Financial Management I - Chapter 3 Time Value Of Money PDF

FINANCIAL MANAGEMENT I CHAPTER THRE 3. TIME VALUE OF MONEY Chapter Objectives: After completing this chapter, a student shall be able to: Understand time value of money concept and why money has time value Und...

FINANCIAL MANAGEMENT I CHAPTER THRE 3. TIME VALUE OF MONEY Chapter Objectives: After completing this chapter, a student shall be able to: Understand time value of money concept and why money has time value Understand simple and compound interest Compute future and present value of simple interest Compute future and present value of single amount and apply in financial management decisions Compute future and present value of single amount and apply in financial management decisions Calculate effective interest rate 3.1. What is time value of money (TVM) and why money has time Value? Time Value of Money (TVM) is an important concept in financial management. It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities. TVM concept stated that a sum of money today is worth more than the same sum of money in the future. Any rational person prefers to receive sum of money now rather than to defer into the future. The time value of money demonstrates that, all things being equal, it is better to have money now rather than later. Why money has time value? One of the most fundamental concepts in finance is that money has a time value. That is to say that money in hand today is worth more than money that is expected to be received in the future. There might be different reasons for this such as: Inflation — the money may lose its purchasing power Risk — the future cash receipt is not certain Preference for consumption The most important reason is straight forward. Money can be invested and earn interest. That is, money that you hold today is worth more because there is an opportunity to invest the money we have now and earn a return on it. For example, if you have Br. 1,000 today, you could save it in a bank and earn interest. This leads to the saying that summarizes the concept of time value as "a birr today is worth more than a birr tomorrow. " Key concepts of TVM are: 1. Interest — interest is a cost of using money, usually stated as a percentage of the amount of money to be used over a specific period of time. There are two basic types of interest: simple interest and compound interest. Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 1  FINANCIAL MANAGEMENT I Simple interest is computed only on the principal amount borrowed. It is the return on the principal amount only for one time period. In simple interest, the interest does not earn interest. Simple interest is used only for short-term financial decisions. Simple Interest = p x i x n where: p =principal (original amount borrowed or loaned) i = interest rate for one period n = number of periods Example 1: You borrow $10,000 for 3 years at 5% simple annual interest. Interest = p x i x n = 10,000 x 0.05 x 3 = 1,500 Example, 2: You borrow $10,000 for 60 days at 5% simple interest per year (assume a 365- day year) Interest = p x i x n = 10,000 x 0.05 x (60/365) = 82.1917 In contrast, compound interest is calculated each period on the original amount borrowed plus all unpaid interest accumulated to date. The interest earned in each period is added to the principal of the previous period to become the principal for the next period. Example: You borrow $10,000 for three years at 5% annual interest compounded annually: interest year 1 = p x i x n = 10,000 x 0.05 x 1 = 500 interest year 2 = (p2 = pl +il) x i x n = (10,000 + 500) x 0.05 x 1= 525 interest year 3= (p3= p2+i2) xix n = (10,500 + 525) x 0.05 x 1 = 551.25 Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25 In financial management we are largely interested on compound interest. So, in the sections that follow we shall discuss the concepts and techniques of the time value of money in the context of compound interest. 2. Number of Periods — refer to regular intervals of time. Each interval must correspond to a compounding period for a single amount or a payment period for an annuity. 3. Payments — is/are a single or a series of equal cash flows. In TVM applications, payments must represent all outflows or all inflows. 4. Present Value — is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate. The future amount can be a single sum that will be received at the end of the last period, or a series of equal payments (an annuity), or both. 5. Future Value — is the amount of money that an investment with a fixed, compounded interest rate will grow to by some future date. The investment can be a single sum deposited at the beginning of the first period, a series of equal payments (an annuity), or both. Since money has time value, we naturally expect the future value to be greater than the present value. The difference Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 2  FINANCIAL MANAGEMENT I between the two depends on the number of compounding periods involved and the going interest rate. A single sum of money or a series of equal payments or receipts promised in the future can be converted to an equivalent value today. Conversely, you can determine the value to which a single sum or a series of future payments will grow to at some future date. You can calculate the fifth value if you are given any four of Interest Rate, Number of Periods, Payments, Present Value, and Future Value. 