Chapter 4
第四章
Future Value, Present Value, and Interest Rates
未來價值、現值及利率
Conceptual and Analytical Problems
概念與分析問題
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? Explain the patterns you see in your answers over different time horizons and at different interest rates. (LO1)
計算未來 5 年、10 年和 15 年後的$100 在 8%的利率下的未來價值。如果利率為 5%，這些時間範圍內的未來價值會是多少？解釋您在答案中看到的不同時間範圍和不同利率下的趨勢。（LO1）
Answer:
回答：
Future value in 5 years = $100 × (1.08)5 = $146.93
未來 5 年的未來價值 = $100 × (1.08)^0# = $146.93
Future value in 10 years = $100 × (1.08)10 = $215.89
未來 10 年的未來價值 = $100 × (1.08) 10 = $215.89
Future value in 15 years = $100 × (1.08)15 = $317.22
未來 15 年的未來價值 = $100 × (1.08) 15 = $317.22
Future value in 5 years = $100 × (1.05)5 = $127.63
未來 5 年的未來價值 = $100 × (1.05)^0# = $127.63
Future value in 10 years = $100 × (1.05)10 = $162.89
未來 10 年的未來價值 = $100 × (1.05)^0# = $162.89
Future value in 15 years = $100 × (1.05)15 = $207.89
未來 15 年的未來價值 = $100 × (1.05)^0# = $207.89
The future value of $100 is higher for any given time horizon when the interest rate is higher, as more is being earned each year
未來值在給定時間範圍內，當利率更高時會更高，因為每年獲得的收益更多.
The future value of $100 at a given interest rate increases as we go further into the future, because longer time horizons provide additional years in which to earn interest.
未來值隨著時間的推移而增加，因為較長的時間範圍提供了更多的年分來獲得利息。
The additional benefit of a longer time horizon is greater when the interest rate is higher. For example, going from a 5 to a 10-year horizon, the future value increases by 47 percent when the interest rate is 8 percent, but only 28 percent when the interest rate is 5 percent. This reflects compounding.
長期限額的額外好處在高利率時更為顯著。例如，從 5 年期升到 10 年期，當利率為 8%時，未來價值增加 47%，但當利率為 5%時，僅增加 28%。這反映了複利現象。
Compute the present value of a $100 investment to be made 6 months, 5 years, and 10 years from now at 4 percent interest. Explain why the present value is lower the further into the future the investment is to be made. (LO1)
計算將於 6 個月、5 年、10 年後進行的 100 美元投資的現值，利率為 4%。解釋為何隨著投資時間的延遲，現值會降低。（LO1）
Answer:
回答：
6 months: Present Value = 100/(1.04)0.5 = $98.06
6 個月：現值 = 100/(1.04)^0# = $98.06
5 years: Present Value = 100/(1.04)5 = $82.19
5 年：現值 = 100/(1.04)^0# = 美元 82.19
10 years: Present Value = 100/(1.04)10 = $67.56
10 年：現值 = 100/(1.04)^0# = $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.
記得，你正在計算未來將進行的投資的現值。注意，6 個月計算的指數為 0.5，代表未來一年的半個年頭。
The further into the future the investment is to be made, the more time that is available to earn interest on a current investment to reach the $100 target, and therefore the lower the present value of the $100.
未來投資越遠，可獲得利息的時間就越多，達到 100 美元目標的當前投資現值就越低。
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? (LO2)
假設當前利率為 3%，計算面值為 1000 美元、五年期、5%票面利率的債券的現值。當利率上升至 4%時，情況如何？當利率下降至 2%時，情況又如何？（LO2）
Answer:
回答：
Present Value for 5-year 5 percent coupon bond with face value of $1,000 (i = 0.03) =
現值為 5 年期、票面利率為 5%、面值為 1000 美元的債券（i = 0.03）=
$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 = 0.04) =
現值為 5 年期、票面利率為 5%、面值為 1000 美元的債券（i = 0.04）=
$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.
現值在利率上升至 4%時下降。
Present Value for 5-year 5 percent coupon bond with face value of $1,000 (i = 0.02) =
現值為 5 年期、票面利率為 5%、面值為 1000 美元的債券（i = 0.02）=
$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.
現值在利率降至 2%時上升。
*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? (LO1)
給您兩個投資選擇，您會選擇一個在五年內總回報率為 30%的投資，還是選擇一個每月回報 0.5%的投資？（LO1）
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
另可將首次投資轉為每月回報:
(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.
