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1. Introduction to Financial Management BBA LLB

2. Overview of Financial Management Definition: Financial Management involves planning, organizing, directing, and controlling financial activities within an organization. Scope: It covers various aspects like financial planning, decision-making, and risk management to achieve organizational goals.

3. Importance in Business Operations Ensures efficient utilization of financial resources. Facilitates strategic decision-making. Enhances profitability and shareholder wealth.

4. Objectives of Financial Management Maximizing Shareholder Wealth Ensuring Liquidity and Solvency Balancing Risk and Return

5. Role of Financial Manager Responsibilities: Financial Planning and Forecasting Capital Budgeting Financing Decisions Risk Management Financial Reporting and Analysis

6. Scope of Financial Management Investment Decision Financial Decision Dividend Decision Liquidity Decision

7. Functions of Financial Manager Financial Planning Investment Decisions Financing Decisions Risk Management Financial Control

8. Profit Maximization vs. Wealth Maximization Profit Maximization: Focuses on maximizing the absolute level of profits. Short-term oriented approach. Ignores the timing and risk associated with cash flows. May lead to decisions that sacrifice long-term growth for short-term gains. Emphasizes on increasing profits regardless of shareholder wealth. Wealth Maximization: Aims at maximizing the net present value of cash flows. Long-term perspective. Considers the timing and risk associated with cash flows. Encourages decisions that enhance the long-term value of the firm. Focuses on increasing shareholder wealth, which includes both dividends and stock price appreciation.

9. Profit maximization may not align with the long-term interests of shareholders. Wealth maximization considers both the timing and risk of cash flows. Wealth maximization leads to decisions that benefit shareholders in the long run.

10. Summary Financial management is crucial for organizational success. It involves planning, decision-making, and control of financial resources. Financial managers play a vital role in managing financial activities effectively.

11. Any questions?

12. Fundamentals of Financial Management: Value Maximization: The primary goal of financial management is to maximize the value of the firm, ensuring that shareholders' wealth is maximized over the long term. Risk-Return Tradeoff: Financial decisions involve a tradeoff between risk and return. Higher returns are generally associated with higher levels of risk. Time Value of Money: A dollar received today is worth more than a dollar received in the future due to the opportunity cost of capital. Financial decisions should consider the time value of money. Diversification: Spreading investments across different assets reduces risk. Diversification is a fundamental principle of risk management in financial management.

13. Propositions in Financial Management: Modigliani-Miller Theorem: In a perfect market, the value of a firm is unaffected by its capital structure. This theorem establishes the irrelevance of capital structure under certain assumptions. Efficient Market Hypothesis (EMH): Prices of securities reflect all available information, making it impossible to consistently achieve above-average returns. This proposition has implications for investment decision-making. Capital Asset Pricing Model (CAPM): This model describes the relationship between risk and expected return in a portfolio. It helps in determining the expected return on an investment based on its risk. Pecking Order Theory: Firms prefer internal financing over external financing and prefer debt over equity when external financing is necessary. This theory explains the financing behavior of firms based on asymmetrical information.

14. Financial Management vs. Financial Accounting Financial Management: Focus: Financial management is concerned with managing the finances of the organization to achieve its objectives and maximize shareholder wealth. Scope: It involves strategic financial planning, investment decisions, financing decisions, and risk management. Objective: The primary goal is to maximize the value of the firm and ensure efficient allocation of resources. Users: Financial managers, executives, investors, creditors, and other stakeholders use financial management information. Financial Accounting: Focus: Financial accounting is primarily concerned with recording, summarizing, and reporting financial transactions of the organization. Scope: It involves preparing financial statements such as the balance sheet, income statement, and cash flow statement in accordance with accounting standards. Objective: The main objective is to provide accurate and reliable financial information to external users for decision-making purposes. Users: External users such as investors, creditors, regulatory authorities, and the general public rely on financial accounting information.

