Running head: THE TIME VALUE OF MONEY 1 The Time Value of Money Ma. Cesarlita G. Josol MBA - Acquisition Strayer University
FIN 534 THE TIME VALUE OF MONEY 2 Abstract The paper presents computations about applications on time value of money, bonds, bonds valuation and bonds interest rates. It is essential for financial managers to have a good understanding of the concept of time value of money (TMV) and its impact on stock prices. Companies do not only sustain their operation through assets, but also through debts. Bond is a form of debt where corporations derive its capital for its operations. Hence, financial managers also give specific attention to bonds, valuation of bonds and how its interest rates will affect their financial decision-making. This paper focuses on time value analysis applications like calculation of present value of uneven cash flow using four different procedures: the step-by-step approach, the formula approach, using the financial calculator and by use of spreadsheets. The calculations will show how to do the discounting method to find the present value. Other applications of the time value concept is also presented here such as the concept of future value, the effective annual rate, compounding and interest rates. The paper will also present the calculation of the bond price. It will also provide how the change in market interest rates would result in a discounted bond and a premium bond. I will also present the calculation of yield to maturity, the total return, current yield and the capital gains yield a bond. There will be nine (9) questions presented in this paper, of which I provided solutions on a step-by-step approach. Keywords: Time value of money, present value, future value, bonds, bond valuation
FIN 534 THE TIME VALUE OF MONEY 3 The Time Value of Money If you are offered a $1,000 now and a $1,100 in two years, what will you choose? In a situation where you have extra money to spare for investment what is your decision on accepting the offer to get money now or in two years. Think you really don t need the money now, and you can afford to save for a rainy day. I am going to take the $1,000 and invest in a fixed rate that grows my money in a certain period, rather than choosing to get the $1,100 offered to me in two years. This is the concept of time value of money (TVM). For me the time value of money is a concept where you are going to decide how you will give value to your cash and decide to invest and get the value of the money over time. In businesses, it is essential for managers financial managers, to have a thorough understanding of the time value of money (TVM). A good analysis and understanding of how money will be valued at a certain future is important in sustaining the operations of a business. Another vital information is to get a good grasp of the concept of debts or borrowing. This paper presents calculations of bond valuations, bond interest rates, and changes of bond valuation over time. There is a relationship that exist between the market interest rate and the annual coupon rate of bonds that would result in either discounted bonds or premium bonds The following nine (9) questions provides scenarios where I calculate solutions by using the applications of the concept time value of money and bonds. What is the present value of the following uneven cash flow stream - $50, $100, $75, and $50 at the end of Years 0 through 3? The appropriate interest rate is 10%, compounded annually? This question involves an annuity with uneven cash flow instead of constant payment. There are two important cases of uneven cash flows: (1) where the cash flow stream consists of a
FIN 534 THE TIME VALUE OF MONEY 4 series of annuity payments plus an additional final payment, and (2) the uneven or irregular cash flow stream (Brigham, & Ehrhardt, 2014). This problem is an example of the uneven cash flow stream. In calculations involving cash flows, payment (PMT) is used in situations where the cash flow are constant and thus an annuity is involved; if cash flows are different in different time periods, the term CFt is used which means cash flow in period t. There are three ways to calculate for the net present value (NPV): (1) the step-by-step, (2) the use of the financial calculator, and (3) Excel spreadsheet Variables: I = 10%; CF0 = -$50; CF1 = $100; CF2 = $75; CF3 = $50; N = 3 Cash Flow Stream 0 1 2 3 ($50) $100 $75 $50 1. The step-by-step method The Step-by-Step Method Periods(N) 0 1 2 3 Cash Flow (CF) $ 50.00 $100.00 $ 75.00 $ 50.00 PVs of the CFs $ 50.00 $ 90.91 $ 61.98 $ 37.57 PVt of the Irregular CF Stream $ 140.46 Calculation of the above step-by-step method: PVt = CF0 / (1 + I) 0 + CF1 / (1 + I) 1 + CF2 / (1 + I) 2 + CF3 / (1 + I) 3 PV0 = CF0 / (1 + I) 0 = ( $50) / (1 + 0.10) 0 = ( $50) / (1) = $50.00 PV1 = CF1 / (1 + I) 1 = ($100) / (1 + 0.10) 1 = ($100) / (1.1) = $90.91 PV2 = CF2 / (1 + I) 2 = ($75) / (1 + 0.10) 2 = ($75) / (1.21) = $61.98 PV3 = CF3 / (1 + I) 3 = ($50) / (1 + 0.10) 3 = ($50) / (1.331) = $37.57 PVt = $50.00 + $90.91 + $61.98 + $37.57 = $140.46
