VIEWS: 25 PAGES: 8 POSTED ON: 2/2/2010 Public Domain
Chapter 2 - The Basics of Time Value of Money The “time value of money” describes the reality that a dollar today is not worth the same as a dollar one year from today when interest rates are positive. This is true because you could invest your dollar today and have more than one dollar in a year. Thus, we typically say that a dollar today is worth more than a dollar at some future point in time. Cash flows from different points in time (different months, years, etc), can not simply be added together because they have different “time values.” Therefore, cash flows from different time periods must all be discounted or compounded to the same point in time before they can be added together. This handout will cover both lump sum and annuity time value of money computations. A lump sum payment is a one time payment while an annuity is a series of equal, periodic payments such as a loan. At the end of the handout, you will find practice problems with answers so you can practice doing time value of money computations. Learning about the time value of money is the basis of many financial concepts. Most importantly, learning how to do time value of money computations will prepare you to perform more complicated valuation calculations such as valuing a share of stock, a bond, or even a business. Abbreviations: PV = present value, the value of a cash flow today (also the principal of a loan) FV = future value, the value of a cash flow at some future point in time r = the rate of interest in decimal form (i.e. 5% equals .05 in decimal form) t = is the time period PMT = an equal, period cash flow such as the payment on a loan Lump Sum Formulas Compounding – Computing Future Value When there is a known cash flow today (PV), and you want to know what it will be worth at some point in the future, you will compound that present value (PV) to a future value (FV) using the following formula: FV = PV(1+r)t Example: Future Value Suppose your great, great grandmother put $1.00 in a savings account 150 years ago. If it earned 4% annual what would the dollar be worth today? HINT: FV=PV (1+r) t Using the formula: FV=PV(1+r)t FV=1(1.04)150 _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 Note: In order to take 1.04 to 150th power on a calculator, you can do the following: 1. Enter 1.04 2. Hit the yx key 3. Enter 150 4. Hit the = key 5. You should get 358.92 which you then multiply by 1 for a final answer of FV=358.92 Using a financial calculator: If you have a financial calculator with the PV, FV, and PMT keys, and you wish to use it for these computations, the key strokes are listed below. 1 PV 4 %i 150 N CPT FV 358.92 Discounting – Computing Present Value If you know that you a certain amount of money at some point in the future, that is the future value (FV). You can figure out how much you would need to invest today (PV) at a particular rate so the money will grow to the desired future value (FV) at the designated future date. We call this “discounting” the future value back to the present. PV= FV (1+r)t Example: Present Value What if you knew that the account had $500 in it today, and it earned 4% annually for 150 years, what did grandma originally put in the account? Using the formula: PV= FV PV= 500 PV= 500 PV=$1.39 (1+r)t (1.04)150 358.923 Using the financial calculator: 500 FV 4 %i 150 N CPT PV 1.39 Example: More Present Value _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 You need $20,000 in 3 years to pay tuition at USD and you can earn 8% per year on your money. How much would you need to put away today to have $20,000 in 3 years? Using the formula: PV= FV PV= 20,000 = 20,000 = $15,876.65 (1+r) t (1.08) 3 1.2597 Using the financial calculator: 20000 FV 8 %i 3 N CPT PV 15,876.65 Example: Want to be a millionaire? If you are currently 45 years old and you can earn 8% on your investments, how much will you have to invest today to accumulate $1 million by age 65? Using the formula: PV= FV PV= 1,000,000 = 1,000,000 = $214,546.23 (1+r)t (1.08)20 4.661 Using the financial calculator: 1,000,000 FV 20 N 8 %i CPT PV $214,548.21 Example: A series of cash flows What is the present value of three payments of $500 each if the first payment occurs in one year, the second payment in two years, and the third payment in three year? Assume the interest rate (or discount rate) is 6%. To solve this problem, you must do three separate present value computations and sum the results of the three computations as illustrated below: PV = 500 + 500 + 500 = 471.70 + 445.00 + 419.81 = $1,336.51 (1.06) 1 (1.06)2 (1.06)3 _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 Annuity Formulas An annuity is a series of equal periodic cash flows. Annuity Present Value The annuity present value is used to discount a series of future, equal, periodic cash payments to a present value. The annuity present value formula can be used to compute the present value of a series of equal, periodic payments. The formula can also be used to compute the payments on a car loan or a mortgage. 1 ( 11r )t PVannuity = PMT x r Example: A series of cash flows revisited What is the present value of three payments of $500 each if the first payment occurs in one year, the second payment in two years, and the third payment in three year? Assume the interest rate (or discount rate) is 6%. This problem is really an annuity problem because the future cash flows are an annuity, a series of equal, annual payments of $500 each. You can solve for the present value using the PVannuity formula above. 