Introduction to Financial Planning

Reading 2 Financial Prudence Chandrakant Soni, CFPCM Financial Prudence is nothing but getting cautious, judicious and thoughtful in matter of finances. It is important for Individual, company and country alike. Those who ignores this, suffers heavily. Individual: In the absence of financial prudence, the suffering is more in case of individual, as he has to manage it alone. The social disintegration, nuclear family, competition at work place and in life and lack of emotional support makes it even worse. Company: Financial prudence finds strong place in the case of company also. A wrong decision could lead to the extinction of the company or change in ownership with lots of hardship to all the stake holders and employees Country: At country level it impacts entire economy. To understand the significance of financial prudence, let’s look at some of the global events that have impacted entire world in last year and half. The world is a changed place today and would see new order emerge going forward. It all started because of excesses: 1. Excess of Money 2. Excess of spending 3. Excess of leveraging 4. No prudence This all starts when you stop being prudent and start thinking that world will move the way you want it to move, the events will unfold the way you want them to unfold and so on. It began with lower interest rate due to abundant liquidity. Easy and cheap money lead to huge demand for goods and services and lead to asset bubble defying the basic fundamentals. Eg. Oil price, Property price, commodity price etc. Excess liquidity lead to heavy lending initially to the borrower with excellent track-record (Prime category) and subsequently to the sub-prime category borrower. Liquidity was getting replenished by bundling of receivables in exotic instruments and selling it across the globe by the investment bankers to the next level of lenders. The fresh liquidity received by the lender through such act was back into the lending business. The story was going on so long as the prices of the assets were going up and the last buyer was able to make money on his investments. It all turned sour, when the last buyer could not exit from the assets and when he realized that the value of the assets was far less than what he thought of. He found himself trapped at the peak of this cycle. 201

The sub-prime backgrounds lead to default on the loan. Since there were various level of lenders involved (i.e. primary lender, lender of lender and so on) the default had cascading effect on the entire world. The defaults of loan at one level lead to the default at second level and so on. The financial system across the world got chocked up. Trust completely disappeared. There was no credit flow across the system. The excess of money suddenly evaporated in the thin air. The risk aversion or “no risk” became a new mantra. What does this indicate??? The lack of financial prudence at all levels. At Central Bank Level: Central bank could not gauge the gravity of the problem and could not reduce the liquidity from the market in timely manner. At Company Level: Companies went on over drive in business expansion, capital expenditure, mergers and acquisitions due to easy availability of money. This lead to excess capacity, high cost, low profitability and so on. Companies had to get into the cost cutting mode by rationalizing/downsizing the manpower, cutting on the expenses. This results in job losses which has gone up to 10% in case of USA at this point in time. At Individual level: Individuals also over leveraged themselves at the back of continuous increase in earnings. When economy got into trouble and companies started shedding flab, the individuals also started facing the problems. Those countries, where social security system is present, people were still able to manage, but in the other countries where there is no such system individuals suffering is very harsh. When situation became extremely grave and difficult, the central back around the world sprang into action and turned into printing press deluging the market with money. It provided some breather to the markets that had got choked up completely. Central Banks had to also bail out many companies/banks around the world. The Govt. and central banks could do all this at the expense of huge fiscal deficit. This has its own implication, as government has lesser money to spend on infrastructure and social sectors. Few Thoughts: Govt. could print money….. Companies could get bail-out package or could merge or could be sold off…. But…. What happened to the individuals?? Huge suffering……. Therefore, he has to better be prudent in financial matters. Why Financial Prudence?  Spending money with gay abundance provide temporary pleasure  Prudent spending & savings provide for rainy days  Helps in capital formation in the country  Leads to economic progress Tools for Financial Prudence 202

 Appropriate budgeting and robust planning  Right spending  Regular saving and prudent investing To conclude, financial prudence is very important for everyone and lack of it could jeopardize wealth creation for sun-set years in case of individuals, invite trouble for companies and countries. Proper planning and thoughtful decision helps in all round prosperity. It is very important for everyone and all the more in case of individuals because of vulnerability of being individual. The family dependence makes it quite compelling on the part of individuals to remain prudent in the matter of finances. A thoughtful and prudent planning lead to happy family life today, tomorrow and forever. 203

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Part - III Financial Mathematics 3.1 Calculation of Returns 3.2 Time Value of Money 3.3 Loan Calculations 3.4 Total Assets, Net Worth and Financial Ratios 205

Part – III Financial Mathematics Introduction Financial planning does not consist of picking investments indiscriminately. Among other things, a financial planner must be technically very competent in money matters. Clients will expect their financial planner to have skills and competencies that enable them to correctly evaluate different investment opportunities. In order to do this you need to understand the time value of money and, consequently, what computations are necessary in order to compare different investment options with different cash flows. In this section we cover some fundamental mathematics of finance. We concentrate on the fundamentals of future value and present value. With these two basic tools and the aid of a financial calculator, we will show how to deal with a number of practical problems in financial planning. The emphasis in this unit is on execution. It is essential that you can use your financial calculator to get the right answer. You will be required to solve similar problems to those presented here in the examination and the only tools available to you will be your financial calculator, pen and paper. Learning Outcomes After studying this topic, you should be able to:  Explain the terms ‘simple interest’ and ‘compound interest’;  Work out accumulated interest and accumulated amounts of term investments;  Explain the notion of present value and the time value of money;  Work out repayment schedules using the present value concept;  Apply the present value concept to complex problems involving annuities and other investments in order to compare their profitability; and  Solve all such problems quickly and efficiently using a financial calculator. The key concept that underlies all of the material presented in this topic is that of the time value of money, that is, a rupee today is worth more than a rupee tomorrow. The content of this topic is oriented towards developing your understanding of this concept and the techniques that might be employed to equate money amounts. The procedures that you are introduced to in this topic may appear very difficult at first. However, if you are prepared to do every problem presented in the notes and review exercises, then you will achieve the required level of competency. This topic is structured as follows: 1. the necessary tools utilised to solve financial problems; 2. basic principles relating to interest; 3. ways to calculate singular cash flows and annuities; 4. methods used to evaluate different investment options; and 5. practical examples and questions to test your understanding. Tools for Solving Financial Problems In order to evaluate investment proposals, basic mathematical processes are applied. These processes can be assisted by the use of a financial calculator or a spreadsheet. A financial calculator and a spreadsheet are 206

tools used by the financial planner to perform basic mathematical tasks. Consequently, the correct use of a financial calculator or spreadsheet reduces the task of the financial planner to one of understanding the nature of the investments to be analysed, the appropriate method of analysis, and then the determination of the appropriate variables to be input into the calculator or spreadsheet. Following a sequence of steps on the calculator or spread-sheet, an answer is then produced. As a financial planner, you should also, of course, have the skill of evaluating whether such an answer is reasonable, or whether you have pressed a wrong key or inserted incorrect information. An additional tool that may be used is the cash flow diagram. This will help you identify the amount and timing of cash flows. Such diagrams can visually represent complex problems in a simple manner. The use of the calculator, spreadsheet and cash flow diagram as tools for solving financial problems will now be outlined. The Financial Calculator You will need a financial calculator, complete with its own user guide, in order to perform the calculations in this topic. There are several models on the market, for example the Sharp EL-733 or 733A or 735; or the Hewlett Packard 10B , FC 100V FC 200V. Any model will do, as long as it can perform index, net present value (NPV) and internal rate of return calculations. It is your responsibility to become thoroughly familiar with the operations of your own financial calculator, particularly the financial functions. Use the user guide that came with your calculator for demonstrations of these. The website http://www.awl.com.au/pmf/UsingCalcs.htm offers an additional guide for the Sharp EL-735 calculator. Note: You will need to have your calculator beside you as you read the topic notes. Be prepared to use it to check every step in any worked problems which appear. This topic is a technical part of the course. Using Spreadsheets in Financial Planning Today a typical financial plan would be constructed with the help of a computer spreadsheet. These are remark-ably powerful tools for solving complicated financial problems that go well beyond the capabilities of even the most sophisticated financial calculator. Knowledge of how these can assist in the construction of a portfolio is part of the essential intellectual inventory of a good financial planner. As well as performing complex calculations with ease, spreadsheets offer storage of solutions, easy modification, and the capacity to provide enhanced documentation to clients detailing different investment scenarios. There are a wide variety of spreadsheet programs available which can be used for financial planning. In this topic we will use the most popular spreadsheet on the market today, Microsoft Excel. Please note, however, that the principles we discuss apply to virtually any other spread- sheet product currently available (e.g. those provided by Computer Associates and Lotus). Also be aware that specialist financial products such as Microsoft’s Money; Intuit’s Quicken series (like the Quick books) and Managing Your Own Money (MYOB) carry the spreadsheet analysis into far more depth than is discussed here. Further, many of these products can link up online via the internet to dedicated financial planning houses. These recent developments have taken financial planning into the high-tech era where the horizons are expanding rapidly. However, a discussion of these developments would take us well beyond what would be appropriate in this topic. Each spreadsheet product can handle routine financial calculations easily enough. However, many spreadsheet products have idiosyncrasies that take some getting used to, and Microsoft Excel is no exception in this regard. The power of the software comes with a price, in that it may be unnecessarily 207

cumbersome to use for simple financial problems. The architects of the Excel spreadsheet’s in-built financial programs obviously presumed its users would have some prior experience with financial problems, because if you read its ‘Help’ section you will find it uses advanced concepts and formulae even there. However, Excel can also do simple requests like converting one cash flow from the future to the present. Cash Flow Diagrams Some people like to use cash flow diagrams to clarify problems. We shall use the convention that, from the investor’s point of view, amounts of money being paid out (outflows) are negative and amounts received (inflows) are posi- tive. In a cash flow diagram, inflows are represented with upward arrows and outflows with downward arrows. Figure : Cash Flow Diagrams Showing Direction of Cash Flows In Diagram A, there is an initial amount being invested or paid out (from the investor’s point of view) and a final amount returned after four periods. This diagram represents compound growth or a savings account. In Diagram B, four payments or investments are made and a final amount is returned. This diagram represents a savings plan or a pension fund. Basics of the Time Value of Money In this section we explore the basic principles underlying the time value of money concept. Would you prefer to have a rupee today rather than a rupee tomorrow? Most of us would prefer to have the rupee today in our pockets than wait until tomorrow. Why? Well, once we have received the money, then we can do something with it. The receipt of the money provides us with the opportunity to put our money to work and earn interest. It is a fact that we can earn interest on money that gives different values to cash flows received or paid at different times. Before we can learn how to adjust the money amounts received or paid so that they may be equated, we need to cover the basics of interest. We will start with a general discussion of interest, followed by an overview of simple interest. After explaining the function of powers and indices, compound interest will then be examined. Interest A lender would expect to be compensated by a borrower for the use of the lender’s capital. This compensation is known as interest. Interest is paid to the lender as compensation for three things: 1. The lender’s loss of utility — the lender does not have access to their money for the period during which the borrower has control of it; 2. The risk involved — the lender is risking the total loss of their money if the borrower does not return the money at the end of the period of the loan; and 208