第三個選項是將第二筆投資的月利率轉換為 5 年期利率：(1.005) 60 – 1 = 0.348850152 或 34.88%，這個數值高於第一筆投資的 30%。
When converted to a common unit of measurement, we see that the second investment gives a higher return.
當換算為常見的度量單位時，我們可以看到第二筆投資的回報更高。
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? (LO3)
金融機構向您提供一張年利率為 5%的一年期定期存款證書。您預期通脹率為 3%。您的存款實際回報率是多少？（LO3）
Answer: The real interest rate equals the nominal rate less the expected rate of inflation; therefore 5% – 3% = 2%
答案：實際利率等於名義利率減去預期的通脹率；因此 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? (LO3)
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) .
A friend has received an unexpected windfall of $10,000 and is considering whether to use
the money to pay down their existing debt, on which they pay 6 percent interest, or invest it in a mutual fund. What advice would you give to your friend? (LO3)
Answer: Advise your friend that, unless they can find a risk-free investment that earns more than 6 percent, they should use the funds to pay down their debt. The opportunity cost of investing the funds is the interest they are paying on the debt. It is as if they are borrowing the funds to invest at an interest rate of 6 percent and so, if the investment doesn’t yield a risk-free interest rate higher than that, it is not worth it.
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. (LO1)
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]
x = [$2,000 – $250/(1 + 0.1)3]/[1/(1 + 0.1) + 1/(1 + 0.1)2 + 1/(1 + 0.1)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. (LO1)
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 5 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. (LO1)
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. (LO3)
*使用費雪方程式詳細解釋當借款人支付名義利率時，他為何要向貸款人補償。(LO3)
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.
答：費雪方程式說明名義利率（i）可以分解為兩個部分，其中 i = r + π e 。將 i 觀為年利率，借款人正在為來年預期的通脹向貸款人支付補償，因為這將降低一定數量的美元的購買力。借款人還在向貸款人支付實際利率，以補償貸款人對其錢款的使 用。貸款人放棄了在貸款期間使用其錢款，因此需要得到這個機會成本的補償。
If the current interest rate increases, what would you expect to happen to bond prices? Explain. (LO2)
如果當前利率上升，您預期債券價格會發生什麼變化？解釋。（LO2）
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.
答案：利率與債券價格呈反比關係，因此當利率上升時，債券價格將下跌。債券價格是與債券相關的未來付款的現值之和。利率越高，這些付款的現值就越低。
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. (LO2)
何者在利率上升的情况下最受影響 - 只在第三、四、五年支付票息的五年期票據債券價格，還是只在第一、二、三年支付票息的五年期票據債券價格，其他條件均相同？解釋之。（LO2）
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.
答：較後期支付的債券價格將相對下降更多。付款時間越遠，利率變動對其現值的影響就越大。從現值公式中我們可以看見，例如，一年後支付的款項除以（1 + i），而五年後支付的款項除以（1 + i） 5 ，所以後者的影響會更大。
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? (LO2)
在什麼情況下，你願意為一張三年到期、票面利率為 10％、面值為 1000 美元的優先券支付超過 1000 美元？(LO2)
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.
答案：如果市場利率低於 10%，與該債券相關的付款流現值將高於 1000 美元。您可以使用現值公式進行驗證。例如，假設利率為 8%。與該債券相關的付款流現值將為 100/1.08 + 100/(1.08)^0# + 100/(1.08)^1# + 1000/(1.08)^2# = 1051.54 美元。
*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? (LO1)
大約需要多長時間，投資 100 美元才能達到 800 美元，如果獲得 5%的回報？如果利率是 10%呢？在 5%的利率下，投資 200 美元需要多長時間才能達到 800 美元？為什麼將利率加倍與將初始投資加倍之間會有差異？（LO1）
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.
答案：使用 72 則，我們知道如果利率為 5%，則投資將需要 72/5 = 14.4 年才能翻倍至 200 美元。再進行兩次練習（從 200 美元翻倍至 400 美元，然後從 400 美元翻倍至 800 美元），我們可以看到投資將需要 43.2 年才能達到 800 美元。
If the interest rate is 10 percent, it will take 72/10 = 7.2 × 3 = 21.6 years to double—exactly half the time.
如果利率為 10%，則需要 72/10 = 7.2 × 3 = 21.6 年才能翻倍——正好是時間的一半。
(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.)
