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Finance School of Management Chapter 4: Time Value of Money Objective Explain the concept of compounding and discounting and to provide examples of real life applications 1 School of Management Finance Chapter 4 Contents Compounding Frequency of Compounding Present Value and Discounting Alternative Discounted Cash Flow Decision Rules Multiple Cash Flows Annuities Perpetual Annuities Loan Amortization Exchange Rates and Time Value of Money Inflation and Discounted Cash Flow Analysis Taxes and Investment Decisions 2 School of Management Finance Financial Decisions – Costs and benefits being spread out over time – The values of sums of money at different dates – The same amounts of money at different dates have different values. 3 School of Management Finance Time Value of Money – Interest – Purchasing power – Uncertainty 4 School of Management Finance Compounding – Present value (PV) – Future value (FV) – Simple interest: the interest on the original principal – Compound interest: the interest on the interest – Future value factor 5 School of Management Finance Value of Investing $1 at an Interest Rate of 10% – Continuing in this manner you will find that the following amounts will be earned: 1 Year $1.1 Simple interest: 0.1 2 Years $1.21 Simple interest: 0.1+0.1=0.2 Compound interest: 0.01 3 Years $1.331 4 Years $1.4641 6 School of Management Finance Value of $5 Invested – More generally, with an investment of $5 at 10% we obtain 1 Year $5*(1+0.10) $5.5 2 years $5.5*(1+0.10) $6.05 3 years $6.05*(1+0.10) $6.655 4 Years $6.655*(1+0.10) $7.3205 7 School of Management Finance Value of $5 Invested – If we can earn 10% interest on the principal $5, then after 4 years FV 5 (1 0.1) 7.2305 4 8 School of Management Finance Future Value of a Lump Sum FV PV * (1 i ) n F V w ith gro w th s fro m -6 % to +6 % F utu re V a lue o f $1 0 0 0 3 ,5 0 0 6% 3 ,0 0 0 2 ,5 0 0 4% 2 ,0 0 0 1 ,5 0 0 2% 1 ,0 0 0 0% -2 % -4 % -6 % 500 0 0 2 4 6 8 10 12 14 16 18 Y ea rs 9 20 School of Management Finance Example: Future Value of a Lump Sum Your bank offers a CD (Certificate of Deposit) with an interest rate of 3% for a 5 year investments. You wish to invest $1,500 for 5 years, how much will your investment be worth? FV PV * (1 i ) n $ 1500 * (1 0 .03 ) 5 $ 1738 .1111145 n i PV FV Result 5 3% 1,500 ? 1738.911111 10 School of Management Finance Example: Reinvesting at a Different Rate You have $10,000 to invest for two years. Two years CDs and one year CDs are paying 7% and 6% per year respectively. What should you do? Reinvestment rate? You are sure the interest rate on one-year CDs will be 8% next year. With the two-year CD FV $10,000 (1.07 ) 2 $11,449 With the sequence of two one-year CDs FV $10,000 1.06 1.08 $11, 448 11 School of Management Finance Frequency of Compounding – Annual percentage rate (APR) – Effective annual rate (EFF) Suppose you invest $1 in a CD, earning interest at a stated APR of 6% per year compounded monthly. FV $1 (1 0.06 / 12)12 1.06168 General formula APR EFF 1 m m 1 12 School of Management Finance Effective Annual Rates of an APR of 18% Annual Percentage Frequency of rate Compounding Annual Effective Rate 18 18 18 18 18 18 18.00 18.81 19.25 19.56 19.68 19.72 1 2 4 12 52 365 13 School of Management Finance The Frequency of Compounding Note that as the frequency of compounding increases, so does the annual effective rate. What occurs as the frequency of compounding rises to infinity? m APR APR EFF Lim 1 e 1 m m 14 School of Management Finance Present Value – In order to reach a target amount of money at a future date, how much should we invest today? – Discounting – Discounted-cash-flow (DCF) 15 School of Management Finance Present Value of a Lump Sum FV PV * (1 i ) n Divide both sides by (1 i ) to obtain : n FV n PV FV * (1 i ) n (1 i ) 16 Finance School of Management Example: Present Value of a Lump Sum You have been offered $40,000 for your printing business, payable in 2 years. Given the risk, you require a return of 8%. What is the present value of the offer? FV PV (1 i ) n 40,000 (1 0.08) 2 34293.55281 $34,293.55 today 17 School of Management Finance Solving Lump Sum Cash Flow for Interest Rate FV PV * (1 i ) n FV (1 i ) n PV FV n (1 i ) PV FV n i 1 PV 18 School of Management Finance Example: Interest Rate on a Lump Sum Investment If you invest $15,000 for ten years, you receive $30,000. What is your annual return? FV i 1 PV 1 30000 10 10 10 1 2 1 2 1 15000 0.071773463 n 7.18% (to the nearest basis point) 19 School of Management Finance Solving Lump Sum Cash Flow for Number of Periods FV PV * (1 i ) n ln FV (1 i ) n PV FV n ln ( 1 i ) n * ln 1 i PV FV ln PV ln FV ln PV n ln 1 i ln 1 i 20 School of Management Finance NPV (Net Present Value) Rule – NPV – NPV rule: Accept a project if its NPV is positive. – Opportunity cost of capital: The rate (of return) we could earn