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Lecture 2 More on Interest Rates

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Lecture 2 More on Interest Rates Lecture 2 More on interest rates Lecture 2 1 / 25 Calculating present values is also known as discounting, and 1 d = k (1 + r)k is the k-year discount factor. Discount factors (1) We have seen that a cash flow xk occuring at time k has present value equal to x 1 PV = k = x · : (1 + r)k k (1 + r)k Lecture 2 2 / 25 Discount factors (1) We have seen that a cash flow xk occuring at time k has present value equal to x 1 PV = k = x · : (1 + r)k k (1 + r)k Calculating present values is also known as discounting, and 1 d = k (1 + r)k is the k-year discount factor. Lecture 2 2 / 25 Discount factors (2) The m-period and continuously compounded discount factors are given by 1 d = k (1 + r=m)k and −rt dt = e respectively. Lecture 2 3 / 25 Theorem (The main theorem on present values) If all cash flows are discounted using the same constant interest rate r, then two cash flow streams are equivalent if they have the same present value. Equivalent streams of cash flows We say that two streams of cash flows are equivalent if we can use the cash flows from one to create the other, and vice versa. Lecture 2 4 / 25 Equivalent streams of cash flows We say that two streams of cash flows are equivalent if we can use the cash flows from one to create the other, and vice versa. Theorem (The main theorem on present values) If all cash flows are discounted using the same constant interest rate r, then two cash flow streams are equivalent if they have the same present value. Lecture 2 4 / 25 The left-hand side of this equation is the present value of x using the interest rate r, and this is indeed the economical intuition behind the IRR: which interets rate makes the total present value of x equal to zero. Recalling The main theorem on present valus we see that the IRR is the interest rate that makes the value of x equivalent to the zero cash flow stream (0; 0;:::; 0). Internal rate of return (1) Given a stream of cash flows x = (x0; x1;:::; xn), the internal rate of return (IRR) is any number r satisfying n X xk = 0: (1 + r)k k=0 Lecture 2 5 / 25 Recalling The main theorem on present valus we see that the IRR is the interest rate that makes the value of x equivalent to the zero cash flow stream (0; 0;:::; 0). Internal rate of return (1) Given a stream of cash flows x = (x0; x1;:::; xn), the internal rate of return (IRR) is any number r satisfying n X xk = 0: (1 + r)k k=0 The left-hand side of this equation is the present value of x using the interest rate r, and this is indeed the economical intuition behind the IRR: which interets rate makes the total present value of x equal to zero. Lecture 2 5 / 25 Internal rate of return (1) Given a stream of cash flows x = (x0; x1;:::; xn), the internal rate of return (IRR) is any number r satisfying n X xk = 0: (1 + r)k k=0 The left-hand side of this equation is the present value of x using the interest rate r, and this is indeed the economical intuition behind the IRR: which interets rate makes the total present value of x equal to zero. Recalling The main theorem on present valus we see that the IRR is the interest rate that makes the value of x equivalent to the zero cash flow stream (0; 0;:::; 0). Lecture 2 5 / 25 Introducing 1 d = 1 + r we can write the IRR equation as 1 d2 + d − 1 = 0: 2 Internal rate of return (2) Example To determine the IRR of the cash flow stream x = (−100; 50; 100) we need to solve the equation 50 100 −100 + + = 0: 1 + r (1 + r)2 Lecture 2 6 / 25 Internal rate of return (2) Example To determine the IRR of the cash flow stream x = (−100; 50; 100) we need to solve the equation 50 100 −100 + + = 0: 1 + r (1 + r)2 Introducing 1 d = 1 + r we can write the IRR equation as 1 d2 + d − 1 = 0: 2 Lecture 2 6 / 25 Since 1 1 d = , r = − 1 1 + r d we get 1 IRR = − 1 = 15:2%: 0:868 Internal rate of return (3) Example (Continued) The solutions to this equation are d1 = 0:868 and d2 = −1:36: From an economic point of view we are interest in solutions that satifies d 2 (0; 1) since this corresponds to r 2 (0; 1), or simply r > 0. Lecture 2 7 / 25 