Study Session 2: Quantitative Methods— Basic Concepts © 2016 Wiley ss02.indd 123 October 2015 7:09 PM The Time Value of Money Reading 5: The Time Value of Money LESSON 1: INTRODUCTION, INTEREST RATES, FUTURE VALUE, AND PRESNT VALUE The Financial Calculator It is very important for you to be able to use a financial calculator when working with TVM problems CFA Institute allows only two types of calculators for the exam—the TI BA II Plus™ (including the TI BA II Plus™ Professional) and the HP 12C (including the HP 12C Platinum) We highly recommend that you choose the TI BA II Plus™ or the TI BA II Plus™ Professional, and the keystrokes defined in our readings cater exclusively to TI BA II Plus™ users However, if you already own an HP 12C and would like to use it, by all means continue to so The TI BA II Plus™ comes preloaded from the factory with the periods per year (P/Y) function set to 12 This feature is not appropriate for most TVM problems, so before moving ahead please set the “P/Y” setting of your calculator to “1” by using the following keystrokes: [2nd] [I/Y] “1” [ENTER] [2nd] [CPT] Your calculator’s P/Y setting will remain at even when you switch it off However, if you replace its batteries you will have to reset the P/Y setting to “1” If you wish to check this setting at any time, simply press [2nd] [I/Y] and the display should read “P/Y = 1.” With these setting in place, you can think of “I/Y” as the interest rate per compounding period, and of “N” as the number of compounding periods Please take the time to familiarize yourself with the following keys on your TI Calculator: N = Number of compounding periods I/Y = Periodic interest rate PV = Present Value FV = Future Value PMT = Constant periodic payment CPT = Compute Timelines To illustrate some examples, we will use timelines to present the information more clearly It is very important for you to recognize that the cash flows occur at the end of the period depicted on the timeline Further, the end of one period is the same as the beginning of the next period For example, a cash flow that occurs at the beginning of Year is equivalent to cash flow that occurs at the end of Year 3, and will appear at t = on the timeline Sign Convention Finally, pay attention to the signs when working through TVM questions Think of inflows as positive numbers and outflows as negative numbers We will continue to emphasize this point through the first few examples in this reading so that you get the hang of it © 2016 Wiley r05.indd 125 125 October 2015 8:51 PM The Time Value of Money Suppose you were offered a choice between receiving $100 today or $110 a year from today If you are indifferent between the two options, you are attaching the same value to receiving $110 a year from today as you are to receiving $100 today It is obvious that the cash flow that will be received in the future must be discounted to account for the passage of time An interest rate, r, is the rate of return that shows the relationship between two differently dated cash flows The interest rate implied in the tradeoff above is 10% Present value (PV) is the current worth of sum of money or stream of cash flows that will be received in the future, given the interest rate For example, given an interest rate of 10%, the PV of $110 that will be received in one year is $100 PV = $100 $110 Future value (FV) is the value of a sum of money or a stream of cash flows at a specified date in the future For example, assuming a 10% interest rate, the FV of $100 received today is $110 $100 FV1 = $110 LOS 5a: Interpret interest rates as required rates of return, discount rates, or opportunity costs Vol 1, pp 278–279 Interest rates can be thought of in three ways: The minimum rate of return that you require to accept a payment at a later date The discount rate that must be applied to a future cash flow in order to determine its present value The opportunity cost of spending the money today as opposed to saving it for a certain period and earning a return on it 126 r05.indd 126 © 2016 Wiley October 2015 8:51 PM The Time Value of Money LOS 5b: Explain an interest rate as the sum of a real risk‐free rate and premiums that compensate investors for bearing distinct types of risk Vol 1, p 279 Interest rates are determined by the demand and supply of funds They are composed of the real risk‐free rate plus compensation for bearing different types of risks: • The real risk‐free rate is the single‐period return on a risk‐free security assuming zero inflation With no inflation, every dollar holds on to its purchasing power, so