FIN 534 THE TIME VALUE OF MONEY 5 Sum of the individual Present Values (PVs) = $140.46 PVt = $140.46 2. Using the financial calculator Texas Instrument BA II Plus 2. Financial Calculator (using Texas Instruments BA II Plus ) Step 1: Press 2nd and CE C Step 2: Press CF Step 3: key in 50 (+/-) enter and press down arrow once, CO1 appears on screen Step 4: Key in 100, press enter and press down arrow twice, C02 appears on screen Step 5: Key in 75 and press enter and press down arrow twice, C03 appears on screen Step 6: Key in 50 press enter Step 7: Press NPV key and the I appears, key in 10, press enter Step 8: Press down arrow once, NPV= 0.00 appears on screen Step 9: Press CPT Screen gives the net present value =NPV= $ 140.46 3. Excel Spreadsheet 1 A B C D E F 2 Inputs: 3 Interest Rate= I = 10% 4 Table of Cash Flows 5 Periods 0 1 2 3 6 Cash Flow -50 100 75 50 Calculation: Using the NPV function = NPV(I, CFS) Fixed Inputs NPV = NPV(0.10,100,75,50) = 190.46 Cell references NPV = NPV(C3,D6:F6) = 190.46 NPV = 190.46 50 = $140.46
FIN 534 THE TIME VALUE OF MONEY 6 The Net Present Value is the present value of the expected future cash flows less the cost of the investment (Microsoft Excel, n.d.). The NPV function in Excel only calculates the present value from period 1 to period 3 in this problem. Then we need to subtract the -$50 which is an outflow (Microsoft Excel, n.d.). The result of the NPV is equal to $140.46. We sometimes need to find out how long it will take a sum of money (or something else, such as earnings, population, or prices) to grow to some specified amount. For example, if a company s sales are growing at a rate of 20%, how long will it take sales to double? In order to know how long will it take for a company s sales to double with the given interest rate of 20%, we need to find the number of years, N. For example the company s sales is $1,000,000; the interest rate given here is 20%. How long will it take for the $1,000,000 to double? In order to find the number of years, N, we can use three procedures: (1) Using the financial calculator, (2) Excel spreadsheet, (3) by working with natural logs Variables: FV = $2,000,000; PV = $1,000,000; I = 20%; N=? Method 1: using the financial calculator Texas Instrument BA II Plus Using the N,I/Y,PV, PMT, FV 1. Clear all values, press 2 nd CE C, press 2 nd FV 2. Key in 2,000,000 then press FV 3. Key in 1,000,000 the + button, then press PV 4. Key in 20, press I Y 5. Press 2 nd P Y, key in 1 and press enter 6. Press 2 nd then QUIT 7. Press CPT and press N key 8. N = 3.80
FIN 534 THE TIME VALUE OF MONEY 7 It will take 3.8 years for the sales of $1,000,000 to double with the given interest rate of 20% Method 2: using the Excel spreadsheet A B C 1 2 Present Value ($1,000,000) 3 Future Value $2,000,000 4 Interest Rate 20% NPER 3.8018 In Excel, the NPER function is used to determine the period, N. NPER returns the number of periods for an investment based on a periodic constant payments and a constant interest rate. NPER in this example is NPER = NPER(I,PMT,PV,FV). Providing the data NPER = NPER(0.20,0,-1,000,000,2,000,000) = 3.8018 or 3.8 years. Method 3: Using the log solution by finding the natural logs using the financial calculator and solve N (Brigham, & Ehrhardt, 2014). Interest Rate, I = 20% Sales = $1,000,000 $2,000,000 = $1,000,000(1+I) N $2,000,000 = $1,000,000(1 + 0.20) N $2,000,000 $1,000,000 = (1 + 0.20) N 2 = (1 + 0.20) N ln 2 = N[ln(1.20)] N = ln(2) / ln(1.20) N = 0.6931471806 / 0.1823215568 N = 3.8018 = 3.80
