs509.52 (B) P = Rs419.08 (C) P = Rs422.75 (D) P = Rs352.75 19. Consider the following positive cash flows: Rs400 at t=0, Rs400 at the end of year 1, Rs400 at the end of year 2, Rs400 at the end of year 3, Rs400 at the end of year 4, Rs500 at the end of year 5, Rs500 at the end of year 6, and Rs500 at the end of year 7. At an interest rate of 12 percent compounded annually, what equivalent cash flow series makes the inflow series equivalent to the outflow series between t=2 to t=9. (A) C = Rs397.45 (B) C = Rs428.99 (C) C = Rs500.63 (D) C = Rs536.17 20. Consider the following cash flow: Rs500 at the end of year 0, Rs1000 at the end of year 1, Rs1000 at the end of year 2, Rs1000 at the end of year 3, Rs1000 at the end of year 4, and Rs1000 at the end of year 5. In computing F at the end of year 5 at an interest rate of 10% compounded annually, which of the following statements is correct? (A) F = 1000(F/A, 10%, 4) + 500(F/P, 10%, 5) (B) F = 500(F/A, 10%, 6) + 500(F/A, 10%, 5) (C) F = [500 + 1000(P/A, 10%, 5)] x (F/P, 10%, 5) (D) F = [5000(A/P, 10%, 5) + 1000] x (F/A, 12%, 5) 21. Using a 12% interest rate compounded annually, solve for the present worth of the following cash flow series: -Rs30 @ t=1, Rs30 @ t=3, Rs60 @ t=4, Rs90 @ t=5, -Rs30 @ t=6, -Rs30 @ t=7, -Rs30 @ t=8, -Rs30 @ t=9, -Rs90 @ t=10, Rs60 @ t=10, -Rs90 @ t=11, Rs60 @ t=11, -Rs90 @ t=12 and Rs60 @ t=12. (A) Rs6 (B) Rs18 (C) Rs13 (D) Rs9 22. Consider the first cash flow series: Rs200 @ t=1, Rs150 @ t=2, Rs400 @ t=3, Rs150 @ t=4 and Rs400 @ t=5; and the second series: Rs100 @ t=1, Rs X at t=2, Rs X @ t=3, -Rs200 @ t=4 and Rs X @ t=5. Find the value of X in the second series so that the two cash flows are equivalent for an interest rate of 12% compounded annually. (A) X = Rs545 (B) X = Rs454 (C) X = Rs465 (D) X = Rs525 23. What single amount at the end of the fourth year is equivalent to a uniform annual series of Rs7000 per year starting at t=1 and ending at t=10, if the interest rate is 7% compounded annually? (A) X = Rs 62,798 (B) X = Rs 46,445 (C) X = Rs 60,564 (D) X = Rs 64,446 24. At an interest rate of 7% compounded annually, which equation from the list below would correctly compute either the equivalent present worth (P) at t=0 or future worth (F) at t=5 for the following cash flow series: Rs A @ t=0, Rs A @ t=1, Rs A @ t=2, Rs A @ t=3, Rs A @ t=4 and Rs A @ t=5. (1) P = A (P/A, 7%, 6); F = A (F/A, 7%, 6) (2) P = A + A (P/A, 7%, 5); F = A (F/A, 7%, 5) + A (F/P, 7%, 6) (3) P = A + A (P/A, 7%, 5); F = A (F/A, 7%, 6) (4) P = A (P/A, 7%, 6); F = A (F/A, 7%, 5) + A (F/F, 7%,5) (A) (1) (B) (2) (C) (3) (D) (4) 25. Consider the first cash flow series: -Rs20 @ t=0, Rs20 @ t=2, Rs40 @ t=3, Rs60 @ t=4 and Rs80 @ t=5; and the second series: Rs A @ t=2, Rs A at t=3, Rs A @ t=4 and Rs A @ t=5. Find the equivalent equal-cash-flow series (A), such that the two aforementioned cash flows are equivalent at 10% compounded annually. (A) A = Rs41 (B) A = Rs81 (C) A = Rs51 (D) A = Rs21 26. How much would you have to deposit in a savings account today, earning 8% compound annual interest, such that you will be able to make 5 equal end of year withdrawals of Rs10,000 beginning 6 years from today, with your last withdrawal bringing your savings account balance to zero? (A) P = Rs 25,321 (B) P = Rs 27,174 (C) P = Rs 29,559 (D) P = Rs 24,785 27. A professional hockey player free agent is trying to decide which of two teams he should play for based on economic considerations. Both teams have offered him a signing bonus, which he will receive today, and an annual salary (assume that the salary is paid out at the end of each year). His total salary from Team A will be Rs 6,000,000 over 3 years