# 10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans

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3 Real return (rr) = [(1 + nominal return (rn)) / (1 + inflation (π))] 1 Problem 3: Real Return (i.e., the Return after Inflation) Paul just graduated from college and landed a job that pays \$23,000 per year. Assume that inflation averages 1.96 percent per year. A. What nominal rate will Paul need to earn in the future to maintain a 2-percent real return rate? B. In nominal terms, what will Paul s salary be in 10 years? Assume that his salary keeps up with inflation and that inflation averages the same 1.96 percent per year. a. To determine the nominal rate of return, remember the formula for real return: rr = ((1 + rn) / (1 + π)) 1. Now plug in the values you know: 0.02 = (1 + x) / ( ) 1. Solving for x results in a nominal return of 4.00 percent. Thus, Paul s nominal return must be 4.00 percent in the future to maintain a real return of 2 percent. The formula for the nominal rate of return is NR = (1 + RR)*(1+I) -1. b. To maintain his current purchasing power 10 years from now, Paul will have to make \$27, in real terms. This problem is very similar to the Future Value we have already discussed. Use the following values to solve this problem: PV = \$23,000 (This is Paul s current salary.) I = 2 (Interest is replaced by inflation.) N = 10 (This is the number of years in the future.) FV =? The formula is FV = PV * (1+I) N Understand How to Solve Problems Related to Annuities An annuity is a series of equal payments that a financial institution makes to an investor; these payments are made at the end of each period (usually a month or a year) for a specific number of years. To set up an annuity, an investor and a financial institution (for example, an insurance company) sign a contract in which the investor agrees to transfer a specific amount of money to the financial institution, and the financial institution, in turn, agrees to pay the investor a set amount of money at the end of each period for a specific number of years. To determine the set amount of each equal payment for a certain investment, you must know the amount of the investment (PV), the interest rate (I), and the number of years the annuity will last (N)

6 interest or discount rate. What is the present value of each offer? If you could take either offer, which person would you sell your house to? First offer: The present value of this offer is \$200,000 because the buyer can pay you all of the money today. Second offer: This offer is a little different because you will not receive all of the money today; therefore, you must calculate the present value. To calculate the present value of the first offer using a financial calculator, clear your calculator s memory, set the number of payments to one annual payment, and make sure your calculator is set to end mode. Then, input the following information: PMT = \$15,000 N = 25 I = 5 PV =? The present value of the second offer is \$211,409. If you do not have a financial calculator, use the following formula to solve for the present value: PVN,I = PMT * (1 (1 / (1 + I) N )) / I PVN,I = \$15,000 * [1 (1 / (1.05) 25 ] / 0.05 = \$211,409 Which is the better offer? The second offer has a higher present value: if we can assume that you don t need the money right away and that you are willing to wait for payments and confident the buyer will pay you on schedule, you should accept the second offer. As you can see from this example, it is very important that you know how to evaluate different cash flows. Problem 7: Future Value of Annuities Just as it is possible to calculate the present value of an annuity, it is also possible to calculate the future value of an annuity. Josephine, age 22, started working full time and plans to deposit \$3,000 annually into an IRA that earns 6 percent interest. How much will be in her IRA in 20 years? 30 years? 40 years? To solve this problem, clear your calculator s memory and set the number of payments to one (for an annual payment). Set I equal to six and the PMT equal to \$3,000. The formula is: PMT * (((1 + I) N )-1) / I. For 20 years: Set N equal to 20 and solve for FV. FV = \$110,357 For 30 years: Set N equal to 30 and solve for FV. FV = \$237,175 For 40 years: Set N equal to 40 and solve for FV. FV = \$464,

7 If Josephine increased her return rate to 10 percent, how much money would she have after each of the three time periods? How does this interest rate compare to the 6 percent interest rate over time? Do the previous problems at 10-percent interest. Begin by clearing the calculator s memory. Set I equal to 10 and the PMT equal to \$3,000. For 20 years: Set N equal to 20 and solve for FV. FV = \$171,825 (\$61,468 more than she would earn at the 6 percent interest rate) For 30 years: Set N equal to 30 and solve for FV. FV = \$493, (\$256, more than at the 6 percent rate) For 40 years: Set N equal to 40 and solve for FV. FV = \$1,327, (\$863, more than at the 6 percent rate) Your rate of return and the length of time you invest make a big difference when you retire. Solve Problems Related to Amortized Loans An amortized loan is paid off in equal installments (payments) made up of both principal and interest. With an amortized loan, the interest payments decrease as your outstanding principal decreases; therefore, with each payment a greater amount of money goes toward the principal of the loan. Examples of amortized loans include car loans and home mortgages. To determine the amount of a payment, you must know the amount borrowed (PV), the number of periods during the life of the loan (N), and the interest rate on the loan (I). Problem 8: Buying a Car You take out a loan for \$36,000 to purchase a new car. If the interest rate on this loan is 15 percent, and you want to repay the loan in four annual payments, how much will each annual payment be? How much interest will you have paid for the car loan at the end of four years? Before solving this problem, clear your calculator s memory and set your calculator to one annual payment. Input the following information into your financial calculator: PV = \$36,000 N = 4 I = 15 PMT =? Solve for your PMT to get \$12, The formula is: PMT = PVN,I / ((1 (1 / (1 + I) N )] / I)

