CHAPTER 3 Basic Math
第3章 基础数学
What you'll want to remember: present value = future value/ (1 + r) t
你需要记住的公式：现值 = 未来值 / (1 + r)t
It is no wonder that the investing public has a difficult time comprehending the bond market when they are told that bonds are issued in denominations of $1,000 but are priced out of $100. This $1,000 is the principal amount, or face value. Adding further confusion is that bonds are priced as a percentage of this amount, as we saw earlier.
当投资者被告知债券面值为1,000元，但交易价格却以100元为单位标示时，难怪投资者很难理解债券市场。这1,000元就是本金或票面价值。更令人困惑的是，债券的定价是以票面价值的百分比来表示，如前文所述。
While the minimum unit of a bond is $1,000 face value, the pricing out of $100 actually makes it easier to understand when you realize that a price of, let's say, $98 represents 98 percent of the face value of the bond, which by extension, makes one bond in this example worth $980. It is an uphill struggle, but one must remember that bond traders are simple folk and pricing out of $100 is easier. We are going to use the terms par, par value, and face value to mean the same thing - the principal amount you originally loaned.
虽然债券的最小单位是1,000元票面值，但以100为基数报价实际上更便于理解，因为你会意识到，例如，当报价为98时，代表票面价值的98%，也就是说在这个例子中，一张债券价值980元。虽然这种理解方式需要花费一点功夫，但必须记住债券交易者都是简单直接的人，而且以100为单位报价更直观。我们将用“平价”、“票面价值”、“本金”等术语来表示同一个概念——你最初借出的本金。
WHAT IS A BASIS POINT?
什么是基点？
This is a phrase commonly used in the fixed-income business and one that I use frequently in this book. Consider what 1 percent in yield represents. Most bonds do not trade at round yields such as 3 percent, 4 percent, or 5 percent. Even though they may have been issued with those yields, once they start to trade, their yield moves up and down and so they will be presented in fractional form. For example, if interest rates move upwards by one-half of 1 percent, a bond issued at 3 percent, will be quoted as yielding 3 1/2 percent. Seems simple enough; however, most of the time, the yield will not fluctuate so evenly and so you might find this same bond quoted to yield 3 37/100 percent. This is a little unwieldy, so what the simple bond folk have done is to divide 1 percent into one hundred even pieces, each one of which is called a basis point. In bond lingo, basis points are known as "beeps." Thus, 25 basis points constitute 25/100, or 1/4 of 1 percent. The bond people then replace fractions with decimal points. So, if a bond moves up in yield by one basis point, they do not quote it as yielding 3 1/100 percent but rather 3.01 percent. The previous example of the bond moving up by 37 basis points would therefore be quoted at 3.37 percent. Got it? It is important to remember the definition of basis points, as that is what we use to compare one bond with another. For example, the British Columbia 3.70 percent December 18, 2020 yield 71 basis points or "beeps" more than the Canada 3.50 percent bonds due June 1, 2020, 3.54 percent versus 2.83 percent.
这是在固定收益领域中经常使用的一个术语，本书中我也经常使用。考虑一下1%的收益率代表什么。大多数债券的交易收益率不会是圆整的3%、4%或5%。即使它们在发行时可能确实有那样的收益率，但一旦开始交易，其收益率会上下波动，因此会以小数形式显示。例如，如果利率上升了半个百分点，一张发行时收益率为3%的债券，其报价可能会变为3.5%。看起来很简单；然而，大部分时间内，收益率的波动不会这么整齐，因此你可能会看到这张债券的报价为3.37%，也就是37/100的变化。这种情况显得有点笨拙，所以债券界的人们将1%分成一百个相等的部分，每一部分称为一个基点。在债券术语中，基点也被称为“beeps”。因此，25个基点构成25/100，或者说1/4个百分点。债券从业者会用小数来代替分数。所以，如果一只债券的收益率上升了一个基点，他们不会报价为3 1/100%，而是写作3.01%。前面提到债券上升37个基点的例子就会写作3.37%。明白了吗？记住基点的定义很重要，因为我们正是用它来比较不同债券。例如，不列颠哥伦比亚省2020年12月18日到期、票息为3.70%的债券，其收益率比2020年6月1日到期、票息为3.50%的加拿大债券多71个基点，分别为3.54%对2.83%。
This chapter is about mathematics, money, and yield. Money invested earns a return. If money is invested in bonds or other fixed-income investments, the return consists of regular interest payments plus the ultimate return of the original amount loaned. If the interest is not needed, it will be reinvested, thus earning interest on interest, a concept we label compounding. This is a key concept that we'll return to later.
本章讨论数学、货币和收益率。投入的资金会产生回报。如果资金投入到债券或其他固定收益类投资中，回报包括定期的利息支付加上最终归还的本金。如果利息不需要用作消费，它将被再投资，从而产生利滚利的效果，这个概念我们称为复利。这是一个关键概念，我们后面还会详细讨论。
The first concept we will discuss is present value. Understanding it will lead you to the full knowledge of how to correctly value a bond or, at least, to fully understand the concept of present value (and which way bond prices go when interest rates go up).
我们首先讨论的概念是现值。理解现值会让你充分了解如何正确估值债券，或至少理解现值的概念（以及当利率上升时债券价格如何变化）。
Obviously, $100 today is worth $100 today; it has no time value. Would you lend someone $100 today for, say, two years, and accept nothing but the $100 back? Unless this was a family member or an extremely close friend, you would likely ask for some interest on the loan; otherwise you would have loaned your money for nothing. Money has time value. Putting it another way, $100 due in two years is not worth $100 today — not unless it were to earn 0 percent and inflation were zero. In the real world, money earns a return and there is inflation, so in two years, $100 will grow by the amount of interest (both simple and compound) earned during that period. The present value formula calculates the present value of that future amount of money using a discount rate, or yield. This concept is vital to a solid understanding of how the bond market functions and is extremely important to retirement planning. That's why I included it at the beginning of this chapter. Here it is again: present value = future value/ (1+r) t, where r is the interest rate per period and t is the number of periods.
显然，今天的100元值100元；它没有时间价值。如果你今天借出100元给某人，比如两年后只收回这100元，除非对方是家人或非常亲近的朋友，你肯定会要求收取一定的利息，否则你就是白白借出钱。钱是有时间价值的。换句话说，两年后的100元今天绝对不值100元——除非它的收益率为0%且没有通货膨胀。在现实中，钱是会产生回报的，而且存在通货膨胀，所以两年后100元会因为这期间赚取的利息（无论是单利还是复利）而增加。现值公式就是用贴现率或收益率计算未来这笔钱在今天的价值。这个概念对于牢固理解债券市场的运作方式以及退休规划都至关重要。这就是我在本章开头就提到它的原因。公式再说一遍：现值 = 未来值 / (1+r)t，其中r是每期利率，t是期数。
Do not despair! Present value tables have already been prepared for your ease of use and enjoyment. To determine the current value of $100,000 due three and a half years from now at a semi-annual yield of 5 percent, you would calculate the answer thusly: $100,000 / (1+2.50 percent)7 = $100,000 / 1.025077 = 1/1.18868.60 = $84,126.52. Stated differently, $84,126.52 today invested at 5 percent compounded semiannually will grow to $100,000 in exactly three and a half years.
别担心！已经为你准备好了现值表，方便你查阅和使用。假设你想计算在半年度收益率5%的情况下，3年半后到期的100,000元的现值，你可以这样计算：100,000元 / (1+2.50% )7 = 100,000元 / 1.025077 = 1/1.18868.60 = 84,126.52元。换句话说，今天投资84,126.52元，以5%的半年度复利计算，正好在3年半后长成100,000元。
Before continuing, we will spend some time discussing compound interest, since the all-important present value is merely the reciprocal of a compounded amount.
在继续之前，我们先花些时间讨论复利，因为这至关重要的现值其实就是复利的倒数。
COMPOUND INTEREST
复利
Baron de Rothschild described compound interest as "the eighth wonder of the world. Compound interest is a concept vital to the understanding of yield to maturity. Compounding means that not only does your original dollar earn interest but interest you receive also earns interest by being reinvested. In this case, we are compounding on a semi-annual basis; although this appears to complicate matters, the domestic bond market functions on a semi-annual basis. Thus, a bond paying semi-annual interest and paying 5 percent pays that 5 percent in two payments of 2.5 percent each, paid six months apart. The following example will use an original amount of $100,000. This amount will be returned at maturity, in this case three and a half years from now. There will be seven interest payments made. How much will each payment be? The original face amount times the interest rate divided by two: $100,000 *.05/2 = $2,500.