15. Investment Decision Definition: Investment decision refers to the process of allocating funds or resources into different assets or projects with the expectation of generating returns in the future. Key Components: Evaluation of Investment Opportunities: Assessing potential investment options based on factors such as expected returns, risk, time horizon, and alignment with organizational goals. Capital Budgeting: Utilizing various techniques like Net Present Value (NPV), Internal Rate of Return (IRR), and Payback Period to evaluate the financial viability of investment projects. Risk Analysis: Conducting thorough risk analysis to identify and mitigate potential risks associated with investment decisions, considering factors such as market risk, liquidity risk, and operational risk. Diversification: Spreading investments across different asset classes or projects to reduce overall risk and optimize portfolio returns. Decision Making Criteria: Establishing clear criteria and thresholds for decision making, considering factors such as investment objectives, risk tolerance, and financial constraints. Importance: Investment decisions play a critical role in shaping the financial performance and long-term growth prospects of an organization. Effective investment decisions can lead to increased profitability, enhanced competitiveness, and sustainable value creation.

16. Financing Decision Definition: Financing decision refers to the process of determining the optimal mix of debt and equity to fund the operations and investment activities of an organization. Key Components: Capital Structure Management: Deciding the proportion of debt and equity in the capital structure based on factors such as cost of capital, risk tolerance, and financial flexibility. Debt Financing: Raising funds by borrowing money from creditors or financial institutions, which includes issuing bonds, loans, or other debt instruments. Equity Financing: Obtaining funds by selling ownership stakes in the company to investors, which may involve issuing shares of stock or seeking venture capital. Cost of Capital Analysis: Evaluating the cost of various financing options and determining the weighted average cost of capital (WACC) to optimize the overall cost of funds. Leverage and Risk Management: Assessing the impact of leverage on the company's risk profile and financial stability, and implementing strategies to mitigate financial risk. Importance: Financing decisions have significant implications for the capital structure, cost of capital, and overall financial health of the organization. Effective financing decisions can enhance liquidity, profitability, and shareholder value, while poor decisions may lead to financial distress and loss of market confidence.

17. Dividend Decision Definition: The dividend decision refers to the process of determining the portion of earnings that will be distributed to shareholders as dividends and the portion that will be retained for reinvestment in the company. Key Components: Dividend Policy Formulation: Establishing a dividend policy that outlines the criteria and guidelines for determining the amount and timing of dividend payments, considering factors such as profitability, cash flow, and growth opportunities. Dividend Payment Methods: Choosing between different methods of dividend payment, including cash dividends, stock dividends, or a combination of both, based on the company's financial position and shareholder preferences. Dividend Stability vs. Growth: Balancing the objectives of providing stable dividends to shareholders with the need to retain earnings for future growth and investment opportunities. Impact on Shareholder Value: Assessing the impact of dividend decisions on shareholder value, stock price, and market perception, and making decisions that maximize long-term shareholder wealth. Importance: The dividend decision plays a crucial role in determining the attractiveness of the company's stock to investors and maintaining shareholder confidence. Effective dividend policies can enhance shareholder value, promote investor loyalty, and support long-term growth and sustainability.

18. Risk-Return Dimension of Financial Decision Making Definition: The risk-return dimension of financial decision making refers to the trade-off between the potential return on an investment and the level of risk associated with that investment. Key Concepts: Risk: Risk refers to the uncertainty or variability of returns associated with an investment. It encompasses factors such as market volatility, economic conditions, and business risks. Return: Return represents the gain or loss generated by an investment over a specific period. It includes income (such as interest or dividends) and capital gains or losses. Risk-Return Tradeoff: The risk-return tradeoff suggests that higher returns are generally associated with higher levels of risk. Investors must balance their risk tolerance with their return objectives when making investment decisions. Types of Risk: Market risk: The risk of losses due to changes in market conditions. Credit risk: The risk of default by borrowers or counterparties. Liquidity risk: The risk of being unable to sell an investment at its fair market value. Operational risk: The risk of losses due to internal processes, systems, or human error. Importance: Understanding the risk-return relationship is essential for making informed investment decisions and achieving a balance between risk and reward. By assessing risk and return characteristics, investors can construct portfolios that align with their investment objectives and risk tolerance.

