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Chapter 4 Future Value, Present Value, and Interest Rates Conceptual and Analytical Problems Compute the future value of $100 at an 8 percent interest rate 5, 10, and 15 years into the future. What would the future value be over these time horizons if the interest rate were 5 percent? Answer: Future value in 5 years = $100 × (1.08)5 = $146.93 Future value in 10 years = $100 × (1.08)10 = $215.89 Future value in 15 years = $100 × (1.08)15 = $317.22 Future value in 5 years = $100 × (1.05)5 = $127.63 Future value in 10 years = $100 × (1.05)10 = $162.89 Future value in 15 years = $100 × (1.05)15 = $207.89 Compute the present value of a $100 investment made 6 months, 5 years, and 10 years from now at 4 percent interest. Answer:6 months: Present Value = 100/(1.04)0.5 = $98.06 5 years: Present Value = 100/(1.04)5 = $82.19 10 years: Present Value = 100/(1.04)10 = $67.56 Remember, you are calculating the present value of an investment to be made in the future. Notice that the exponent for the 6-months calculation is 0.5, representing one-half of one year into the future. Assuming that the current interest rate is 3 percent, compute the present value of a five-year, 5 percent coupon bond with a face value of $1,000. What happens when the interest rate goes to 4 percent? What happens when the interest rate goes to 2 percent? Answer: Present Value for 5-year 5 percent coupon bond with face value of $1,000 (i = 3%) = $50/(1.03) + $50/(1.03)2 + $50/(1.03)3 + $50/(1.03)4 + $1,050/(1.03)5 = $1,091.59 Present Value for 5-year 5 percent coupon bond with face value of $1,000 (i = 4%) = $50/(1.04) + $50/(1.04)2 + $50/(1.04)3 + $50/(1.04)4 + $1,050/(1.04)5 = $1,044.52 The present value falls when the interest rate rises to 4 percent. Present Value for 5-year 5 percent coupon bond with face value of $1,000 (i = 2%) = $50/(1.02) + $50/(1.02)2 + $50/(1.02)3 + $50/(1.02)4 + $1,050/(1.02)5 = $1,141.40 The present value rises when the interest rate falls to 2 percent. Given a choice of two investments, would you choose one that pays a total return of 30 percent over five years or one that pays 0.5 percent per month for five years? Answer: To compare the investments, you need to measure their returns in the same units. One option would be to convert both these returns to annual rates. The first investment gives us an annual increase of (1.30)1/5 – 1 = .053873952 or 5.39 percent per year. The second one gives (1.005)12 –1 = .061677812 or 6.17 percent. Therefore, choose this investment that pays 0.5 percent per month for five years. Alternatively, you could convert the first investment to a monthly return: (1.30) 1/60 – 1 = .004382312 or 0.44 percent per month– which is lower than the 0.5 percent on the second investment. A third option would be to convert the monthly rate on the second investment into a 5-year rate: (1.005)60 – 1 = 0.348850152 or 34.88 percent, which is higher than the 30 percent on the first investment. When converted to a common unit of measurement, we see that the second investment gives a higher return. A financial institution offers you a one-year certificate of deposit with an interest rate of 5 percent. You expect the inflation rate to be 3 percent. What is the real return on your deposit? Answer: The real interest rate equals the nominal rate less the expected rate of inflation; therefore 5% - 3% = 2% Consider two scenarios. In the first, the nominal interest rate is 6 percent and the expected rate of inflation is 4 percent. In the second, the nominal interest rate is 5 percent and the expected rate of inflation is 2 percent. In which situation would you rather be a lender? In which would you rather be a borrower? Answer: In the first scenario the real interest rate is 2 percent (the difference between the nominal interest rate and the expected inflation rate) and in the second the real interest rate is 3 percent. As a lender you want a high real return and so would rather lend with the real interest rate at 3 percent (when the nominal rate is 5 percent). As a borrower, you want a low real interest rate and so would rather borrow when the real rate is 2 percent (even though the nominal interest rate is 6 percent). You decide you would like to retire at age 65, and expect to live until you are 85 (assume there is no chance you will die younger or live longer). You figure that you can live nicely on $50,000 per year. Describe the calculation you need to make to determine how much you must save to purchase an annuity paying $50,000 per year for the rest your life. Assume the interest rate is 7 percent. If you want to keep your purchasing power constant, how would your calculation change if you expected inflation to average 2 percent for the rest of your life? Answer: a. $50,000/(1.07)1 + $50,000/(1.07)2 + … + $50,000/(1.07)20 b. If you want to have $50,000 in purchasing power for each year of your retirement, you would need to calculate: $50,000/ (1.07) + $50,000 × (1.02)/(1.07)2 + $50,000 × (1.02)2/(1.07)3 + … + $50,000 × (1.02)19/(1.07)20 Most businesses replace their computers every two to three years. Assume that a computer costs $2,000 and that it fully depreciates in 3 years, at which point it has no resale value and is thrown away. If the interest rate for financing the equipment is equal to i, show how to compute the minimum annual cash flow that a computer must generate to be worth the purchase. Your answer will depend on i. Suppose the computer did not fully depreciate but still had a $250 value at the time it was replaced. Show how you would adjust the calculation given in your answer to part (a). What if financing can only be had at a 10 percent interest rate? Calculate the minimum cash flow the computer must generate to be worth the purchase using your answer to part (a). Answer: a. If x = minimum annual cash flow: $2,000 = x/(1 + i) + x/(1 + i)2 + x/(1 + i)3 x = $2,000/[1/(1 + i) + 1/(1 + i)2 + 1/(1 + i)3] b. $2,000 = x/(1 + i) + x/(1 + i)2 + x/(1 + i)3 + $250/(1 + i)3 x = [$2,000 - $250/(1 + i)3]/[1/(1 + i) + 1/(1 + i)2 + 1/(1 + i)3] c. x = $2,000/[1/(1 + 0.1) + 1/(1 + 0.1)2 + 1/(1 + 0.1)3] = $804.23 Some friends of yours have just had a child. Thinking ahead, and realizing the power of compound interest, they are considering investing for their child’s college education, which will begin in 18 years. Assume that the cost of a college education today is $125,000. Also assume there is no inflation and no tax on interest income used to pay college tuition and expenses. If the interest rate is 5 percent, how much money will your friends need to put into their savings account today to have $125,000 in 18 years? What if the interest rate were 10 percent? The chance that the price of a college education will be the same 18 years from now as it is today seems remote. Assuming that the price will rise 3 percent per year, and that today’s interest rate is 8 percent, what will your friend’s investment need to be? Return to part (a), the case with a 5 percent interest rate and no inflation. Assume that your friends don’t have enough financial resources to make the entire investment at the beginning. Instead, they think they will be able to split their investment into two equal parts, one invested immediately and the second invested in five years. Describe how you would compute the required size of the two equal investments, made five years apart. Answer: PV = $125,000/(1.05)18 = $51,940.08 PV = $125,000/(1.10)18 = $22,482.35 If the price rises 3 percent per year, the cost of a college education in 18 years will be: $125,000 × (1.03)18 = $212,804.13 PV = $212,804.13/(1.08)18 = $53,254.03 If x is the size of each investment: $125,000 = x(1.05)18 + x(1.05)13 x = $125,000/[(1.05)18 + (1.05)13] = $29,122.13 You are considering buying a new house, and have found that a $100,000, 30-year fixed-rate mortgage is available with an interest rate of 7 percent. This mortgage requires 360 monthly payments of approximately $651 each. If the interest rate rises to 8 percent, what will happen to your monthly payment? Compare the percentage change in the monthly payment with the percentage change in the interest rate. Answer: If the annual interest rate is 8 percent, then the monthly rate is (1.08)1/12 – 1 = 0.006434 Using the equation from Appendix 4A: C = ($100,000 × 0.006434)/[1 – (1/(1.006434)360)] = $714. Monthly payments have risen by ($714 – $651)/$651 = 9.7% and the interest rate has risen by (8% – 7%) / 7% = 14.3%. Use the Fisher equation to explain in detail what a borrower is compensating a lender for when he pays her a nominal rate of interest. Answer: The Fisher equation illustrates that the nominal interest rate (i) can be broken down into two components where i = r + пe. Taking i to be an annual interest rate, the borrower is compensating the lender for the inflation that is expected over the coming year, as this will reduce the purchasing power of a given number of dollars. The borrower is also paying the lender a real interest rate to compensate the lender for the use of her money. The lender is foregoing the use of her money for the duration of the loan and so needs to be compensated for this opportunity cost. If the current interest rate increases, what would you expect to happen to bond prices? Explain. Answer: Interest rates and bond prices are inversely related so bond prices will fall when interest rates increase. Bond prices are the sum of the present values of the future payments associated with the bond. The higher the interest rate, the lower the present value of these payments. Which would be most affected in the event of an interest rate increase– the price of a five-year coupon bond that paid coupons only in years 3, 4, and 5 or the price of a five-year coupon bond that paid coupons only in years 1, 2, and 3, everything else being equal? Explain. Answer: The price of the bond with the later payments will fall by relatively more. The payments are made further into