3.2) Future Value of Single Amount (Compound Interest)__________________ It is the amount of money that an investment made will grow to by some future date. If the money is lent at compound interest, the interest is added each year to the principal and earns interest in the subsequent periods. The formula for Future Value of Single Amount (FVSA) for n periods is as follows: FVSA= PV (1 + i)n OR; FVSA= PV (FVIFi,n) Where; PV is the principal amount or the present value, i is the interest per period (i is the annual interest rate divided the number of periods per annum), and n is the number of periods (n is number compounding per year times number of years). Example 1: You can afford to put $10,000 in a savings account today that pays 6% interest compounded annually. How much will you have 5 years from now if you make no withdrawals? FVSA= PV (1 + i)n FVSA = 10,000 (1.06)5 FVSA = 10, 000(1.3382) FVSA = Br 13,382 Example 2: Bontu deposited Br 10,000 at Awash Bank which was paying 6 percent interest rate, compounded semi-annually, on a ten-year time deposit. How much would the deposit have grown at the end of ten years? i = 6% / 2 = 3%, since it is compounded semiannually n = 10 x 2 = 20 periods FVSA= PV (1 + i)n FVSA= 10,000 (1.03)20 FVSA = 10, 000(1.8061) FVSA =18,061.11 Exercise 1: Fekadu deposited Br 8,000 at Dashen Bank which pays 4% interest rate compounded quarterly on an 8 year time deposit. How much the deposit would grow at the end of 8 years? Checking Figure: = Br 10,999.53 Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 3  FINANCIAL MANAGEMENT I Rate of return Example 4: Ato Gemeda deposited Br 22,000 at Commercial Bank of Ethiopia at certain interest rate, compounded annually so that the deposit would grow to Br 29,440.96 in 5 years time. What is the applicable interest rate? FVSA= PV (1 + i)n 29,440.96 = 22,000 (1 + i)5 29,440.96 = 22,000 (1 + i)5 22,000 22,000 (29,440.96 / 22,000) = (1 + i)5 1.338225 = (1 + i)5 (1.338225)1/5 =(1 + i) 1.05999 = 1 + i i=1.05999-1 i= 6% Number of Periods Example 5: How many years will it take for Br 2,000 to grow to Br 4,025 at 6% interest, compounded annually? FVSA= PV (1 + i)n 4,025 = 2,000 (1.06)n (4,025 / 2,000) =1.06n 2.0125 =1.06 n log 2.0125 = log 1.06n log 2.0125 = n log 1.06 (log 2.0125 / log 1.06) n= 12 periods Or FVIF (10%, n) = 2.0125 n is close to 12 period 3.4) Present Value of Single Amount (Compound Interest) The present value of a sum of money to be received at a future date is determined by discounting the future value at the interest rate that the money could earn over the period. In other words, Present Value is an amount today that is equivalent to a future payment that has been discounted by an appropriate interest rate. Since money has time value, the present value of a future amount is worth less the longer the period that we wait to receive it. The difference between the two depends on the number of compounding periods involved and the interest rate. The Present Value of single amount is calculated by using the following formula: PVsA = FV = FV(PVIFi,n) n (1+i) PVsA = FV(1 +i)-n Where: PV = Present Value or Principal amount FV = Future Value of single amount Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 4  FINANCIAL MANAGEMENT I i = Interest Rate per Period n = Number of Compounding Periods Example 1: Suppose that an investor wants to have Br 20,000 to be received after 5 years at 12% interest rate, compounded annually. How much shall be deposited now? PVsA = FV(1 +i)-n PVsA = 20,000(1.12)-5 PVsA = 20,000 (0.5674) PVsA = Br 11,348.54 Example 2: How much must be deposited now in an account paying 12% compounded monthly in order to pay Br 10,000 down payment on a home 5 years from now? PVsA = FV(1 +i)-n, i=12%/ 12 = 1% and n=5 *12 = 60 PVsA = 10,000 (1+1%)-60 PVsA = 10,000 (1.01) -60 PVsA = 10,000 (0.55045) PVsA =5,504.50 Example 3: a business wants to buy a building 5 years from now for Br 1,500,000. Assuming a 6% interest rate compounded annually, how much should the business invest today to yield Br 1,500,000 in 5 years? PVsA = 1,500,000 (1+6%)-5 PVsA = 1,500,000 (1.06)-5 PVsA = 1,500,000 (0.74726) PVSA=Br 1120887.26 Example 4: taking example 3 above, assume the business finds another financial institution that offers an interest rate of 6% compounded semiannually. How much money shall the business deposit today to yield Br 1,500,000 in five years? Which alternative is better? Interest is compounded twice per year so you must divide the annual interest rate by two to obtain a rate per period of 3%. Since there are two compounding periods per year, you must multiply the number of years by two to obtain the total number of periods. PVsa= FV (1 + i)-n; 6%/2 = 3; and n=5*2=10 PVsA=1,500,000 (1 +3%)-10 PVsA= 1,500,000 (1.03) -10 PVsA= 1,500,000 (0.74409) PVsA=Br1,116,140.87 The second alternative (6% compounded semiannually) has a benefit of Br 4746.39 (Br 1,120,887.26 - Br 1,116,140.87). Thus, the second alternative is the better. Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 5  FINANCIAL MANAGEMENT I Exercise 1: sum of money deposit now at 8% compounded quarterly will provide just enough money to pay a birr 10,000 debt due 7 years from now? Checking Figure: PVSA = 5,743.40 3.5. Future Value of an Annuity(Compound interest) An annuity is a series of equal payments or receipts over a fixed(regular) time interval at a constant interest rate. The periodic payments, deposits, withdrawals, receipts, etc in annuity are called rents (R). For a series of cash flows to be an annuity four conditions should be fulfilled. First, the cash flows must be equal. Second, the interval between any two cash flows must be fixed. Third, the interest rate applied for each period must be constant. Last but not least, interest should be compounded during each period. If any one of these conditions is missing, the cash flows cannot be an annuity. Annuity can be: Ordinary annuity (OA) — is an annuity in which each payment or receipt is made at the end of the period. It is also called annuity in arrears. Annuity due (AD) — is an annuity in which each payment or receipt is made at the beginning of the period. 1. Future Value of Ordinary Annuity (FVOA) An Ordinary Annuity is a series of equal periodic rents in which each payment is made at the end of the period. The Future Value of ordinary annuity is calculated by using the following formula: (1+𝑖)𝑛 −1 FVOA = PMT ቂ ቃ 𝑖 Where: FVOA = Future value of an ordinary annuity PMT = Periodic payments i = Interest rate per period n = Number of periods Or FVAn = PMT(FVIFAi,n) Where: (FVIFAi, n) = the future value interest factor for an annuity (1+𝑖)𝑛 −1 = 𝑖 Example 1: Suppose a business decided to deposit Br 10,000 at the end of each year for next 10 years at 10% interest. How much would this annuity accumulate at the end of the ten years? FVOA = 10,000 [(1 + 10%) 10-1] 10% Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 6  FINANCIAL MANAGEMENT I FVOA = 10,000 [(1.1)10-1] 0.1 FVOA = 10,000 (15.93742) FVOA = Br 159,374.25 Example 2: You are 40 years old and have accumulated Br50,000 in your savings account. You can add Br 100 at the end of each month to your account which pays an interest rate of 6% per year. How much will you have in 20 years? You can treat this as the sum of two separate calculations: 1. the future value of 240 monthly payments of $100 Plus 2. the future value of the $50,000 now in your account PMT = $100 per period Interest per period = i= 6% / 12 = 0.06 /12 =0.005 n = 20x12 = 240 periods FVoa = 100 [(1+0.005)240 -1] 0.005 =100(462.0409) = 46,204.09 FV = 50,000 (1 +i) 240 =50,000 (1.005)240 = 50,000(3.3102) = Br 165,510 Atter 20 years you will have accumulated Br 211,714.09 (46,204.09 + 165,510). Example 3: If Br 100 is deposited in an account at the end of every quarter for the next 5 years, how much will be in the amount by the time the final deposit is made if the interest is 8 % compounded quarterly? Solution: i= 8%/4 =2%, and n=5*4=20 period FVoa = 100 [(1.02)20 -1] 0.02 FVoa =100(24.2974) = 2,429,74 Example 4: In 10 years, you will need $50,000 to pay for college tuition, Your savings account pays 8% interest compounded monthly. How much should you save each month to reach your goal? Solution: FVoa = 50,000, the future savings goal Interest per month = i = 0.08 / 12 = 0.0067 N= 10x12= 120 periods PMT? FVoa = PMT [(1+i)n -l] i Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 7  FINANCIAL MANAGEMENT I 50,000 = PMT [(1+0.0067)120 -1] 0.0067 50,000 = PMT (183.3548) PMT = 50,000 = 272.70 183.3548 2. Future Value of Annuity Due (FVAD) Annuity due is a series of equal periodic payments in which each payment is made at the beginning of the period. The Future Value of an Annuity Due is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, the future value of an annuity due is calculated by multiplying the result of future value of ordinary of annuity by. FVAD = FVOA (1+i) = PMT [(1+i)n -l] (l+i) i Where: FVad = Future Value of an Annuity Due FVoa = Future Value of an Ordinary Annuity Example 1: What amount will accumulate if we deposit $5,000 at the beginning