Fundamentals of Finance PDF 2024

FIN 1302 Fundamentals of Finance Mr. Jethusan. J Lecturer (Temp) Department of Finance Faculty of Management and Finance University of Colombo 1  Level I Semester II – 2024 02 Credit Points...

FIN 1302 Fundamentals of Finance Mr. Jethusan. J Lecturer (Temp) Department of Finance Faculty of Management and Finance University of Colombo 1  Level I Semester II – 2024 02 Credit Points 2 Time Value of Money (Part 01) 3  Learning Outcomes Understanding of the concepts of time value of money, opportunity cost of capital and required rate of return. 4  Lesson Outline 1. Introduction 2. Future Value 3. Present Value 4. Net Present Value 5. Effective Annual Rate of Interest 6. Opportunity Cost of Capital 7. Required Rate of Return 5  01. Introduction “A Rupee today is more valuable than a year hence”. Thus Rs. 1000 today or Rs. 1000 ten years from today? – Which would you prefer? A rational person would take the Rs. 1000 today, because there is a time value to money. The immediate receipt of Rs.1000 provides us with the opportunity to put the money to work and earn interest. 6  Introduction… In a world in which all cash flows are certain, the rate of interest can be used to express the time value of money. The rate of interest will allow us to adjust the value of cash flows, whenever they occur, to a particular point in time. Then we can answer more difficult questions, such as: which should you prefer- Rs. 1000 today or Rs. 2000 ten years from today? 7  Introduction… The reason behind your decision is that the value of a certain amount of money today is more valuable than its value tomorrow. Or else the value of money varies with different points in time. 8 02. Reasons for Time Value of Money Risk and Uncertainty- Future is always uncertain and risky. Even though the cash outflows are in our control, no certainty for future cash inflows. Inflation- In an inflationary economy, the money received today has more purchasing power than the money to be received in future. Consumption- Individuals generally prefer current consumption to future consumption. Investment opportunities – An investor can profitably employ a rupee received today, to give him a higher value to be received tomorrow. 9  03. Required Rate of Return The minimum annual percentage earned by an investment that will induce individuals or companies to put money into a particular security or project. This is the opportunity cost of capital in comparable risk. To calculate the required rate, the following factors need to be considered;  Return of the market as a whole  The rate you could get if you took on no risk (the risk- free rate of return)  The volatility of the stock or the overall cost of funding the project. 10  Required Rate of Return… RRR = Risk Free Rate + Risk Premium Time Risk RRR Risk Premium RF 11 04. The Opportunity Cost of Capital Cost of capital refers to the opportunity cost of making a specific investment. It is the rate of return that could have been earned by putting the same money into a different investment with equal risk. Thus, the cost of capital is the rate of return required to persuade the investor to make a given investment. Required rate of return is the opportunity cost of capital in comparable risk. 12  Introduction … Notations of TVM: Y0: The beginning of Year 1 (Today) Y1: The end of year 1 Y2: The end of year 2 Y3: The end of year 3 The difference in the value of money today and tomorrow ( present and future) is referred to as Time Value of Money (TVM). 13  Introduction … Option 1 ( Future Value/ Compounding) Y0 Y1 Y2 Y3 Rs. 1000 Mn Rs. 1000 Mn + Interest Option 2 ( Present Value/ Discounting) Y0 Y1 Y2 Y3 Rs. 1000 Mn - Rs. 1000 Mn Interest 14 Future Value 15  Future Value Future value is the value of a present sum of money or stream of cash flows at a specified date in the future under a specified rate of return. PV 𝑭𝑽𝟑 16  Simple Interest The interest that is paid/ earned on only the original amount (principal) that is borrowed/lent. formula for Simple Interest calculation : Simple Interest (Rs.)