Module 2 Basic Concepts and Application of Time Value of Money PDF

Module 2: Basic Concepts and Application of Time Value of Money After this lecture, you should be able to: 1. Explain the mechanics of compounding and bringing the value of money back to the present. 2. Understand annuities. 3. Determine the future or present value of a sum when the...

Module 2: Basic Concepts and Application of Time Value of Money After this lecture, you should be able to: 1. Explain the mechanics of compounding and bringing the value of money back to the present. 2. Understand annuities. 3. Determine the future or present value of a sum when there are nonannual compounding periods. 4. Determine the present value of an uneven stream of payments and understand perpetuities. 1 Lecture Outline: 1. The concept of time value of money 2. Using Timelines to Visualize Cashflows 3. Compounding and Future Value 4. Discounting and Present Value 5. Computing Number of Periods 6. PV or FV of Annuities, Annuities Due and Perpetuities 7. Computing Number of Periods or Interest Rates or Payment given Relevant Variables 8. Future and Present Value of Uneven Cash Flows 9. Comparing Interest Rates: APRs vs EARs 10. Some Applications of TVM 2  2.1 The concept of time value of money Money Has a Time Value: A dollar in hand today is worth more than a dollar in hand tomorrow. Why Money Has a Time Value? – I could buy something today and thus get the use today of what I buy. – I could invest today and gain the return from that investment. – I could avoid the loss of value due to inflation in costs. – I could lend the money today and gain the interest on that loan. Interest rate – “exchange rate” between earlier money and later money – There needs to be a return, given the value today vs. tomorrow. – The loss of value from the other potential uses must be recognized. – There are risks that the loan may not be repaid. 3  2.1 The concept of time value of money There are therefore four relevant variables in dealing with the time value of money: – The initial amount lent, called the principal amount – The time period of the loan – The interest rate – The time period to which the interest rate applies Note that there are two separate and potentially different (in fact, usually different) time periods involved: (1) the time period of the loan, and (2) the time period to which the interest rate applies. 4  2.1 The concept of time value of money Simple vs. Compound Interest Simple interest is applied to the initial amount, called the principal, for a given time period for interest. If the period of the loan is greater than the time period for interest, the simple interest will be repeated, at the same amount, and accumulate during successive time periods for interest until the end of the time period of loan. Compound interest occurs when interest paid on the investment during the first period is added to the principal; then, during the second period, interest is earned on this new sum. The rationale for compound interest is that the interest is in fact money that should be in hand at the end of the time period for interest, i.e., at the time it is due. Therefore, if that interest is not received, it is, in effect, also lent and therefore should also bear interest. 5  2.1 The concept of time value of money Simple vs. Compound Interest An Example: Suppose that you deposit $500 in your savings account that earns 5% annual interest. How much will you have in your account after two years using (a) simple interest and (b) compound interest? Simple Interest – Interest earned = 5% of $500 =.05×500 = $25 per year – Total interest earned = $25×2 = $50 – Balance in your savings account = Principal + accumulated interest = $500 + $50 = $550 Compound interest (assuming compounding once a year) – Interest earned in Year 1 = 5% of $500 = $25 – Interest earned in Year 2 = 5% of ($500 + accumulated interest) = 5% of ($500 + 25) =.05×525 = $26.25 – Balance in your savings account: = Principal + interest earned = $500 + $25 + $26.25 = $551.25 6  2.2 Using Timelines to Visualize Cashflows 0 1 2 3 r% CF0 CF1 CF2 CF3 A timeline identifies the timing and amount of a stream of cash flows along with the interest rate. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. A timeline is typically expressed in years, but it can also be expressed in months, days, or any other unit of time. 7  2.2 Using Timelines to Visualize Cashflows Uneven cash flow stream 0 1 2 3 r=10% -$100 $100 $75 $50 The 3-year timeline illustrates the following: The interest rate is 10%. A cash outflow of $100 occurs at the beginning of the first year (at time 0), followed by cash inflows of $100, $75 and $50 in years 1, 2 and 3. 8  2.3 Compounding and Future Value Future value: what a cash flow will be worth in the future. Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year? ▪ Interest in first year= 1,000(.05) = 50 ▪ Value in one year = principal + interest = 1,000 + 50 = 1,050 ▪ Future Value (FV) = 1,000(1 +.05) = 1,050 Suppose you leave the money in for another year. How much will you have two years from now? ▪ Interest in second year=1050(0.05)=52.50 ▪ Value in two years=Principal + interest=1000+50+52.5=1102.50 ▪ FV = 1,000(1.05)(1.05) = 1,000(1.05)2 = 1,102.50 9  2.3 Compounding and Future Value What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? 0 1 2 3 10% 100 FV = ? After 1 year: FV1 = PV(1 + I) = $100(1.10) = $110.00 After 2 years: FV2 = PV(1 + I)2 = $100(1.10)2 = $121.00 After 3 years: FV3 = PV(1 + I)3 = $100(1.10)3 = $133.10 10  2.3 Compounding and Future Value General Formula for finding future value FV = PV(1 + r)t where FV = future value , PV = present value, r = period interest rate, expressed as a decimal, t = number of periods ▪ Compounding factor= (1 + r)t. The effect of compounding is small for a small number of periods, but increases as the number of periods increases. To find future value of a sum, we multiply by compounding factor. ▪ In Excel: Use FV(rate, nper, pmt, pv) to compute future value. ▪ An Example: Suppose you invest the $1,000 from the previous example for 5 years. How much would you have if the interest rate is 5.75%? 11  2.4 Discounting and Present Value Future value: If you deposit today, how much will you have in the future? Present value: what a cash flow would be worth to you today. If you are promised money in the future, how much is it worth today Present value and Future value are mirror images! – We have : FVt = PV(1 + r)t – Rearrange to solve for PV = FVt / (1 + r)t – 1 / (1 + r)t is discounting factor – To find the present value of a future sum we multiply by discounting factor. In Excel: Use PV(rate, nper,pmt,fv) to calculate present value. 12  2.4 Discounting and Present Value Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? – PV = 10,000 / (1.07)1 = 9,345.79 Dan wants to begin saving for his daughter’s college education and he estimates that she will need $150,000 in 17 years. If he feels confident that he can earn 8% per year, how much does he need to invest today? – PV = 150,000 / (1.08)17 = 40,540.34 Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? – PV = 19,671.51 / (1.07)10 = 10,000 13  2.4 Discounting and Present Value For a given interest rate – the longer the time period, the lower the present value ▪ What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% ▪ 5 years: PV = 500 / (1.1)5 = 310.46 ▪ 10 years: PV = 500 / (1.1)10 = 192.77 ▪ For a given time period – the higher the interest rate, the smaller the present value ▪ What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.59 14  2.5 Computing Number of Periods Start with the basic equation and solve for t (remember your logs) ▪ FV = PV(1 + r)t ▪ T=number of period = ln(FV / PV) / ln(1 + r) You want to purchase a new car, and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years In Excel: Use NPER(rate,pmt,pv,fv) to calculate number of periods. 