the future, so the change in the interest rate has a greater impact on their present value. From the present value formula we can see, for example, that a payment made in one year is divided by (1 + i) while a payment made in five years is divided by (1 + i)5, so the impact will be bigger in the latter case. Under what circumstances might you be willing to pay more than $1,000 for a coupon bond that matures in three years, has a coupon rate of 10 percent, and a face value of $1,000? Answer: If the interest rate in the market were less than 10 percent, the present value of the payment flows associated with the bond would be higher than $1,000. You can use the present value formula to verify this. For example, suppose the interest rate were 8 percent. The present value of the payment flows associated with the bond would be 100/1.08 + 100/(1.08)2 + 100/(1.08)3 + 1,000/(1.08)3 = $1,051.54. Approximately how long would it take for an investment of $100 to reach $800 if you earned 5 percent? What if the interest rate were 10 percent? How long would it take an investment of $200 to reach $800 at an interest rate of 5 percent? Why is there a difference between doubling the interest rate and doubling the initial investment? Answer: Using the rule of 72, we know that if the interest rate is 5 percent, it will take 72/5 = 14.4 years for the investment to double to $200. Repeating the exercise twice more (doubling from $200 to $400 and then from $400 to $800), we see that the investment will take 43.2 years to reach $800. If the interest rate is 10 percent, it will take 72/10 = 7.2 × 3 = 21.6 years to double – exactly half the time. (You can check that your calculations are approximately correct using the future value formula. Alternatively, you could have directly solved for n in the future value formula to find the number of years needed to get to $800.) If $200 is invested at 5 percent, it will take 72/5 = 14.4 × 2 = 28.8 years reach $800 – which is more than half the time it took for $100 to reach $800 at the same interest rate. The reason lies in the compounding – the greater interest earnings having interest paid on them in subsequent years has a bigger impact than the interest being calculated from a larger initial investment. Rather than spending $100 today on paint today, you decide to save the money until next year, at which point you will use it to paint your room. If a can of paint costs $10 today, how many cans will you be able to buy next year if the nominal interest rate is 21 percent and the expected inflation rate is 10 percent? Answer: Saving $100 today means forgoing 10 cans of paint. Since the funds grow in nominal terms by 21 percent, $121 will be available in one year. Expected inflation is 10 percent, so the anticipation is that can of paint in one year will cost $11. Thus, the number of cans that can be bought in one year will be 11 = $121 / $11. So, even though the nominal interest rate is 21 percent, forgoing the purchase of 10 cans of paint today allows purchase of only 10 percent more next year; because the real interest rate is 10 percent. Note that, in this example, the Fisher equation approximation of the real interest rate (the nominal rate less expected inflation, or 11 percent in this example) is overstated. Because the nominal interest rate and the inflation rate are both high, the additional interaction term (described in text footnote 4) subtracts 1 percent (r × π = .10 × .10 = .01) from the approximation. Recently, some lucky person won the lottery. The lottery winnings were reported to be $85.5 million. In reality, the winner got a choice of $2.85 million per year for 30 years or $46 million today. Explain briefly why winning $2.85 million per year for 30 years is not equivalent to winning $85.5 million. The evening news interviewed a group of people the day after the winner was announced. When asked, most of them responded that, if they were the lucky winner, they would take the $46 million up-front payment. Suppose (just for a moment) that you were that lucky winner. How would you decide between the annual installments or the up-front payment? Answer: $2.85 million per year is not equivalent to winning $85.5 million because of the time value of money. If you received all the money today, you could invest it and earn interest on it. Given that you don’t receive most of the money until sometime in the future, the value is less because of the foregone interest. The equivalent amount today is the sum of the present values of the sequence of payments. b. I would calculate which payment option gave me the highest present value. I would look at the market to determine the appropriate interest rate to use and calculate the PV of the installments over 30 years. I would compare this with $46 million to see what is highest. Another factor to consider would be whether the tax treatment was the same for both options. You are considering going to graduate school for a one-year master’s program. You