somewhere else if we did not invest in the project under evaluation. – Yield to maturity or Internal Rate of Return (IRR) 21 School of Management Finance Example: Evaluate a Project A five-year savings bond with face value $100 is selling for a price of $75. Your next-best alternative for investing is an 8% bank account. Is the savings bond a good project? NPV $100 $75 1.08 5 $68 .06 $75 $6.94 FV $75 1.08 5 $110 .20 22 School of Management Finance Example: Evaluate a Project IRR (100 / 75) 1/ 5 1 5.92% 100 n ln ln 1.08 3.74 75 23 School of Management Finance Example: Borrowing You need to borrow $5,000 to buy a car. A bank can offer you a loan at an interest rate of 12%. A friend says he will lend the $5,000 if you pay him $9,000 in four years. Should you borrow from the bank or the friend? $9,000 NPV $5,000 1.124 $719.66 24 School of Management Finance PV of Annuity Formula PV PMT (1 i )1 PMT (1 i ) 2 PMT 1 (1 i ) n i PMT (1 i ) n 25 Finance School of Management Example: Buying an Annuity You are 65 years old and NPV $10,000 expect to live until age 80. $1,000 1 (1 8%) 15 For a cost of $10,000, an 8% insurance company will pay $1,440 .52 you $1,000 per year for the rest of your life. i 5.56% You can earn 8% per year on your money in a bank account. n 21 Does it pay to buy the insurance policy? 26 School of Management Finance Perpetual Annuities / Perpetuities Recall the annuity formula: pmt 1 PV * 1 n i 1 i Let n -> ∞ with i > 0: pmt PV i 27 School of Management Finance Loan Amortization Home mortgage loans or car loans are repaid in equal periodic installments. Part of each payment is interest on the outstanding balance of the loan. Part is repayment of principal. The portion of the payment that goes toward the payment of interest is lower than the previous period’s interest payment. The portion that goes toward repayment of principal is greater than the previous period’s. 28 School of Management Finance Calculator Solution n i PV 3 9% 100,000 FV PMT 0 ? Result -39,505.48 This is the yearly repayment 29 School of Management Finance Amortization Schedule for 3-Year Loan at 9% Beginning Year Banlance 1 2 3 100,000 69,495 36,244 Total Payment Interest Paid Principal Paid Remaining Balance 39,505 39,505 39,505 9,000 6,255 3,262 30,505 33,252 36,244 69,495 36,244 0 30 Finance School of Management Example: Exchange Rates Investing $10,000 in dollar-denominated bonds offering an interest rate of 10% per year Investing in yen-denominated bonds offering an interest rate of 3% per year The exchange rate for the yen is now $0.01 per yen. Which is the better investment for the next year? 31 School of Management Finance Time U.S.A. $10,000 Japan 0.01 $/¥ 10% $/$ (direct) $11,000 ¥ 1,000,000¥ 3% ¥ / ¥ ? $/¥ 1,030,000¥ 32 School of Management Finance Time U.S.A. $10,000 Japan 0.01 $/¥ 10% $/$ (direct) $11,124 $11,000 ¥ 1,000,000¥ 3% ¥/¥ 0.0108 $/¥ 1,030,000¥ 33 School of Management Finance Time U.S.A. $10,000 Japan 0.01 $/¥ 10% $/$ (direct) $10,918 ¥ $11,000 ¥ 1,000,000¥ 3% ¥ / ¥ 0.0106 $/¥ 1,030,000¥ 34 School of Management Finance Time U.S.A. $10,000 Japan 0.01 $/¥ 10% $/$ (direct) $11,000 ¥ $11,000 ¥ 1,000,000¥ 3% ¥ / ¥ 0.01068 $/¥ 1,030,000¥ 35 Finance School of Management The Real Rate of Interest 1 Nominal interest rate 1 Real interest rate 1 Rate of inflation Nominal interest rate - Rate of inflation Real interest rate 1 Rate of inflation 36 Finance School of Management Switch to a Gas Heat ? You currently heat your house with oil and your annual heating bill is $2,000. By converting to gas heat, you estimate that this year you could cut your heating bill by $500. You think the cost differential between gas and oil is likely to remain the same for many years. The cost of installing a gas heating system is $10,000. Your alternative use of the money is to leave it in a bank account earning an interest rate of 8% per year. Is the conversion worthwhile? 37 School of Management Finance Switch to a Gas Heat ? Assume that the $500 cost differential will remain forever. The investment in switch of heating is a perpetuity, i.e. paying $10,000 now for getting $500 per year forever. i $500 / $10,000 5% If the $500 cost differential will increase over time with the general rate of inflation, then the 5% rate of return is a real rate of return. The conversion is not worthwhile unless the rate of inflation is greater than 2.875% per year. (0.08 0.05) /1.05 2.875% 38 School of Management Finance Taxes After tax interest rate (1 Tax rate) Before tax interest rate 39 Finance School of Management Taking Advantage of a Tax Loophole You are in a 40% tax bracket and currently have $100,000 invested in municipal bonds earning a tax-exempt rate of interest of 6% per year. Now you buy a house at a cost of $100,000. A bank offers a loan for you at an interest rate of 8% per year. Does it pays for you to borrow? After tax interest rate (1 0.4) 8% 4.8% 40