(您可以使用未來價值公式來檢查您的計算是否大約正確。或者，您可以直接在未來價值公式中解出 n，以找到達到 800 美元所需的年數。)
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.
如果投資 200 元於 5%的利率，將需要 72/5 = 14.4 × 2 = 28.8 年才能達到 800 元——這比以相同的利率將 100 元增至 800 元所需時間多出一半。原因在於複利——較大的利息收入在後續年分別計算利息時產生的影響比從較大的初始投資計算的利息更大。
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? (LO3)
不過於今天花 100 美元購買油漆，你決定將錢存到明年，到時你將用這筆錢來油漆你的房間。如果現在油漆罐價格為 10 美元，假設名義利率為 21%，預期通脹率為 10%，明年你能買多少罐油漆？（LO3）
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 = 0.10 × 0.10 = 0.01) from the approximation.
回答：今天儲存 100 美元等於放棄 10 罐油漆。由於資金名義上增長 21%，一年後將有 121 美元。預期通脹為 10%，因此預期一年後那罐油漆將成本 11 美元。因此，一年後可以購買的油漆罐數將是 11 = 121 / 11。所以，即使名義利率為 21%，今天放棄購買 10 罐油漆也只允許明年購買多 10%；因為實際利率為 10%。注意，在這個例子中，實際利率的 Fisher 方程近似（名義利率減去預期通脹，或本例中的 11%）被高估了，因為名義利率和通脹率都很高，額外的交互項（在文字註腳 4 中描述）從近似中減去了 1%（r × π = 0.10 × 0.10 = 0.01）。
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. (LO1)
最近，有個幸運的人中獎了。報導稱獎金為 8,550 萬美元。實際上，獲勝者有選擇每年 2,850 萬美元，連續 30 年，或者現金 46 萬美元的選擇。（LO1）
Explain briefly why winning $2.85 million per year for 30 years is not equivalent to winning $85.5 million.
解釋一下為什麼每年贏得 285 萬美元，持續 30 年不等於贏得 8550 萬美元。
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?
晚上新聞在勝利者公佈的第二天採訪了一群人。當被問及時，他們中的大多數人回答說，如果他們是幸運的勝利者，他們會選擇領取 4600 萬美元的現金預付款。假設（只是瞬間）你是那個幸運的勝利者。你會如何決定是選擇年償還還是現金預付款？
Answer:
回答：
$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.
每年二百八十五萬美元不等同於贏得八千五百五十萬美元，因為錢的時間價值。如果您今天收到所有的錢，您可以投資並獲得利息。考慮到您大部分的錢要到未來某時才能收到，由於放棄的利息，其價值較低。今天的等價金額是這個付款序列的現值之和。
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.
我會計算哪種付款選項給我帶來最高的現值。我會查看市場以確定適用的利率，並計算 30 年分期付款的 PV。我會將這個數字與 4600 萬美元相比，看看哪個更高。另一個需要考慮的因素是，兩種選項的稅務處理是否相同。
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? (LO1)
您正在考慮去上研究生院讀一年的碩士學位課程。您已經做了一些研究，並相信碩士學位將在您未來 10 年的工作生涯中每年為您增加 5,000 美元的薪資，從今年年底開始計算。從那時起，在接下來的 10 年後，這將無所謂。完成碩士學位課程將花費您 35,000 美元，您將需要以 6%的利率借錢。您將如何決定這項對您教育的投資是否具有利潤？（LO1）
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
回答：您應該計算完成碩士學位課程的內部收益率。如果內部收益率超過 6%，則將會獲利。計算方法是
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.
使用試算表或財務計算器，我們發現內部收益率（IRR）大約為 7％。由於 IRR 高於利率，投資於碩士課程是有利可圖的。
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. (LO3)
假設還款的機會相同，10%的名義利率是否總是比 5%的名義利率對貸款人更具吸引力？解釋之。（LO3）
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.
答：貸款人關注他們實際獲得的回報。如果較高的名義利率代表更高的實際利率，則貸款人會發現這更吸引人。另一方面，如果較高的名義利率僅僅反映更高的預期通脹，這可能對貸款人沒有好处。
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.
