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Module I CORPORATE FINANCE 1. the TIME VALUE of MONEY An Module I CORPORATE FINANCE 1. THE TIME VALUE OF MONEY An enterprise must select the best combination on investment, financing and dividends. The decision to purchase new plants and equipments and to introduce a new product in the market requires the use of capital allocating techniques. The firm must determine whether future benefits are sufficiently large to justify current outlays. The first step towards making capital allocating decisions is to develop the mathematical tools of the time value of money. The passage of time between the outflows and inflows in a typical investment situation results in different current values associated with cash flows that occur at different points in time. It is not rational to assess an investment by adding up all the cash inflows and outflows and by comparing the values without considering when the cash flows occur. A monetary unit received in the future is worth less than a monetary unit received at the present for four primary reasons: a). the presence of positive rates of inflation reduces the purchasing power of a monetary unit through time. b). the opportunity cost of lost earnings as the monetary unit could have been invested and earned a return between now and a certain time point in the future. c). the uncertainty of future values due to the risk of default or nonperformance of investments. d). human preferences typically involve impatience or the preference to consume goods and services now rather than in the future. Interest rates represent the price paid to use money for some period of time. Interest rates are meant to compensate lenders and savers for foregoing the use of money for some interval of time. Lenders of capital receive interest, and borrowers pay interest due to the positive time value of money. For example a lender who provides 1000 lei today at a 10% interest per year is paid back 1100 lei at the end of the year. The 100 lei compensate the lender for not making an alternative investment, for giving up personal consumption or for the risk that the money might not have been repaid. Managers are often confronted with investment options with different length lives, different sized investments, differing financing terms, differing tax implications, etc. In all cases the cash flows associated with an investment are converted to similar terms and then converted to their equivalent values at a common point in time by using tools and techniques that collectively comprise the concepts known as the Time Value of Money. 1). Simple and compound interest. Future Values of Present Sums Consider an initial value V0 deposited in an accumulating account at an annual interest rate r. Assuming that the interest earnings are never withdrawn, after one period the account will be worth the initial principal plus interest earnings V1 = V0 + r×V0 = V0×(1 + r) For the second period of time the amount will be worth its initial value at the beginning of the period V0×(1 + r) plus the interest r×V0×(1 + r). 2 V2 = V0×(1 + r) + r×V0×(1 + r) = V0×(1 + r) …………………………………. n Vn = V0×(1 + r) FV = PV×(1 + r)n n = the final period in time FV = future value PV = present value r = the interest rate per period of time 2). Present Values of Future Sums This is the first basic principle in finance. The present value of a delayed payoff may be found by multiplying the payoff by a discount factor which is less than 1. Calculate the present value of 100 lei to be received 1,2,3,4 and 5 years from now at 7% interest. Year 1 2 3 4 5 Discount factor 1/1.07 = 1/1.072 = 1/1.073 = 1/1.074 = 1/1.075 = 0.934 0.873 0.816 0.763 0.713 Present value 93.4 87.3 81.6 76.3 71.3 Wrap-up for 1). & 2) and examples: Investing means spending money now (t0) to buy assets that will yield cash flow(s) in the future (t1), (t2), (t3) … Timing of the cash flow(s) matters! Interest is a key factor affecting the time value of money, for example: investing £100 for 10 years at 8% yields £216 and investing £100 for 10 years at 2% yields £122. One-period investment: present value (PV), future value (FV), rate of return (r) You are investing a given amount (present value, PV) now (t0) at the rate of return (r) for one year. After a year (t1) you receive the amount invested (PV) plus the income (r • PV): this is the future value (FV) of the investment. Hence: (1) FV = PV + r • PV = PV • (1+r) And by rearranging and solving for PV yields (2) PV = FV • [1/(1+r)] And by rearranging and solving for r yields (3) (1+r) = FV/PV or r = [(FV/PV) – 1] EXAMPLES: Q: What is the FV of £150 invested at 7% for one year? A: FV = £150 • (1 + 0.07) = £160.5 Q: What is the rate of return (r) on £150 invested for one year if the FV (value in t1) is £180? A: £180 = £150 • (1 + r) hence: r = [(£180/£150) – 1] = 0.2 = 20% Q: What is the PV of £180 paid next year (t1) if r = 5%? A: £180 = PV • (1 + 0.05) hence: PV = £180 • [1/(1+ 0.05)] = £180 • 0.952 = £171.4 Multi-period investment with compounding of interest You are investing an amount at the rate r for three years. Annual interest payments (PV • r) are reinvested at the rate r for the remainder of the three-year period (compounding). After three years you get: FV = PV•(1 + r)•(1 + r)•(1 + r) = PV•(1 + r)3. Hence, for n-period investments with compounding: (1) FV = PV • (1+r)n (1+r)n is called compound factor (2) PV = FV • [1/(1+r)n] 1/(1+r)n is called discount factor (3) (1+r)n = FV/PV and r = [(FV/PV)1/n – 1] EXAMPLES Q: What is the FV of £150 invested for 4 years at 7%? A: FV = £150 • (1 + 0.07)4 = £150 • 1.311 = £196.6 Q: What is the rate of return (r) on £150 invested for 4 years, if the FV is £180? A: £180 = £150 • (1 + r)4 hence: r = [(£180/£150)1/4 – 1] = (1.20.25 -1) = 0.047 = 4.7% Q: What is the PV of £180 paid after 10 years (t10) if r = 5%? A: £180 = PV • (1 + 0.05)10 hence: PV = £180 • [1/(1+ 0.05)10] = £180 • 0.614 = £110.5 3). Analyzing Investments Money is invested now for an expected return sometime in the future. Net cash flows for three hypothetical investments are shown in the next table. Each investment has a life of 4 years and brings a total net cash flow of 120000 lei. The discount rate is 8%. Year Investment A Investment B Investment C Net Discount Present Net Discoun Presen Net Discoun Presen t t t t cash flow factor value cash cash flow factor value flow factor value 1 30000 0.925 27777 15000 0.925 13889 45000 0.925 41666 2 30000 0.857 25720 25000 0.857 21433 35000 0.857 30007 3 30000 0.793 23815 35000 0.793 27784 25000 0.793 19846 4 30000 0.735 22051 45000 0.735 33076 15000 0.735 11025 Total 120000 - 99363 120000 - 96182 120000 - 10254 4 These three situations illustrate the importance of the flow of funds for an investment. All three investments show a total undiscounted return of 120000 lei. Though the total sum is the same, investment C receives most of its flow in the early years, investment A receives the same amount each year and investment B receives most of its money in the later years. 4). Significance of the discount rate The discount rate or the normal rate of return for a project is determined according to the formula: r = Rf + the risk premium The risk free rate Rf is composed of the minimum real rate of return in the economy Rr and the inflation premium Rr ri, where ri is the rate of inflation. Rf = Rr + Rr ri The initial cost of an investment project is of 50000 lei. The project generates in the following year a cash flow of 70000 lei with a probability of 30% and of 48000 lei with a probability of 70%. The average discounted cash flow equals the cost of investment. Determine the risk premium of the project knowing that the risk free rate is 4%. I = 50000 lei CF = 70000×30% + 48000×70% = 54600 lei 54600 50000 = → r = 9.2%; Rf = 4%; risk premium = 5.2% 1+ r 2. INVESTMENT CRITERIA Capital budgeting is the planning process used to determine whether a firm’s long term investments such as new machinery, replacement machinery, new plants, new products and research and development projects are worth pursuing. Capital budgeting methods are divided into two categories: a). simplified or traditional methods such as the accounting rate of return, return on investment or payback period. b). discount based methods such as net present value, internal rate of return, profitability index, modified internal rate of return, modified net present value, equivalent annuity, discounted payback period. There are two equivalent decision rules for capital investment: 1). Net present value rule: accept investments with positive net present values. 2). Rate of return rule: accept investments that offer rates of return in excess of their opportunity cost of capital. The rate of return of an investment is simply the profit as a proportion of the initial outlay: 1.
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