= P0 (i) (n) Where, P0= Original amount borrowed/lent at time period 0 i = Interest rate per time period n= Number of time periods 17  Simple Interest and Future Value The future value (accumulated value) of an account at the end of n periods is; 𝐹𝑣𝑛 = 𝑃0 + 𝑃0 𝑖 × 𝑛 𝐹𝑣𝑛 = 𝑃0 (1 + 𝑖𝑛) 26/09/2016 18  Simple Interest and Future Value Example 01: Akila deposits Rs.1000/= for 5 years in a bank account paying 10% simple interest. - Find the total interest amount he earns at the end of 5 years? - How much the account will be worth at the end of 5 years? 19 Difference between Simple Interest and Compounding Interest Years At Simple Interest At Compound Interest 2 1.16 1.17 20 2.60 4.66 200 17.00 4,838,949.59 It is the interest on interest, or compounding, effect that accounts for the dramatic difference between simple and compound interest. 20  Compounding Interest The interest paid / earned on a loan / an investment is periodically added to the principal. Interest is earned on interest as well as the initial principal. (𝑃0 =100 and i=10%) 𝐹𝑉𝑛 ??? 𝑌0 𝑌1 𝑌2 𝑌3 𝑌𝑛 PV=10 𝐹𝑉1 𝐹𝑉2 𝐹𝑉3 𝐹𝑉𝑛 0 100+100 × 0.1 110+110 × 0.1 121+121 × 0.1 100(1 + 0.1) 121 𝑃0 (1 + 𝑖)𝑛 110 133 21  Compounding Interest The interest paid / earned on a loan / an investment is periodically added to the principal. Interest is earned on interest as well as the initial principal. (𝑃0 =100 and i=10%) 𝐹𝑉𝑛 ??? 𝑌0 𝑌1 𝑌2 𝑌3 𝑌𝑛 𝐹𝑉1 𝐹𝑉2 𝐹𝑉3 𝐹𝑉𝑛 𝑃0 𝑃0 + 𝑃0 × 𝑖 𝑃0 1 + 𝑖 + 𝑃0 1 + 𝑖 × 𝑖 𝑃0 (1 + 𝑖) 𝑃0 (1 + 𝑖)(1 + 𝑖) 𝑃0 (1 + 𝑖)𝑛 22  Compounding Interest The future value (accumulated value) of an account at the end of n periods is; 𝐹𝑉𝑛 = 𝑃0 (1 + 𝑖)𝑛 FVn = Future value (compounded value) at the end of n number of periods i = Interest rate per time period n = Number of time periods P0 = Present value / Original amount 26/09/2016 23 Compounding Interest and Future Value Example 02: Suppose you place Rs.100 in a savings account that earns 6% interest compounded annually. How much you can get at the end of the each period? Time Receivable Amount at the end of the each period 0 / Today 100 = Present Value (PV) Year 1 106.00 = PV (1+0.06) Year 2 112.36 = PV (1+0.06) (1+0.06) Year 3 119.10 = PV (1+0.06) (1+0.06) (1+0.06) Year 4 126.25 = PV (1+0.06)^4 Year 5 133.82 = PV (1+0.06)^5 Year 6 141.85 = PV (1+0.06)^6 …………… ……………………………………………………………………………………. 100 years 33,930.20 =PV (1+0.06)^100 24 Compounding Interest and Future Value… Example 03: Akash deposits Rs 1000/= in to his bank account for 5 years with an interest rate of 10%, compounded annually. - How much will the Rs 1000/= be worth at the end of a year? - How much the account would be worth at the end of 5 years? 25 Compounding Interest and Future Value… Long process of compounding Interest on Principal Year Opening Interest Closing Balance Balance 1 1,000.00 100.00 1,100.00 2 1,100.00 110.00 1,210.00 3 1,210.00 121.00 1,331.00 4 1,331.00 133.10 1,464.10 5 1,464.10 146.41 1,610.51 26  Interest tables for Compounding Future value interest factor of Rs.1/= at i% at the end of n periods can be obtained using tables which have been constructed for values of (1+i)^n. FV equation can be written as follows. FVn = PV0(FVIF i,n) 𝑭𝑽𝑰𝑭 = (𝟏 + 𝒊)𝒏 At i=10%; FV5 = 1000 * 1.6105 = 1,610.5 27  Compounding more than once a Year Compounding once a year 𝑃0 =100 and i=10% 𝑌0 𝑌1 𝑌2 𝟏𝟏𝟎 𝟏𝟐𝟏 PV 100 + 100 × 0.1 110 + 110 × 0.1 FV Compounding may occur more frequently as quarterly, monthly or daily interest payments in real life. 28  Compounding more than once a Year… 𝑃0 =100 and i=10% Compounding Semi- annually 𝑌0 𝑌1 𝑌2 FV PV 𝟏𝟎𝟓 𝟏𝟏𝟎. 𝟐𝟓 𝟏𝟏𝟓. 𝟕𝟔 𝟏𝟐𝟏. 𝟓𝟓 0.1 100 + 100 × 0.1 0.1 2 105 + 105 × 115.76 + 115.76 × 2 2 0.1 110.25 + 110.25 × 2 29 Compounding more than once a Year… Compounding may occur more frequently as quarterly, monthly or daily interest payments in real life. Adjustment to FV formula; 𝒊 𝒎𝒏 𝑭𝑽𝒏 = 𝑷𝟎 (𝟏 + ) 𝒎 Where, P0 = Present Value/Original Amount i = Nominal interest rate per annum m= Number of times the interest is compounded per year {Quarterly (m=4), Monthly (m=12), Daily (m=365) } n = Number of years 30 Compounding more than once a Year… Example 04: A bank pays a nominal 10% annual interest that gets compounded semi-annually. Calculate the end of year wealth, if you deposit Rs.1000/= today? = 1,102.50 The future