例如，一個名義利率為 10%，反映預期通脹為 8%以及實際利率為 2%的利率，不會被貸款人比一個名義利率為 5%，反映預期通脹為 1%以及實際利率為 4%的利率所偏好。重要的是實際利率，而非名義利率。
*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? (LO2)
您的公司有機會購買一台永恆運動機器以用於您的業務。該機器價值 100 萬美元，並將每年增加 75,000 美元的利潤。內部收益率是多少？（LO2）
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 / PV = .075, or 7.5 percent
答案：使用附錄方程式（A5）中的結果，當 n 變得任意大時，我們有 PV = C / i。機器的價格為$1,000,000，每年固定利潤流為$75,000，因此內部收益率為 i = C / PV = .075，或 7.5%
*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? (LO3)
假設兩方同意，明年預期的通脹率為 3%。基於此，他們簽訂了一項貸款協議，其中應收取的名義利率為 7%。如果該年的通脹率實際為 2%，誰獲利，誰損失？（LO3）
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.
答：事前實際利率為 4%。這是借款人認為他或她正在支付的，以及貸款人認為他或她正在獲得的。如果通脹實際發生率低於預期，例如 1%，則事後實際利率將為 5%。這對貸款人有利，因為他或她在實際條件下獲得的收益超過了他或她預期的收益。當通脹實際發生率低於預期時，借款人會受到損失，因為他或她支付的實際利率高於他或她預期的利率。
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?
22. 2007-2009 年金融危機後的一個不尋常的發展是，某些金融工具的名義利率變為負值。以下哪個例子中的名義利率會是負值？
a. The real interest rate is 2 percent and the expected inflation rate is 1 percent.
a.實際利率為 2%，預期通脹率為 1%。
b. The real interest rate is zero and the expected inflation rate is 2 percent.
b.實際利率為零，預期通脹率為 2%。
c. The real interest rate is 1 percent and the expected inflation rate is -2 percent.
c.實際利率為 1%，預期通脹率為-2%。
d. The real interest rate is minus 2 percent and the expected inflation rate is 3 percent
d.實際利率為負 2%，預期通脹率為 3%
Explain your choice. (LO3)
解釋你的選擇。（LO3）
Answer:
回答：
Using the Fisher equation, i = r + πe, we can calculate the associated nominal interest rate for each of the four examples above.
使用費雪方程式，i = r + π e ，我們可以計算上述四個例子中相關的名義利率。
i = 3 percent
i = 3 百分比
i = 2 percent
i = 2 百分比
i = -1 percent
i = -1 百分比
i = 1 percent
i = 1 百分比
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.
我們看到，在 c)中，名義利率為負值，因為通脹率（負通脹）的絕對值大於實際利率，導致名義利率為負值。
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? (LO1)
23. 假設分析師們認為，由於氣候變遷造成的損失將在 100 年後達到 x 美元。使用現值概念解釋為何對今天為了對抗這些損失需要投入多少的估計可能會有很大的差異。如果相關損失是在 200 年而非 100 年後發生，您預期變異會縮小還是擴大？（LO1）
Answer:
回答：
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.
從現值公式，PV = FV/(1 + i) n ，我們可以看出，給定未來價值的 PV 會隨著使用的折現率（i）而變化。未來成本被評估的年數（n）越多，使用不同利率（折現率）對 PV 的影響就越大。因此，在指定的利率下，隨著 n 的增加，來自此來源的 PV 估計變化將會變得越寬。
24. If a climate change analyst applies a discount rate of 2 percent to losses expected in 300 years’ time, how much, per $1 of expected loss, might she be willing to spend today to avoid those losses? Would your answer be higher or lower if the losses were expected sooner? Explain your answer. (LO1)
24 若氣候變遷分析師將 2%的折現率應用於預計 300 年後的損失，她為了避免這些損失，每 1 美元的預計損失願意現在花費多少？如果預計損失發生得更早，你的答案會更高還是更低？解釋你的答案。（LO1）
Answer: The present value of $1 in 300 years’ time using a discount rate of 2 percent is
答：以 2%的折现率計算，300 年後的 1 美元的現值是
PV = 1/(1.02)300 = 0.26%.
PV = 1/(1.02) 300 = 0.26%
If the losses were expected sooner, you would expect her to be willing to spend more today to avoid those losses, as she would weight them higher. In other words, the further into the future a loss is expected, the more it is discounted, for the same reason as the sooner payments are expected to be received, the more valuable they are, everything else equal
如果損失預期得早，你會預期她願意今天花更多錢來避免這些損失，因為她會將其評估得更高。換句話說，預期損失發生得越遠，其折扣就越大，原因與預期收到的付款越早，其價值就越高相同，其他條件相等。.