FIN 534 THE TIME VALUE OF MONEY 8 Will the future value be larger or smaller if we compound an initial amount more often than annually for example, every 6 months, or semiannually the stated interest rate constant? Why? The future value of an investment would be large if the initial amount will compounded more than annually. There will be higher future values when an initial investment is compounded more frequently. Interest will be earned on interest more often the more frequent compounding occurs (Brigham, & Ehrhardt, 2014). The effective annual rate (EAR) also known as the effective percentage rate (EFF%) will increase due to frequent compounding; hence the future value and the EFF% will increase as the frequency of the compounding increase (Brigham, & Ehrhardt, 2014). The biggest increase occurs when compounding goes from annual to semi-annual (Brigham, & Ehrhardt, 2014). Comparison of annual compounding and semi-annual compounding: Variables: PV = $100; Interest rate = 8%; N = 1; M = 2 Compounding: FVN = PV(1 + IPER) Number of Periods = PV(1 + INOM / M) MN Compounding annually: FV1 = $100(1 + 0.08 / 1) = $108 Compounding semi-annually: FV2 = $100(1 + 0.08 / 2) 2 = $108.16 What is the effective annual rate (EAR or EFF%) for a nominal rate of 12%, compounded semi-annually? Compounded quarterly? Compounded monthly? Compounded daily? The effective (equivalent) annual rate (EAR or EFF%) is the annual (interest once a year) rate that produces the same final result as compounding at the periodic rate for M times per year (Brigham, & Ehrhardt, 2014). Given a nominal rate of 12%. The EAR, also known as EFF% is found from the following equation:
FIN 534 THE TIME VALUE OF MONEY 9 EAR = EFF% = (1 + IPER ) M 1.0 EAR = EFF% = (1 + INOM / M) M 1.0 Where: INOM is the nominal rate; IPER is the periodic rate; M is the number of periods per year; N is the number of years and the INOM = 12% 1. Compounded semi-annually: M = 2 EAR = EFF% = (1 + IPER) M 1.0 EAR = EFF% = (1 + INOM / M) M 1.0 EAR = EFF% = (1 + 0.12 / 2) 2 1.0 EAR = EFF% = (1 + 0.06) 2 1.0 EAR = EFF% = (1.06) 2 1.0 EAR = EFF% = 1.1236 1.0 EAR = EFF% = 12.36% 2. Compounded quarterly: M = 4 EAR = EFF% = (1 + 0.12/4) 4 1.0 EAR = EFF% = (1 + 0.03) 4 1.0 EAR = EFF% = (1.03) 4 1.0 EAR = EFF% = 12.5509% 3. Compounded monthly: M = 12 EAR = EFF% = (1 + 0.12/12) 12 1.0 EAR = EFF% = (1 + 0.01) 12 1.0
FIN 534 THE TIME VALUE OF MONEY 10 EAR = EFF% = (1.01) 12 1.0 EAR = EFF% = 1.12682503 1.0 EAR = EFF% = 12.6825% 4. Compounded daily M = 365 EAR = EFF% = (1+0.12/365) 365 1.0 EAR = EFF% = (1+0.000327671233) 365 1.0 EAR = EFF% = 1.127475 1.0 EAR = EFF% = 12.7475% Suppose the on January 1 you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later? To solve this problem we focus on fractional time periods. Here I assume a 365 days in a year so M = 365 Given: Nominal interest rate = 11.33463%; Period = 9 months or 9/12 1. Computing for the Periodic Rate IPER = INOM / M = 0.1133463/365 = 0.0003105378082 per day Computing for the number of days = (9/12)(365) = 273.75 = 274 days Amount in account on October 1 Number of Periods = $100 (1 + IPER) = $100(1.0003105378082) 274 = $108.8797799 Amount in account on October 1 = $108.88 What would be the value of the bond described below if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing investors to require a 13% return?
FIN 534 THE TIME VALUE OF MONEY 11 Would we now have a discount or a premium bond? A firm issues a 10-year par value bond with a 10% annual coupon and a required rate of return is 10%. N = 10; I YR = 13%; PMT = $1,000(10%)= $100 First I will calculate the bond price on what is originally given which is $1,000. Input Output 10.00 10.00 100.00 1000.00 N I YR PV PMT FV ($1,000.00) Then calculating using raised interest rate of 13% using the financial calculator returns a PV = $837.21, therefore the bond price is $837.21, which is lower than the original bond price. Input Output 10.00 13.00 100.00 1000.00 N I YR PV PMT FV ($837.21) This is a discount bond. The coupon rate remains after the issuance of the bond however the interest rates in the market move up and down (Brigham, & Ehrhardt, 2014). An increase in the market interest rates (rd) will cause the price of the bond to fall (Brigham, & Ehrhardt, 2014). This bond is a discount bond as this fixed-rate bond s price fell below its par value due to the increased market interest rate over the annual coupon rate. What would happen to the bond s value if inflation fell and rd declined to 7%? Would we now have a premium or a discount bond? In the same situation, that a firm issues a 10-year, $1,000 par value bond with a 10% annual coupon and a required rate of return is 10%. So the original bond price is $1,000. First calculating for the bond price at the given 10% interest rate.