罗斯柴尔德男爵曾称复利为“世界第八大奇迹”。复利是理解到期收益率（Yield to Maturity）的重要概念。复利意味着不仅你的原始资金能赚取利息，连你收到的利息再被再投资后也能产生利息。在本例中，我们按照半年度进行复利；虽然这看似让计算变得复杂，但国内债券市场就是以半年度运作的。所以，一只支付半年度利息、票息为5%的债券会在一年内支付两次利息，每次为2.5%。以下例子以最初的投资100,000元为例，这笔金额将在3年半后到期归还。期间将支付7次利息。每次支付金额是多少？计算方法为：本金乘以利率再除以2，即100,000元 * 0.05 / 2 = 2,500元。
We will assume that all of these payments (except the last one, since it is paid at the same time the $100,000 is repaid) will be reinvested or compounded at the same rate (2.5 percent) for the remaining term of the investment, and so on with each successive payment. This might be a difficult concept to grasp initially, but it is how bond calculators work. It provides a common yield measuring device for us simple bond folk, since it is obviously not a real world calculation. In the real world, every interest payment will be invested at a different yield, if reinvested at all.
我们假设所有这些支付（最后一次支付除外，因为它与100,000元的还本在同一时间支付）都将以同样的2.5%的利率再投资或复利，且每笔后续支付都如此处理。这个概念初看可能有点难以理解，但这正是债券计算器的工作原理。它为我们这些简单的债券投资者提供了一种统一的收益率衡量方法，尽管这显然不是一个现实中的计算方式。在现实中，每笔利息支付如果再投资，其收益率可能各不相同。
How $100,000 will grow at 5 percent compounded semi-annually:
100,000元在5%半年度复利下如何增值：
| Interest | Compound Interest |
Total Interest |
Amount | Total |
|---|---|---|---|---|
| $ 2,500.00 | $ 0.00 | $ 2,500.00 | $ 100,000.00 | $ 102,500.00 |
| $ 2,500.00 | $ 62.50 | $ 5,062.50 | $ 100,000.00 | $ 105,062.50 |
| $ 2,500.00 | $ 126.56 | $ 7,689.06 | $ 100,000.00 | $ 107,689.06 |
| $ 2,500.00 | $ 192.23 | $ 10,381.29 | $ 100,000.00 | $ 110,381.29 |
| $ 2,500.00 | $ 259.53 | $ 13,140.82 | $ 100,000.00 | $ 113,140.82 |
| $ 2,500.00 | $ 328.52 | $ 15,969.34 | $ 100,000.00 | $ 115,969.34 |
| $ 2,500.00 | $ 399.23 | $ 18,868.58 | $ 100,000.00 | $ 118,868.58 |
| $ 17,500.00 | $ 1,368.58 | $ 18,868.58 | $ 100,000.00 | $ 118,868.58 |
Thus, $100,000 invested today at 5 percent compounded semi-annually will grow to $118,868.58 in three and a half years.
因此，今天投资100,000元，在5%半年度复利下，3年半后将增长到118,868.58元。
Explaining this table in more detail, the first interest payment of $2,500 is received after six months. At twelve months, the next $2,500 is paid, but since the payments are being reinvested, the first $2,500 payment will have earned 2.5 percent, or $62.50, during those six months; this is the interest on the interest, or what we call compound interest. For the next payment, the $2,500 is received plus the interest earned on the first two payments plus the compound interest ($2,500 + 2,500 + 62.50) *.025 = $126.56. So, the total is now the three payments of $2,500 each plus $62.50 plus $126.56, for a total of $7,689.06, and so on, until the compounding period ends and the total future value totals the original face value plus the interest payments received (7 * $2,500 = $17,500) plus the total compound interest of $1,368.58: $100,000 + $17,500 + $1,368.58 = $118,868.58.
对上表的详细解释：第一笔2,500元的利息在六个月后收到。在十二个月时，再支付2,500元，但由于前期支付已被再投资，第一笔2,500元在这六个月内赚取了2.5%的利息，即62.50元，这就是所谓的“利滚利”或者复利。对于下一次支付，收到2,500元加上前两次支付所赚取的利息以及复利部分计算为 (2,500 + 2,500 + 62.50) * 0.025 = 126.56元。所以，总额现在是三次2,500元支付加上62.50元和126.56元，共计7,689.06元，依此类推，直到复利期结束，未来值总额等于原始票面金额加上所有收到的利息（7 * 2,500 = 17,500元）再加上累计复利1,368.58元：100,000元 + 17,500元 + 1,368.58元 = 118,868.58元。
Let us now go back to the present value formula since it will be referred to many times and is a very important concept. Since the above calculation shows the future value of a sum invested today at a certain interest rate for a specified period of time, we should be able to calculate the present value of a sum in the future discounted by an interest rate for a specified period of time. All we have to do is take the reciprocal of the future value formula. In other words, $118,868.58 discounted at an interest rate or yield of 5 percent semi-annually for three and a half years is worth $100,000 today. If you wished to express this out of a round amount like $100,000, simply divide the $100,000 by $118,868.58 to get an answer of .841265. Thus, $84,126.50 invested today at 5 percent semi-annually compounded will grow to exactly $100,000 in three and a half years. This calculation will be very important when we discuss stripped bonds in a later chapter. A present value table, calculated for a wide number of terms to maturity and interest rates, may be found in the stripped bond chapter.
现在让我们回到现值公式，因为这一公式会被反复引用，是一个非常重要的概念。上面的计算显示了以某一利率在特定期限内今天投资一定金额未来的增长情况，我们同样可以计算未来某金额的现值，只需对未来值公式取倒数。换句话说，将118,868.58元以5%半年度贴现3年半后的现值，就是今天的100,000元。如果你希望以一个整齐的数字表示，比如100,000元，只要将100,000元除以118,868.58元，得到0.841265。这样，今天投资84,126.50元，在5%半年度复利下，正好在3年半后增长为100,000元。当我们在后面的章节讨论分离式债券（stripped bonds）时，这种计算会非常重要。有关不同到期期限和利率的现值表可以在分离式债券的章节中找到。
Let us now return to the bond used earlier, the Canada 3.50 percent bond maturing June 1, 2020. This is a benchmark issue with some $13.1 billion outstanding. We will demonstrate how the value of this bond is really determined by calculating the present value of each of the interest payments and the principal. For this example we will assume a yield or discount rate of 2.883 percent for each of the components and that the bond was bought at a price of $104.861. In addition, we will assume that the bond was purchased on June 1, 2011, so that the first payment is in exactly six months. You can calculate all these present values yourself by using a present values table, the calculator found on my website (www.inyourbestinterest.ca), or Excel, which has bond math in its Analysis Tool Pack. (See Appendix C to see how to use Excel for bond yields)
现在让我们回到前面讨论的债券，即到2020年6月1日到期、票息为3.50%的加拿大债券。这是一只基准债券，约有131亿元在外流通。我们将通过计算每笔利息支付和本金的现值来展示这只债券真正的价值是如何确定的。在这个例子中，我们假设每个部分的贴现率或收益率为2.883%，而且债券的购买价格为104.861元。再者，我们假设债券是在2011年6月1日购买的，因此第一笔利息支付正好在六个月后。你可以利用现值表、我网站上提供的计算器（www.inyourbestinterest.ca），或者Excel中的债券数学工具包（见附录C介绍如何使用Excel计算债券收益率）自行计算所有这些现值。
| Date | Interest Payment |
Principal | Discount Rate |
Present Value of $1 |
Value |
|---|---|---|---|---|---|
| Dec. 1, 2011 | $ 1,750 | 2.883 | 0.98579 | $ 1,725.13 | |
| June 1, 2012 | $ 1,750 | 2.883 | 0.97178 | $ 1,700.61 | |
| Dec. 1, 2012 | $ 1,750 | 2.883 | 0.95797 | $ 1,676.45 | |
| June 1, 2013 | $ 1,750 | 2.883 | 0.94436 | $ 1,652.63 | |
| Dec. 1, 2013 | $ 1,750 | 2.883 | 0.93094 | $ 1,629.15 | |
| June 1, 2014 | $ 1,750 | 2.883 | 0.91771 | $ 1,605.99 | |
| Dec. 1, 2014 | $ 1,750 | 2.883 | 0.90467 | $ 1,583.17 | |
| June 1, 2015 | $ 1,750 | 2.883 | 0.89181 | $ 1,560.67 | |
| Dec. 1, 2015 | $ 1,750 | 2.883 | 0.87914 | $ 1,538.50 | |
| June 1, 2016 | $ 1,750 | 2.883 | 0.86665 | $ 1,516.64 | |
| Dec. 1, 2016 | $ 1,750 | 2.883 | 0.85433 | $ 1,495.08 | |
| June 1, 2017 | $ 1,750 | 2.883 | 0.84219 | $ 1,473.83 | |
| Dec. 1, 2017 | $ 1,750 | 2.883 | 0.83023 | $ 1,452.90 | |
| June 1, 2018 | $ 1,750 | 2.883 | 0.81843 | $ 1,432.25 | |
| Dec.1, 2018 | $ 1,750 | 2.883 | 0.80680 | $ 1,411.90 | |
| June 1, 2019 | $ 1,750 | 2.883 | 0.79533 | $ 1,391.83 | |
| Dec. 1, 2019 | $ 1,750 | 2.883 | 0.78403 | $ 1,372.05 | |
| June 1, 2020 | $ 1,750 | 2.883 | 0.77289 | $ 1,352.56 | |
| June 1, 2020 | $ 1,750 | $100,000 | 2.883 | 0.77289 | $ 77,289.00 |
| Total | $ 31,500 | ||||
| Grand Total | $ 131,500 | $ 104,861.00 | |||
I am assuming a cost of $104.861 for a semi-annual yield of 2.883 percent as of June 1, 2011. Price per $100 is therefore $104.861, exactly the cost of the bond.