3. The loss of purchasing power — the amount that the lender receives back from the borrower at the end of the period of the loan will purchase fewer goods in a period of inflation. The rate of interest applicable invariably depends upon the term for which money is required (among other factors), so that at any point in time, interest rates have a term structure. On the financial planning level, we are primarily concerned with rates offered by banks and other institutions operating in the retail market. In the first part of this section we consider only fixed rates of interest. That is, we will assume that the annual effective rate of interest is fixed over the entire term of the loan. Simple Interest If Jai lends you Rs. 1,000 for a period of three years and you agree that you will pay Jai Rs. 100 at the end of each of the three years and return the principal or capital to him at the end of three years, then you will be paying Jai 10% per annum simple interest. This is because 10% of Rs. 1,000 is Rs. 100, or because: So, over the three years Jai would receive Rs. 300 in interest. Thus, the total amount you pay is easy to work out. You pay Rs. 1,300 in total for the opportunity to use Jai’s Rs. 1000 for three years. In general, we can work out the rate of interest as being: If the rate of interest and the principal amount are known, then the amount of interest can be calculated by simply transposing the above equation as follows: Amount of interest = principal x rate of interest Question 1 (a) A person borrows Rs. 2500 and agrees to pay the lender Rs. 60 per year at the end of each year for the period of the loan. Calculate the rate of interest p.a. (B) The lender is unhappy with this arrangement and requests that the borrower pay an 8.5 per cent interest rate. How much interest will the borrower pay each year? (C) Based on the information in part (b), if the loan was for four years, how much interest would the borrower expect to pay? Consider now the case where the period of the loan is less than one year. For instance, if Rs. 1,000 was borrowed for six months at 10% per annum interest. Since at this rate of interest the interest bill for one year would be Rs.100, for six months it would be Rs. 50. If the Rs. 1,000 were borrowed for three months, then the interest bill would be Rs. 25. These are very easy calculations, but think about what we have done to arrive at the six month and three month interest bill. We have calculated the interest cost annually first, and then multiplied that answer by the portion of the year 209

that the loan period occupies. So, if the interest bill is Rs. 100 for one year, and the money is only borrowed for six months (half of a year), then the interest bill is obviously Rs. 100 x 0.5, which is Rs. 50. This simple calculation can be formally represented by the following equation: If =P Amount borrowed =r Rate of interest =t Time period (in years) Interest= i Then i=PXrXt Example This formula can be used for the earlier example, where Rs. 1,000 was lent for a period of three years, with Rs. 100 being paid at the end of each of the three years. In this example t was equal to 1. P = Rs. 1,000 r = 10% t = 230 days (However, it needs to be presented in terms of years. So what portion of a year is 230 days? Ignoring leap years, a year has 365 days. Therefore the portion of a year is 230/365.) Therefore I = 1000 X (10/100) X (230/365) = Rs. 63.01 Question 2 If you borrowed Rs. 2,500 at 9% per annum, how much interest would you expect to pay for each of the following periods: (a) one year (b) six months (c) three months (d) 29 days (Note that the value of t is calculated based on the time period specified. So if the time period is six months, then calculate t based on 12 months in a year. If the time period is days, then calculate t based on 365 days in a year.) Indices or Powers Before we proceed to discuss compound interest you should be aware of certain conventions about powers or indices. For example, if we were to multiply 2 X 2 X 2, an easier way to represent this is 2 to the power of 3, represented by 23. The answer to 2 X 2 X 2 is 8. Rather than keying in 2 X 2 X 2 into your calculator, a simpler way of calculating the answer is to use the ^ function. First push the number 2, then push the shift button, after that push button 6; this tells the calculator that 2 is the x value. Then push the number 3, and close bracket then press EXE button. This tells the calculator that the Y value is 3. Your calculator should then give you the answer 8. Try something harder: use your calculator to show that 1.053 = 1.157625. 210

Example Use the ^ function on your calculator to evaluate the expressions below. Check all answers against the values provided. Expression Value (to six decimal places) 1.074 1.310796 215 32,768 1.063.4 1.219102 (1.1)1 /12 1.007974 (1.1)1 /13 1.007358 Notice, though, that while it is important to know the meaning of a negative power, the ^ function produces the result in a single operation. The ^ ‘accepts’ numbers like ‘-2.4’ ‘1)12’ for power. Example Check the value of each of the expressions below which contain negative powers, using a single operation of the XY function on your calculator for each. Expression Value (to six decimal places) (i) 1.1-4 0.683013 (ii) 1.08-4.7 0.696480 (iii) 1.045-20.5 0.405617 Compound Interest If Rs. 1,000 is borrowed for three years and 10% per annum in interest is to be paid, then the lender will receive Rs. 100 at the end of each year as interest payment. The total amount in interest received for the three years will be Rs. 300. However, if the lender requests that the borrower adds the interest payment on to the principal at the end of each year and then pays interest on the new principal in the next year, then the effect is slightly different. First Year Interest = 10% of Rs. 1,000 = Rs. 100 Principal = Rs.1,000 + Rs.100 = Rs.1,100 The new principal could also be calculated by multiplying the initial principal amount by 1 + the interest rate. Therefore: New Principal = Rs. 1,000 X(1 + 10%) Rs.1,000 X (1.1) Rs. 1,100 Second Year In the second year the interest will be earned on the new principal amount of Rs.1,100 rather than the Rs. 1,000. 211

The interest received in the second year will therefore be: Interest = 10% of Rs.1100 = Rs.110 Principal = Rs.1,100 + Rs.110 = Rs.1,210 The new principal at the end of the second year can also be calculated by multiplying the original amount by 1 + the interest rate. This can be represented by: New Principal = Rs. 1,100 X (1.1) = Rs. 1,210 Or alternatively, using the original principal amount of Rs.1000, after the second year the new principal could be calculated: New Principal = Rs. 1000 X (1.1) X (1.1) = Rs. 1,210 From our earlier work on powers and indices, you should realise that this can be represented as: New Principal = Rs.1000 X(1.1)² You may wish to check this on your calculator. Third Year In the third year, interest will be earned on the new principal amount of Rs. 1,210. The interest received in the third year can be calculated as follows: Interest = 10% of Rs.1,210 = Rs.121 Principal = Rs. 1,210 + Rs.121 = Rs.1,331 How can the answer of Rs.1,331 be obtained using the original principal amount of Rs.1,000 and using your knowledge of indices and powers? It can be simply calculated by: New Principal = Rs. 1,000 X(1.1)3 = Rs. 1,331 In this case, the total amount of interest received over the three years will be Rs. 331. This is Rs. 31 more than if simple interest were being paid. This difference is explained by the fact that with compound interest, the lender is receiving interest on their interest. Let’s now try representing these calculations with a basic formula. This formula can then be applied to any calculation that involves interest compounded over time where the accumulated value is being calculated. Where: P = the principal amount invested (in this example, P = Rs. 1,000) i = the interest for the period (in this example, i = 10%) n = the number of periods of compounding (in this example, n = 3 years) 212

A= the accumulated amount at the end of the time period Given our earlier calculations, the formula should be as follows: A = P(1 + i)ⁿ We can infer that a sum or principal (P) invested at a rate (i) per interest period for (n) such periods will have grown to P (1+i) ⁿ at the end of the term. To check the formula, we will use the values above for P, i and n: = Rs. 1,000 (1 + 0.1)³ Rs. 1,000 × 1.331 Rs.1,331 You may wish to apply the formula to the earlier calculation when n = 2. Now note the difference between the compounding interest formula and the simple interest formula: Compounding Interest Formula: A = P (1 + i) ⁿ Simple interest formula: A = P (1 + ni) Because there is no compounding effect, there is no need to have any powers or indices in the simple interest formula. Using the simple interest formula and the information from the previous example: Where: i = 10% or 0.1 A P = Rs. 1,000 n=3 = Rs. 1,000 [1 + (3 ×0.1)] Rs. 1,000 ×1.3 Rs. 1,300 Question 3 Calculate the accumulated amounts for the following investments: (a) Rs.8,000 invested at 5% for five years, compounded annually. (b) Rs. 24,000 invested at 7 per cent for 12 years, compounded annually. Question 4 You invest Rs.10,000 for three years in a fixed term deposit offering 6% p.a. At the end of three years you then hope to reinvest the balance in another fixed term deposit for a further two years, with an annual compounding interest rate of 8%. How much will your savings have grown at the end of the five years? 213