19. Concept and Relevance of Time Value of Money Concept: Time Value of Money (TVM): TVM is the concept that money available today is worth more than the same amount in the future due to its potential earning capacity. It is based on the principle that a dollar received today can be invested and earn interest, making it worth more than a dollar received in the future. Relevance: Investment Decision Making: TVM is fundamental in evaluating investment opportunities by comparing the present value of cash flows with their future value. Techniques like Net Present Value (NPV) and Internal Rate of Return (IRR) incorporate TVM principles to assess the profitability of investments. Financial Planning: TVM guides financial planning by helping individuals and organizations determine the amount needed to achieve future financial goals. It enables the calculation of future savings requirements, retirement planning, and debt repayment schedules. Loan and Mortgage Analysis: TVM is essential in analyzing loans and mortgages by determining the total interest cost and monthly payments over the loan term. It helps borrowers understand the true cost of borrowing and make informed decisions about loan terms. Inflation and Purchasing Power: TVM considers the effects of inflation on the purchasing power of money over time. It helps individuals and businesses adjust for inflation when making financial decisions to ensure that future cash flows maintain their value.

20. Reasons for preference of current money are as follows : 1. Future Uncertainties : One of the reason for preference for current money is that there is a certainty about it whereas the future money has an uncertainty. There may be an apprehension that the other party (the creditor) may become insolvent or untraceable. 2. Preference for Present Consumption : Besides certainty, every person also has a preference for present consump tion , though this preference may be subjective and differ from one person to another. The present money may be required for some specific purpose e.g. to buy a consumer durable or otherwise. Moreover, in an inflationary situa tion , the money received today has a greater purchasing power. 3. Reinvestment opportunities : Both the individuals and the firm have preference for present money because they have reinvestment opportunities available to them. If they have got the money, they can invest this money to get further returns on this. This opportunity to get returns will not be available if the money is not invested now. The existence of reinvestment opportunities and the urge to earn a return by investing this current money seem to be the obvious reason for the time preference for money. This expected return which can be earned by investing the present money is in fact the TVM.

21. Compounding Technique Compounding: Compounding is the process of calculating the future value of an investment by adding interest earned to the initial principal amount, and then earning interest on both the principal and accumulated interest over time. Techniques: Simple Interest: Simple interest is calculated only on the initial principal amount. The interest remains constant throughout the investment period. Compound Interest: Compound interest involves earning interest on both the initial principal and any accumulated interest. The interest is added to the principal amount, resulting in exponential growth over time. Formula: The formula to calculate the future value of an investment with compound interest is: FV = PV ×(1+ r ) n Where: FV = Future value PV = Present value (initial investment) r = Interest rate per period n = Number of periods

22. Non-annual compounding FV = PV (1 + r/m)mn if the compounding is made every 6 months, then the time period ‘n’ will become 2 times in a single year. Similarly, the interest rate is also to be adjusted, because the rate of interest will remain same but the interest amount of any 6 months will be compounded in the next 6 months and so on. The more frequently the interest is compounded, the faster a FV grows. (I) the exponent has been increased from ‘n’ to ‘ m.n ’ to reflect the increased number of compounding periods, (II) the interest rate per annum has also been adjusted by dividing ‘m’, to correspond to the shorter compounding periods.

23. THE EFFECTIVE RATE OF INTEREST Definition: Effective Rate of Interest: The effective rate of interest, also known as the annual equivalent rate (AER) or annual percentage yield (APY), is the true annual interest rate earned or paid on an investment or loan, taking into account the effects of compounding over a specified period. Calculation: The formula to calculate the effective rate of interest is: Effective Rate=(1+ nr ​) n −1 Where: r = Nominal interest rate per compounding period n = Number of compounding periods per year Example: Suppose you have an investment with a nominal interest rate of 6% compounded semi-annually. What is the effective rate of interest? Effective Rate=(1+0.062)2−1Effective Rate=(1+20.06​)2−1 Effective Rate≈0.06136Effective Rate≈0.06136 Effective Rate≈6.136%Effective Rate≈6.136%

24. Accounting for Compounding: The effective rate of interest accounts for the compounding effect, providing a more accurate measure of the true annual interest rate. Comparative Analysis: Comparing effective rates allows investors to evaluate investment opportunities or loan offers accurately, considering both nominal interest rates and compounding frequencies. Financial Transparency: Banks and financial institutions are required to disclose the effective rate of interest on financial products to ensure transparency and enable consumers to make informed decisions. Decision Making: Understanding the effective rate of interest helps individuals and businesses make better financial decisions, such as choosing the most advantageous savings account or loan option.