我假设按2.883%的半年度收益率，2011年6月1日购入的成本为104.861元。因此，每100元的价格即为104.861元，这正是该债券的成本。
You will notice that by calculating the present value of each component of the bond at the purchase yield, we arrive at the exact cost of the bond as seen above: $104,861. Thus, a bond is demonstrated to be the sum of its parts.
你会注意到，通过以购买时的收益率计算债券每一部分的现值，得到的债券总价值正好是上面显示的104,861元。这证明了一只债券只是它各组成部分现值的总和。
As you can see, a bond is merely a sum of its parts; put another way, yield to maturity is the discount rate that equates the future cash flows to today's price.
如你所见，一只债券仅仅是它各部分的总和；换句话说，到期收益率是那个能将未来现金流折合到今天价格的贴现率。
Now you see how artificial the yield to maturity calculation is, since yields at different maturities will not be identical. Put another way, the odds of each coupon payment being reinvested at the purchase yield are very, very low. For that to be the case, yields would have to be the absolute same at all the interest payment dates as at the time of purchase. We will see when we discuss yield curves that this is a very rare event. Another calculation produces the realized yield, which takes into account the effect of different reinvestment rates on the yield to maturity. This calculation is best left to calculators. What you need to remember is the longer the term of the bond, the more important the interest payments and, therefore, the more important the interest on the interest.
现在你明白了，到期收益率的计算是多么不符合现实，因为不同期限的债券其收益率永远不会完全相同。换句话说，每笔息票支付都能以购买时的收益率再投资的可能性是非常非常低的。若要实现这一点，各个利息支付日的收益率必须与购买时完全一致。我们在讨论收益率曲线时会看到这种情况极为罕见。另一种计算方法得到的是实际收益率，它考虑了不同再投资利率对到期收益率的影响。这个计算最适合交给计算器来处理。你需要记住的是，债券期限越长，利息支付就越重要，因此“利上加利”效应就越显著。
Following is an example of a longer-term bond, the Canada 3.5 percent due December 1, 2045. Showing the effect of different reinvestment rates illustrates how important interest is when assessing the value of a bond. These numbers represent the total amounts received if this bond was held to its maturity date with the reinvestment rate varying. The importance of reinvestment rates is also striking. In all three cases the simple interest received over the life of the bond exceeds the final principal repayment, while in two of the cases the compound interest also exceeds the principal payment.
下面是一个长期债券的例子：加拿大3.5%债券到期日为2045年12月1日。展示不同再投资利率的影响可以说明在评估债券价值时利息有多么重要。这些数字表示如果持有该债券至到期日，在不同的再投资利率下所收到的总金额。再投资利率的重要性也非常明显。在所有三种情况下，债券整个生命周期中获得的单利总和都超过了最终的本金偿付，而在其中两种情况下，复利部分甚至超过了本金。
At June 12, 2011:
在2011年6月12日：
| Reinvestment Rate | 2% | 3.35% | 6% |
| Principal | $100,000 | $100,000 | $100,000 |
| Simple Interest | $120,640 | $120,460 | $120,460 |
| Compound Interest | $ 51,850 | $103,409 | $268,624 |
| Total | $272,490 | $324,049 | $489,264 |
"Thrift is a wonderful virtue — especially in an ancestor," said Mark Twain. Compound interest is one of the strongest forces in the investment business and one of the keys to investing success. For example, the following table indicates how long it takes for $1 to double at different semi-annual compounding yields (pre-tax, of course).
马克·吐温曾说：“节俭是一种美德——尤其是对祖先而言。”复利是投资领域中最强大的力量之一，也是投资成功的关键之一。例如，下面这张表显示了在不同半年度复利条件下，1元翻倍所需的时间（当然，这是税前数据）。
| Rate | Years to Double |
|---|---|
| 4% | 17.5 |
| 6% | 11.7 |
| 8% | 8.8 |
| 10% | 7.1 |
| 12% | 5.9 |
You may have noticed that multiplying the rate times the number of years produces an answer close to seventy-two. The "Rule of 72" says that if you divide seventy-two by the interest rate, the answer is the number of years (approximately) that it takes for money to double.
你可能已经注意到，利率乘以年数会得出一个接近72的数值。“72法则”说，如果你用72除以利率，所得结果就是资金翻倍大约需要的年数。
So you want to make a million? The present value of $1 million due in twenty years discounted at a yield of 10 percent semi-annually is $142,000. Therefore, $142,000 invested at 10 percent will equal $1 million in 20 years (pre-tax, but this is possible in RRSPs). At 5 percent, that same $142,000 would grow to only $381,279 in the same period, while at 15 percent it would grow to $2,562,281! These dramatic differences underscore how important it is to take advantage of high-yields when they are available. At 8 percent compounded semi-annually, $1,000 would grow to more than $2,000 in nine years, $7,106 in twenty-five years, $50,504 in fifty years, and $2,550,749 in one hundred years.
那么，你想赚到一百万吗？按10%半年度复利计算，20年后到期的100万元的现值是142,000元。因此，用142,000元以10%的利率投资，20年后将变成100万元（税前，但这在RRSP账户中是可能的）。若是按5%计算，同样的142,000元在同一时期只会增至381,279元，而按15%的利率则会增至2,562,281元！这些戏剧性的差异强调了在有机会获得高收益率时抓住它的重要性。在8%半年度复利下，1,000元在九年内可增值至超过2,000元，二十五年达到7,106元，五十年达到50,504元，而一百年则达到2,550,749元。
You can see that compounding is a very powerful force. Consider the tale of the numerically naive king and his mathematical daughter who happened to be a whiz at checkers. He offered her monthly prizes for scholastic achievements. His daughter told him that she wanted just one dollar to be put on the first square of her checkerboard after the first month and then to have that amount doubled every succeeding month. So, on month two it would be $2 on the second square, $4 on the third month, and so on, until the sixty-four squares were full. Well, the king ran out of money. By the tenth square, he put down $512, and by the twentieth, it had become $524,288. The amount would have been in the trillions by the time he reached square sixty-four! She was earning a 100 percent return per month.
可见，复利是一种非常强大的力量。试想一个在数字上天真的国王和他在跳棋方面颇有天赋的数学家女儿的故事。他给她提供每月的奖品以鼓励她的学术成就。女儿告诉国王，她只要求第一格放上一元，然后每个月的金额翻倍。也就是说，第二个月第二格放2元，第三个月第三格放4元，如此下去，直到填满64个格子。结果，国王的钱很快就花完了。到了第十格时，已经放下512元，第十格以后，到了第二十格时，金额变成了524,288元。等到第六十四格时，金额早已达到万亿级别！换句话说，她每个月获得的是100%的回报率。
REINVESTMENT RISK - OR, WHAT YOU SEE YOU MAY NOT GET!