Before we complete this section on the basics of the time value of money, there are two further terms relating to interest that need to be considered: nominal rate of interest and effective rate of interest. Nominal Rate of Interest The nominal rate of interest is a term used in the finance sector to represent the rate of interest on a per annum (yearly) basis. With many investments, an interest rate is specified which does not match the time period of interest being calculated. For example, an investment may have a life span of less than 12 months, yet the interest rate is normally quoted as a nominal rate p.a.. Also, an investment that has a compounding period of less than 12 months is also normally quoted with a nominal interest rate. For both types of investments this nominal interest rate will, however, be different from the actual or real interest rate for the period. As with the earlier section on simple interest, when the compounding period is less than one year, then the interest rate is adjusted accordingly. Thus, where the nominal interest rate is 12% p.a., then: If the compounding period is one year, the interest rate for the period is 12% If the compounding period is six months, the interest rate for the period is 6%; If the compounding period is three months, the interest rate for the period is 3%. When undertaking compounding interest calculations, always ensure that the value of the interest rate (i) used in the calculation reflects the compounding period. Therefore, the nominal (or per annum) rate of interest should be adjusted so that the value of i matches the compounding period. If the compounding period is six months, the value of i should be for six months. Example Suppose Rs. 1,000 is borrowed over four years at 12 per cent p.a. nominal interest. 1. If the amount is compounded annually: Then = 12% i =4 n So A = Rs. 1,000 X (1 + 0.12)4 = Rs. 1,573.52 2. If the amount is compounded semi-annually: Then = 6% i =8 n So A = Rs. 1,000 X(1 + 0.06)8 = Rs. 1,593.85 3. If the amount is compounded quarterly: Then i = 3% 214

n = 16 So = Rs. 1,000 X (1 + 0.03)16 = Rs. 1,604.71 Effective Annual Interest Rate The effective annual interest rate is the real or actual interest rate that is earned or paid on an investment on an annual basis. Therefore, with an investment that is compounded semi-annually, the effective annual interest rate would reflect the rate of interest earned on that investment on an annual basis The formula for calculating the effective annual rate of interest is: Effective annual rate = (1 + I) –1 (where m = the number of compounding periods per year) For example Number 3 above, where the interest was compounded quarterly, we could work out the effective annual interest rate by calculating: Effective annual rate (1 + i) – 1 (1+0.03) 4 - 1 1.1255 - 1 0.1255 12.55% Question 5 If you had a Rs. 100,000 investment loan at 9% p.a. nominal interest, what would be the interest paid at the end of a year if the loan was compounded: (a) Annually? (b) Semi-annually? (c) Monthly? In each case, also calculate the effective annual interest rate. The effective annual interest rate is a very important tool in financial planning as it provides a consistent base with which to compare interest rates. Investments with different compounding periods can therefore be compared by calculating their effective rates of interest. Review This is the end of the section titled ‘Basics of the time value of money’. In this section we have endeavoured to examine the concepts of simple and compound interest, along with the terms nominal and effective annual interest rates. Take time before you proceed to the next section to reflect on what you have learned. If there are any aspects that you find confusing, try to work through those particular areas again, as it is important that you have a good understanding of these basic concepts and computations before you 215

proceed to the next section. The next section builds on these basic concepts and computations, introducing you to the specific methods used to equate and then compare rupee values. Present and Future Values To evaluate and compare investment proposals, it is necessary to consider the size and timing of their cash flows. Given that the value of a rupee changes depending on the time when it is received, to equate the cash flows received at different times for different investment proposals, the cash flows need to be converted to reflect their value at a common point in time. This common point in time can either be the present point in time or a future point in time. This section will examine the methods by which cash flows can be converted to both present and future values. We will also examine two types of cash flows. First, we will deal with singular cash flows; then we will discuss how to convert a series of regular cash flows of the same amount, often referred to as an annuity. You will shortly be introduced to various mathematical formulae which can be used to evaluate and compare investment proposals. However, a financial calculator will enable you to perform many of the required calculations without you having to use these formulae. Singular Cash Flows Any investment will include at least one cash flow. All of the cash flows within an investment can potentially be treated as singular cash flows. If these individual cash flows are then converted to a common basis, such as present or future values, they can be added together. The present and future values of such investments can then be used as a basis for their evaluation. Future Value You are already quite familiar with the formula for calculating the future value of an amount. We used this formula earlier when we calculated the accumulated value of an investment with compounding interest. The formula we used was: Where: A = the accumulated value P = the principal amount A = P(1 + i) ⁿ To use the terms associated with the time value of money, we simply replace ‘the accumulated value’ (A) with the term ‘future value’ (FV), and ‘the principal amount’ (P) with ‘present value’ (PV). FV = PV(1 + i) ⁿ From this point onwards, we will use the terms future value and present value when examining investments. Example If you place Rs. 10,000 in a savings account with an annual interest rate of 6%, how much will you have in your savings account at the end of 10 years? FV = Rs.10,000(1 + 0.06)10 Rs.10,000(1.7908) Rs.17,908 216

Present Value Calculating the future value means working out how much a certain sum will grow to in the future. In contrast, the calculation of a present value requires us to work out the value of a future sum in today’s rupees. Calculating the present value of future cash flows allows us to compare and evaluate cash flows of many different types, as all of the cash flows are converted to today’s rupees. Example The example we used above can be turned around to ask what amount would need to be invested to become Rs.17,908 in 10 years’ time, given an annual interest rate of 6%? As we stated earlier, the amount invested is known as a present value (PV). We have already seen how we can work out the future value by using: FV = PV (1 + i)n We can reformulate this equation to make the present value (PV) the subject: PV=FV/(1+i)n Therefore PV = Rs. 17,908/(1. 06)10 = Rs. 10,000 Example Would you prefer to receive Rs. 5,000 today or Rs. 7,000 in six years’ time, assuming that the Rs.5,000 can be invested at 5% for six years with interest compounded annually? First, we will try to answer the question by converting the appropriate cash flows to future values. Evaluation based on future value: Option 1 FV = Rs. 5,000 (1 + 0.05)6 = Rs. 6,700 Option 2 FV = Rs. 7,000 Secondly, we will try to answer the question by converting the appropriate cash flows to present rupee values. Evaluation based on present value: Option 1 PV = Rs. 5,000 Option 2 PV = Rs. 7,000/(1 + 0.05)6 = Rs. 5,226 Which option would you prefer? Both methods should indicate that you should choose to receive Rs. 7,000 in six year time. Take a moment to work these calculations through on your calculator, to check that you understand how the formula changes work. After working through this question you should have an appreciation of the need to convert monetary amounts to similar rupee values so that they may be compared. The Rs. 5,000 and Rs. 7,000 cannot be 217

compared unless they are adjusted to take into account the timing of the cash flows, along with the compounding interest rate. In any calculations about the present value or future value of a single amount there are four variables: Present value (PV); Future value (FV); The interest rate for each period (i); and The number of periods (n). Normally, any question will involve being given three of these variables and having to find the fourth. We have already used two of the four formulae above. We would like to introduce you to the other two formulae, but please note that it is not necessary to remember or even apply these formulae. One benefit of using a financial calculator is that you follow a similar sequence of keystrokes for the three variables that are known, regardless of the composition of those three variables, and the answer for the fourth variable is then provided. Formula to Calculate i If we need to calculate the interest, given PV, FV and n, we use: i = (FV/PV)1/n - 1 Formula to calculate n If we need to calculate the number of periods, given PV, FV and i, we use: n=ln(FV/PV) ÷ ln(1+i) Where: In = Natural log (see below) In the formula to calculate the value of n we make use of logarithms. Logarithms enable us to perform difficult calculations more easily. For example: the equation ax = b x = log b / log a This is probably the only time you will need to use logarithms, as a financial calculator can assist you to solve most financial mathematics problems. We could use either logarithms to the base 10, normally the log button on a calculator, or natural logarithms, normally the in button on a calculator. Please note that if you choose to use a financial calculator you do not need to use logarithms to arrive at a value for n. Financial calculators might appear intimidating at first glance, but they all have an underlying logic which comes from the present value and future value calculations discussed above. Remember that no matter how complicated the calculations might seem, they all are capable of being decomposed ultimately into simple PV formulas. So you will see on every financial calculator the four keys: PV, FV, i for interest (or discounting) rate, and n for the number of compounding periods. All you need to do is enter three of these and the fourth will be predetermined by the fundamental equation FV = PV (1+i)n 218

Please further note that when you use either a financial calculator (using the special function keys) or spreadsheet for present value calculations, the answer is presented as a negative. The calculator or spreadsheet represents the present value as a cash outflow and thus makes it a negative value. It is assumed that if you are to receive the future value (a cash inflow) in n years’ time, then this is equivalent to investing the present value (a cash outflow) today. This cash inflow/outflow assumption will not hold true for all present value calculations but your calculator or spreadsheet will not recognise this. Despite this rather mechanistic approach, the normal convention for written present value calculations is to recognise the present value as a positive number. It is therefore expected that your answers to present value questions will be positive. Let’s now do some practical questions. You may wish to use the formulae provided earlier or you may wish to use a financial calculator. For each of the examples below, we have provided the relevant formula as well as the set of key strokes you would use if you had a Casio FC -200V financial calculator. If you do not have either of these models of calculator you may need a different set of key strokes—you will need to consult the user guide that came with your calculator. Example 1 What will Rs. 1,000 become if it is invested at 8% p.a. for three years compounded semi-annually? In this case we want to know the future value (FV): If PV = Rs. 1,000 n = 6 (two compounding periods in each year) i = 4% Then = Rs.1,000 ×(1 + 0.04)6 FV = Rs.1,265.32 Using a financial calculator To clear all data currently in calculator Press Display Memory ? 1. Shift 9 Clear : : EXE : EXE Set up : EXE Memory All 2. All : EXE ↓ ↑ then EXE 3. EXE YES 4. AC. Using Financial Calculator Press Display CMPD Compount Int. Set : End n=0 219

I% = 0 Use ↓ ↑ to select 1 “Set” and then press ‘EXE’ Button Press 2 to select ‘End’ Use ↓ ↑ to select “n” input 3 then press ‘EXE’ Button Use ↓ ↑ to select “I%” input 8 then press “EXE” Button Use ↓ ↑ to select “PV” input 1000, then press “EXE” Button Use ↓ ↑ to select “e/y” input 2 then press “EXE” Button Use ↓ ↑ to select “FV” then press “Solve” Button you will get = 1,265.32 Example 2 What amount must be invested at 8% per annum for four years compounded quarterly to grow to Rs. 1,000? In this case we want to know the present value (PV): If FV = Rs.1,000 n = 16 (four compounding periods in each year) I i= 2% Then PV = Rs.1,000 /(1 + 0.02)16 = Rs.728.45 Using a financial calculator To clear all data currently in calculator Press Display Memory ? 1. Shift 9 Clear : EXE : EXE Set up : EXE : Memory All 2. All : EXE ↓ ↑ then EXE 3. EXE YES 4. AC. Using a financial calculator Press Display CMPD Compount Int. Set : End n=4 I% = 0 Use ↓ ↑ to select 1 “Set” and then press ‘EXE’ Button Press 2 to select ‘End’ Use ↓ ↑ to select “n” input 4 then press ‘EXE’ Button Use ↓ ↑ to select “I%” input 8 then press “EXE” Button Use ↓ ↑ to select “e/y” input 4 then press “EXE” Button Use ↓ ↑ to select “FV” input 1000, then press “EXE” Button Use ↓ ↑ to select “PV” then press “Solve” Button you will get = 728.45 Example 3 If Rs.1000 is invested and grows to Rs.1127.16 in 24 monthly compounding periods, calculate the monthly rate of interest and the nominal interest rate required for this to take place. In this case we want to know the rate of interest (i): If 220