value at end of a year is greater with semiannual compounding than with yearly compounding; Annual compounding- Original value (base value) remains as Rs. 1,000 throughout the entire year. 31 Compounding more than once a Year… However in semi-annual compounding- Original Rs. 1,000 will remain only for first 6 months as the investment base and the base value for the second six months would be Rs. 1,050. The wealth at the end of second six months; FV= [1050*(1+0.1/2)^1]. Hence the Rs.2.50/= difference is caused by interest being earned in the second six months on the Rs.50/= in interest paid at the end of the first six months. The more times during the year that interest is paid, the greater the future value at the end of a given year. When m approaches infinity, we achieve continuous compounding. 32 Future value of a series of cash flows Example 05: Your father will be retiring after five years from today. He is planning to start a business after his retirement. He expects to make continuous deposits at the beginning of each year with the purpose of obtaining the required capital investment for his new business. - How much he will have for the new business at time of his retirement, if the bank grants a 12% nominal interest on his deposits? Year Cash Deposits (Rs) 1 125,000 2 90,000 3 105,000 4 123,500 5 87,500 33 Future value of a series of cash flows Year Value of the No: of periods FVIF n,12% Future Value deposit until Retirement 0 125,000 5 FVIF 5,12% 1.7623 220,292.71 1 90,000 4 FVIF 4,12% 1.5735 141,616.74 2 105,000 3 FVIF 3,12% 1.4049 147,517.44 3 123,500 2 FVIF 2,12% 1.2544 154,918.40 4 87,500 1 FVIF 1,12% 1.1200 98,000.00 Value of the deposit at the end of 5th year 762,345.29 34 Usage of Timelines- Future Value 35 Future values with different interest rates 36 Present Value 37  Present Value PV FV Present Value is the current worth of a future sum of money or stream of cash flows, under a specified rate of return. 38 Simple Interest and Present Value Present Value is the current worth of a future sum of money or stream of cash flows, under a specified rate of return. Under simple interest, present value ( Original investment) of an account at the beginning of n period is; 𝐹𝑉𝑛 𝑃0 = (1 + 𝑖𝑛) 39  Compounding interest and Present Value FV=133 and i=10% PV ???? 𝑌0 𝑌1 𝑌2 𝑌3 PV 𝐹𝑉𝑛 100 110 121 133 121 133 110 (1.1) (1.1) (1.1) 40 Compounding Interest and Present Value This is exactly the opposite of future value formula. PV Formula; 𝐹𝑉𝑛 𝑃𝑉 = (1 + 𝑖)𝑛 Where; PV = Present Value FVn = Future value at interest rate i = Interest Rate n = Number of compounding periods 41 Compounding Interest and Present Value Example 06: Ajith will receive Rs. 420,000, a year from now. Applicable interest rate is 5% per annum. What is the present value of this sum as at today? Method 1: Using formula = 400,000 42  Interest Tables for Discounting Present value interest factor of Rs.1/= at i% at the beginning of n periods can be obtained using tables which have been constructed for values of (1+i)^-n. PV equation can be written as follows. PV = FV * PVIF i,n Method 2: Using Interest tables Rs. 400,000 =Rs. 420,000 * 0.9524 Where; (i=0.05) (n=1) 1 𝑃𝑉𝐼𝐹 = 𝑛 1+𝑖 43  Discounting more than one year FV=121 and i=10% once a year PV ???? 𝑌0 𝑌1 𝑌2 FV PV 𝟏𝟏𝟎 𝟏𝟐𝟏 𝟏𝟎𝟎 110 121 (1 + 0.1) (1 + 0.1) 44  Discounting more than once a Year FV=121.55 and i=10% Compounding Semi- annually 𝑌0 𝑌1 𝑌2 FV PV 𝟏𝟎𝟓 𝟏𝟏𝟎. 𝟐𝟓 𝟏𝟏𝟓. 𝟕𝟔 𝟏𝟐𝟏. 𝟓𝟓 𝟏𝟎0 105 110.25 115.76 121.55 0.1 0.1 0.1 0.1 (1 + 2 ) (1 + 2 ) (1 + 2 ) (1 + 2 ) 45  Discounting more than once a Year Instead of dividing the future cash flow by (1+i)^n as we do when annual compounding is involved, we determine the present value by; 𝑭𝑽𝒏 𝑷𝑽 = 𝒊 (𝟏 + 𝒎)𝒎𝒏 Where, FVn = Future cash flow to be received at the end of year n m = Number of times a year interest is compounded i = Interest rate n = Number of years 46 Discounting more than once a Year.. Example 07 What is the present value of Rs.1000 to be received at the end of year 3 for a nominal discount rate of 8% compounded quarterly. PV0 = Rs.1000 / (1+ [0.08/4])^ (4)(3) = Rs.788.85 If the interest rate is compounded only annually, we have PV0 = Rs.1000 / (1+0.08) ^ (3) = Rs.793.80 Thus, the fewer times a year that the nominal discount rate is compounded, the greater the present value. 