再投资风险 —— 或者，你未必能得到你看到的！
It is vital that after gaining the knowledge of how important compound interest is, investors understand how much their future retirement stakes are at risk. There are two types of reinvestment risk: interest rate risk and maturity risk. The first has to do with the risk of reinvesting interest payments at lower interest rates than existed at the time of investment, and the second means having too much of your portfolio invested in one maturity, exposing yourself to the possibility of having to reinvest most or all of your money at lower interest rates (or the opposite - owning too much of a long maturity bond and watching the value of your bond decline as yields rise with no money to reinvest).
在认识到复利的重要性之后，投资者必须了解他们未来退休资金面临的风险有多大。再投资风险主要有两种：利率风险和到期风险。第一种与以低于投资时的利率再投资利息支付的风险有关；第二种则指投资组合中过分集中于某一到期日，可能导致大部分甚至全部资金不得不以较低的利率再投资（或反过来——持有过多长期债券，在利率上升时看着债券价值下跌，却没有资金用于再投资）。
Let us again review the concept of yield to maturity of a bond. On a typical semi-annual bond, the quoted yield to maturity assumes that each and every interest payment is reinvested at that yield. They would have to remain constant at the purchase yield for the entire life of that bond! This is not likely in the real world, since interest rates are dynamic; yields are rising and falling all the time. So, this yield to maturity calculation is artificial. As we have seen, the longer the term of the bond, the more important the total interest received. For example, a 3.5 percent bond paying semi-annually and due in 29 years will produce 1.20 times in simple interest payments and 1.03 times in compound interest the original principal amount, assuming a constant reinvestment rate of 3.35 percent. The following table, repeated from above, underscores the importance of this reinvestment rate.
让我们再一次回顾债券到期收益率的概念。在一只典型的半年度付息债券上，所引用的到期收益率假设每一笔利息支付都以该收益率再投资。实际上，这种情况要求整个债券生命周期内所有支付的利率都与购买时相同！这在现实中是不可能的，因为利率是动态变化的，总是在不断上升和下降。所以这种到期收益率的计算是人为的。如我们所见，债券期限越长，累计收到的利息越重要。例如，一只票息3.5%、半年度付息、到期29年的债券，其单利支付总额将是本金的1.20倍，而复利部分则为1.03倍，前提是再投资利率保持在3.35%。下表（与前文重复）强调了再投资利率的重要性。
| Reinvestment Rate | 2% | 3.35% | 6% |
| Principal | $100,000 | $100,000 | $100,000 |
| Simple Interest | $120,640 | $120,460 | $120,460 |
| Compound Interest | $ 51,850 | $103,409 | $268,624 |
| Total | $272,490 | $324,049 | $489,264 |
Since a quoted yield to maturity on a bond is far removed from the real world, you might wonder why this yield calculation is done at all. It offers a consistent way to value all bonds and helps in valuing one bond in relation to another one. Investors are at greater risk than they realize unless they are invested in a product with no reinvestment risk. Yes, Virginia, there is such a product, and it gets its own chapter later on: the zero coupon or stripped bond. To combat this reinvestment risk, I advocate a laddered portfolio of stripped bonds in an RRSP account, as stripped bonds have no interest to reinvest. I advocate the laddered approach for taxable portfolios, since the staggered maturities of a ladder allow investors to avoid the reinvestment risk mentioned above. More on this later.
由于债券所报出的到期收益率与现实存在较大差距，你可能会疑惑为何还要进行这种收益率计算。它为评估所有债券提供了一种统一的方法，并有助于在比较债券价值时进行衡量。除非投资于没有再投资风险的产品，否则投资者面临的风险比他们想象的要大。是的，Virginia，确实有这种产品，而且它在后面的章节中会单独讨论：零息债券或分离式债券。为应对这种再投资风险，我主张在RRSP账户中采用梯形投资组合，因为分离式债券没有利息需要再投资。而对于应税账户，我也主张采用梯形投资，因为梯形结构中不同到期日的分散使得投资者能够避免上面提到的再投资风险。更多内容将在后文讨论。
The other form of reinvestment risk involves maturity selection. In day-to-day terms, planning your retirement with short-term investments is like timing a marathon runner after 100 yards and extrapolating the result over the ensuing 26 miles, 285 yards. The following high-tech chart outlines the direction of interest rates since the Second World War.
另一种再投资风险涉及到期选择。从日常角度看，以短期投资规划退休就像在马拉松比赛中只测量100码的成绩，然后将其外推至接下来的26英里285码。下图展示了自第二次世界大战以来利率变化的高科技趋势图：
While somewhat simplified, it underscores the reinvestment risk of maturity selection. No one knows where interest rates are going, so why take a chance? Since the peak in yields in 1982, anyone who has invested retirement funds in treasury bills and GICs has discovered the unpleasant experience of having a high-yielding investment mature with much lower yields available. Another way of addressing maturity reinvestment risk is to use the analogy of real estate. Suppose you owned a property with five units for rent. The property is your capital. Would you rent all your units for the same term? This would be imprudent as when all the leases came due at the same time, there could be a recession underway and your rents could fall and/or you might lose some tenants entirely. Thus, you would be wise to rent your units out for different terms. Also, you might want some diversity in your tenants; renting to several people in the same business runs the risk of an economic setback. Applying this approach to bond portfolios means that you should diversify by maturity so that your money does not mature at an inopportune time and you should also diversify by credit to spread your risk out.
虽然这幅图略显简化，但它突显了到期选择上的再投资风险。没有人能准确预测利率的走向，那为何要冒险呢？自1982年收益率见顶以来，任何将退休资金投资于国库券和定期存单（GICs）的人都经历过高收益投资到期后，却只能获得大幅下降的收益率的尴尬局面。另一种解决到期再投资风险的方法是用房地产类比。假设你拥有一处带有五套出租单元的房产，这处房产代表你的资本。你会将所有单元都签订相同期限的租约吗？这显然是不明智的，因为当所有租约同时到期时，很可能正值经济衰退期，租金会下降，或者你可能会失去部分租客。因此，你应该明智地将各单元的租期分散开来。同样，你可能希望租客多样化；如果全都租给同一类行业，当经济下行时风险就会更大。将这种方法应用到债券投资组合中，就意味着你应按到期日分散投资，以避免所有资金同时到期于不利时期，同时在信用风险上也需多样化以分散风险。
If only I had bought those twenty-year bonds yielding 16.3 percent! As you can see, five-year GIC rates have moved progressively lower. In 1982, you could have invested in a twenty-year Government of Canada bond at 16.33 percent. But, buying the five-year GICs and renewing for five years when they matured produced progressively lower returns.
“要是我当初买了那些票息16.3%的20年期债券就好了！”正如你所见，五年期GIC的收益率已经逐步下降。1982年，你可以投资于票息为16.33%的20年期加拿大政府债券，但购买五年期GIC并在到期后续期五年，其收益却逐步降低。
| Year | Five-year GICs | Long-term Canadas |
|---|---|---|
| 1982 | 16.14% | 16.33% |
| 1987 | 8.79% | 8.51% |
| 1992 | 7.92% | 8.92% |
| 1997 | 4.91% | 7.38% |
| 2002 | 3.93% | 5.51% |
| 2007 | 3.73% | 4.57% |
| Today | 3.00% | 3.35% |
For a typical RRSP plan, the previous table underscores the long-term effects of investing in short-term investments at a time of declining yields. I am writing this book in such a period. Obviously, in a period of generally rising rates, staying short is a winning process since yields will be higher as each bond matures. The purpose of this chapter is to stress the fact that, since even the great Templeton could not find anyone who can consistently forecast the trend in interest rates, it is important to find an approach that removes or significantly reduces reinvestment risk. It is the laddered approach, and it is fully discussed in Chapter 8.
对于典型的RRSP计划，上表突显了在收益率下降时期，将资金投资于短期产品的长期影响。我写这本书正处于这种时期。显然，在普遍利率上升的时期，保持短期投资是一种成功策略，因为随着每笔债券到期，收益率会相应提高。本章的目的是强调，即使连著名的Templeton也找不到任何能够始终如一预测利率走势的人，因此找到一种能消除或显著降低再投资风险的方法非常重要。这正是梯形结构投资法，其内容将在第八章中详细讨论。
Since the domestic bond market is built around semi-annual yield and the issuers (banks and trust companies) of deposit instruments (GICs, mainly) typically quote their yields in annual terms, it is important to know the "apples to apples" comparison, since the difference can be meaningful.
