**TI82** TxtView file generated by CalcText - Kouri2 #chap4#!˙chap 44.1 The Timeline 4.2 The Three Rules of Time Travel 4.3 Valuing a Stream of Cash Flows 4.4 Calculating the Net Present Value 4.5 Perpetuities and Annuities 4.6 Using an Annuity Spreadsheet or Calculator 4.7 Non-Annual Cash Flows 4.8 Solving for the Cash Payments 4.9 _The Internal Rate of Return A timeline is a linear representation of the timing of potential cash flows. Drawing a timeline of the cash flows will help you visualize the financial problem. time line Differentiate between two types of cash flows Inflows are positive cash flows. Outflows are negative cash flows, which are indicated with a – (minus) sign. Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments. The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow. Timelines can represent cash flows that take place at the end of any time period – a month, a week, a day, etc. Financial decisions often require combining cash flows or comparing values. Three rules govern these processes. 1st rule= only value at the same point in time can be compared or combined 2nd= to move a cash flow forward, you must compound it formula: FV of cash flow = C x (1+r)^n 3rd= to move a cash flow baxkward, you must discount it forumla: PV= C/(1+R)^n A dollar today and a dollar in one year are not equivalent. It is only possible to compare or combine values at the same point in time. Which would you prefer: A gift of $1000 today or $1210 at a later date? To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?” To move a cash flow forward in time, you must compound it. Suppose you have a choice between receiving $1000 today or $1210 in two years. You believe you can earn 10% on the $1000 today but want to know what the $1000 will be worth in two years. Recall the first rule: It is only possible to compare or combine values at the same point in time. So far we’ve only looked at comparing. Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today? Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows: if we want to find the present value of a stream of cash flows, we simply add up the present values of each. Future Value of a Cash Flow Stream with a Present Value of PV FVn= PV x (1+r)^n Calculating the NPV of future cash flows allows us to evaluate an investment decision. Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs). Perpetuities When a constant cash flow will occur at regular intervals forever it is called a perpetuity. The value of a perpetuity is simply the cash flow divided by the interest rate. Present Value of a Perpetuity PV(C in perpetuity) = C/r Annuities When a constant cash flow will occur at regular intervals for a finite number of N periods, it is called an annuity. Present Value of an Annuity To find a simpler formula, suppose you invest $100 in a bank account paying 5% interest. As with the perpetuity, suppose you withdraw the interest each year. Instead of leaving the $100 in forever, you close the account and withdraw the principal in 20 years. You have created a 20-year annuity of $5 per year, plus you will receive your $100 back in 20 years. So 100$ = PV(20-year annuity 5$ per year) + PV(100$in 20 years) Re-arranging terms resolver l'équation For the general formula, substitute P for the principal value and PV(annuity of C for n periods) = P-PV(PinperiodsN) = C/r (1-1/(1+r)^n) Future Value of an Annuity FV (annuity) = PV x (1+r)^n = C x 1/r ((1+r)^n - 1) Growing Perpetuity Assume you expect the amount of your perpetual payment to increase at a constant rate, g. Present Value of a Growing Perpetuity Formula: PV(growing perpetuity) = C/r-g Growing Annuity The present value of a growing annuity with the initial cash flow c, growth rate g, and interest rate r is defined as: Present Value of a Growing Annuity Formula: PV= C x (1/r-g)(1 - (1+g)/(1+r)^N) The same time value of money concepts apply if the cash flows occur at intervals other than annually. The interest and number of periods must be adjusted to reflect the new time period. Sometimes we know the present value or future value, but we do not know one of the variables we have previously been given as an input. For example, when you take out a loan you may know the amount you would like to borrow but may not know the loan payments that will be required to repay it. C = Payment/1/r(1- 1/(1+r)^N) IRR = In some situations, you know the present value and cash flows of an investment opportunity, but you do not know the internal rate of return (IRR), the interest rate that sets the net present value of the cash flows equal to zero. Solve pour si par le SOLVE r selon le PV ou FV ˙-Ó