47  Present Value of a series of cash flows Example 08: Mr. Perera started up a new business. He estimated his expected future business income for next five years as follows. Assume all cash flows are received to be at the end of each year. Expected Return on Investment (ROI) is 12%. Calculate the present value of his estimated future income. Year Expected Cash Income (Rs) 1 150,000 2 224,000 3 266,650 4 178,600 5 156,000 48 Present Value of a series of cash flows Year Expected Cash Present PVIF n, 12% PV of Income Value expected Interest cash flows Factor 1 150,000 PVIF 1, 12% 0.8929 133,928.57 2 224,000 PVIF 2, 12% 0.7972 178,571.43 3 266,650 PVIF 3,12% 0.7118 189,796.20 4 178,600 PVIF 4, 12% 0.6355 113,503.53 5 156,000 PVIF 5, 12% 0.5674 88,518.59 Present value of expected cash flows 704,318.32 49  Rationalization of Present Value Huge amount of uncertainty exists with future cash flows. Due to the risk involved with these future cash flows, the value of a given amount of cash in a future period cannot be equalized to the value of the same amount today. Hence future cash flows are discounted at a discount rate which is equal to the cost of capital to identify its present value. Higher the discount rate, lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly value future cash flows that are to be used in financial decision making. 50  Continuous Compounding The general formula for FV; 𝒊 𝒎𝒏 𝑭𝑽𝒏 = 𝑷𝑽𝟎 (𝟏 + ) 𝒎 𝒊 As m= ∞, the term (𝟏 + 𝒎)𝒎𝒏 approaches e^in, whereas e is approximately 2.71828. Hence the FV at the end of n years of an initial deposit of PV0 where interest is compounded continuously at a rate of i percent is; FV 𝒏=PV0 (e)^in 51  Continuous Compounding… The future value of a Rs.1000/= deposit at the end of three years with continuous compounding at 10% would be; FV3= Rs.1000 (e)^ (0.10)(3) = Rs.1349.85 This compares with a future value with annual compounding of; FV3 = Rs.1000 (1 + 0.10) ^ 3 = Rs. 1331 Continuous compounding results in the maximum possible future value at the end of n periods for a given nominal rate of interest. 52  Continuous Compounding… The general formula for PV; PV0 = FVn / (e)^in The present value of Rs.1000/= to be received at the end of 10 years with a discount rate of 20%, compounded continuously, is; PV0 = Rs.1000 / (e)^ (0.20) (10) = Rs.1000 / (2.71828)^2 = Rs. 135.34 Although continuous compounding results in the maximum possible future value, it results in the minimum possible present value. 53  Net Present Value (NPV) One of the key decision criteria used in financial decision making especially with regard to project appraisal of a firm. NPV matches the present value of future cash flows expected from a project with the present value of the cash flows invested in that project. Example 09: As per example 08, calculate NPV if the initial investment is Rs. 650,000? 54  Effective Annual Rate of Interest Different Investments may provide returns based on various compounding periods. If we want to compare alternative investments that have different compounding periods, we need to state their interest on some common, or standardized basis. This leads us to make a difference between nominal(stated) interest and the effective annual interest rate. This is the interest rate compounded annually that provides the same annual interest as the nominal rate does when compounded m times per year. 55  Effective Annual Rate of Interest Formula: 𝑟 𝑚 𝐸𝐴𝑅 = (1 + ) − 1 𝑚 Where, r = Annual interest rate (Nominal rate) m = Number of compounding periods per year Example 10: A company offers 12% rate of interest on deposits. Calculate the effective rates of interest if they are compounded on; Yearly, Half-yearly, Quarterly and Monthly bases. 56  The Opportunity Cost of Capital Opportunity Cost of capital refers to the opportunity cost of making a specific investment. It is the rate of return that could have been earned by putting the same money into a different investment with equal risk. Thus, the cost of capital is the rate of return required to persuade the investor to make a given investment. 