25. Annual Compounding: Nominal interest rate: 6% per annum Compounding frequency: Annually (1 time per year) Using the formula: Effective Rate=(1+0.061)1−1Effective Rate=(1+10.06​)1−1 Calculation: Effective Rate=(1+0.06)1−1Effective Rate=(1+0.06)1−1 Effective Rate=1.06−1Effective Rate=1.06−1 Effective Rate=0.06Effective Rate=0.06 Effective Rate = 6% Semi-Annual Compounding: Nominal interest rate: 6% per annum Compounding frequency: Semi-annually (2 times per year) Using the formula: Effective Rate=(1+0.062)2−1Effective Rate=(1+20.06​)2−1 Calculation: Effective Rate=(1+0.03)2−1Effective Rate=(1+0.03)2−1 Effective Rate=(1.03)2−1Effective Rate=(1.03)2−1 Effective Rate=1.0609−1Effective Rate=1.0609−1 Effective Rate=0.0609Effective Rate=0.0609 Effective Rate ≈ 6.09% Quarterly Compounding: Nominal interest rate: 6% per annum Compounding frequency: Quarterly (4 times per year) Using the formula: Effective Rate=(1+0.064)4−1Effective Rate=(1+40.06​)4−1 Calculation: Effective Rate=(1+0.015)4−1Effective Rate=(1+0.015)4−1 Effective Rate=(1.015)4−1Effective Rate=(1.015)4−1 Effective Rate=1.06136−1Effective Rate=1.06136−1 Effective Rate=0.06136Effective Rate=0.06136 Effective Rate ≈ 6.136% Monthly Compounding: Nominal interest rate: 6% per annum Compounding frequency: Monthly (12 times per year) Using the formula: Effective Rate=(1+0.0612)12−1Effective Rate=(1+120.06​)12−1 Calculation: Effective Rate=(1+0.005)12−1Effective Rate=(1+0.005)12−1 Effective Rate=(1.005)12−1Effective Rate=(1.005)12−1 Effective Rate=1.06168−1Effective Rate=1.06168−1 Effective Rate=0.06168Effective Rate=0.06168 Effective Rate ≈ 6.168% These examples illustrate how the effective rate of interest increases as the compounding frequency becomes more frequent, even though the nominal interest rate remains constant. This highlights the impact of compounding on the overall return or cost of an investment or loan.

26. Future Value of Single Cash Flow Definition: Future Value (FV): Future value represents the value of a cash flow or investment at a specified future date, assuming a certain rate of return or interest earned over time. Formula: The formula to calculate the future value of a single cash flow (FV) is: FV = PV ×(1+ r ) n Where: FV = Future value PV = Present value (initial investment) r = Interest rate or rate of return per period (expressed as a decimal) n = Number of periods Example: Suppose you invest $1,000 today in a savings account that earns an annual interest rate of 5% compounded annually. What will be the future value of this investment in 5 years? FV = $1,000 \times (1 + 0.05)^5 FV = $1,000 \times (1.05)^5 FV ≈ $1,276.28

27. Future Value of Series of Cash Flows Definition: Future Value of an Annuity (FV): Future value of an annuity represents the total value of a series of equal cash flows or payments at a specified future date, assuming a certain rate of return or interest earned over time. Formula: The formula to calculate the future value of a series of cash flows or an annuity (FV) is: FV = PMT × r (1+ r ) n −1​ Where: FV = Future value PMT = Payment per period (cash flow) r = Interest rate per period (expressed as a decimal) n = Number of periods Example: Suppose you deposit $500 at the end of each year into a retirement account that earns an annual interest rate of 6% compounded annually. What will be the future value of these deposits after 10 years? FV = $500 \times \frac{{(1 + 0.06)^{10} - 1}}{0.06} FV ≈ $7,228.92

28. Key Points: Regular Cash Flows: Future value of a series of cash flows calculates the total value of periodic payments or receipts over a specified period. Compound Interest Effect: Similar to the future value of a single cash flow, the future value of an annuity considers the compounding effect of interest over time on each cash flow. Financial Planning: Calculating the future value of series of cash flows helps individuals and businesses plan for future financial goals such as retirement savings, loan repayments, or investment growth. Considerations: The formula assumes that the cash flows are made at the end of each period, and the interest rate remains constant throughout the period.