have done some research and believe that the master’s degree will add $5,000 per year to your salary for the next 10 years of your working life, starting at the end of this year. From then on, after the next 10 years, it makes no difference. Completing the master’s program will cost you $35,000, which you would have to borrow at an interest rate of 6 percent. How would you decide if this investment in your education were profitable? Answer: You should calculate the internal rate of return from completing the master’s program. If the IRR is greater than 6 percent, then it will be profitable. The calculation is 35,000 = 5,000/(1 + i) + 5,000/(1 + i)2 + 5,000/(1 + i)3 5,000/(1 + i)4 5,000/(1 + i)5 5,000/(1 + i)6 + 5,000/(1 + i)7 + 5,000/(1 + i)8 + 5,000/(1 + i)9 + 5,000/(1 + i)10. Using a spreadsheet or financial calculator, we find the IRR is around 7 percent. As the IRR is greater than the interest rate, it is profitable to invest in the master’s program. Assuming the chances of being paid back are the same, would a nominal interest rate of 10 percent always be more attractive to a lender than a nominal rate of 5 percent? Explain. Answer: Lenders are concerned with the real return they receive. If the higher nominal interest rate represents a higher real interest rate, then the lender will find it more attractive. If, on the other hand, the higher nominal interest rate merely reflects higher expected inflation, this may not be to the benefit of the lender. For example, a nominal interest rate of 10 percent reflecting expected inflation of 8 percent and a real interest rate of 2 percent would not be preferred by lenders over a nominal interest rate of 5 percent reflecting expected inflation of 1 percent and a real interest rate of 4 percent. It is the real, not the nominal, interest rate that matters. Your firm has the opportunity to buy a perpetual motion machine to use in your business. The machine costs $1,000,000 and will increase your profits by $75,000 per year. What is the internal rate of return? Answer: Using the result in the appendix equation (A5) as n becomes arbitrarily large, we have PV = C / i. The price of the machine is $1,000,000 and the constant stream of profits is $75,000 per year, so the internal rate of return is i = C / P = .075, or 7.5 percent Suppose two parties agree that the expected inflation rate for the next year is 3 percent. Based on this, they enter into a loan agreement where the nominal interest rate to be charged is 7 percent. If inflation for the year turns out to be 2 percent, who gains and who loses? Answer: The ex ante real interest rate is 4 percent. This is what the borrower thinks he or she is paying and the lender thinks he or she is earning. If inflation turns out to be lower than expected, say 1 percent, the ex post real interest rate will be 5 percent. This benefits the lender, as he or she is earning more in real terms than he or she anticipated. The borrower loses when inflation is lower than expected, as he or she is paying a higher real interest rate than he or she anticipated. 22. An unusual development in the wake of the 2007-2009 financial crisis was that nominal interest rates on some financial instruments turned negative. In which of the following examples would the nominal interest rate be negative? a. The real interest rate is 2 percent and the expected inflation rate is 1 percent. b. The real interest rate is zero and the expected inflation rate is 2 percent. c. The real interest rate is 1 percent and the expected inflation rate is minus 2 percent. d. The real interest rate is minus 2 percent and the expected inflation rate is 3 percent Explain your choice. Answer: Using the Fisher equation, i = r + πe, we can calculate the associated nominal interest rate for each of the four examples above. i = 3 percent i = 2 percent i = -1 percent i = 1 percent We see that the nominal interest is negative only in c), where the rate of deflation (negative inflation) is larger in absolute terms than the real interest rate resulted in a negative nominal interest rate. 23. Suppose analysts agree that the losses resulting from climate change will reach x dollars 100 years from now. Use the concept of present value to explain why estimates of what needs to be spent today to combat those losses may vary widely. Would you expect the variation to narrow or get wider if the relevant losses were 200, rather than 100, years into the future? Answer: From the present value formula, PV = FV/(1 + i)n, we can see that the PV of a given future value will vary with the discount rate (i) used. The higher the number of years (n) into the future the cost that is being valued is, the larger the impact on the PV of using different interest rates (discount rates). Therefore, the variation in PV estimates from this source will get wider as n increases at specified interest rates. Data Exploration How does inflation affect nominal interest rates? Plot the three-month U.S. Treasury bill rate (FRED