由于国内债券市场是围绕半年度收益率建立的，而发行存款工具（主要是定期存单）的机构（银行和信托公司）通常以年化收益率报价，因此了解“同类比较”非常重要，因为这种差异可能非常明显。
Consider a one-year investment of $100,000 offered at a yield of 5.0625 percent annually and compare this with another one-year investment paying 5 percent semi-annually. Which one would you buy? On the surface you would buy the one yielding 5.0625 percent, but assuming we invest $100,000 in each of them, let us see what the return:
试想一个100,000元的一年期投资，其年化收益率为5.0625%，再与另一项一年期投资，其半年度收益率为5%进行比较。你会选择哪一个？表面上看，你可能会选择收益率为5.0625%的那个，但假设我们各投资100,000元，我们来看看回报：
| Annual 5.0625 Percent | Semi-Annual 5 Percent | |
|---|---|---|
| Interest (6 months) | $0 | $2,500 |
| Interest (12 months) | $5,062.50 | $5,000 |
| Compound Interest | $0 | $62.50 (2,500*.025) |
| Total Interest | $5,062.50 | $5,062.50 |
Thus, we can say that 5 percent compounded semi-annually is equivalent to 5.0625 percent annually. Obviously, the higher the yield, the greater the difference:
因此，我们可以说，5%半年度复利等同于年化5.0625%。显然，收益率越高，这种差异就越大：
| Semi-annual | Annual Equivalent |
|---|---|
| 4 percent | 4.04 percent |
| 5 percent | 5.06 percent |
| 6 percent | 6.09 percent |
| 7 percent | 7.12 percent |
| 8 percent | 8.16 percent |
| 9 percent | 9.20 percent |
| 10 percent | 10.25 percent |
Investment advisors worth their salt will provide both yields so there can be no misunderstanding. Most investment dealers now include these yields on the contracts that their clients receive. Since this is not typically the case, and since investment products are seldom offered at a nice round yield, how can you work this out yourself? You could use a simple formula. To convert from semi-annual to annual, take the semiannual yield and square it, then divide by four to calculate the number of basis points to add to arrive at the annual yield equivalent. Taking 10 percent semi-annually and squaring it produces 100, and dividing by 4 produces 25 basis points, which when added to 10 percent produces a 10.25 percent annual yield equivalent.
有见识的投资顾问会同时提供两种收益率，确保信息不出现歧义。大多数投资经纪公司现在在合同中为客户列出这两种收益率。由于通常情况并非如此，而且投资产品很少以一个整齐的收益率提供，你如何自己计算呢？你可以采用一个简单的公式：要将半年度收益率转换为年化收益率，就先将半年度收益率平方，然后除以4，算出需增加的基点数，再加到半年度收益率上得到年化等效收益率。以10%的半年度收益率为例，10%的平方是100，除以4得25个基点，加上10%就得到了10.25%的年化收益率。
For a semi-annual yield such as 7.56 percent, follow the same process: 7.56 squared equals 57, then divide by 4 to equal 14.3 basis points, which added to 7.56 percent produces an annual yield equivalent of 7.703 percent.
对于7.56%的半年度收益率，按同样方法：7.56的平方约等于57，除以4约为14.3个基点，加上7.56%后得到的年化等效收益率约为7.703%。
YIELD
收益率
Yield is what an investment returns to an investor, whether expressed as dividends, capital gains, interest, or compound interest. As the compound section indicates, the rate at which money or capital grows is crucial to investment results. This brings us to the yield on bonds.
收益率是指投资给投资者带来的回报，无论这种回报以股息、资本利得、利息或复利的形式表现出来。正如复利部分所指出的，资金或资本的增长速度对于投资结果至关重要。这就引出了债券收益率的问题。
Investing in a bond paying 5 percent produces 5 percent a year in interest payments, whether payable semi-annually or not. If that interest is used or spent, a simple yield of 5 percent (pre-tax) is realized. When the interest is reinvested (say in an estate, an RRSP, etc.), then this yield becomes a different matter. When you buy a fixed-income security that offers a 5 percent yield to maturity, it implies that every interest payment will be reinvested or compounded at 5 percent to produce that yield. This is not a real-world calculation. It is merely a benchmark number to measure different bond yields: you may calculate it by buying appropriate software or downloading it from your FI. Or you could get it free from my website. The formula for converting the price of a bond to its yield to maturity is a long, iterative formula best left to calculators. Calculating the price of a bond given its yield is far easier, as it involves adding up all the present values as we have seen.
投资一只票息为5%的债券，每年会获得5%的利息支付，无论是半年度支付还是其他方式。如果这些利息被使用或花掉，就实现了5%的简单收益率（税前）。然而，如果这些利息被再投资（例如投资到遗产、RRSP账户等），收益率的情况就完全不同。当你购买一只固定收益证券，其到期收益率为5%时，这意味着每一笔利息支付都会以5%的利率再投资或复利，从而实现该收益率。这并不是现实中的计算方式，而只是衡量不同债券收益率的一个基准数字：你可以通过购买相应软件或从你的金融机构下载，甚至从我的网站上免费获取。将债券价格转换为其到期收益率的公式相当冗长且需反复迭代，最好交给计算器来计算。相对于此，给定收益率计算债券价格要简单得多，因为只需要将所有现值加总即可，如前所示。
DURATION, OR WHAT YOU SEE IS NOT WHAT YOU HAVE
久期 —— 你看到的不一定就是你拥有的
A ten-year bond is a ten-year bond, right? Sorry! The term or maturity of the bond may be ten years but it would only have a duration of ten years if that were the only payment you received. Returning to the Canada 3.5 percent due June 1, 2020, we observe that there are eighteen interest payments spaced six months apart plus the return of the principal. These payments are a part of your bond, and since they range in term from six months to nine and a half years, they have an impact on the average life or average term of your investment. Yields and bond prices move inversely. Perhaps I should take a minute and explain that last sentence.
十年期债券就等于十年期债券，对吧？不好意思！债券的期限或到期日可能是十年，但如果你只收到这一笔付款，它的久期才会是十年。回到那只到2020年6月1日到期、票息为3.5%的加拿大债券，我们注意到它有18次六个月间隔的利息支付，加上本金的偿还。这些支付都是债券的一部分，由于这些支付的到期期限从六个月到九年半不等，它们会影响你投资的平均寿命或平均期限。正如我们已经知道的，收益率与债券价格呈反向关系。也许我应该花点时间解释一下这句话的意思。
Not a week goes by without someone asking me if bond prices go up when yields or interest rates go up. After working through the present value calculations, you can see that if interest rates rise, then the present value of the pieces of the bond will be worth less and therefore the total value of the bond will fall. The converse is equally true. If yields fall, the present value of all the parts of a bond rise and the total value of the bond rises.
几乎每周都会有人问我，当收益率或利率上升时债券价格是否会上涨。经过现值计算后，你可以看出，如果利率上升，债券各组成部分的现值就会降低，因此债券的总价值会下跌；反之亦然，如果收益率下降，债券各部分的现值会上升，总价值也会随之上升。
The same applies for present values of future payments: the higher the discount rate or yield, the lower the present value (and vice versa). Let us take the December 1, 2015, interest payment for example. At 2.883 percent, as we have seen, this payment has a present value of $1,538.50. Suppose we raise the discount rate to 8 percent. The present value now works out to be $1,229.38. Lowering the discount rate to 2 percent produces a higher present value of $1,600.03. Thinking about it logically, and reversing the formula, at a higher yield, you need less capital today to arrive at a specified future value (and of course the converse is true for a lower yield). Thus, to produce $1,750 in four and a half years at 2 percent you need $1,600.03 today, but if you could invest at 8 percent you would only need $1,229.38.