29. Single Cash Flow vs. Annuity Single Cash Flow: Definition: A single cash flow refers to a one-time payment or receipt of money at a specific point in time. Example: A lump-sum investment, such as purchasing a car or receiving a bonus payment. Future Value Formula: FV = PV ×(1+ r ) n FV = Future value PV = Present value (initial investment) r = Interest rate per period n = Number of periods Annuity: Definition: An annuity is a series of equal periodic payments or receipts made at regular intervals over a specified period. Example: Monthly mortgage payments, quarterly rental income, or annual retirement contributions. Future Value Formula: FV = PMT × r (1+ r ) n −1​ FV = Future value PMT = Payment per period (cash flow) r = Interest rate per period n = Number of periods Key Differences: Timing of Cash Flows: Single cash flow: Occurs at a specific point in time. Annuity: Consists of regular payments or receipts made at equal intervals over time. Calculation of Future Value: Single cash flow: Future value calculated using the present value and compounding formula. Annuity: Future value calculated using the annuity formula, which accounts for multiple cash flows over time. Financial Planning: Single cash flow: Used for one-time investments or receipts. Annuity: Used for recurring payments or receipts, such as retirement savings or loan repayments.

30. Present value of single cash flow Present Value (PV): The present value of a single cash flow represents the current worth of a future sum of money, discounted at a specific rate of return or interest rate. Formula: The formula to calculate the present value of a single cash flow (PV) is: PV =(1+ r ) nFV ​ Where: PV = Present value FV = Future value (amount to be received) r = Interest rate per period (discount rate) n = Number of periods Example: Suppose you will receive $1,000 in 3 years, and the discount rate (interest rate) is 5% per year. What is the present value of this amount? PV = \frac{{$1,000}}{{(1 + 0.05)^3}} PV ≈ \frac{{$1,000}}{{1.157625}} PV ≈ $863.84

31. Key Points: Time Value of Money: The present value calculation considers the time value of money, recognizing that a dollar received in the future is worth less than a dollar received today. Discounting: Present value involves discounting future cash flows at a specified discount rate to determine their current value. Financial Decision Making: Present value calculations are used in investment analysis, loan pricing, valuation of assets, and other financial decision-making processes. Net Present Value (NPV): NPV is a common application of present value, used to evaluate investment projects by comparing the present value of cash inflows with the present value of cash outflows.

32. Present Value of Annuity Present Value (PV) of Annuity: The present value of an annuity represents the current worth of a series of equal periodic payments or receipts, discounted at a specific rate of return or interest rate. Formula: The formula to calculate the present value of an annuity (PV) is: PV = PMT × r 1−(1+ r )− n ​ Where: PV = Present value PMT = Payment per period (cash flow) r = Interest rate per period (discount rate) n = Number of periods Example: Suppose you plan to receive annual payments of $1,000 for the next 5 years, and the discount rate (interest rate) is 6% per year. What is the present value of this annuity? PV = $1,000 \times \frac{{1 - (1 + 0.06)^{-5}}}{{0.06}} PV ≈ $4,111.59

33. Regular Cash Flows: The present value of an annuity calculates the current value of a series of equal periodic payments or receipts made at regular intervals over a specified period. Discounting: Present value involves discounting each cash flow in the annuity at the specified discount rate to determine its current worth. Financial Decision Making: Present value of annuity calculations are used in evaluating investments, pension plans, loan payments, and other financial planning decisions. Annuity Due vs. Ordinary Annuity: Annuity due involves payments made at the beginning of each period, while an ordinary annuity involves payments made at the end of each period. Adjustments may be needed in the formula for annuity due calculations.