1 ( 11r )t PVannuity = PMT x r PVannuity = 1 ( 1 .1 )3 500 x 06 = $1,336.41 .06 Interest Rates Interest can be compounded (paid) on an annual basis (annual compounding) or more frequently, such as on a monthly basis. The Annual Percentage Rate (APR) is the annual rate that this charged by the lender or promised by the borrower. If interest is compounded monthly (car loans, home loans, credit card payments, etc), we convert the APR to a monthly rate. To convert the APR to a monthly rate, simply divide the APR by 12. _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 In addition to using the PVannuity formula to solve for a present value, we can use it solve for the annual or monthly payment on a loan if the present value (principal) is known. Example: Car Loan You want to take advantage of excellent cash rebates GM is currently offering on their Yukon XL’s. After rebate, the cost of the vehicle is $32,000. If you put no money down on a 5-year, 8% interest loan, how much will your monthly payments be? Because car payments and mortgage payments are typically made on a monthly basis, we need to adjust r and t to do this computation. We need to divide r by 12 to get a monthly interest rate, and we need to multiply t by 12 to get a number of months. Using the formula: 1 1 ( 1 12 )tx12 rr PVannuity = PMT x r 12 This time, we will solve for PMT because the principal, PV annuity, is already known. 1 1 = PMT x 1 1.4899 = PMT x 1 0.6712 = PMT x .3288 ( 1 .12 )5 x 12 08 1 32,000 = PMT x .006667 .08 12 .006667 .006667 32,000 = PMT x 49.318 PMT = 32,000/49.318 = $648.85 Using the financial calculator: 32000 PV 5*12 = 60 N 8/12 =.6667 %i CPT PMT 648.85 _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 Example: Home Mortgage You have just decided to purchase a new home for $200,000. If the interest rate on a 30- year mortgage is 6%, and you put 5% down, how much will your monthly payments be? The principal that you will actually borrow is 200,000*(1.-05) = 190,000. Using the formula: 1 1 ( 1 .12 )30x 12 06 = 1 6.0226 1 1 0.16604 .83396 190,000 = PMT x .005 = PMT x PMT x = PMT x .06 12 .005 .005 190,000 = PMT x 166.792 PMT = 190,000/166.792 = $1,139.14 Using the financial calculator: 190000 PV 30*12 = 360 N 6/12 = 0.5 %i CPT PMT $1,139.15 After you have computed the payment on a mortgage or loan, you can multiply the dollar amount of the monthly payment by the number of months the payment will be made to compute the total of all of the payments. If you subtract the principal from that total, you will get an estimate of the total dollars of interest that will be paid on the loan. Total payments = 360 * 1,139.14 = 410,090.40 Interest = Total payments – principal = 410,090.40 – 190,000 = 220,090.40 Shocking, isn’t it! Most of us would probably prefer not to know the total interest that will be paid on a 30 year mortgage loan. However, you can significantly decrease the interest paid over the life of the loan by making extra principal payments when at all possible. _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 Time Value of Money Practice Problems These problems are designed to enable you to practice solving the time value of money problems discussed on the previous pages. The answer for each problem is giving in order that you might check your work. 1. You are to receive $6,000 five years from now. At an annual rate of 10%, what is this worth today? 3,725.53 2. You receive $4,000 today. If the account you invest it in earns 10% per year, what will the balance in the account be at the end of five years? 6,442.04 3. You are offered the option of receiving $4,000 today or $6,000 five years from now. Assuming an annual interest rate of 10%, which of the two would you choose? (Base your choice only on the TVM.) 4,000 today 4. You need $15,000 each year for the next 3 years. If you earn 9% interest, how much must you invest today? 37,969.42 5. You want to buy a car that costs $14,000, you will put 10% down, and a bank will finance the balance at a rate of 1% per month for 60 months. What is your monthly payment? 280.28 6. A car dealer says a car will cost you only $179.99 per month. He won't tell you the price of the car so you resort to computing it yourself. To make you a well-informed buyer, he tells you the payment of $179.99 per month is based on a sixty-month loan with an 8% APR. Based on this, the cost of the car is ____? 8,876.82 7. A car dealership offers you a car for $8,000. If you can finance it for 60 months at an APR of 12.6%, what is your monthly payment? 180.39 8. You want to buy a house. You can finance $148,000 of the purchase price with a local bank at a fixed 7% APR with monthly payments for 30 years. Compute your monthly payment on this loan. 984.65 9. Using the information from the previous question, if you make all 360 _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004 payments on time, what are your total payments on the mortgage? How much is interest? How much is principal? Total payments = 354,474 Principal = 148,000 Interest = 206,474 10. Use the information from the previous two questions but assume an APR of 6% instead. What is your monthly payment on the 30 year loan? How much do your total payments amount to? How much is interest? How much is principal? Monthly payment = 887.33 Total payments = 319,440.52 Principal = 148,000 Interest = 171,440.52 11. You borrow $12,000 on a car loan with an APR of 8.75% and monthly payments over 5 years. A. What is your monthly payment? 247.65 B. If you make all of your payments on time, how much interest do you pay over the life of the loan? 2,859 _____________________________________________________________________________________________ Prepared by Angeline M. Lavin, Ph.D., CFA May 2004