of each year for the next 5 years? Assume an interest of 6% compounded annually. PMT = 5,000, i= 0.06, n=5 FVAD = FVOA (1+i) FVOA = 5,000 [(1+0.06) 5 -1] 0.06 FVOA = 5,000 (5.6371) = 28,185.50 FVAD= FVOA (1+i) = 28,185.50 (1.06) = 29,876.63 Example 2: If birr 100 is deposited in an account at the beginning of eve/" quarter for the next 5 years, how much will be in the account by the time of the final deposit is made assuming 8% interest rate compounded quarterly? Solution: i = 8%/4 = 2%; and n=5 * 20 periods FVAD =100 [(1.02)20- 1] x (1.02) 0.02 FVAD = 100 (24.7833) FVAD = Br 2,478.33 3.6. Present Value of an Annuity (compound interest) i) Present value of an Ordinary Annuity is a single amount of money that should be invested now at a given interest rate in order to provide for an annuity for a certain number of future periods. 𝟏−(𝟏+𝒊)−𝒏 PVOA = PMT ቂ ቃ = PMT (PVIFAi, n) 𝒊 Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 8  FINANCIAL MANAGEMENT I Where: PVOA = The present value of an ordinary annuity (PVIFA i, n) = The present value interest factor for an annuity Example 1: What amount must you invest today at 6% compounded annually so that you can withdraw Br5,000 at the end of each year for the next 5 years? PMT =5,000, i=0.06 and n=5 𝟏−(𝟏+𝒊)−𝒏 PVOA = PMT ቂ ቃ 𝒊 𝟏−(𝟏+𝟎.𝟎𝟔)−𝟓 PVOA = 5000ቂ ቃ 𝟎.𝟎𝟔 = 5,000 (4.2124) = 21,062 Example 2: A computer dealer offers to lease a system to you for Br 50 per month for two years. At the end of two years, you have the option to buy the system for Br 500. You will pay at the end of each month. OR He will sell the same system to you for Br 1,200 cash. If the going interest rate is 12%, which is the better offer? You can treat this as the sum of two separate calculations: 1. the present value of an ordinary annuity of 24 payments at Br 50 per monthly period Plus 2. the present value of Br 500 paid as a single amount in two years 1. solution: PMT = 50 per period, i=0.12/12=0.01, and n = 24 number of periods PVoa = 50 [1-(1.01)-24)] 0.01 = 50(PVIFA 1%, 24) = 50(21.2434) = 1,062.17 2. Solution: FV = 500 Future value (the lease buy out), i = 0.01 Interest per period and n =24 Number of periods PV=FV[1/(1+i)n] = FV (l+i)n =500 [1 / (1.01)24] = 500(PVIF 1%, 24) = 500(0.7876) = 393.80 Example 3: You can get a $150,000 home mortgage at 7% annual interest rate for 30 years. Payments are due at the end of each month and interest. is compounded monthly. How much will your payments be? PVoa = 150,000, the loan amount, i = 0.07/12 =0.0058 and n= 12x30 = 360 periods 150000 = PMT [(1-(1.0058)-360] 0.0058 150000 = PMT [150.9165] PMT = 150,000/150.9165 = 993.93 Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 9  FINANCIAL MANAGEMENT I ii). Present Value of Annuity Due (PVAD) The Present Value of an Annuity Due is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (l + i). Thus, Present Value of Annuity Due is computed by the following formula: PVAd = PMT [ 1-(1+i)-n] (1+i) or PVOA (1+i) i Where: PVAd = The present value of an annuity due Example 1: What amount must you invest today a 6% interest rate compounded annually so that you can withdraw $5,000 at the beginning of each year for the next 5 years PMT =5,000, i=0.06 and n=5 PVAD = PVOA ( 1+i) PVOA = PMT [1-(1+i)-n] (l+i) i PVOA = 5,000 [1 - (1 +0.06)-5] = PVOA = 5,000(PVIFA 6%, 5) 0.06 PVOA = 5,000 (4.2124) = 21,062 PVAD = PVOA (1+i) = 21,062 (1.06) = 22,325.72 Example 2: On September l, 2002 Dinkayehu Pvt. Ltd. Company has borrowed Br 500,000 to pay the down payment on the lease of the land for a plant site. The company will amortize the loan by making a monthly payment of some amount starting from September l, 2002 over a period of 5 years. shall be the monthly payment if the interest rate is 12% compounded monthly? Solution: 12%/ 12 =1% and n= 5 years * 12 = 60 periods; PVAD= 500,000; PMT? 500,000 = PMT [1-(1.01)-60] (1.01) 0.01 500,000 = PMT (45.40459) PMT = 500,000/45.40459 PMT = Br 11,012.10 3.6. Special Case Annuities 3.6.1 Present Value of a Perpetuity A perpetuity is an annuity with an indefinite cash flows. In a perpetuity payments are made continuously forever. The present value of a perpetuity is found by using the following formula: PV (Perpetuity) = Cash Payment = PMT Interest rate i Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 10  FINANCIAL MANAGEMENT I Example: What is the present value of a perpetuity of Br. 7,000 per year if the appropriate discount rate is 7%? Given: PMT = Br. 7,000; i = 7%;, PV (Perpetuity) = ? PV (Perpetuity) = PMT = Br. 7,000 = Br. 100,000. i 7% This means that receiving Br. 7,000 every year forever is equal to receiving Br. 100,000 now. Future value of single amount Table Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 11  FINANCIAL MANAGEMENT I Present value of single amount Chapter three, Time value of Money Complied by Lencho M. (MSc) Page 12