FIN 534 THE TIME VALUE OF MONEY 12 Input Output 10.00 10.00 100.00 1000.00 N I YR PV PMT FV ($1,000.00) Then computing for the bond price with a rd = 7% Input Output 10.00 7 100.00 1000.00 N I YR PV PMT FV ($1,210.71) The bond price is $1,210.71 which is above the original bond price. Bond prices rise when the market interest falls or when rd falls. When a going interest rate falls below the coupon rate, a fixed-rate bond s price will rise above its par value and it is called a premium bond (Brigham, & Ehrhardt, 2014). This bond that has a 7% market interest rate returns a bond price of $1,210.71 is a premium bond. What is the yield to maturity on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887.00? That sells for $1,134.20? What does a bond selling at a discount or a premium tell you about the relationship between rd and the bond s coupon rate? In this problem the given for the described bond is N=10 years, Coupon rate = 9%, Par Value = $1,000, Bond Price = $887.00. The yield to maturity (YTM) is the same as the market rate of interest, rd. To solve for the yield to maturity (YTM) for the bond price of $887.00, using the financial calculator, I entered all the required values in the financial calculator to solve the yield to maturity (YTM)
FIN 534 THE TIME VALUE OF MONEY 13 Input Output 10 ($887.00) 90.00 1000.00 N I YR PV PMT FV 10.91 The yield to maturity (YTM) is 10.91% To solve for the yield to maturity (YTM) for the bond price of $1,134.20, using the financial calculator: Input Output 10 ($1,134.20) 90.00 1000.00 N I YR PV PMT FV 7.08 The yield to maturity (YTM) is 7.08% A bond selling at a discount means that the rate of interest (rd) is above the bond s coupon rate that would result in the bond s fixed-rate price falling below the par value. While a bond selling at a premium means that the rate of interest (rd) falls below the bond s coupon rate, as a result the fixed-rate s bond price will rise above its par value. When rd goes up above the coupon rate the bond price goes down below the par value, and when rd goes down, below the bond s coupon rate, the bond price goes up above the par value. Whenever the rd equals the coupon rate, a fixed-rate bond will sell at its par value. What are the total return, the current yield, and the capital gains yield for the discount bond in the previous question #8 at $887.00? At $1,134.20 (Assume the bond is held to maturity and the company does not default on the bond). The bond described in this problem is a 10-year, 9% annual coupon, $1,000 par value bond. As I stated before when rd rose above the coupon rate,
FIN 534 THE TIME VALUE OF MONEY 14 the bond would sell below the par value hence the bond will sell at a discount, this bond is called the discount bond. The discount bond price of $887.00 is a result of the rise of the rd to 10.91%. Using this bond price, I will compute the total return, the current yield, and the capital gains for this discount bond from Year 2(with N=9 years) and Year 1 (with N=10 years). The question did not specify as to what year or what period the total return, current yield, and the capital gains be calculated, I will provide two periods. First I will calculate for the required data for year 2 and then I will calculate at the maturity date which is the year 10, where N = 0. This is assuming the bond is held to maturity and company does not default on the bond. This calculation will hold through for the discount bond and the premium bond. 1. Calculations for the Discount bond of $887.00 a) Computing for the total return, current yield, and the capital gains for the discount bond of $887 for year 2 where N = 9 Variables: N = 10 years; Par = 1,000; Annual coupon rate = 9%; YTM for discount bond or rd = 10.91% Computing for the bond price with N = 9 years to maturity Discount bond price for year 2 = $893.87 Input Output 9 10.91% 90.00 1000.00 N I YR PV PMT FV ($893.87) Computing for the capital gain for the year for Year 2 and Year 1: Capital Gain = $893.87 - $887.0 = $6.87 Computing for the following: Current yield = PMT / Discount Bond Price at Year 1