8 The amount of interest you will have paid after four years is equal to the total amount of the payments (\$12, * 4 = \$50,438.20) minus the cost of your automobile (\$36,000); the total comes to \$14, That is one expensive loan! In fact, the interest alone is more than the cost of another less-expensive car. If you want to buy this car, go ahead, but don t buy it on credit save for it! Problem 9: Buying a House What are the monthly payments on each of the following mortgage loans? Which loan is the best option for a homeowner who can afford payments of \$875 per month? What is the total amount that will be paid for each loan? Assume each mortgage is \$100,000. Loan A: 30-year loan with a fixed interest rate of 8.5 percent Loan B: 15-year loan with a fixed interest rate of 7.75 percent Loan C: 20-year loan with a fixed interest rate of percent Loan A. To determine the monthly payment for a 30-year loan with an 8.5-percent fixed interest rate, clear your calculator s memory, then set your calculator to 12 monthly payments and end mode. Input the following to solve this equation: PV = \$100,000 N = 360 (Calculate the number of monthly periods by multiplying the length of the loan by the number of months in a year: 30 * 12 = 360.) I = 8.5/12 PMT =? Your monthly payment for this loan would be \$768.91, and the total amount of all payments would be \$ * 360, or \$276, The formula is: PV/((1-(1/(1+(I/P))^(N*P)))/(I/P)) Loan B. For a 15-year loan at 7.75 percent interest, follow the same steps explained above. This time, input the information listed below: PV = \$100,000 N = 15 * 12 = 180 I = 7.75 PMT =? The monthly payment for this loan would be \$941.28, and the total amount of all payments would be \$ * 180, or \$169, Loan C. For a 20-year loan at percent interest, the calculations are still the same. Input the following in your financial calculator: PV = \$100,

11 Review Materials Review Questions of a financial calculator, including the functions of present value, future value, payments, interest rates, and number of periods. 1. What is an annuity? 2. How do you set up an annuity? 3. What is a compound annuity? 4. What is the relationship between interest rate and present value? 5. What is inflation? How does it impact investments? Case Studies Case Study 1 Data Lee is 35 years old and makes a \$4,000 payment every year into a Roth Individual Retirement Account (IRA) (this is an annuity) for 30 years. Calculations Assuming the discount, or interest, rate Lee will earn is 6 percent, what will be the value of his Roth IRA investment when he retires in 30 years (this is future value)? Note: The formula is a bit tricky. It is FVN = Payment * (((1 + I) N ) 1)/I (This is the future value of an annuity factor N,I) Case Study 1 Answer Case Study 2 There are two ways for Lee to solve the problem. Using the formula, the problem is solved this way: FVN,I = Payment * (((1 + I) N ) 1) / I = FV = \$4,000 * [(1.06) 30 1] /.06 = \$316, If you are using a financial calculator, clear the calculator s memory and solve: 1 = P/Y (payments per year) 4,000 = PMT (payment) 6 = I (interest rate) 30 = N (number of years) Solve for FV = \$316, Data Janice will make a yearly \$2,000 payment for 40 years into a traditional IRA account. Calculations

12 Given that the discount, or interest, rate is 6 percent, what is the current value of Janice s investment in today s dollars? The formula is: PVN,I = Payment * [1 (1 / (1 + I) N )] / I (the present value of an annuity factor N,I) Case Study 2 Answer Case Study 3 Using the formula, the calculation is PVN,I = Payment * [1 (1 / (1 + I) N )] / I = PV = 2,000 * [1 (1 / (1.06) 40 )] /.06 = \$30, Using the financial calculator, the calculation is Clear memories and use the following: 1 = P/Y 2,000 = PMT 6 = I 40 = N Solve for PV = \$30, Data Brady wants to borrow \$20,000 dollars for a new car at 13 percent interest. Calculations He wants to repay the loan in five annual payments. How much will he have to pay each year (this indicates present value)? The formula is the same formula that was used in the previous problem: PVN = Payment * (PVIFAI,N) Case Study 3 Answer Case Study 4 Data Using the formula, put Brady s borrowed amount into the equation and solve for your payment. PVN,I = Payment * [1 (1 / (1 + I) N )] / I = PV = 20,000 = Payment * [1 (1 / (1.13) 5 )] /.13 = \$5, per year. Using a financial calculator, clear the calculator s memory and use the following: 1 = P/Y = PV 13 = I 5 = N Solve for PMT = \$5,

13 Kaili has reviewed the impact of inflation in the late 1970s. She reviewed one of her parent s investments during that time period and discovered that inflation was 20 percent and that her parent s investment made a 30 percent return. Calculations What was her parent s real return on this investment during that period? Case Study 4 Answers The traditional (and incorrect) method for calculating real returns is Nominal return inflation = real return. This formula would give you a real return of 10%: 30% 20% = 10%. The correct method is (1 + nominal return) / (1 + inflation) 1 = real return (1.30 / 1.20) 1 = 8.33%. In this example, the traditional method overstates return by 20 percent ((10% / 8.33%) 1). Be very careful of inflation, especially high inflation!

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