Lecture 3. Chapter 4: Allocating Resources Over Time

Transcription

1 Lecture 3 Chapter 4: Allocating Resources Over Time 1

2 Introduction: Time Value of Money (TVM) $20 today is worth more than the expectation of $20 tomorrow because: a bank would pay interest on the $20 inflation makes tomorrow s $20 less valuable than today s uncertainty of receiving tomorrow s $20 2

3 3.1 Compounding Assume that the interest rate is 10% p.a. What this means is that if you invest $5 for one year, you have been promised $5*(1+10/100) or $5.50 next year. Investing $5 for yet another year promises to produce 5.50 *(1+10/100) or $6.05 in 2-years. Definitions: Simple interest is the interest on the original principal. Compound interest is the interest earned on interest already paid. 3

4 Generalizing the method Generalizing the method requires some definitions: i is the interest rate, usually expressed in percent per year. n is the number of years the account will earn interest. PV is the present value or beginning amount in your account. FV is the future value at the end of n years. 4

5 3.2 Frequency of Compounding You have a credit card that carries a rate of interest of 18% per year compounded monthly. What is the interest rate compounded annually? That is, if you borrowed $1 with the card, what would you owe at the end of a year? If the credit card pays an APR of 18% per year compounded monthly, the (real) monthly rate is 18%/12 = 1.5% and the real annual rate is ( ) 12-1 = 19.56% The two equal APR with different frequency of compounding have different effective annual rates. 5

6 APR v. EFF Interest rates on loans and saving accounts are usually stated in the form of an annual percentage rate (APR), (e.g., 6% per year) with a certain frequency of compounding (e.g., monthly). Effective annual rate (EFF) is used for comparison of interest rates, when they have different frequency of compounding. EFF is defined as the equivalent interest rate, if compounding were only once per year. m APR = 1 + 1, lim m APR EFF 1 + = m m m e APR 6

7 3.3 Present Value and Discounting FV: How much will we have in n years if we invest PV today at an interest rate of i percent per year? PV: How much should we invest today in order to reach some target amount at a date in the future? FV = PV *(1 + i) n Divide both sides by (1 + i) n to obtain : PV = FV (1 + i) n = FV *(1 + i) n 7

8 3.4 Alternative Discounted Cash Flow Decision Rules The discounted cash flow concepts provide a powerful set of tools for making investment decisions. NPV rule: The NPV is the difference between the present value of all future cash inflows minus the present value of all current and future cash outflows. Accept a project if its NPV is positive. Reject a project if its NPV is negative. For the NPV calculation of any investment, we use the opportunity cost of capital as the interest rate. The opportunity cost of capital is simply the rate that we could earn somewhere else if we did not invest it in the project under evaluation. It is also called the market capitalization rate. 8

9 Future Value rule: Invest in the project if its FV is greater than the FV that will obtain in the next best alternative. YTM (yield to maturity) or IRR (internal rate of return): The discount rate that makes the present value of the future cash inflows equal to the PV of cash outflows. In other words, the IRR is exactly that interest rate at which the NPV is equal to zero. Accept an investment if its YTM or IRR is greater than the opportunity cost of capital. 9

10 3.5 Multiple Cash Flows A useful tool in analyzing the timing of cash flows is a diagram known as a time line. Time Cash Flow

11 3.6 Annuities An annuity is a level stream of cash inflows or payments. Time Cash Flow (immediate) (ordinary) An immediate annuity would have an FV equal to that of the ordinary annuity multiplied by 1 + i. For an ordinary annuity of $1 per year the FV will be: FV = n ( 1+ i) 1 i 11

12 PV of Annuity Formula PV = = pmt *{1 pmt i i * i ( ) 1 n ( ) n 1+ i } 12

13 3.7 Perpetual Annuities (Perpetuity) A perpetuity is a stream of cash flows that lasts forever. Recall the annuity formula: Let n -> infinity with i > 0: PV PV = When the cash flows from an investment grow at a constant rate: pmt * 1 i pmt i 1 = n PV = ( 1+ i) pmt i g 13

14 3.8 Loan Amortization The process of paying off a loan s principal gradually over its term is called loan amortization. For example, suppose you take a $100,000 home mortgage loan at an interest rate of 9% per year to be repaid with interest in three annual installments. What will the annual PMT be? In the first year, how much of the PMT is interest and how much is repayment of principal? You are buying a car and thinking of taking a one-year installment loan of $1,000 at an APR of 12% per year to be repaid in 12 equal monthly payments. What will the monthly PMT be? 14

15 3.9 Exchange Rate and Time Value of Money To avoid confusion when making financial decisions with different currencies there is a simple rule that one must observe: In any time value of money calculation, the cash flows and the interest rate must be denominated in the same currencies. 15

16 3.10 Inflation and Discounted Cash Flow Analysis The nominal interest rate is the rate denominated in dollars or in some other currency, and the real interest rate is denominated in units of consumer goods. (1 + NominalRate) = (1 + RealRate)*(1 + InflationRate) NominalRate InflationRate RealRate = 1+ InflationRate With continuous compounding, RealRate = NominalRate InflationRate 16

17 Inflation and Investment Decisions It is essential to take account of inflation in investment decisions as well as in saving decisions. A simple rule: When comparing investment alternatives, never compare a real rate of return to a nominal opportunity cost of money. The above rule is just a slightly different version of the following caution: Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows. 17

18 3.11 Taxes and Investment Decisions After - Tax Interest Rate = (1 Tax Rate) Before- Tax Interest Rate The rule for investing: Invest so as to maximize the net present value of your after-tax cash flows. This is not necessarily the same thing as to minimize the taxes you pay. 18