57  The Opportunity Cost of Capital… How it works; Cost of capital is determined by the market and represents the degree of perceived risk by investors. When given the choice between two investments of equal risk, investors will generally choose the one providing the higher return. Let's assume Company XYZ is considering whether to renovate its warehouse systems. The renovation will cost Rs.50 million and is expected to save Rs.10 million per year over the next 5 years. There is some risk that the renovation will not save Company XYZ a full Rs.10 million per year. Alternatively, Company XYZ could use the Rs.50 million to buy equally risky 5- year bonds in ABC Co., which return 12% per year. 58  The Opportunity Cost of Capital… Because the renovation is expected to return 20% per year (Rs.10,000,000 / Rs.50,000,000), the renovation is a good use of capital, because the 20% return exceeds the 12% required return XYZ could have gotten by taking the same risk elsewhere. The return an investor receives on a company security is the cost of that security to the company that issued it. A company's overall cost of capital is a mixture of returns needed to compensate all creditors and stockholders. This is often called the weighted average cost of capital and refers to the weighted average costs of the company's debt and equity. 59  The Opportunity Cost of Capital… Why it matters; Cost of capital is an important component of business valuation work. Because an investor expects his or her investment to grow by at least the cost of capital, cost of capital can be used as a discount rate to calculate the fair value (a sale price agreed to by a willing buyer and seller, assuming both parties enter the transaction freely) of an investment's cash flows. Investors frequently borrow money to make investments, and analysts commonly make the mistake of equating cost of capital with the interest rate on that money. It is important to remember that cost of capital is not dependent upon how and where the capital was raised. Put another way, cost of capital is dependent on the use of funds, not the source of funds. 60  Additional Questions 1. If the stated rate of interest, 8% is compounded quarterly, what is the effective rate of interest? 2. If you deposit Rs. 5000 into an account paying 6% annual interest compounded monthly, how long until there is Rs. 8000 in the account? 3. In 1790, Amal bought approximately an acre of land on the east side of Sea Shells Island. He made many such purchases. How much would he have in 2009, if instead of buying the land, Amal had invested the $58 at 5% compound annual interest? (Use on DCF tables). 61  Additional Questions 4. On a contract you have a choice of receiving Rs.25,000 six years from now or Rs.50,000 twelve years from now. At what implied compound annual interest rate should you be indifferent between the two contracts? 5. The following cash-flow streams need to be analyzed. Cash-flow stream End of Year 1 2 3 4 5 W 100 200 200 300 300 X 600 - - - - Y - - - - 1200 Z 200 - 500 - 300 62  Additional Questions a. Calculate the future value of each stream at the end of year 5 with a compound annual interest rate of 10%. b. Compute the present value of each stream if the discount rate is 14% 6. Senaya is considering two different savings plans. The first plan would have her deposit Rs.500 every six months, and she would receive interest at a 7% annual rate, compounded semi annually. Under the second plan she would deposit Rs.1000 every year with a rate of ineterst of 7.5%, compounded annually. The initial deposit with Plan 1 would be made six months from now and, with Plan 2, one year hence. 63  Additional Questions a. What is the future value of the first plan at the end of 10 years? b. What is the future value of the second plan at the end of 10 years? c. Which plan should she use, assuming that her only concern is with the value of her savings at the end of 10 years? d. Would your answer change if the rate of interest on the Plan 2 were 7%? Thank You 64  Any Questions?? “All your dreams can come true f you have the courage to purse them.” —Walt Disney 65