whereas the total salary from Team B will be Rs 6,250,000 also over 3 years. The structure of each team's offer is summarized below. Team A: Rs 500,000 initial signing bonus, Rs 1,500,000 for year 1, Rs 2,000,000 for year 2 and Rs 2,000,000 for year 3.Total = Rs 6,000,000. Team B: Rs 350,000 initial signing bonus, Rs 400,000 for year 1, Rs 2,000,000 for year 2 and Rs 3,500,000 for year 3. Total = Rs 6,250,000. Assuming the player uses a 15% interest rate compounded annually to evaluate his options, which team offers do you recommend? (A) Team A (B) Team B 28. Woods Manufacturing Company, a small toothpick fabricator, needs to purchase a molding machine for Rs 200,000. Woods will borrow money from a bank at an interest rate of 9% compounded annually over 5 years. Woods expects its product sales to be slow during the first year but to increase subsequently at an annual rate of 10%. Woods therefore arranges with the bank to pay off the loan with increasing payments, with the lowest payment at the end of first year, each subsequent payment to be just 10% more than the previous one. Determine the fifth annual payment. (A) A = Rs 52,660 (B) A = Rs 62,660 (C) A = Rs 72,660 (D) A = Rs 82,760 29. ACB Inc. has invested Rs1.5 million in new technology. The entire investment was financed with a loan bearing interest of 15% compounded annually. The new technology will increase the net cash flow per unit of product sold by Rs250. Assuming that the same number of units will be sold each year over the six year life (assume end of year sales) of the technology, how many units have to be sold each year to recover the Rs1.5 million investment and interest on the loan? (A) X = Rs 1,285 units per year (B) X = Rs 1,385 units per year (C) X = Rs 1,485 units per year (D) X = Rs 1,585 units per year 30. You have Rs 10,000 to invest and you expect it to double in 8 years. Using the Rule of 72, what compound annual interest rate do you have to earn on your investment. (A) i = 11% (B) i = 8% (C) i = 10% (D) i = 9% ANSWERS TO ABOVE Question 1: 'b' is the correct answer! Simple interest: I = iPN = (0.06)(Rs 750)(8) = Rs 360; Compound interest: I = P[(1+ i)^N - 1] = Rs 750[(1.06)^8 - 1] = Rs 445.39 Question 2: Rs 'a' is the correct answer. Compound interest: F = Rs 1,500(1 + 0.05)^2 = Rs 1653.75; Simple interest: F = Rs 1,500(1 + 0.07(2)) = Rs 1710 Question 3: Rs 'd' is the correct answer.: P = Rs 4,000/(1 + 0.07)^2 = Rs 3493.75 Question 4: Rs 'd' is the correct answer.: F = Rs 6,000(1 + 0.1)^3 = Rs 7986 Question 5: Rs 'b' is the correct answer.: F = Rs 9,000(1 + 0.09)^9 = Rs 19547.04 Question 6: Rs 'a' is the correct answer.: P = Rs 17,000/(1 + 0.11)^7 = Rs 8188.19 Question 7: Rs 'c' is the correct answer.: P = Rs 4,000/(1 + 0.07)^4 = Rs 3051.58 Question 8: Rs 'b' is the correct answer.: F = Rs 3,000(1 + 0.07)^3 = Rs 3675.13 Question 9: 'd' is the correct answer!: P = [Rs 1,000/(1 + 0.09)^2 ] + [Rs 1,500/(1 + 0.09)^3 ] + [Rs 3,000/(1 + 0.09)^4 ] + [Rs 5,000/(1 + 0.09)^5 ] = Rs 7374.89 Question 10: Rs 'a' is the correct answer.: F = Rs 300(1 + 0.03)^10 + Rs 500(1 + 0.03)^8 + Rs 700(1 + 0.03)^6 = Rs 1872.40 Question 11: 'd' is the correct answer.: P = Rs 1500(P/F,7%,3) + Rs 3000(P/F,7%,6) + Rs 5000(P/F,7%,8) = Rs 6133.35 Question 12: 'b' is the correct answer!: F = Rs 750(F/A,8%,7) = Rs 6692.10 Question 13: 'c' is the correct answer!: A = Rs 13,000(A/F,8%,20) = Rs 284.70 Question 14: 'b' is the correct answer.: F = F1 + F2 = Rs 250(F/A,5%,5) + Rs 200(F/G,5%,5) = Rs 250(F/A,5%,5) + Rs 200(A/G,5%,5)(F/A,5%,5) = Rs 3483.89 Question 15: 'c' is the correct answer.: F = Rs 800(F/A,10%,5) - Rs 100(F/G,10%,5) = Rs 800(F/A,10%,5) - Rs 100(P/G,10%,5)(F/P,10%,5) = Rs 3778.99 