34. Present Value of Perpetuity Definition: Perpetuity: A perpetuity is a financial instrument or investment that promises to pay a fixed amount of money at regular intervals indefinitely. Formula: The formula to calculate the present value of a perpetuity (PV) is: PV = rPMT ​ Where: PV = Present value PMT = Payment per period (cash flow) r = Interest rate per period (discount rate) Example: Suppose you invest in a perpetuity that pays $100 annually, and the discount rate (interest rate) is 5% per year. What is the present value of this perpetuity? PV = \frac{{$100}}{{0.05}} PV = $2,000

35. Infinite Cash Flows: A perpetuity promises to make regular payments forever, resulting in an infinite stream of cash flows. Simple Calculation: The present value of a perpetuity is calculated using a simple formula that divides the payment per period by the discount rate. Financial Applications: Perpetuities are commonly used in valuation models, such as the Gordon Growth Model for stock valuation, and in determining the value of certain bonds or securities. Limitations: While perpetuities provide a straightforward valuation method, they may not accurately reflect real-world financial instruments with finite lifespans or changing cash flows.

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37. Special cash flow Perpetuity A perpetuity may be defined as an infinite Series of equal cash flows occurring at regular intervals. It has indefinitely long life. PVp = Annual Cash flow/r Find out the present value of an investment which is expected to give a return of 2,500 p.a. indefinitely and the rate of interest is 12% p.a. Solution : Using the Equation 2.5A, PVp = Annual Cash flow/r = 2,500/.12 = 20,833.33

38. Annuity due FV or the PV of an annuity was based on the presumption that the cash flows occur at the end of each of the periods starting from now. However, in practice the cashflow may also occur in the beginning of each period. Such a situation is known as annuity due. In annuity due, the first cashflow occurs now and the last cashflow will occur in the beginning of the nth year i.e. at time n – 1. FV of an Annuity Due : The FV of an annuity is given by the formula : FV = Annuity Amount × CVAF( r,n ) × (1 + r) For example, a recurring deposit of 100 is made in the beginning of each of 4 years starting now at 6% p.a. What will be the total deposit at the end of 4 years? This can be calculated as follows : FV = Annuity Amount × CVAF( r,n ) × (1 + r) = 100 (4.375) (1 + .06) = 463.75.

39. Present Value of Annuity Due Definition: Annuity Due: An annuity due is a series of equal periodic payments or receipts made at the beginning of each period over a specified period. Formula: The formula to calculate the present value of an annuity due (PV) is: PV = Annuity Amount × PVAF(r,n) × (1+ r) Where: PV = Present value PMT = Payment per period (cash flow) r = Interest rate per period (discount rate) n = Number of periods

40. For example, if 1,000 is receivable in the beginning of next 4 years starting from now and the rate of interest is 6% then the PV may be calculated as follows : PV = Annuity Amount × PVAF( r,n ) × (1 + r) = 1,000 (3.465) (1 + .06) = 3,673.

41. Growing Perpetuity A growing perpetuity may be defined as an infinite series of periodic cash flows which grow at a constant rate per period. For example, an amount is receivable indefinitely in such a way that the amount of a particular period is 10% more than the amount for the preceding period. PV = Cash flow1 /(r – g) where cash flow1 = The cash flow at the end of the first period, r = rate of interest, and, g = growth rate in perpetuity amount. However, it may be noted that above formula can be used only if the rate of interest is more than the rate of growth i.e. r > g

42. For example, a company is expected to declare a dividend of 2 at the end of first year from now and this dividend is expected to grow 10% every year. What is the PV of this stream of dividend if the rate of interest is 15%? The PV for this dividend stream can be calculated as follows : PV = Cashflow1 /(r – g) = 2/(.15–.10) = 40.