15  2.5 Computing Number of Periods Suppose you want to buy a new house. You currently have $15,000, and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year. How long will it be before you have enough money for the down payment and closing costs? ⁻ Down payment =.1(150,000) = 15,000 ⁻ Closing costs =.05(150,000 – 15,000) = 6,750 ⁻ Total needed = 15,000 + 6,750 = 21,750 – Compute the number of periods – Using the formula: t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years 16  2.6 PV or FV of Annuities, Annuities Due and Perpetuities Annuity: a sequence of equal cash flows, occurring at the end of each period. This is known as an ordinary annuity. – If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. – If you borrow money to buy a house or a car, you will re-pay the loan with a stream of equal payments. Annuity-due: sequence of periodic cash flows occurring at the beginning of each period. – Monthly Rent payments: due at the beginning of each month. – Car lease payments. – Cable & Satellite TV and most internet service bills. Perpetuity – infinite series of equal payments 17  2.6 PV or FV of Annuities, Annuities Due and Perpetuities Annuities and Perpetuities Basic Formulas [ (1 + ) ] 1 1 PV of ordinary annuity = PMT × × 1 − 1 FV of ordinary annuity = PMT × × [(1 + ) − 1] PV of Perpetuities = How to Calculate PV or FV of annuity due: (works for both present and future values) Calculate the present or future value as though it were an ordinary annuity and then multiply your answer by (1+r). 𝑟 𝑟 18 𝑡 𝑟 𝑟 𝑟 𝑡 𝑃 𝑀 𝑇  2.6 PV or FV of Annuities, Annuities Due and Perpetuities: Few Examples 1. After carefully going over your budget, you have determined you can afford to pay $632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow to finance the car? – To determine how much you can borrow, we need to calculate the present value of $632 per month for 48 months at 1 percent per month. 0.01 ( 48 ) 1 1 – ( ) = 632 × 1− = 23,999. 54 (1 + 0.01) 2. Ellen is 35 years old, and she has decided it is me to plan seriously for her re rement. At the end of each year un l she is 65, she will save $10,000 in a re rement account. If the account earns 10% per year, how much will Ellen have saved at age 6 1 0.01 ( –F ( ) = 10,000 × (1 + 0.01)30 − 1) = $1.645 19 𝑃 𝑉 𝑎 𝑛 𝑛 𝑢 𝑖 𝑡 𝑦 𝑉 𝑎 𝑛 𝑛 𝑢 𝑖 𝑡 𝑦 𝑀 𝑖 𝑙 𝑙 𝑖 𝑜 𝑛 ti ti  2.6 PV or FV of Annuities, Annuities Due and Perpetuities: Few Examples 3. You want to endow an annual graduation party at your alma mater. You want the event to be a memorable one, so you budget $30,000 per year forever for the party. If the university earns 8% per year on its investments, and if the first party is in one year’s time, how much will you need to donate to endow the party? – The timeline of the cash flows you want to provide is: – This is a standard perpetuity of $30,000 per year. The funding you would need to give the university in perpetuity is the present value of this cash flow stream $30000 = = = $375,000 today – 0.08 20 𝑟 𝑃 𝑉 𝑃 𝑀 𝑇 2.7 Using Excel’s Functions to compute one variable given Relevant Variables I encourage you not to use formula. Instead, use following excel functions: Calculations Arguments Future Value =FV(rate,NPER,PMT,FV, type) Present Value =PV(rate, NPER,PMT,FV, type) No of Periods =NPER(rate,PMT,PV,FV, type) Interest Rates =RATE(NPER,PMT,PV,FV, type, guess) Payment =PMT(rate, NPER,PV,FV, type) Can you solve all the following problems using Excel’s function? Please Try. 21  2.7 Using Excel’s Functions to compute one variable given Relevant Variables: Examples 1. Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 =.66667% per month). If you take a 4 year loan, what is your monthly payment? – h = (0.00667, 48,20000, 0,0) = $488.30 2. You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can afford only the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000? – = (0.015, − 20,1000, 0,0) = 93.11 months =7.76 years 3. Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long before you pay off the loan? 22 𝑀 𝑇 𝑖 𝑚 𝑜 𝑛 𝑒 𝑡 𝑙 𝑦 𝑁 𝑃 𝑃 𝑎 𝐸 𝑦 𝑅 𝑚 𝑒 𝑛 𝑡 𝑃 𝑀 𝑇  2.7 Using Excel’s Functions to compute one variable given Relevant Variables: Examples 4. Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? = (60, − 207.58, 10000,0, 0) =. 75% 5. Your firm plans to buy a warehouse for $100,000. The bank offers you a 30-year loan with equal annual payments and an interest rate of 8% per year. The bank requires that your firm pay 20% of the purchase price as a down payment, so you can borrow only $80,000. What is the annual loan payment? 23 𝐼 𝑛 𝑡 𝑒 𝑟 𝑒 𝑠 𝑡 𝑅 𝑎 𝑡 𝑒 𝑅 𝐴 𝑇 𝐸  2.8 Future and Present Value of Uneven Cash Flows Excel’s NPV function computes the present value for uneven cash flows Use PV to compute the present value of an annuity stream— all the payments are equal. Use NPV to compute the present value of unequal payments over time. Net present value (NPV) of series of future cash flows is the pv of the cash flows minus the initial investment required to obtain them. Later on we will encounter the net present value in capital budgeting. Using the NPV function = ( , 1, 2,... , )+ 0 24 𝐹 𝑁 𝑁 𝑃 𝑉 𝑟 𝑎 𝑡 𝑒 𝐶 𝐹 𝐶 𝐶 𝐹 𝐶 𝐹 2.8 Future and Present Value of Uneven Cash Flows : Few Examples 1. Assume that we are going to deposit $1500, $1,000 and $500 at the end of each year for the next 3 years in a bank where it will earn 6 percent interest. How much does it worth today? Solution: =NPV(6%, 1500,1000,5000)=$2,724.90 2. Assume that we are going to deposit $1500, $1,000 and $500 at the end of each year for the next 3 years a bank where it will earn 6 percent interest. How much will we have at the end of 5 years? Solution: Step 1: Find Present Value as =NPV(6%,1500,1000,5000)=$2,724.90 Step 2: Find the future value as =FV(6%,3,0,2724.90)= ($3,245.40) 25 2.8 Future and Present Value of Uneven Cash Flows : Few Examples 3. What is the present value of an investment that yields $1,000 to be received in 7 years and $1,000 to be received in 10 years if the discount rate is 6 percent? Solution: =NPV(6%, 0,0,0,0,0,0,1000,0,0,1000) =$1,223.45 26  2.9 Comparing Interest Rates: APRs vs EARs Effective Annual Rate (EAR): This is the actual rate paid (or received) after accounting for compounding that occurs during the year Annual Percentage Rate (APR): This is the annual rate that is quoted by law. Not used in calculations or shown on time lines. – By definition APR = period rate times the number of periods per year – Consequently, to get the period rate we rearrange the APR equation: ▪ Period rate = APR / number of periods per year. Used in calculations and shown on time lines You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate 27 2.9 Comparing Interest Rates: APRs vs EARs ( ( )) Effective Annual Rate = = 1+ −1 1 Annual Percentage Rate = = (1 + ) −1 Remember that the APR is the quoted rate, and m is the number of compounding periods per year In Excel: Effective interest=EFFECT(Nominal_rate, npery) Nominal interest rate=NOMINAL(Effective_rate, npery) 𝐶 𝑜 𝑚 𝑝 𝑜 𝑢 𝑛 𝑑 𝑖 𝑛 𝑔 𝑃 𝑒 𝑟 𝑖 𝑜 𝑑 𝑝 𝑒 𝑟 𝑦 𝑒 𝑎 𝑟 𝑚 𝐸 𝐴 𝑅 28 𝐴 𝑃 𝑅 𝐴 𝐸 𝑅 𝑛 𝐴 𝑃 𝑅 𝑚  2.9 Comparing Interest Rates: APRs vs EARs Some Examples 1. You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you choose and why? ▪ First account: EAR = (1 +.0525/365)365 – 1 = 5.39% ▪ Second account: EAR = (1 +.053/2)2 – 1 = 5.37% 2. Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? APR = 12[(1 +.12)1/12 − 1]=.1138655152 or 11.39% 29  2.9 Comparing Interest Rates: APRs vs EARs Some Examples You’ve just received your first credit card and the problem is the rate. It looks pretty high to you. The quoted rate, or APR, is 21.7 percent, and when you look closer, you notice that the interest is compounded daily. What’s the EAR, or effective annual rate, on your credit card? 30