FV = Rs. 1,127.16 PV = Rs. 1,000 24 = Then = (Rs. 1127.16/Rs.1,000)1/ 24 – 1 1.005 – 1 0.5% per month or 6% per annum compounded monthly Using a financial calculator To clear all data currently in calculator Using a financial calculator To clear all data currently in calculator Press Display Memory ? 1. Shift 9 Clear : EXE :EXE Set up : EXE Memory All : 2. All : EXE ↓ ↑ then EXE 3. EXE YES 4. AC. Using Financial Calculator Press Display CMPD Compount Int. Set : End n=0 I% = 0 Use ↓ ↑ to select 1 “Set” and then press ‘EXE’ Button Press 2 to select ‘End’ Use ↓ ↑ to select “n” input 2 then press ‘EXE’ Button Use ↓ ↑ to select “PV” input 1000 then press “EXE” Button Use ↓ ↑ to select “FV” input 1127.16 then press “EXE” Button Use ↓ ↑ to select “e/y” input 12 then press “EXE” Button Use ↓ ↑ to select “I” then press solve, you will get 6%. This is interest rate per annum monthly interest rate will be 6 ) 12 = 0.5% Example 4 If Rs. 1,000 is invested and grows to Rs.1,257.16 when it is compounded monthly at an annual nominal rate of 12%, how many monthly periods would this take? In this case we are interested in finding the number of periods (n): If FV = Rs. 1,257.16 PV = Rs. 1,000 i =1% In = natural log (see above) 221

Then ln (Rs. 1,257.6/Rs. 1000) ÷ ln (1 + 0.01) = In (1.2576)/In (1.01) 0.23 ÷ 0.01 (note: natural logs have been used) 23 months To clear all data currently in calculator Press Display Memory 1. Shift 9 Clear :? Set up : EXE Memory : EXE All : EXE 2. All : EXE ↓ ↑ then EXE 3. EXE YES 4. AC. Using Financial Calculator Press Display CMPD Compount Int. Set : End n=0 I% = 0 Use ↓ ↑ to select 1 “Set” and then pess ‘EXE’ Button Press 1 to select ‘begin’ 2 to select ‘End’. Use ↓ ↑ to select “I%” input 12 then press ‘EXE’ Button Use ↓ ↑ to Select “PV” input 1000 then press “EXE” Button Use ↓ ↑ to Select “FV” input -1257.16 then prss “EXE” Button Use ↓ ↑ to Select “n” then press “solve” Button You will get 1.92 years or 23 months. You can now further test your understanding of calculating the values of PV, FV, i and n by answering Question 2.6. It is your choice as to whether you use a financial calculator or formulae to answer these questions — but you should try to use the method that you intend to use in the exam. Question 6 1. What would Rs. 150 grow to if it was invested for five years at 8% p.a. compounded quarterly? 2. What amount would need to be invested to grow to Rs. 250 if the amount was invested for four years at 10% compounded semi-annually? 3. If Rs. 500 is invested at a certain rate of interest and is compounded monthly, it grows to Rs. 598.21 in two years. Calculate the monthly rate of interest. Calculate the effective rate of interest. 4. If Rs. 750 is invested at 10% p.a. compounded quarterly, how many years would it take for this amount to grow to Rs. 1,059.73? 222

This concludes the section on the calculation of the present value and future value of singular amounts. You should now have an understanding of how basic financial problems involving singular cash flows can be solved. We will now examine annuities, which are comprised of regular, even cash flows. Annuities An annuity is a series of regular payments over a specified number of periods. Although the term derives from the Latin annum, a year, it is also used to describe cash flows which occur daily, weekly, monthly or whatever. Payments may be level, increasing, decreasing or stochastic (i.e. random). When the payments are sure to be made, for instance contractual, annuities are termed annuities certain. This is to distinguish them from life annuities, in which payments are only made for as long as the annuitant lives. Some Annuities are Familiar: The salaries of employees are designated in nominal annual terms which actually represent a series of regular weekly, fortnightly or monthly payments. From such a salary, a regular amount is paid in tax, forming an annuity of regular tax payments to the income tax department. Payments on home loans, industrial land purchased by firms, loan repayments for plant and so on form annuities. Semi-annual coupon payments on government bonds form annuities which terminate upon redemption of the principal. Retirement fund payments are regular deductions from salaries of employees to funds designated by employers or the government. Subscriptions to professional societies, unions and clubs are regular payments which are often annual or monthly, periodically adjusted for inflation, so they also form annuities. Lease or rent payments also form annuities over the term of the lease. Doubtless you can think of other examples. Annuity payments can be made at the start of the time period when the investment is termed an annuity due, or at the end of the time period when it is called an ordinary annuity. A simple annuity has a period of payment which is the same as the compounding period. Other Types of Annuities Include: A deferred annuity, in which payments are made or received two or more periods after the initial loan is taken out or the initial investment is made; and A perpetuity, which is an annuity where the payments do not terminate after a specific length of time but continue forever. 223

Present Value of an Annuity To find the total present value of a series of payments which occur at different points in time, we need to add the present values of each payment. These then, represent values at the same point in time (the present) and can be legitimately added together. This is called ‘discounting each cash flow back to the present’. Example You might wish to know whether the asking price for an annuity which promised to pay Rs.10,000 p.a. for three years at the end of each year was good value, assuming a rate of interest of 7%. Formula 1 You can arrive at the answer by calculating the present value of each cash flow in turn, using the PV formula we have already used, and then add all of these present values together: Where: i = Rate of interest for each period N = Number of periods PMT = Payment each period If money can be invested at 7% p.a., then the present value of the cash flows is: = Rs. 10,000/(1+.07) + Rs. 10,000/(1+.07)2 + Rs. 10,000/(1+.07)3 = Rs. 10,000/1.07 + Rs. 10,000/1.1449 + Rs. 10,000/1.2250 = Rs. 9,345.7944 + Rs. 8,734.3873 + Rs. 8,163.2653 = Rs. 26,243.45 This can be represented by the following diagram: PV Rs. 10,000 Rs. 10,000 Rs. 10,000 01 2 3 Time (Yrs) Rs. 9,345 Rs. 8,734 Rs. 8,163 Rs. 26,243 Formula 2 (simple ordinary annuities) The above calculation is cumbersome and there is an easier one that can be used for annuities where the cash flows are regular, even and occur at the end of each period (these annuities are termed simple ordinary annuities). To calculate the present value (PV) of a simple ordinary annuity: PV = PMT X (1- (1+i)-n)/i Where: i = Rate of interest for each period 224

N = Number of periods PMT = Payment each period Using the formula, the present value of our annuity can be calculated as follows: PV = Rs. 10,000 (1- (1+.07) -3)/.07 =Rs. 26,243.45 Note that this formula gives us the same answer as Formula 1. Before going on to another example, consider what the answer means. You have calculated the present value of an annuity promising to pay Rs.10,000 each year, for three years. The interest rate is assumed to be 7%, which acts as a benchmark in the evaluation. If you can acquire such an annuity for Rs.26,243 then your return on the annuity will be 7%. If, however, you can acquire such an annuity for less than Rs.26,243, then your return will be greater than 7%, and vice versa. Now try another example: Example How much should be invested at 8% per annum compounded quarterly so that Rs. 500 will be returned each quarter for five years? We are interested in finding the present value (PV). If PMT = Rs. 500 n = 20 (since there are four payments a year for five years) i = 2% or 0.02 (for each payment period) Then PV = Rs. 500 X ((1- (1+.02)-20)/.02 = Rs. 8,175.72 Future Value of an Annuity As we stated above, an annuity is a series of equal rupee payments for a specified number of periods. Sometimes these regular payments will be made so that a sum can be accumulated. Such an investment can be particularly appropriate for individuals wishing to save money for education, for retirement or for a holiday. In any case, they will want to know how much their savings will have grown at that future point in time. To calculate the final or future value of such an annuity we can use either Formula 1 or Formula 2 below: Formula 1 FV = PMT x (1+i) + PMT x (1+i)2 + ….. + PMT X (1+i)n-1 Formula 2 (single ordinary annuities) FV = PMT X ((1+i)n - 1)/i Example A couple invests Rs. 500 every six months towards a fund to pay for their children’s education. If the investment pays 9% per annum compounded semiannually, what will the fund be worth after 10 years? 225

We can answer this question using Formula 1 above, treating each cash flow as a single amount. However, such a calculation would be very tedious given the number of cash flows. Alternatively, we can also use Formula 2. If PMT = Rs. 500 n = 20 (since there are two compounding periods a year for 10 years) i = 4.5% Then FV = Rs. 500 x ((1+.045) 20 - 1)/0.045 = 15,685.71 Using a Financial Calculator Steps Clean all memory Using Financial Calculator Press Display CMPD Compount Int. Set : End n=0 I% = 0 Use ↓ ↑ to select “n” input 20 & press “EXE” Button Use ↓ ↑ to select “I%” input 4.5 & press “EXE” Button Use ↓ ↑ to select “PMT” input 500 & press “EXE” Button Use ↓ ↑ to select “FV “ & press “Solve” Button you will get -15685.72 Question 7 (a) A return of Rs. 700 per year is required (payments in arrears) for five years. If the investment will return 10% p.a. compounded annually, how much should be invested? (b) A sum of Rs. 1,000 is invested each year for four years (in arrears). If the investment returns 8% p.a. com-pounded annually, to what amount will the payments accumulate? (c ) Ram Anand wishes to have a lump sum retirement fund payment of Rs. 250,000 in 30 years time. Assuming he can earn 12% p.a. compounded after tax and expenses, what would be the size of the yearly payment into the fund (in arrears) to accrue this desired lump sum? Annuities in Perpetuity Let’s now look at another type of annuity. Perpetuity is an annuity where the term is infinite. In other words, the number of periods, n, is infinite: they continue forever. For an ordinary simple annuity, the present value is given by: PV = PMT x (1- (1 + i)-n)/i As n increases to infinity, the term (1 + i)-n tends to zero. Hence, the present value of a perpetuity is given by: PV = PMT/i Test this formula by answering the following question. First, answer the question using the perpetuity formula, then answer the question assuming that n is equal to a large number such as 100, using the PV of a 226

simple annuity formula. Does each method give a similar answer? The perpetuity formula is much easier to remember, however! Question 8 Vikas has been asked to sponsor a prize at his children’s secondary college. Vikas would like to give a prize of Rs. 100 in perpetuity. If it is assumed that the sum invested will earn 10%, how much will Vikas have to donate? Timing Issues with Annuities It is important when working with annuities that the timing of cash flows is considered. The present value and future value annuity formulae given earlier assume the cash flows occur in arrears, which means at the end of each period. However, this will not always be the case and consequently some adjustments may be required. The present and future value formulae also calculate the value of an annuity based on the first cash flow occurring at the end of the first period. Again, this will not necessarily be the case with all annuities. One such example is a deferred annuity (an annuity in which payments are deferred). To illustrate these timing issues, consider the following example. Example You have just inherited Rs.20,000 and wish to put some of it aside for your daughter ’s university education. You estimate that when she is at university, 10 years from now, she will need Rs.7,000 at the beginning of each year for the four years of her university tuition. You wish to establish a fund which earns 6% annually. How much do you need to deposit in the fund today to meet your daughter ’s educational expenses? To help us to understand the timing of the cash flows in this example, the cash flow diagram is a valuable tool: Start of year: 10 11 12 13 Cash flow required Rs. 7,000 Rs. 7,000 Rs. 7,000 Rs. 7,000 Initial required investment unknown This is an example of a deferred annuity, where there are regular, even cash flows, but the receipt or payment is deferred. In this example, the receipt of the cash flows is deferred until the start of the 10th year. In order to use the present value of an annuity formula, we need to view the cash flows in terms of being at the end of each period (in arrears). It is reasonable to assume that the start of the 10th year is the same as the end of the ninth year, and so on. Based on this assumption, we can now solve the problem. Solution 1 227