43. Growing Annuity A growing annuity may be defined as a finite series of periodic cash flows growing at a constant rate every period. Since, an annuity is nothing but a truncated growing perpetuity, the growing annuity can also be viewed as a truncated growing perpetuity. The valuation of growing annuity is akin to the valuation of growing perpetu ity . Mathematically, the valuation of a glowing annuity can be arrived at as follows:

44. PV = n 1 CF 1 g 1 r g 1+r ⎡ ⎤ + ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ − ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ (2.9) where, CF1 = Cashflow at the end of the period 1, r = Rate of interest, g = Growth rate, and n = Life of annuity However, the above formula cannot be used if r = g because in this case, CF1 /r – g becomes CF1 /0 which is not allowed. If r = g, then the PV of a growing annuity is calculated as follows: PV = CF1 × n/(1 + r)

45. For example, a person opens a recurring deposit account for a period of 10 years earning 12% interest and accepts the scheme under the condition that for the first year the deposit is 3,150 and for subsequent years the deposit amount will increase by 5% every year. What is the PV of this scheme? The present value of this scheme of deposit may be ascertained by using Equation 2.9 as follows : PV = n 1 CF 1 g 1 r–g 1+r ⎡ ⎤ + ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ = ⎡ ⎤ + ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ 10 3150 1 .05 1 .12–.05 1+.12 = 45,000 [1 – (.937)10] = 45,000(.478) = 21,510.

46. However, if the rate of interest is only 5% i.e., equal to the growth rate, g, then the present value may be calculated by using Equation 2.9A as follows : PV = CF1 × n/(1 + r) = 3,150 × 10/(1 + .05) = 30,000.

47. Application of TVM Finding out the Implicit Rate of Interest Finding out the Number of periods Sinking Funds: Quite often, one may be interested to accumulate a target amount over a given period inclusive of interest for the period in such a way that the annual amount being subscribed over the period is same for all years. For example, an amount of 1,00,000 is required at the end of 5 years from now to repay a debenture liability. What amount should be accumulated every year at 10% rate of interest so that it ultimately becomes 1,00,000 after 5 years? This can be ascertained by finding out the value of ‘Annuity Amount’ in Equation 2.1B. FV = Annuity Amount × CVAF( r,n ) or, Annuity Amount = FV/CVAF( r,n ) (2.10) From the Table A-2, the value of CVAF(10%,5y) is 6.105. There fore, Annuity Amount= 1,00,000/6.105 = 16,380.

48. Capital Recovery: Sometimes, one may be interested to find out the equal annual amount paid in order to redeem a loan of a specified amount over a specified period together with the interest at a given rate for that period. For example, 1,00,000 borrowed today is to be repaid in five equal instalments payable at the end of each of next 5 years in such a way that the interest at 10% p.a. for the intervening period is also repaid. The annuity amount in this case can be ascer tained from the Equation 2.2B as follows : PV = Annuity Amount × PVAF( r,n ) or, Annuity Amount = PV/PVAF( r,n ) From the Table A-4, the value of PVAF(10%,5y) is 3.791. There fore, Annuity Amount = 1,00,000/3.791 = 26,378.

49. Deferred Payments : Suppose a person takes a loan of a specified amount at a given rate of interest. He wants to repay this loan together with interest in such a way that the annual amount being paid is same and further that the first payment be made a few years from now. In this case, the interest for the period for which the payment has been delayed (i.e. the period from the date of loan to the date of first payment) should also be considered in finding out the annual payment for the repayment of loan together with the interest. For example, a loan of 1,00,000 is taken on which interest is payable @ 10%. However, the repayment is to start only at the end of third year from now. What should be the annual payment if the total loan and interest is to be repaid in six instalments ?

50. Doubling the Amount The Rule of 72 is a commonly used "thumb rule" to estimate the time it takes for an investment to double in value, assuming a fixed annual interest rate. It's a quick and simple way to estimate the doubling time without the need for complex calculations. Divide 72 by the Annual Interest Rate: To estimate the doubling time in years, divide 72 by the annual interest rate (expressed as a percentage). The formula is: Doubling Time (in years) ≈ 72 / Annual Interest Rate (as a percentage)

51. This rule is often called the Rule of 114 Divide 114 by the Annual Interest Rate: To estimate the time it takes for an investment to triple in value, divide 114 by the annual interest rate (expressed as a percentage). The formula is: Tripling Time (in years) ≈ 114 / Annual Interest Rate (as a percentage) Suppose you have an investment with an annual interest rate of 8%. Using the Rule of 114: Tripling Time ≈ 114 / 8 ≈ 14.25 years This means it would take approximately 14.25 years for the investment to triple in value at an 8% annual interest rate.