未来支付的现值同样遵循这一规律：贴现率或收益率越高，现值越低（反之亦然）。例如，以2015年12月1日的利息支付为例。如前所述，在2.883%的情况下，该笔支付的现值为1,538.50元。假设我们将贴现率提高到8%，现值计算得出为1,229.38元；若将贴现率降低至2%，则现值为1,600.03元。逻辑上思考并反推公式，高收益率时，你今天所需的资本更少以达成未来特定金额（反之，低收益率则需要更多资本）。因此，要在四年半后产生1,750元的支付，若贴现率为2%，你今天需准备1,600.03元；如果能以8%的收益率投资，则只需要1,229.38元即可。
Each of the payments of a bond has a present value, and taking their relative weights into account along with the present value of the principal payment produces an average life or term of your bond (the average time it takes to receive all the cash flows or payments). This calculation is referred to as the duration of the bond. In our example, the average duration works out to be 7.72 years at a yield of 2.883 percent. Duration is not static: as time passes, the terms of all the payments become shorter and so does the duration. As well, changes in market yields will affect the present value of the payment stream and thus affect the duration. Let us assume we have a 6 percent bond and a 3 percent bond, both with a term to maturity or principal repayment of ten years. Which one has the shorter duration? The present value of the interest payments from the 6 percent bond will have greater value than those from a 3 percent bond at the same yield. Using a face value of $100,000, a starting yield or discount rate of 2.883 percent, and the interest payment at four and a half years produces the following:
债券的每一次支付都有其现值，将这些支付的相对权重与本金的现值结合起来，会得出债券的平均寿命或称久期（即获得所有现金流的平均时间）。这一计算称为债券的久期。在我们的例子中，久期在2.883%的收益率下约为7.72年。久期不是一成不变的：随着时间推移，每笔支付的剩余期限都会缩短，久期也会相应缩短。同样，市场收益率的变化也会影响支付序列的现值，从而影响久期。假设我们有一只票息为6%的债券和一只票息为3%的债券，它们的到期日（本金偿还时间）均为十年。哪一只债券的久期更短？6%债券的利息支付现值会比3%债券在相同收益率下更高。以100,000元票面价值、2.883%的起始收益率计算，四年半处的利息支付如下：
| Interest Payment | Present Value |
|---|---|
| 6 percent bond: | $100,000 * 0.06/2 = $3,000 at 2.883 percent = $2,637.42 |
| 3 percent bond: | $100,000 * 0.03/2 = $1,500 at 2.883 percent = $1,318.70 |
The interest payments from the 6 percent bond have greater present value than those from the 3 percent bond, and so the 6 percent bond will have a shorter duration. In this case the 6 percent bond has a duration of 7.85 years, versus 8.88 years for the 3 percent bond. Another way of looking at this is to look at the present value of the two payments above. Each of the interest payments for the 6 percent bond will have a greater present value than will the same maturity interest payment for the 3 percent bond. Adding all the present values up will show that the weight of the interest payments is greater compared with the principal payment for the higher coupon bond, and this will produce a shorter average term or duration.
因此，6%债券的利息支付现值高于3%债券的对应支付，6%债券的久期较短。在本例中，6%债券的久期为7.85年，而3%债券的久期为8.88年。另一种看法是直接比较上述两笔支付的现值。6%债券的每笔利息支付现值都高于3%债券相同期限的支付。将所有现值加总后，较高票息债券中利息支付的权重相对本金支付更大，这将导致其平均期限（久期）较短。
The reason for stressing this calculation becomes evident when selecting bonds. Duration is used as a measurement of risk. The longer the duration of a bond, the larger the price change for a given yield change. Thus if you believe that interest rates are trending lower, you would invest in bonds of long duration. IAs worth their salt will know what duration is and which bonds are appropriate for you. A shorter duration bond is less sensitive to yield changes.
强调这种计算的意义在于选择债券时，久期可以作为衡量风险的指标。债券的久期越长，在相同收益率变化的情况下，其价格波动就越大。因此，如果你认为利率正趋于下降，你会倾向于投资于久期较长的债券。优秀的投资顾问会了解久期的含义，并知道哪些债券适合你的情况。久期较短的债券对收益率变化的敏感性较低。
The following illustrates how duration affects the price change of bonds of different maturities and durations. What I have done is take bonds of different coupon rates at two different maturity dates (five and twenty years) and measured the impact of a 1 percent (100 basis points) shift in the yield from 6 percent to 5 percent. Within the two maturities, the durations change depending on the coupon rate.
下表说明了不同到期日及不同久期的债券在收益率从6%降至5%（即下降100个基点）时价格变化的影响。我选取了不同票息的债券，在五年期和二十年期两个到期日下进行比较，观察在收益率变化1%时价格的变化。
| Bond Type | Coupon | Duration (Years) | % Increase in Price |
|---|---|---|---|
| Five-year Bonds | 12% | 3.9 | 4.00 |
| 12% | 3.9 | 4.00 | |
| 8% | 4.2 | 4.20 | |
| 6% | 4.4 | 4.40 | |
| 0% | 5.0 | 5.00 | |
| Twenty-year Bonds | 12% | 10.4 | 11.80 |
| 12% | 10.4 | 11.80 | |
| 8% | 11.2 | 11.80 | |
| 6% | 12.7 | 12.60 | |
| 0% | 20.0 | 21.50 |
You can see the bigger price changes for lower coupon bonds of the same maturities. I included a zero coupon bond at each maturity to show that without any cash flows, these bonds have a duration equal to their maturity and have the greatest price change.
你可以看到，相同到期日的低票息债券价格变动更大。我还在每个到期日中都加入了零息债券，以表明没有任何现金流的债券其久期等于其到期日，因此价格变化最大。
BOND PRICE DYNAMICS AND VOLATILITY
债券价格动态与波动性
Why do bond prices change? After all, there is a fixed level of interest payments and a final repayment of principal. So, what is the point of reporting price changes for bonds daily? The daily trading volume for bonds is enormous, averaging $38.4 billion, or some five times that of the stock market. Various participants in the bond market have reasons to trade their fixed-income investments. The price of money changes frequently, as anyone who borrows or lends money knows. The reasons for this are numerous, including the changing supply and demand for money as the economic cycle changes, changes in the rate of inflation, government deficits, political turmoil, foreign exchange crises, or changes in interest rates in other countries. As well, someone investing in a bond may have circumstances change and need to sell it before maturity. Investors' expectations to inflation or other factors may also change. As well, a large pool of speculative capital trades the bond market aggressively, seeking capital gains. Time is also a factor. Consider someone who buys a ten-year bond. Five years later, that investor has a five-year bond, and new ten-year bonds may be offering a much higher yield, prompting the investor to consider exchanging the five-year bond for the ten-year.
债券价格为何会变化？毕竟，债券有固定水平的利息支付和最终的本金偿还。那么，为什么每天都要报道债券价格的变化？债券的日均交易量非常巨大，平均达到384亿元，大约是股票市场的五倍。债券市场的各类参与者都有理由交易他们的固定收益投资。正如每个借贷者都知道的那样，货币的价格会经常变化。这种变化的原因很多，包括经济周期中货币供求的变化、通胀率的变化、政府赤字、政治动荡、外汇危机或其他国家利率的变化。此外，投资债券的某人可能会因个人情况变化而需要在到期前出售其债券。投资者对通胀或其他因素的预期也可能发生变化。与此同时，有大量投机性资本积极参与债券市场，追求资本利得。时间也是一个因素。试想某人购买了一只十年期债券。五年后，该投资者手中的债券就变成了一只五年期债券，而新发行的十年期债券可能提供更高的收益率，这促使投资者考虑将五年期债券换成十年期债券。
There are a whole slew of institutional portfolio managers who are constantly analyzing the market for inefficiencies and arbitrage opportunities. They will examine relationships among thousands of bonds. For example, they may decide that the spread between ten-year Canadas and Ontarios has become too wide based on historical analysis and their own projections and thus they would sell the Canadas and buy the Ontarios, with the aim of reversing the trade if and when the yield spread narrows. Constant analysis by portfolio managers everywhere leads to an efficient market most of the time, but as with all markets, surprises occur. Astute analysts stand to benefit from knowing when there is an anomaly between two different bonds or sectors.
有无数机构投资组合经理一直在分析市场中存在的低效定价和套利机会。他们会研究成千上万只债券之间的关系。例如，他们可能会认为基于历史分析和自己的预测，十年期加拿大债券与安大略省债券之间的息差过大，因此他们会卖出加拿大债券并买入安大略省债券，并计划在息差缩小时逆转交易。各地投资组合经理的不断分析大部分时间使市场趋于高效，但和所有市场一样，总会有意外发生。精明的分析师能从发现两种不同债券或不同领域之间的异常定价中获利。
HOW DO BOND PRICES CHANGE AND HOW VOLATILE ARE THEY?
债券价格如何变化及其波动性有多大？
The following are the basic rules of bond price volatility. Bond price volatility increases as:
以下是债券价格波动性的基本规则。债券价格的波动性随着以下情况增加：
The maturity lengthens. Typically, as maturity lengthens, duration does too, and we already know that the longer the duration of a bond, the more its price will change for a given yield movement.