The analysis involves two steps. 1. Calculate the present value of the annuity. PV = Rs. 7,000 x ((1-(1+0.06)-4)/0.06 = 24,255.74 This present value represents the value of the annuity at the beginning of the ninth year (which is the same as the end of the eighth year). This can be represented by the following diagram: PV Rs. 7,000 Rs. 7,000 Rs. 7,000 Rs. 7,000 0 8 9 10 11 12 Time Rs. 24,255 (end of ? each year) The second stage of this analysis is to calculate the present value, at time 0, of the present value of the annuity at the end of the eighth year. To do this we can use the present value of an amount formula. 2. Discount the present value of the annuity back to time 0. Using the present value of a singular cash flow formula: PV = FV /(1+i)n = 24,255.74/(1+.06)8 = 15,218.35 Solution 2 An alternate way of answering this question is to convert each cash flow to present value (treating each cash flow as a separate amount): PV = PMT/(1+i) n + PMT/(1+i) n+1 + PMT/(1+i) n+2 + PMT/(1+i)n+3 PV Rs. 7,000 Rs. 7,000 Rs. 7,000 Rs. 7,000 0 9 10 11 12 Time Rs. 4,43.28 (end of each year) Rs. 3,908.76 Rs. 3,687.61 Rs. 3,478.78 228

Rs. 15,218.33 PV = Rs. 7,000/(1+.06)9 + Rs. 7,000/(1+.06) 10 + Rs. 7,000/(1+.06) 11 + Rs. 7,000/(1+.06) 12 Rs. 4,143.28 + Rs. 3,908.76 + Rs. 3,687.51 + Rs. 3,478.78 Rs. 15,218.33 Both methods of answering the question are equally valid — you just need to choose which method you are most comfortable with. It is beyond the scope of this topic to cover all of the possible timing issues with annuities. However, you should be aware that there are some issues related to timing and by carefully thinking about the timing of the cash flows it should then be possible to correctly calculate the present or future value of the annuity in question. A cash flow diagram is often a valuable tool when confronted with a timing of cash flows problem. While you are now equipped with the necessary tools to solve financial mathematics problems, the application of these tools can prove to be daunting in practice. Some simple steps that may assist with the solving of financial problems include: 1. Identify the cash flows. Draw a cash flow diagram labelling the cash inflows and outflows. 2. Determine what needs to be known. What is required to answer the question at hand? There are only so many values that can be determined. It must be one of PV, FV, i, n or PMT. 3. Identify the type of problem. Is it an annuity or a single amount? It is an annuity if there is a regular, even stream of cash flows. This will help you to identify which formula to use or which sequence of buttons on the financial calculator you should apply. 4. Consider whether there are any timing issues that may affect the treatment of the cash flows. 5. Now solve the problem by applying the appropriate formula or financial calculator steps. You may find these steps rather basic but they do provide a good initial structure upon which solutions to financial problems can be based. You may wish to use these steps when you answer the following question. Question 9 (a) On receipt of a redundancy payment of Rs. 40,000, Karishma decides to go to university and complete a three- year degree. She invests her money in a fixed interest account earning 5% p.a. compounded quarterly. She intends to withdraw equal sums at the end of each quarter during the three years of her university study. How much can Karishma withdraw each quarter so that her fund is exactly exhausted at the end of the three-year period? (b) Madhu has the choice of investing Rs. 1,000 with a friend who guarantees that he can double her money in seven years. Alternatively, she can put her money in a fixed rate term deposit paying 12%. Which investment should she choose? (c) Maneka needs Rs. 25,000 five years from now. She would like to make equal payments at the end of each year into an account that pays an annual interest rate of 7%. What are her annual payments? (d) Rashmi and Biswas purchase a bond for Rs. 10,000 that will pay no interest during its 10-year life and have a value of Rs. 31,060 at maturity. What interest rate does this bond pay? 229

Comparing Investment Returns In this section, we will examine different methods that may be employed in order to evaluate investment opportunities. In doing so, we will be building on the work covered in the previous section ‘Present and future values’, where we examined the tools for comparing cash flows that occur in different periods. In this section we will further utilise these tools, in conjunction with additional techniques, to compare investment returns. There are two basic types of investment comparison methods. One type, the ‘discounted cash flow’ methods, take into account the time value of money. The other type do not take into account the time value of money and, thus, are labeled ‘non-discounted cash flow’ methods. Discounted Cash Flow Methods The discounted cash flow methods are based on cash flows that are adjusted for the time value of money. The two primary discounted cash flow methods are net present value and internal rate of return. We will discuss each of these methods in turn. Net Present Value The preferred basis for selecting investments is to identify the future cash flows that an investment is expected to generate and, using an appropriate discount rate, compare the present value of expected cash flows across different investments. This is termed the net present value calculation. Basically, the net present value is equal to the present value of these future cash flows, less the initial investment. It is used to decide whether the investment is worth undertaking and better than other investment proposals. The present value of the net cash inflows or outflows is determined by multiplying those flows by a discount rate. The Discount rate Previously we calculated the present value and future value of various cash flows using the interest rate that applied to the investment being considered. However, you may wish to use present value and future value to analyse and compare cash flows over a period of time where no particular interest rate is involved or where different interest rates apply. An example of a situation where there is no particular interest rate to use, is where a client wishes to know the present value of payments to be received under an employment contract, so the contract can be compared to other opportunities available to him or her. In such cases, you will need to select what is called a discount rate to use in your calculations, instead of an ordinary interest rate. Such a discount rate is the rate of return which you consider appropriate for analysing and comparing specific types of investments, based on current market and economic conditions. An appropriate discount rate may be based on rates of return currently available on investments, interest rates currently charged on loans, or other factors such as the inflation rate. To put it another way, it is the rate of return we would ordinarily expect to receive for the type of investment we are considering. The discount rate is the investor’s required rate of return. 230

Incorporating Investment and Financing Decisions So far in this topic, investment and financing decisions have been discussed separately, and therefore the cash flows pertaining to any borrowed money are excluded from the investment decision. No allowance has been made for the cash flows associated with the financing decision (i.e. the cost of the debt). This is because the link between the investment and cost of financing the investment is made through the discount rate. The discount rate incorporates the cost of funds into the investment decision-making processes, so in this way the financing decision can be related to the investment decision. This will be demonstrated later. The preferred basis for selecting investments using the net present value method is to identify the future cash flows that an investment is expected to generate, excluding repayment of interest and principal of any borrowed funds, and then to show these cash flows in present value rupees. Cash flows, not accounting profits, are used. This is because accounting profits often do not match with cash flows because of timing differences and policy decisions regarding depreciation, write-offs, provisions and so on. Cash flows reflect the timing of costs and benefits, when the cash is received and when it can be invested. It is important to exclude the interest payments and loan principal repayments as we separate the investment decision from the financing decision. See the example below for a full explanation of the concept underlying this approach. Please spend some time working through this example and be sure that you grasp this concept and are not left with the misconception that the interest and principal repayments are simply being ignored! Example Mr. Jatin purchases a property for Rs.100,000, with money borrowed from the bank at 12 per cent per annum over 20 years, repayable in equal annual instalments in arrears. The cash flow for this might be either of the following two: Option 1 Year 0: – Rs.100,000 Option 2 Year 1 2 ... 20 Cash outflow – Rs.13,388 – Rs.13,388 – Rs.13,388 (capital and interest) The present value of Option 2 is Rs.100 000. Check this on your calculator, using the borrowing rate of 12% p.a. Therefore, both options provide a PV of Rs. 100,000. If the cash flow included both the initial cash flow of Rs. 100,000 and the cash payments over 20 years, then this would be double counting and show up as an initial cash flow of Rs. 200,000, which is clearly incorrect. Therefore, for any investment or purchase of equipment, the purchase price must be shown as the initial outflow in Year 0, whether the money is borrowed or not, and no repayments of any borrowed money should be shown. 231

In summary, the up-front investment, which includes any borrowed funds, is shown as a cash outflow, and is already in present value terms. It therefore, encompasses the future interest and principal repayments of any borrowed funds. After calculating the net present value of various investments, it is those investments with the highest net present value that generally should be selected. The higher the net present value of an investment, the greater the potential return for the investor. Example An investor has the choice between two mutually exclusive projects, A and B. The projects will initially cost Rs.10,000 and Rs. 11,000 respectively. Project A will provide the investor with Rs. 4,000 for the next three years. Project B will provide the investor with Rs. 3,000, Rs. 4,000 and Rs. 6,000 over the next three years. The discount rate chosen is 6% p.a. Which is the more profitable investment using the net present value method? NPV A = - Rs.10,000 + Rs. 4,000/(1.06) + Rs. 4,000/(1.06)2 + Rs. 4,000/(1.06) 3 = Rs. 692 NPV B = - Rs.11,000 + Rs. 3,000/(1.06) + Rs. 4,000/(1.06) 2+ Rs.6,000/(1.06) 3 = Rs. 428 On the basis of the above calculation, we would choose Project A, as it has a higher net present value. Case Study: Arindam Chatterjee Arindam Chatterjee has been offered two investments, Investment A and Investment B. Both require that Rs.100,000 be invested. The corresponding cash flows are shown below. Year Investment A Investment B 0 - Rs. 100,000 - Rs. 100,000 1 Rs. 12,000 2 Rs. 12,000 Rs. 15,000 3 Rs. 12,000 Rs. 20,000 4 Rs. 12,000 Rs. 25,000 5 Rs. 15,000 Rs. 30,000 6 Rs. 15,000 Rs. 30,000 7 Rs. 15,000 Rs. 25,000 8 Rs.115,000 Rs. 20,000 Rs.15,000 At the beginning of Year 1 (or the end of Year 0), Rs. 100,000 is invested. Investment A returns Rs.12,000 at the end of each of the first four years, then Rs.15,000 at the end of each of the subsequent years with the original Rs.100,000 being returned at the end of the eighth year. Investment B returns the amounts indicated at the end of each of the eight years. What advice would you give Arindam? We could try and assist him by performing some net present value calculations. In order to calculate the present value of the investment, we would need to decide on a suitable discount rate. This is obtained by deciding what sort of return we would expect on our capital. Let us assume a discount rate of 12%. Then the present value of the cash flows in Investment A can be calculated. 232