code: TB3MS) from 1960 to the present. What long-run pattern do you observe? What may have caused this pattern? Plot the inflation rate based on the percent change from a year ago of the U.S. consumer price index (FRED code: CPIAUCSL) from 1960 to the present. How does U.S. inflation history reflect your explanation in part (a)? Answer: The data plot for the U.S. three- month Treasury bill is: Notice that this rate trended higher until peaking above 15 percent in the early 1980s and then trended lower. The rate dropped close to zero after the financial crisis of 2007-2009. If the Fisher equation is correct, we would expect that inflation should follow roughly the same pattern. The CPI plot is: Notice that the pattern of CPI inflation is roughly the same, but not identical, suggesting that the real interest rate varied over time. In Data Exploration Problem 1, you saw the impact of inflation in the U.S. on short-term U.S. Treasury bill rates. Now examine similar data for Brazil. Plot the Brazilian Treasury bill rate (FRED code: INTGSTBRM193N). Notice the range of values and compare them with the range in the U.S. Treasury bill plot from Data Exploration Problem 1. Plot the inflation rate based on the percent change from a year ago of the Brazilian consumer price index (FRED code: BRACPIALLMINMEI). Comment on the inflation rate in Brazil. Download the data at a quarterly frequency to a spreadsheet (You may need to widen the spreadsheet column to see the data.) What happens to the index in the 1990–1994 period? Answer: The plot for the Brazilian interest rate is: Note the high nominal interest rate in the mid-1990s. The plot for Brazilian inflation is: Note the inflation in the mid-1990s approached 5,000 percent per year (following even higher inflation several years earlier). In comparison, the nominal interest rate in the previous graph was much lower, so the expected real interest rate was highly negative. With regard to the price level data in the spreadsheet, the quarterly figures from 1990 to 1994 are: 1990-01-01 0.000345 1990-04-01 0.000703 1990-07-01 0.000996 1990-10-01 0.001527 1991-01-01 0.002556 1991-04-01 0.003331 1991-07-01 0.004779 1991-10-01 0.008359 1992-01-01 0.016099 1992-04-01 0.029117 1992-07-01 0.053159 1992-10-01 0.101760 1993-01-01 0.207886 1993-04-01 0.433116 1993-07-01 0.987128 1993-10-01 2.429347 1994-01-01 6.662270 1994-04-01 19.719672 1994-07-01 29.926005 1994-10-01 31.978231 As the inflation plot implies, the values of the price index show explosive growth. The expected real interest rate is the rate which people use in making decisions about the future. It is the difference between the nominal interest rate and the expected inflation rate, not the actual inflation rate. How does expected inflation over the coming year compare with actual inflation over the past year? Plot the inflation rate since 1978 based on the percent change from a year ago of the U.S consumer price index (FRED code: CPIAUCSL). Add to this figure as a second line the expected inflation rate from the University of Michigan survey of consumers (FRED code: MICH). Is expected inflation always in line with actual inflation? Which is more stable? Answer: Expected inflation tends to move with actual inflation but varies somewhat less. One reason is that actual inflation often includes temporary price disturbances (such as energy price changes) that are not expected to persist. Here is the figure: Plot the expected real interest rate since 1979 by subtracting the Michigan survey inflation measure (FRED code: MICH) from the three-month Treasury bill rate (FRED code: TB3MS). Plot as a second line the ex post or realized real interest rate by subtracting from the three-month Treasury bill rate (FRED code: TB3MS) the actual inflation rate based on the percent change from a year ago of the consumer price index (FRED code: CPIAUCSL). What does it mean when these two measures are different? Answer: The data plots are: The expected real interest rate is computed by subtracting from the nominal three-month Treasury bill interest rate the Michigan survey expectations. The actual or ex post real interest rate subtracts from the same nominal Treasury bill rate the actual CPI inflation rate. As the plots show, these are similar in much of the time period. However, notable and sustained differences are evident after 2008 and in the period up to 1982. These deviations represent inflation surprises (either high or low compared to expectations). When the actual real interest rate is above the expected real rate (as in the period after 2008), inflation has surprised on the low side. As a result, borrowers pay a higher real rate than they had planned, while savers receive a larger real return than they had anticipated. The opposite surprise occurred in the late 1970s indicates more difficult problems. Solution Manual for Money, Banking and Financial Markets Stephen G. Cecchetti, Kermit L. Schoenholtz 9781259746741, 9780078021749, 9780077473075