Question 16: 'd' is the correct answer.: A = Rs 15,000 - Rs 300(A/G,9%,10) = Rs 13,860.66 Question 17: 'c' is the correct answer: g = -10% and P = Rs 300,000(P/A1, -10%, 15%,7) = Rs 984,228.58 Question 18: 'c' is the correct answer.: P = Rs 75(P/A,8%,2) + Rs 100(P/A,8%,2)(P/F,8%,2) + Rs 200(P/F,8%,5) = Rs 422.75 Question 19: 'd' is the correct answer: P (Inflow at t=0) = Rs 400 + Rs 400(P/A, 12%, 4) + Rs 500(P/A, 12%, 3)(P/F, 12%, 4) = Rs 2,378.14 P (Inflow at t=1) = Rs 2,378.14(F/P,12%,1) = Rs 2,663.52 = P (Outflow at t=1) Outflow series: A at t=2, A at t=3, ..., A at t=9. A = P (A/P, 12%,8) = 2,663.52 (A/P, 12%,8) = Rs 536.17 Note: N=8 since between t=2 and t=9 there are only 8 consecutive cash outflow of Rs A Question 20: 'c' is the correct answer.: F = [500 + 1000(P/A,10%,5)] x (F/P,10%,5) Question 21: 'a' is the correct answer!: P = [ Rs 30(P/G,12%,3) + Rs 30(P/A,12%,3)](P/F,12%,2) - Rs 30(P/F,12%,1) - Rs 30(P/A,12%,7)(P/F,12%,5) P = [ Rs 30(2.2208) + Rs 30(2.4018)](0.7972) - Rs 30(0.8929) - Rs 30(4.5638)(0.5674) P = Rs 110.55 - Rs 26.79 - Rs 77.68 = Rs 6.08 Question 22: 'b' is the correct answer.: P = Rs 150(P/A,12%,5) + Rs 50(P/F,12%,1) + Rs 250(P/F,12%,3) + Rs 250(P/F,12%,5) = Rs 540.72 + Rs 44.65 + Rs 177.95 + Rs 141.85 = Rs 905.17 P = Rs 100(P/F,12%,1) + X(P/A,12%,2)(P/F,12%,1) - Rs 200(P/F,12%,4) + X(P/F,12%,5) Rs 905.17 = Rs 89.29 + 1.5091X - Rs 127.10 + 0.5674X X = Rs 454.12 Question 23: 'd' is the correct answer.: Computing the equivalent worth at n = 4, X = Rs 7000(F/A,7%,4) + Rs 7000(P/A,7%,6) = Rs 64,446 Question 24: 'b' and 'c' are both correct answers!: (2) P = A + A(P/A,7%,5); F = A(F/A,7%,5) + A(F/P,7%,5) or (3) P = A + A(P/A,7%,5); F = A(F/A,7%,6) Question 25: 'a' is the correct answer.: P0 = -Rs 20 + [ Rs 20(P/G,10%,4) + Rs 20(P/A,10%,4)](P/F,10%,1) = -Rs 20 + [Rs 20(4.3781) + Rs 20(3.1699)]0.9091 = Rs 117.24 A = Rs 117.24(F/P,10%,1)(A/P,10%,4) = Rs 117.24(1.1)(0.3155) = Rs 40.49 Question 26: 'b' is the correct answer!: P = Rs 10,000(P/A,8%,5)(P/F,8%,5) = Rs 27,174 Question 27: 'a' is the correct answer!: Team A; P = Rs 500,000 + Rs 1,500,00(P/F,15%,1) + Rs 2,000,000(P/F,15%,2) + Rs 2,000,000(P/F,15%,3) = Rs 4,631,6000 Team B; P = Rs 350,000 + Rs 400,000(P/F,15%,1) + Rs 2,000,000(P/F,15%,2) + Rs 3,500,000(P/F,15%,3) = Rs 4,511,230 Question 28: 'b' is the correct answer.: Rs 200,000 = A1(P/A1,10%,9%,5) Rs 200,000 = A1[(1 - (1 + g)^N(1 + i)^(-N))/(i - g)] Rs 200,000 = A1[(1 - (1 + 0.1)^5(1 + 0.09)^(-5))/(0.09 - 0.10)] A1 = Rs 42,798 The fifth payment = Rs 42,798(1 + 0.1)^4 = Rs 62,660 Question 29: 'd' is the correct answer.: Rs 1,500,000 = Rs 250X(P/A,15%,6) Rs 6000 = X(3.7845) X = Rs 1,585 units/year Question 30: 'd' is the correct answer.: Rule of 72: 72/i = N to double Therefore, i = 72/N = 72/8 = 9% Some more practice problems 1.Complete the following, solving for the present value, PV: Case Future value Interestrate Number ofperiods Presentvalue A Rs 10,000 5%5 B Rs 563,0004%20 CRs 5,0005.5%3 Factors that affect stock price Projected cash flows to shareholders Timing of the cash flow stream Riskiness of the cash flowsBasic Valuation Model=+=++ ++++=n1 tttnn2211.k) (1CF k) (1CFk) (1CFk) (1CFValue To estimate an assets value, one estimates the cash flow for each period t (CFt), the life of the asset (n), and the appropriate discount rate (k) Throughout the course, we discuss how to estimate the inputs and how financial management is used to improve them and thus maximize a firms value.Factors that Affect the Level and Riskiness of Cash Flows Decisions made by financial managers: Investment decisions Financing decisions (the relative use of debt financing) Dividend policy decisions The external environment