Investment A PV = Rs. 12,000/1.121 + Rs. 12,000/1.122 + Rs. 12,000/1.123 + Rs. 12,000/1.124 + Rs. 15,000/1.125 + Rs. 15,000/1.126 + Rs. 15,000/1.127 + Rs. 115,000/1.128 PV = 105,790.87 To calculate the net present value (NPV), we then deduct the original amount invested, in this case Rs. 100,000. NPV = Rs.105,790.87 – Rs.100,000 = Rs.5,790.87 Using a Financial Calculator Clean all memory Press shift 9 ↓ ↑ select All then “EXE” again “EXE” Press AC Press Display Cash Cash Flow 1% = 0 csh = D Editor X. NPV = solve Use ↓ ↑ to select “I” input 12 then press “EXE” Button. Use ↓ ↑ to select “Csh = D. Editor X” & then press “EXE” Display | 1| X | 2| | 3| | 4| Input — 1,00,000 then press “EXE” Input 12,000 then press “EXE” Do this step 3 more times Input 15,000 then press “EXE” (two more times) Input 115,000 then press “EXE” Press ESC Use ↓ ↑ to select “NPV” & Press “Solve” Button You will get 5790.87. Question 10 Perform similar calculations for Investment B to find the present value and the net present value. Then recommend to Arindam Chatterjee which investment proposal he should accept, assuming that he can only choose one proposal (i.e. he only has Rs. 100,000 to invest). Internal Rate of Return, or Yield The internal rate of return (IRR), or yield is a measure, similar to an interest rate, which expresses the degree of profitability. It can be thought of as a discount rate which, when applied to the cash flows from an 233

investment, would produce a present value exactly equal to the initial investment amount. In other words, it is the effective rate of return from the investment. The concept of IRR differs from discount rate in that the IRR is the return you will make from an investment, whereas (as we explained above) the discount rate is the rate of return which is appropriate for analysing and comparing specific types of investments, based on current market and economic conditions. If the IRR for an investment is greater than the discount rate you would normally use to assess similar investments, then the investment is worthwhile since it provides greater than the required rate of return. Since the net present values for both Investment A and Investment B above are greater than zero, we can deduce that the internal rate of return, or yield, on each investment is more than 12% (since that was the discount rate used). If the net present value was less than zero we could deduce that the IRR or yield was less than 12%. To find the internal rate of return (r) on Investment A, we need to solve the following equation. [Note that this is the same formula as used to calculate present value for an investment, with the initial cost of the investment (Rs.100,000) treated as the present value and the internal rate of return, (r), used in place of the discount rate.] Rs. 100,000 = Rs. 12,000/(1+r) 1 + Rs. 12,000/(1+r) 2 + ……. +Rs. 15,000/(1+r) 7 + Rs. 115,000/(1+r) 8 Question 11 Write down the equation that would have to be solved for r to find the internal rate of return, or actual yield, on Investment B. The IRR can be very difficult to calculate. Without the use of a financial calculator or spreadsheet, it is only possible to solve these equations by trial and error. For instance with Investment A, if we substitute 0.12 for r, we expect to get Rs.105 790.87 for the right hand side of the equation. We could then try other values of r until the right hand side of the equation is equal to Rs.100 000. Investment A PV = Rs. 100,674.13 Where r = 0.13 PV = Rs. 100,181.63 Where r = 0.131 PV = Rs. 99,692.52 Where r = 0.132 The yield is clearly between 13.1 and 13.2%. To be even more accurate: Where r = 0.1313 PV = Rs.100,034.50 Where r = 0.1314 PV = Rs. 99,985.58 Where r = 0.1315 PV = Rs. 99,936.65 The findings above indicate that to two decimal places the yield or internal rate of return for Investment A is 13.14%, since for this discount rate the present value is closest to Rs.100,000. Using a Financial Calculator After following procedure same as adopted for NPV last step will be ↓ ↑ use to select “IRR” then press solve button. Display will be 13.14. 234

Question 12 Use a financial calculator to calculate the yield on Investment B in the Arindam Chatterjee case study to two decimal places. Going back to your advice to Arindam Chatterjee in Question 10, using 12% as a discount rate for comparing Investment A and Investment B has shown that Investment B is the best choice, since its net present value is higher (Rs. 10,990.60 compared to Rs. 5,790.87). If Arindam agrees that 12% is an acceptable minimum rate of return for investments of this type, either investment is acceptable but Investment B is better. Calculating the internal rate of return for each investment tells you just how much better: Investment A returns 13.14% while Investment B returns 14.97%. Inflation and Tax When using discounted cash flow (DCF) techniques such as the ones we have discussed in this topic, it is important to be consistent in the treatment of inflation rates and tax. If the future cash flows incorporate an inflation component then the discount rate should incorporate the inflation rate. Alternatively, if the cash flows are in today’s rupees then the discount rate should exclude inflation. It is important also to decide whether the cash flows are pre-tax or after-tax as this will be reflected in the discount rate. For example, a 12% discount rate before tax, with a tax rate of 40%, is equivalent to a 7.2% (12% ×, 0.6) discount rate after tax. Example ABC Ltd is investigating a project with the following details. Projected earnings, after paying interest of Rs. 30,000, are Rs. 500,000 per year for three years. The initial outlay is Rs. 720,000. The earnings are after interest and before tax. The discount rate, a nominal rate to be consistent with the cost of funds for the company, is 10%; and the tax rate is 36%. Year 0Rs. Year 1Rs. Year 2Rs. Year 3Rs. Initial outlay 720,000 Earnings after interest, before tax 500,000 500,000 500,000 Add back interest 30,000 30,000 30,000 NCF before interest and tax 530,000 530,000 530,000 Tax (36% of earnings) 180,000 180,000 180,000 NCF before interest, after tax 350,000 350,000 350,000 NPV (discount rate 10%) 150,398.20 IRR 21.53% We add back the interest as the financing costs are included in the discount rate. We then calculate the net cash flows after tax and complete the example by calculating the NPV and IRR. Non-discounted cash flow methods The non-discounted methods used for the evaluation of investments are less sophisticated than the discounted cash flow methods as they do not take into account the time value of money. Nonetheless, these methods are still used in practice to evaluate investment proposals. Due to their unsophisticated nature they can be used to quickly assess the viability of an investment. 235

Payback Period The payback period is the length of time it takes to recover the initial cash outlay of an investment. The emphasis, therefore, is on how quickly the investment will return its original investment. It uses net cash inflows that are unadjusted for the time value of money. Whether an investment is accepted is dependant on whether the payback period is less than or equal to the desired payback period. Example An investment requires an initial cash outlay of Rs. 15,000, and yields cash outflows over the next five years. The cash flows generated from this investment are: Year 1 Rs. 2,000 Year 2 Rs. 5,000 Year 3 Rs. 4,000 Year 4 Rs. 8,000 Year 5 Rs. 6,000 The cash outlay will be recovered sometime in the fourth year. The cash flows from years one to three total Rs. 11,000. Therefore, a further Rs. 4,000 is required to meet the initial outlay. During the fourth year a total of Rs. 8,000 will be returned from the investment. Assuming the Rs. 8,000 will flow into the firm at a constant rate over the year, it will take half of the year (Rs.4,000/Rs.8,000) to recover the remaining Rs.4,000. Thus the payback period for this investment is three and a half years. If the desired payback period for the investment was three years then this investment would not be accepted. A major deficiency of the method is obviously its failure to consider cash flows that occur after the payback period. In the example given, this method may lead to an investment project being rejected when the investment has a positive net present value. This major deficiency aside, the payback period is a simple, quickly understood and easy-to-calculate method which may be used as a rough guide to evaluate investments. Accounting Rate of Return The accounting rate of return is calculated by dividing the average annual profit by either the initial or average outlay. If this rate is above the minimum acceptable level for the firm, the project is considered acceptable. Importantly, this method uses accounting profit as the basis for acceptance or rejection. This is in contrast to the other methods detailed in this section which use cash flows as the basis for acceptance or rejection, and represents a significant deficiency with this method. Example The proposed investment requires an initial outlay of Rs. 10,000. The accounting profit for years one to three of the investment are: Year 1 Rs. 2,000 Year 2 Rs. 3,000 Year 3 Rs. 4,000 The average return is Rs. 3,000. The accounting rate of return (based on the initial outlay) is therefore: Rs.3,000/ Rs. 10,000 = 30%. 236