FIN 534 THE TIME VALUE OF MONEY 15 = $90 / $887.00 = 0.1015 Current yield = 10.15% Capital gains yield = Capital Gain between Year 2 and Year 1 / Discount Bond Price at Year 1 Capital gains yield = $6.87 / $887.00 = 0.0077 = 0.77% Total rate of return, or yield = Current yield + Capital gains yield Total rate of return, or yield = 0.1015 + 0.0077 = 0.1092 Total rate of return, or yield = 10.15% + 0.77% = 10.92% Or Total rate of return, or yield = PMT + Capital gain / Discount bond price Total rate of return, or yield = ($90 + $6.87) / $877.00 = $96.87 / $887.00 = 10.92% b) Computing for the total return, current yield, and the capital gains for the discount bond of $887.00 at maturity which is year 10 where N = 0 Discounted bond price at maturity = $1,000 Input Output 0 10.91% 90.00 1000.00 N I YR PV PMT FV ($1,000.00) Computing for the capital gain for the year for Year 10 and Year 1: Capital Gain at maturity = $1,000.00 - $887.00 = $113.00 Computing for the following: Current yield = PMT / Discount Bond Price at Year 1 = $90 / $887.00
FIN 534 THE TIME VALUE OF MONEY 16 = 0.1015 Current yield = 10.15% Capital gains yield = Capital Gain between Year 10 and Year 1 / Discount Bond Price at Year 1 = ($1,000.00 $887.00) / $887.00 Capital gains yield = $113.00 / $887.00 = 0.1274 Capital gains yield = 12.74% Total rate of return, or yield = Current yield + Capital gains yield Total rate of return, or yield = 0.1015 + 0.1274 = 0.2289 Total rate of return, or yield = 10.15% + 12.74% Total rate of return, or yield = 22.89% Or Total rate of return, or yield = PMT + Capital gain / Discount bond price Total rate or return, or yield = ($90.00 + $113.00) / $887.00 = $203.00 / $887.00 Total rate of return, or yield = 22.89% 2. Calculations for the Premium bond of $1,134.20 a) Calculating the total rate of return, the current yield, and the capital gains yield with the premium bond price of $ 1,134.20: Variables: N = 10; Annual coupon rate = 9%; YTM or rd = 7.08%; Par value = $1,000 Computing for the premium bond price in Year 2 where N = 9 Premium bond price in Year 2 = $1,124.67
FIN 534 THE TIME VALUE OF MONEY 17 Input Output 9 7.08% 90.00 1000.00 N I YR PV PMT FV ($1,124.67) Computing for the capital gain for the year for Year 2 and Year 1: Capital Gain = $ $1,124.67 $1,134.20 = $9.53 Computing for the following: Current yield = PMT / Premium Bond Price at Year 1 = $90 / $1,134.20 = 0.0794 Current yield = 7.94% Capital gains yield = Capital Gain between Year 2 and Year 1 / Premium Bond Price at Year 1 Capital gains yield = $9.53 / $1,134.20 = 0.01 Capital gains yield = 1.00% Total rate of return, or yield = Current yield + Capital gains yield Total rate of return, or yield = 0.0794 + ( 0.01) = 0.07 Total rate of return, or yield = 7% Or Total rate of return, or yield = PMT + Capital gain / Discount bond price Total rate or return, or yield = [$90 + ( $9.53)] / $1,134.20 = $80.47 / $1,134.20 = 0.07
FIN 534 THE TIME VALUE OF MONEY 18 Total rate of return, or yield = 7% b) Computing for the total return, current yield, and the capital gains for the premium bond of $1,134.20 at its maturity in year 10 where N = 0 Premium bond price in Year 10 or at its maturity = $1,000.00 Input Output 0 7.08% 90.00 1000.00 N I YR PV PMT FV ($1,000.00) Computing for the capital gain at maturity which is Year 10 and Year 1: Capital Gain = $ $1,000 $1,134.20 = $134.20 Computing for the following: Current yield = PMT / Premium Bond Price at Year 1 = $90 / $1,134.20 = 0.0794 Current yield = 7.94% Capital gains yield = Capital Gain between Year 10 and Year 1 / Premium Bond Price at Year 1 = $1,000 $1,134.20 / $1,134.20 Capital gains yield = $134.20 / $1,134.20 = 0.1183 = 11.83% Total rate of return, or yield = Current yield + Capital gains yield Total rate of return, or yield = 0.0794 + ( 0.1183) = 0.0389
FIN 534 THE TIME VALUE OF MONEY 19 Total rate of return, or yield = 3.89% Or Total rate of return, or yield = PMT + Capital gain / Discount bond price Total rate or return, or yield = [$90 + ( $134.20)] / $1,134.20 = $44.20/ $1,134.20 = 0.0389 Total rate of return, or yield = 3.89%
FIN 534 THE TIME VALUE OF MONEY 20 References Brigham, E., & Ehrhardt, M. (2014). Financial Management (14 th ed.). Mason, OH: Cengage Learning Microsoft Excel as a Financial Calculator Part III (n.d.). Time Value of Money & Financial Calculator Tutorials.com. Retrieved from http://www.tvmcalcs.com/ calculators/excel_tvm_functions/excel_tvm_functions_page3