债券期限越长。通常，期限越长，久期也越长，而我们已经知道，债券久期越长，在相同收益率波动下价格变动越大。
The coupon rate declines. Again referring back to duration and assuming two bonds of identical maturities, one with a 10 percent coupon and the other with one of 5 percent, the latter one has a longer duration and therefore more volatility since the present value of its interest payment stream is not as great as that of the 10 percent bond.
票息率降低。同样，假设两只债券具有相同到期时间，一只票息为10%，另一只为5%，票息较低的债券其久期较长，因此波动性更高，因为其利息支付流的现值不如票息高的那只大。
The starting yield increases. Let us ensure that we are all familiar with the basic tenet of bond prices and yields: Bond prices and yields move in an inverse pattern; that is, as interest rates or yields rise, the price of bonds falls (and vice versa). The longer the maturity, the greater the price movement for a particular bond.
初始收益率越高。让我们明确债券价格与收益率之间的基本关系：债券价格与收益率呈反比，即当利率或收益率上升时，债券价格会下降（反之亦然）。期限越长，给定债券的价格变动就越大。
Imagine yourself on a schoolyard teeter-totter. One end of the teeter-totter represents bond prices, the other end, yields. If someone larger than you jumps on the yield end, it will go down and you will go up to the highest level (price) that the teeter-totter will go. That is what happens to long-term bonds. If you are a little chicken and sit closer to the centre of the board and the same large bully jumps on, what happens? You do not go as far.
试想你在学校操场上玩跷跷板。跷跷板一端代表债券价格，另一端代表收益率。如果有个比你体型更大的人跳到收益率一侧，该侧便会下沉，而你所在的债券价格那一端便会上升到跷跷板能达到的最高点——这就是长期债券会发生的情况。如果你胆小一些，坐得离跷跷板中心更近，那么当同一个大个子跳上去时，你就不会有太大幅度的升高。
The following table illustrates this teeter-totter effect.
下表说明了这种跷跷板效应：
| Yield Decrease | Price Increase* | |||
|---|---|---|---|---|
| 3-year | 5-year | 10-year | 30-year | |
| -0.50 | 1.40% | 2.30% | 4.20% | 7.10% |
| -1.00 | 2.80% | 4.60% | 8.60% | 15.00% |
| -1.50 | 4.30% | 7.00% | 13.20% | 23.50% |
| -2.00 | 5.80% | 9.50% | 18.00% | 32.80% |
| * Assuming a coupon of 4% and a startinig yield of 4% | ||||
THE YIELD CURVE: THE SHAPE OF THINGS TO COME
收益率曲线：未来走势的形态
The yield curve compares time and yield by connecting a series of dots indicating the yield at different maturities. Practical-minded investors would expect that the longer the term they are willing to lend or invest their money, the higher the return or yield they should receive. The higher yield compensates the investor for the extra risk in lending for a longer period of time. Translating that into a graph would produce the following:
收益率曲线通过连接一系列表示不同到期期限收益率的点来比较时间与收益率。务实的投资者会认为，他们愿意贷款或投资的期限越长，应该获得的回报或收益率也越高。更高的收益率补偿了投资者因更长贷款期限而承受的额外风险。用图形表示这一关系会得到如下图形：
Of course, this curve is not constant; it may be steeper, that is, the upward slope may be more vertical, with long-term yields sharply higher than short-term yields. It could also be flatter, with only gradual increases in yield along the curve. (Note: All the curves in this chapter represent yields available in benchmark Government of Canada obligations, the reference points for all other securities.) The yield curve assumes different shapes at different times in the economic cycle. It can be flat, with yields identical at all maturities:
当然，这条曲线并非固定不变；它可能更陡峭，即上升角度更大，长期收益率明显高于短期收益率；也可能更平缓，收益率沿曲线仅有逐步增加。（注意：本章中所有曲线均代表以加拿大政府基准债券为标准的收益率，这些债券是其他所有证券的参考基准。）收益率曲线在不同经济周期中呈现不同形状。它可以是一条平直的曲线，各到期期限的收益率完全相同：
Or it can be inverted, with the shortest maturities offering the highest yields and the longest maturities the least:
或者，它可以是倒挂的，即最短期限的收益率最高，而最长期限的收益率最低：
You can draw your own. On a piece of paper, draw a vertical line and a horizontal line. Using the daily newspaper or your favourite website, find the yields at the shortest maturity for Government of Canada instruments (say, three months) and, using the horizontal axis for time and the vertical for yield, place a dot where each yield belongs. You could use maturities such as six months, one year, two years, etc., all the way to thirty years. Then connect all the dots to view your curve.
你可以自己绘制。取一张纸，画一条垂直线和一条水平线。利用每天的报纸或你最喜爱的网站，找出加拿大政府证券中最短期限（例如三个月）的收益率，然后将各收益率标记在水平轴（时间）和垂直轴（收益率）的适当位置。你可以选择如6个月、1年、2年，直至30年等不同到期期限。然后将所有点连起来，就能看到你的收益率曲线。
Why dwell on this? What do these shapes mean? How do you get from one shape to another? What's in it for me? Dwelling on this increases your awareness of where Canada (in this case) is in its economic cycle. The shapes and the changes in the shapes assist in this analysis. When the yield curve moves from very steep to less steep, it is a sign that the Bank of Canada (or the market participants) believes that it is an appropriate time to be tightening monetary policy, which is a fancy way of saying that the economy needs to be cooled down before inflation gets out of hand. Some observers liken this approach by central banks to taking away the punch bowl when the party gets going. What is in it for you, then, is that you will have an important gauge at your disposal to be used when contemplating decisions such as whether to stay floating with your mortgage or fix it and whether to buy long-term or short-term bonds. To acquire a good working knowledge of the importance of yield curves is to understand interest cycles, to get along without an economist, and to enhance long-term returns. I really mean what I say about not needing an economist. Economists have a dismal track record in predicting economic turning points; they are about equal to your guesses and mine! Many prominent fixed-income fund managers have also admitted that they basically are guessing which way interest rates are going!
为什么要关注这个问题？这些形状意味着什么？如何由一种形状转变为另一种？这对我有什么好处？关注这些可以提高你对加拿大（在本例中）所处经济周期的认识。曲线的形态及其变化有助于进行这种分析。当收益率曲线从非常陡峭转为较缓时，就说明加拿大央行（或市场参与者）认为现在是收紧货币政策的适当时机，也就是说，为了防止通胀失控，经济需要降温。有些观察家将中央银行的这种做法比喻为当派对进行得正酣时撤走酒水碗。对你而言，这意味着你将拥有一个重要的指标，在思考诸如是否浮动按揭或者锁定按揭、以及买长期还是短期债券等决策时可供参考。掌握收益率曲线的重要性，有助于理解利率周期，能在没有经济学家的情况下做出决策，并提高长期收益。我说“无需依赖经济学家”绝非空口说白话——经济学家在预测经济转折点方面的记录并不理想，他们的预测水平大致和你我猜测的一样！许多著名的固定收益基金经理也承认，他们基本上都是在猜测利率会怎样变化！
The yield curve is dynamic, with the yield relationship between one maturity and another in a constant state of flux. Why is this? For one thing, changes in the shape of the yield curve represent changing inflationary and economic expectations of market participants. At one end of the curve (the very short maturity end) is our central bank, the Bank of Canada, whose duties involve the maintenance of a stable currency, the conduct of monetary policy compatible with economic growth, and a vigilant, anti-inflationary stance. The Bank of Canada may deem it necessary to tighten monetary policy, restricting the growth of money supply and raising short-term interest rates. It can thus be responsible for pushing up treasury bill yields and the prime rate, but can it directly affect long-term bond yields and mortgage yields? The short answer is no. Beyond the shortest of maturity dates, the Bank of Canada cannot dictate where interest rates go. This is because market participants of all types, borrowers and lenders alike, combine to set market rates. For example, when the Bank of Canada begins to raise short-term interest rates, market participants may view this as anti-inflationary and thus they may feel more comfortable investing in longer term securities, whose yields then fall. This is one way that the yield curve flattens (short-term yields rise even as long-term yields fall). This happened in the first half of 2007.