Assuming the benchmark rate of return for the investor is 15%, this investment would be accepted. The deficiencies in this method are significant. First, there is no consideration of the time value of money. Second, it deals with accounting numbers, often affected by choice of particular techniques for recording items such as depreciation and inventory values, rather than cash flows. Similar to the payback period, however, it is relatively simple to use and quick to calculate as a rough guide to evaluate investments. Time-Weighted Rate of Return A further method that may be used by a financial planner to evaluate investment performance is the time- eighted rate of return. This method is useful for calculating the return on a particular fund or asset group with cash flows timed at different intervals to those of other funds or asset groups. It thus provides an effective standard for comparing the performance of different funds in which cash flows can vary considerably. This method makes an allowance for the time value of money but not in the same manner as the discounted cash flow methods outlined earlier. Rather, it uses the relative timing of the cash flows as weights: the earlier the cash flows are received, the greater their relative importance. The time-weighted rate of return is a useful means of appraising a fund manager’s ability to make the fund’s assets perform. This is a relatively recent method of evaluating fund performance. Our objective is to introduce you to this concept so that if you encounter it in practice, you will already have a base level of understanding of the method. The time-weighted rate of return formula assumes that all cash flows occur at the end of the day, and the weights are based on the transaction date. The cash flows included in the calculation include contributions (which are regarded as positive) and fees, taxes, charges and distributions to the owner of the portfolio (which are regarded as negative). Example You begin the year with an investment portfolio worth Rs. 100,000 (B in the formula below). During the year you add a further Rs. 4,000 (C) to your portfolio, and at the end of the year it is worth Rs. 110,000 (A). You might report the gain as being Rs. 10,000 on Rs. 100,000 or 10%, but in actual fact the gain is only Rs. 6,000, since Rs. 110,000 – Rs. 100,000 – Rs. 4,000 = Rs. 6,000. So to calculate the amount of return, we use the expression A – B – C. The question then is what was the amount invested — was it Rs.100,000 (if the Rs. 4,000 was invested at the end of the year) or Rs. 104,000 (if the Rs. 4,000 was invested at the beginning of the year)? The amount invested will clearly depend on the time of the year that the Rs. 4,000 was added to the portfolio. For example, if the Rs. 4,000 was invested at the middle of the year, then the ‘average’ amount of the investment during the year was Rs. 102,000, and the return is given by: (110,000 - 100,000 - 4,000)/(100,000 + 2,000) X 100% or 5.88%. The formula to calculate the rate of return is as follows: Performance =(A - B - C)/ B + S (Wn) X 365/n X 100% Where: A = End of period market value 237

B = Initial market value C = Sum of all cash flows S = ‘the sum of’ Wn = ((N-J)/N) ×cash flow N = Number of days in the period J = Date of each cash flow You should work slowly through the following example to develop an understanding of this rate of return. Whilst it may appear quite complex initially, it is relatively simple in practice. Example An investment had an initial value of Rs. 150,000. Its market value is Rs. 175,000. The following cash flows occurred during the time period of the investment: Date J Cash flow Weighting% Money weighted Rs. Rs. 30/6/96 1 1,000.00 99.497 994.97 30/9/96 92 1,200.00 53.769 645.23 31/12/96 184 -456.75 7.538 -34.43 15/01/97 199 1,000.00 0.000 0.00 2,743.25 1,605.77 Performance = (175,000 - 150,000 - 2,743.25) / (150,000 + 1,605.77) x (365/199) x 100% (22,256.75 / 151,605.77) x (365/199) x 100% 26.93% The time-weighted rate of return of about 27% can then be compared to the performance of similar funds so that the performance of this particular fund can be evaluated. Often we come across loans of a client that need to be repaid and the client may not be clear as to how the repayment schedule works. Therefore, it is essential for the planner to be clear how EMIs are for example calculated. With this objective in mind, we are briefly touching upon loan repayment schedules. Effective Interest Rate To calculate the effects of compounding on a nominal interest rate over a number of compounding periods, use the formula iₑff = ((1+i/n) n) – 1 where iₑff – effective interest rate i – nominal interest rate n – the number of compounding periods For Example: If nominal interest rate = 10% Compounding period n Effective Rate (%) Annual 1 10 238

Semi-annual 2 10.25 Quarterly 4 10.38 Monthly 12 10.47 Daily 365 10.52 Continuous Compounding If the interest is compounded continuously then the amount is as follows: A=P (e)x where x = ni Where P – principal e – 2.7183 i – interest rate n – number of compounding periods Rule of 72 To calculate quickly the number of years, a sum of money will be doubled in, is to divide 72 by the rate of interest to get the number of years. For example, at 6% compounded annually, it will take Rs. 1,000 approximately 12 years (72/6) to become Rs. 2,000. Growth with Discounting Let us assume an annuity of Rs. 66 which grows at 10% each year. What is its present value after n years if the discount rate is 21%. Steps 1. Calculate A/(1+g) where A= Rs. 66 and g = 10% 2. Calculate i* = (i-g)/(1+g) where i = 21% 3. The first step gives you Rs. 60 4. The second step gives you 0.1 or 10% 5. Now calculate the present value of annuity of Rs. 60 discounted at 10% which gives you the final answer. 2.2.11 Loan Repayment Schedule Another topic under financial mathematics is the loan repayment schedule which is useful in working out your cash flow liabilities under a loan taken and which is to be repaid. The loan repayment schedule is also known as the loan amortisation schedule. It consists of repayment of both principal and interest. Interest expense is tax deductible in the hands of the business entity which pays it. In the case of term loans, usually the practice is to repay principal in equal instalments and pay interest on the outstanding amount of principal. In such a case, the payment of interest will decline over the years and the loan repayment instalments will not be equal. Such instalments are called balloon instalments. For example let us consider a Rs. 30,000 loan repayable over five years at an interest rate of 15%. The loan repayment schedule will be as follows: 239

Year Loan outstanding at Principal Interest Loan Loan at the beginning of year repayment instalment end of the year (1) (2) (3) (4) (5) (6) 1 30,000 6,000 4,500 10,500 24,000 2 24,000 6,000 3,600 9,600 18,000 3 18,000 6,000 2,700 8,700 12,000 4 12,000 6,000 1,800 7,800 6,000 5 6,000 6,000 900 6,900 0 An alternative way of repayment or amortisation of loan is to have equal instalments including both interest and principal repayments. For example, if we consider a five year loan of Rs. 30,000 at 10%, the answer is to find a five year annuity at a 10% rate whose present value will be equal to Rs. 30,000. In other words we have to find the solution to the following: 30,000 = A/(1.10) 1 + A/(1.10) 2 + A/(1.10) 3 + A/(1.10) 4 + A/(1.10) 5 Therefore A = Rs. 7,914 The loan amortisation or loan repayment schedule is as follows: Year Loan Outstanding at Principal Interest Principal Loan at the beginning of year repayment instalment end of the year (3)-(4) (1) (2) (3) (4) (5) (6) 1 30,000 7,914 3,000 4,914 25,086 2 25,086 7,914 2,509 5,405 19,681 3 19,681 7,914 1,968 5,946 13,735 4 13,735 7,914 1,374 6,540 7,195 5 7,195 7,914 720 7,195 0 Example: Rohan, aged 30 has purchased a flat worth Rs.1crore and has taken Rs.60,00,000 home loan from ICICI Bank. Loan is taken for a period of 25 years @11% p.a. fixed rate of return. Answer the following: a) Calculate the amount of EMI b) Calculate total amount of interest paid after 1 year c) Calculate total amount of principal paid after 3 years d) Calculate outstanding loan amount at the end of 5 years Solution: In FC 200 V Calculator : a) Go to CMPD: Set: End N= 25*12 I%= 11/12 PV= -6000000 FV= 0 PMT SOLVE 58806.78 per month b) Now go to AMRT PM1= 1 240

PM2= 12 “ INT: SOLVE = 657625.04 (Out of total EMI in 1 year (58806.78*12=705681.36) In the same question if I want to see how much principal has been pain in 1 year “PRN: SOLVE= 48056.36 Total amount paid through EMI in 1 year = 705681 Total interest paid in 1 year = 657625 Total principal paid in 1 year = 48056 c) Now go to AMRT PM1= 1 PM2= 36 “PRN: SOLVE= 161495.69 d) Now go to AMRT PM1= 1 PM2= 60 BAL: SOLVE= -5697291.79 If in this question we wish to calculate the amount of principal paid in 5 years, it is 302708.20 Total loan amount Rs.6000000 Minus amount paid Rs.302708 Rs.5697292 which is outstanding loan amount after 5 years Suggested Answers Question 1 = 60/Rs. 2,500 ×100 = 2.4% (a) Simple interest rate = 0.085 × Rs. 2,500 = Rs. 212.50 (b) Annual interest = 4 × Rs. 212.50 = Rs. 850 (c) Total interest Rs. 2,500 × 0.09 ×1 Question 2 = Rs. 225 = Rs. 2,500 × 0.09 × 1/2 (a) Interest = Rs. 112.50 = Rs. 2,500 × 0.09 × 1/4 (b) Interest = Rs. 56.25 = Rs. 2,500 × 0.09 × 29/365 (c) Interest = Rs. 17.87 = (d) Interest 241

Question 3 = P(1 + i)n = Rs. 8,000 (1 + 0.05)5 (a) A A = Rs.10,210.25 = P(1 + i)n (b) A = Rs. 24,000 (1 + 0.07)12 A = Rs. 54,052.60 Question 4 A= P(1 + i)n = Rs. 10,000 (1 + 0.06)3 = Rs. 11,910.16 then reinvested for a further 2 years: Rs. 11,910.16 (1 + 0.08)2 A= Rs. 13,892.01 = Rs. 100,000 ×(1 + 0.09)1 – Rs. 100,000 Question 5 Rs. 109,000 – Rs. 100,000 Rs. 9,000 (a) Interest = (1 + 0.09) 1–1 = 0.09 or 9% Effective rate = Rs. 100,000 ×(1 + 0.045)2– Rs. 100,000 (b) Interest = Rs. 109,202.5 – Rs. 100,000 = Rs. 9,202.50 Effective rate = (1 + 0.045) 2– 1 (c) Interest = 0.0920 or 9.20% = Rs. 100,000 ×(1 + 0.0075)12– Rs. 100,000 Effective rate = Rs. 109,380.68 – Rs. 100,000 = Rs. 9,380.68 = (1 + 0.0075)12– 1 = 0.0938 or 9.38% = = = Question 6 = PV X (1+I)n Rs. 150 × (1 + 0.02)20 (a) FV Rs. 222.89 Using a financial calculator = FV X (1+i) -n Go to CMPD N = 20 I% = 2 PV = -150 Solve FV= 222.89 (b) PV Rs. 250 × (1 + 0.05) -8 242