Internal rate of return (3) Example (Continued) The solutions to this equation are d1 = 0:868 and d2 = −1:36: From an economic point of view we are interest in solutions that satifies d 2 (0; 1) since this corresponds to r 2 (0; 1), or simply r > 0. Since 1 1 d = , r = − 1 1 + r d we get 1 IRR = − 1 = 15:2%: 0:868 Lecture 2 7 / 25 The answer to these three questions are in general no. But cash flow streams often have a structure that implies that we can often find a unique and strictly positive IRR. Internal rate of return (4) Questions Does these always exist an IRR? If an IRR exist, is it unique? If an IRR exist, is it always strictly positive Lecture 2 8 / 25 But cash flow streams often have a structure that implies that we can often find a unique and strictly positive IRR. Internal rate of return (4) Questions Does these always exist an IRR? If an IRR exist, is it unique? If an IRR exist, is it always strictly positive The answer to these three questions are in general no. Lecture 2 8 / 25 Internal rate of return (4) Questions Does these always exist an IRR? If an IRR exist, is it unique? If an IRR exist, is it always strictly positive The answer to these three questions are in general no. But cash flow streams often have a structure that implies that we can often find a unique and strictly positive IRR. Lecture 2 8 / 25 Internal rate of return (5) Theorem (Main theorem of IRR) Assume that the stream of cash flows x = (x0; x1;:::; xn) fulfills x0 < 0 xk ≥ 0 for every k with at least one xk > 0 Pn k=0 xk > 0 Then there exists a unique strictly positive IRR. Lecture 2 9 / 25 Internal rate of return (6) Proof. Define the function n X k f (d) = xk d : k=0 We have f (0) = x0 < 0 and n 0 X k−1 f (d) = kxk d > 0 when d > 0: k=1 Hence there exists a unique d0 > 0 such that f (d0) = 0. Since Pn f (1) = k=0 xk > 0 we know also that d0 < 1. The IRR is now given by 1 IRR = − 1 > 0: d0 Lecture 2 10 / 25 It is not uncommon to divide a stream of cash flows x = (x0; x1;:::; xn) into two parts: x0 and (x1; x2;:::; xn). In an investment situation the initial cash flow x0 is often negative, and one writes x0 = −I where I is the amount we need to invest today. Net present value (1) The expression net present value (NPV) is often used as a synonym for the present value of a cash flow stream, when discussing an investment opportunity. Lecture 2 11 / 25 In an investment situation the initial cash flow x0 is often negative, and one writes x0 = −I where I is the amount we need to invest today. Net present value (1) The expression net present value (NPV) is often used as a synonym for the present value of a cash flow stream, when discussing an investment opportunity. It is not uncommon to divide a stream of cash flows x = (x0; x1;:::; xn) into two parts: x0 and (x1; x2;:::; xn). Lecture 2 11 / 25 Net present value (1) The expression net present value (NPV) is often used as a synonym for the present value of a cash flow stream, when discussing an investment opportunity. It is not uncommon to divide a stream of cash flows x = (x0; x1;:::; xn) into two parts: x0 and (x1; x2;:::; xn). In an investment situation the initial cash flow x0 is often negative, and one writes x0 = −I where I is the amount we need to invest today. Lecture 2 11 / 25 We see that he NPV is the difference of the present value of the future cash flows minus the investment today { hence it is a net present value. Net present value (2) If we let n X xk PV = ; (1 + r)k k=1 then we can write NPV = PV − I : Lecture 2 12 / 25 Net present value (2) If we let n X xk PV = ; (1 + r)k k=1 then we can write NPV = PV − I : We see that he NPV is the difference of the present value of the future cash flows minus the investment today { hence it is a net present value. Lecture 2 12 / 25 The basic principle is the following: (1) Calculate a numerical value. (2) If the calculated numerical value satisfies some prespecified condition, then invest, otherwise do not invest. Evaluation criteria (1) Given the prospect of investing in a project that generates the cash flow stream x, we need to have some way (or ways) to choose if we should start the investment or not.
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