this rate purely reflects individuals’ preferences for current versus future consumption • An inflation premium is added to the real risk‐free rate to reflect the expected loss in purchasing power over the term of a loan The real risk‐free rate plus the inflation premium equals the nominal risk‐free rate • The default risk premium compensates investors for the risk that the borrower might fail to make promised payments in full in a timely manner • The liquidity premium compensates investors for any difficulty that they might face in converting their holdings readily into cash at their fair value Securities that trade infrequently or with low volumes require a higher liquidity premium than those that trade frequently with high volumes • The maturity premium compensates investors for the higher sensitivity of the market values of longer term debt instruments to changes in interest rates LOS 5e: Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows Vol 1, pp 280–284, 289–303 The Future Value of a Single Cash Flow We shall go through LOS 5c after LOS 5e Let’s start off with a relatively simple concept If you had $100 in your pocket right now, and interest rates were 6%, what would be the future value of your money in one year, and in two years? FVN = PV(1 + r) N In one year the value of $100 will be: $100 × (1 + 0.06)1 = $106 Or using your calculator: PV = −100; I/Y = 6; N = 1; CPT FV → FV = $106 TI calculator keystrokes: [2nd ] [FV] “−100” [PV] “6” [I/Y] “1” [N] [CPT] [FV] © 2016 Wiley r05.indd 127 We have shown the PV as a negative number so that the resulting FV is positive Basically, an investment (outflow) of $100 today at 6% would result in the receipt (inflow) of $106 in one year 127 October 2015 8:51 PM The Time Value of Money In solving time value of money problems remember that the stated interest rate, I/Y, and the number of compounding periods, N, should be compatible For example if N is stated in days, I/Y must be the unannualized daily interest rate In two years the value of $100 will be: 100 × (1 + 0.06)2 = $112.36 Or using your calculator: PV = − 100; I/Y = 6; N = 2; CPT FV → FV = $112.36 TI calculator keystrokes: [2nd ][FV] “−100” [PV] “6” [I/Y] “2” [N] [CPT] [FV] On your investment of $100 you earn 0.06 × 100 = $6 in simple interest each year In the second year, the $6 simple interest earned in Year also earns interest in addition to the principal This $6 × 0.06 = $0.36 of additional interest earned is compound interest Over the two years, total interest earned equals $6 + $6 + $0.36 = $12.36 Drawing up timelines will help you avoid careless mistakes when handling TVM questions A general timeline for the future value concept looks like this: N–1 N FVN = PV (1+ r)N If we wanted to determine the future value after 15 periods, the PV and FV would be separated by a future value factor of (1 + r)15, where r would be the interest rate corresponding to the length of each period Since PV and FV are separated in time, remember the following: • We can add sums of money only if they are being valued at exactly the same point in time • For a given interest rate, the future value increases as the number of periods increases • For a given number of periods, the future value increases as the interest rate increases An investment (outflow) of $750 today at 7% would result in the receipt (inflow) of $1,689.14 in 12 years Example 1-1: Calculate the FV of $750 at the end of 12 years if the annual interest rate is 7% Important: After each problem, get into the habit of clearing your TI calculator’s memory by pressing [2nd] [FV] and [2nd] [CE|E] Example 1-2: Calculate the value after 20 years of an investment of $500, which will be made after years The expected annual rate of return is 8% 128 r05.indd 128 Solution PV = −$750; N = 12; I/Y = 7; CPT FV → FV = $1,689.14 Solution PV = −$500; N = 13; I/Y = 8; CPT FV → FV = $1,359.81 Note: The investment will earn interest for 13 periods © 2016 Wiley October 2015 8:51 PM ... PV (1 + r) N In one year the value of $10 0 will be: $10 0 × (1 + 0.06 )1 = $10 6 Or using your calculator: PV = 10 0; I/Y = 6; N = 1; CPT FV → FV = $10 6 TI calculator keystrokes: [2nd ] [FV] “ 10 0”... assuming a 10 % interest rate, the FV of $10 0 received today is $11 0 $10 0 FV1 = $11 0 LOS 5a: Interpret interest rates as required rates of return, discount rates, or opportunity costs Vol 1, pp 278–279... years the value of $10 0 will be: 10 0 × (1 + 0.06)2 = $11 2.36 Or using your calculator: PV = − 10 0; I/Y = 6; N = 2; CPT FV → FV = $11 2.36 TI calculator keystrokes: [2nd ][FV] “ 10 0” [PV] “6” [I/Y]