Rs. 169.21 Using a financial calculator Go to CMPD N = 8 I% = 5 FV = 250 Solve FV= -169.20 (c) Interest = (FV/PV) n/1 - 1 = (598.21/500) 1/24 - 1 = 0.0075 or 0.75% per compounding period Using a financial calculator Go to CMPD N = 24 PV = -500 FV = 598.21 Solve I%= .75% .0075*12=9% annual rate Effective rate is 9.38% (d) Number of periods = In (FV/PV) +ln (1+I) In (Rs. 1,059.73/Rs. 750) ÷ In (1 + 0.025) 14 compounding periods or 3.5 years Using a financial calculator Go to CMPD PV = -750 FV = 1059. 73 I%=2.5 Solve N=14 (Number of quarters) Question 7 (a) PV = Rs. 700 X (1 - (1+ 0.10)-5)/0.10 = Rs. 2,653.55 Therefore, the initial investment is Rs. 2,653.55. Using a financial calculator Go to CMPD PMT= 700 N= 5 I%=10 Solve PV=-2653.55 (b) FV = Rs. 1,000 X ((1+0.08)4 -1)/0.08 Rs. 4,506.11 The final sum is Rs. 4,506.11. Using a financial calculator Go to CMPD PMT= -1000 N= 4 I% = 8 Solve FV= 4506.11 Using a Financial Calculator Go to CMPD FV=250000 N = 30 I% = 12 Solve PMT= -1035.91 243

Question 8 PV = PMT/i = Rs. 100/0.1 = Rs. 1,000 Hence, Vikas must donate Rs. 1,000 so that a prize of Rs. 100 may be paid every year in perpetuity. Question 9 a) (ii) Need to calculate PMT Annuity No timing issues—cash flows occur at the end of each period To calculate PMT: PV = Rs. 40,000 i = 1.25% per quarter n =12 PMT = Rs. 3,610.33 Using a financial calculator Go to CMPD PV=-40000 N=12 I%=1.25 Solve = 3610.33 b) (ii) Need to calculate FV (iii) Singular cash flow (iv) No timing issues (v) To calculate FV: PV=-1000 N= 7 I%=12 Solve = 2210.68 (Maneka should choose the fixed rate term deposit.) c) (ii) Need to calculate PMT (iii) Annuity (iv) No timing issues (v) To calculate PMT: Go to CMPD FV=25000 N=5 I%=7 Solve PMT=4347.27 d) (ii) Need to calculate i (iii) Singular cash flow (iv) No timing issues (v) To calculate i: FV = Rs. 31,060 PV = Rs. 10,000 n = 10 i = 12% Question 10 Present value of inflows is Rs. 110,990.60 Net present value is Rs. 10,990.60 Arindam should accept investment B due to the higher NPV. Using a financial calculator 244

Go to Cash I=12 Execute Cash=D.Editor x will appear —- Execute Feed the value like this -100000 15000 20000 25000 30000 30000 25000 20000 15000 After feeding value Press ESC Solve NPV = 10990.60 Question 11 Rs. 100,000 = Rs. 15,000 / (1+r) + Rs. 20,000 / (1+r)2 + ..... + Rs. 20,000 / (1+r)7 + Rs. 15,000 / (1+r)8 Question 12 Present value Rs. 100,039.30 r Rs. 100,005.10 0.1496 Rs. 99,970.91 0.1497 0.1498 245

Summary This topic has introduced a number of fundamental financial mathematical concepts in order to help you calculate accumulated and present values which may be needed to provide clients with investment advice. Specifically, we have:  calculated and discussed the meaning of simple and compound interest;  calculated and discussed the significance of effective annual rate of interest;  discussed what is meant by present value and net present value, and  calculated them so that cash sums occurring on different dates may be valued effectively;  discussed the meaning of the terms annuity and discount rate;  calculated and discussed the significance of internal rate of return (or yield);  outlined non-discounted cash flow methods of evaluating investments; and  used a financial calculator to perform financial calculations. If you had no prior knowledge of how to perform complex financial calculations and have worked through all the examples and questions so far in this topic, you will have come a long way. Review questions are provided below to further enable you to self-test your understanding of financial mathematics. Review questions 1. After two years a Rs. 10,000 investment earning 8% simple interest will accumulate to: (a) Rs. 10,800 (b) Rs. 11,664 (c) Rs. 11,600 (d) Rs. 12,597 2. After two years a Rs.10,000 investment earning 8% p.a. compounded every six months will accumulate to: (a) Rs. 11,600 (b) Rs. 11,664 (c) Rs. 11,698 (d) Rs. 12,597 3. Which is the highest rate of interest? (a) 4.125% compounded annually (b) 4% compounded semi-annually (c) 4% compounded quarterly (d) 4% compounded monthly 4. An annuity in which payments are made two or more periods after the initial investment is made is called: (a) a perpetuity (b) a deferred annuity (c) a simple or ordinary annuity (d) an annuity due 5. Which is the best measure of the cost of money? (a) exact simple interest (b) effective annual rate of interest 246

(c) exact compound interest (d) ordinary simple interest 6. What amount would need to be invested to grow to Rs. 5,000 if the amount was invested for eight years at 10% compounded monthly? (a) Rs. 2,332.54 (b) Rs. 1,1090.88 (c) Rs. 2,254.10 (d) Rs. 5,282.68 7. If Rs. 1,500 is invested at a certain rate of interest and is compounded quarterly, it grows to Rs. 2,138.64 in three years. The effective rate of interest p.a. is: a) 4.00% b) 3.00% c) 16.12% d) 12.55% 8. If a sum of Rs. 2,000 is invested at 8% p.a. compounded semi-annually, how many years would it take for this amount to grow to Rs. 3,202.06? (a) 12 (b) six (c) five (d) four 9. A return of Rs. 300 is required at the end of each year for six years. If the investment will return 8% p.a. compounded annually, how much must be invested? (a) Rs. 1,500.00 (b) Rs. 1,428.79 (c) Rs. 1,386.86 (d) Rs. 1,658.24 10. A sum of Rs. 2,000 is invested at the beginning of every year over three years (total amount invested Rs.6,000). If the investment returns seven per cent compounded annually, what will be the accumulated balance at the end of the third year? (a) Rs. 6,429.80 (b) Rs. 6,879.89 (c) Rs. 9,501.48 (d) Rs. 7,562.91 11. Jai Kishen wishes to have a lump sum retirement fund payment of Rs. 200,000 in 25 years time. Assuming he can earn 11% p.a. compounded after tax and expenses, what would be the size of the yearly payments into the fund to accrue the desired lump sum, if the payments are made at the end of each year? (a) Rs. 8,000.00 (b) Rs. 6,642.15 (c) Rs. 1,748.05 (d) Rs. 1,827.56 247

12. Janki inherits Rs. 50,000 which she wishes to draw on for the next seven years for her daughter ’s education. She invests the money in a fixed interest account earning 6% p.a. compounded semi- annually. She intends to withdraw equal sums twice a year over the next seven years. How much can Janki withdraw each period so that her fund is exactly exhausted at the end of the seven-year time period? (a) Rs. 7,231.98 (b) Rs. 6,245.65 (c) Rs. 4,598.12 (d) Rs. 4,426.32 13. Professor David wants to allow a research grant at his university of Rs.1,000 per year in perpetuity. If it is assumed that the one sum invested will earn 8% p.a., how large must the sum be? (a) Rs. 12,500 (b) Rs. 12,000 (c) Rs. 11,500 (d) Rs. 8,000 14. Geet Sethi has been offered an investment costing Rs. 200,000 that returns Rs. 40,000 for the first four years and Rs. 60,000 for the remaining three years. Using a discount rate of 13%, what is the net present value of the investment? (a) Rs. 6,254.91 (b) Rs. 5,321.62 (c) Rs. 5,867.20 (d) Rs. 4,215.61 15. From the information in Question 14 above, what is the internal rate of return for the investment? (a) 13.85% (b) 14.02% (c) 3.40% (d) 7.30% 16. Mohan Kumar has calculated his profit after tax for a project to be Rs. 200,000 in a particular year. If interest expense is Rs. 24,000 and the tax rate is 36%, calculate the earnings after interest and before tax. (a) Rs. 312,500 (b) Rs. 336, 500 (c) Rs. 136,000 (d) Rs. 143,336 17. How are financing costs included in NPV and IRR calculations? (a) by including them in the interest payments (b) by considering the interest rate in the setting of the discount rate (c) as a tax deduction (d) by including them in the earnings 248

18. When the discount rate used to calculate net present value is the same as the internal rate of return for the investment, the net present value you calculate will be: (a) greater than 0 (b) less than 0 (c) it bears no relationship to the net present value (d) equal to 0 19. An investment with an initial outlay of Rs. 100,000 is estimated to produce annual net cash flows of Rs.30,000 at the end of the next five years. The payback period of the investment is: (a) 2.5 years (b) 3 years (c) 3.3 years (d) 4 years 20. Using the following information, calculate the accounting rate of return. Investment outlay: Rs. 40,000 Length of investment: five years Accounting profits: Year 1 Rs. 5,000 Year 2 Rs. 3,000 Year 3 Rs. 2,000 Year 4 Rs. 3,000 Year 5 Rs. 1,000 (a) 7% (b) 10% (c) 5% (d) 8% 249

Review Questions Answer 1. c Accumulation of Rs.10,000 for two years at 8% p.a. simple interest is: A = (10,000 × 0.08× 2) = Rs. 11,600 from the simple interest accumulation formula i = P × r × t 2. c The two-year accumulation at 8% p.a. compounded semi-annually is: A = 10,000 (1 + 0.04)4 = Rs. 11, 698.59 from the compound interest accumulation formula A = P(1+i)n 3. a We need to determine the effective annual rate of interest. (a) Effective annual rate is 4.125%. (b) Effective annual rate of 4% p.a. compounded semi-annually is: (1 + 0.02) 2–1 1.022–1 4.04% (c) Effective annual rate of 4% p.a. compounded quarterly is: (1 + 0.01) 4–1 1.014–1 4.06 per cent (d) Effective annual rate of 4% p.a. compounded monthly is: (1 + 0.00333)12–1 1.0033312–1 4.07% 4. b 5. b 6. c Rs. 5,000 = P (1+(0.10 /12))96 = P × 2.21818 So P = 5,000/2.21818 = Rs.2,254.10. 7. d We are only asked for an effective annual rate of interest, despite the fact that it is a nominal annual rate compounded quarterly. This makes life easier. The formula for calculating interest rate is: i = (FV/PV)1/n - 1 so i = (2,138.64 / 1,500) 1/3-1 = 0.125508 or 12.55% 8. Using the formula for n: n = ln(FV/PV)/ln(1+i) n = ln(3,202.06/2,000) / ln (1+(0.08/2)) = 12 periods The answer is b, 6 years. 9. c 10. b 11. c 12. d Use PMT function (0.03, 14, 50.000) = Rs. 4,426.32 13. a 14. c 15. a 16. a Rs. 200,000/ (1 – t) = Rs. 312,500 17. b 18. d 19. c 20. a 250