收益率曲线是动态的，不同到期期限之间的收益率关系始终处于不断变化之中。这是为什么呢？首先，收益率曲线形态的变化反映了市场参与者对通胀和经济前景预期的变化。在曲线的一端（非常短期的一侧）是我们的中央银行——加拿大银行，其职责包括维护货币稳定、实施与经济增长相适应的货币政策，并保持反通胀的警惕性。加拿大银行可能认为有必要收紧货币政策，从而限制货币供应增长并提高短期利率。它因此可能推动国库券收益率和基本利率上升，但它能否直接影响长期债券和抵押贷款的收益率呢？答案很短：不能。超过最短期限后，加拿大银行无法决定利率走向。这是因为各种类型的市场参与者——借款人和贷方——共同决定了市场利率。例如，当加拿大银行开始上调短期利率时，市场参与者可能会将此视为抗通胀信号，从而更愿意投资于长期证券，导致其收益率下降。这就是收益率曲线变平的一种方式（短期收益率上升而长期收益率下降）。这种情况发生在2007年上半年。
At the outset there are a few theorems to advance. Almost always invest in the longest-term investments (for your particular circumstances) when it costs you the most in current income to do so. For example, ten-year bonds may be yielding 6 percent while ninety-day treasury bills yield 8 percent. This inversion of yields normally occurs at a time of maximum monetary tightness, close to interest rate peaks. When rates begin to fall, superior total returns are obtained from investing in long-term securities even though treasury bills appear to yield more. The converse of this is equally true. An investor should invest in the shortest maturities when it again costs the most in current income to do so, that is, when the yield curve is in an extremely steep configuration. In this case, ninety-day treasury bills may be 2 percent and long-term yields 6 percent. This typically occurs at a time of maximum monetary ease by central banks, when the economy is expanding rapidly and is beginning to generate inflationary pressures and when credit demand is strong. At times like this, the next major event is monetary tightening when the central bank begins to ratchet interest rates higher. Interest rates begin to rise at all maturities, so even though current income is higher in long-term investments, total returns will be worse, Your capital is at risk. In other words, capital preservation should become more important at cyclical extremes than extra current income.
首先提出几个基本原理。当你需要牺牲较多当前收益以获得更高未来回报时，应选择符合你个人情况的最长投资期限。例如，当十年期债券的收益率为6%，而90天国库券的收益率却高达8%时，就出现了收益率倒挂。这种现象通常出现在货币政策极度紧缩、利率接近峰值的时期。随着利率开始下行，长期证券往往能获得更高的总回报，即使短期产品（如国库券）表面上的收益更高。反之，当收益率曲线变得极其陡峭，也就是当前收益相对较高时，投资者应选择最短期限的产品。比如，此时90天国库券的收益率可能仅为2%，而长期证券可能达到6%。这种状况通常发生在中央银行实施极度宽松政策时，此时经济迅速扩张，开始面临通胀压力且信贷需求旺盛。随后，中央银行往往会逐步上调利率以收紧货币政策，导致各期限利率普遍上升。这样一来，尽管长期投资的当前收益较高，但总回报反而可能更低，资本也面临风险。换句话说，在经济周期的极端阶段，与追求额外当前收益相比，保护资本安全更为重要。
One of the principal driving forces of all markets is greed. By always being greedy and reaching for higher apparent yields, investors in fixed-income products may find only short-term benefits. For instance, in the first case above, once the monetary tightening ends and inflation subsides, short-term and long-term rates will both fall. When the ninety-day treasury bill matures, ninety-day bills might then be only 4 percent while ten-year bonds may have fallen to 5 percent. Now the investor must decide whether to take the lower yield on the bills or invest in the ten-year bond at 5 percent that they could have bought at 6 percent. An opportunity lost. The converse is typically true in the other example. After a long period of monetary ease, the Bank of Canada decides to tighten monetary policy and yields rise, so that investors reaching for higher yield at the longer maturities see their principal value decline while short-term yields begin to rise.
市场的主要推动力之一是贪婪。总是贪婪地追求表面上更高的收益率，固定收益产品的投资者可能只会获得短期利益。比如，在前述例子中，一旦货币紧缩结束且通胀消退，短期和长期利率都会下降。当90天国库券到期时，90天国库券的利率可能仅有4%，而十年期债券可能已降至5%。此时，投资者必须决定是接受较低的国库券收益率，还是以5%的收益率投资于原本能以6%购买的十年期债券，结果就是错失了机会。另一种情况通常也是如此。在经历了长期宽松之后，加拿大银行开始收紧货币政策，收益率上升，于是那些盲目追求长期更高收益的投资者看到他们的本金价值下降，而短期收益率却开始上升。
In a coming chapter, I discuss the laddered approach to fixed-income investing, as it takes the guesswork out of the equation. Suffice it to say for now that it spreads your fixed-income investments over a variety of maturity dates. Until everyone is converted to ladders, it is important to pay attention to both the shape of the yield curve and how it is shifting. It is educational, and it demonstrates that greed (seeking higher yield) may be a false economy if in reaching for that apparent higher yield, investors are putting their principal at risk. In other words, buying a three-month treasury bill because it offers a higher yield than does a ten-year bond can be a very short-sighted decision. A similar analogy is the GIC investor who never considers anything else beyond five years when, in fact, there are times when significant gains in total returns are possible through purchasing bonds of ten- and twenty-year maturities.
在后续章节中，我将讨论梯形投资法，这种方法能消除很多主观猜测。暂且可以说，它将你的固定收益投资分散到不同到期期限上。直到大家都采用梯形投资法之前，关注收益率曲线的形态及其变化是非常重要的。这不仅具有教育意义，也证明了贪图高收益可能是一种虚假的节约，因为追求表面上更高的收益率可能会使投资者将本金置于风险之中。换句话说，购买一只三个月期国库券仅仅因为它的收益率高于一只十年期债券，可能是一种非常短视的决策。类似的类比还有那些只看五年期限的定期存单投资者，事实上，通过购买十年或二十年期债券，有时可以获得显著更高的总回报。
In the practical advice section, I mention that a large percentage of investments become concentrated in the one-year and five-year terms. This has a lot to do with the whole GIC business, where one- and five-year terms are most common and so that is where the bulk of term matching takes place, as institutions constantly attempt to keep their assets (loans, mortgages) and liabilities (deposits, GICs) matched as to maturity. At these specific maturities, yield spreads between different issuers (Canadas, treasury bills, GICs, strips, and corporates) tend to become compressed. What occurs — and what therefore becomes an opportunity to astute advisors and investors — are yield anomalies. For example, eighteen-month investments typically offer an above average yield pickup over one-year investments. As a result, those investors willing to stick their necks out for an extra six months will be rewarded when their investment becomes a one-year investment and the yield spreads compress for the reasons advanced above. They may then sell and buy another eighteen-month investment or hold to maturity. Either way, there is a relative yield pickup. A similar anomaly often exists at five years, when a one-year extension to six years may offer a higher yield than the slope of the yield curve would suggest. In other words, the investor is rewarded with a higher yield than expected for that extra year. One year hence, the investor will have a five-year investment trading at a far narrower spread than at the time of the purchase. As with the one-year/ eighteen-month anomaly, the investor may sell and reinvest in another six-year maturity or hold the investment and enjoy the extra yield.
在实际建议部分，我提到大部分投资往往集中在一年期和五年期产品上。这在很大程度上与定期存单市场有关，因为一年和五年期产品最为常见，而资产负债匹配主要也是围绕这两个期限展开的，因为机构总是试图使其资产（贷款、按揭）和负债（存款、GICs）的到期日相匹配。在这些特定的到期日上，不同发行者之间的收益率差距（包括加拿大政府债券、国库券、定期存单、分离式债券和公司债）往往会变得非常狭窄。由此产生的——也是对精明顾问和投资者的一个机会——就是收益率异常。举例来说，18个月期限的投资通常会比一年期投资提供更高的收益率补偿。因此，那些敢于冒险延长6个月投资期的投资者，在其投资转变为一年期时，收益率差距压缩会带来相对的收益率提升。他们可以选择出售并买入另一只18个月期投资，或者持有至到期。无论哪种方式，都能获得相对的收益率提升。类似的异常情况常出现在五年期上，当将到期延长一年（从五年延长到六年）时，所获得的收益率可能比收益率曲线所暗示的更高。换句话说，投资者为额外的一年得到了比预期更高的收益率。一年后，投资者手中的五年期产品可能会以比购买时明显更窄的息差交易。与一年期/18个月期的异常情况类似，投资者可以选择出售并重新投资另一只六年期产品，或持有该产品享受额外收益。
Within the money market itself, the money market yield curve (twelve months and less) offers clues all the time about the near-term direction of short-term interest rates, a subject everyone cares about. The Bank of Canada directly influences short-term interest rates through its actions in the overnight market and the three-month treasury bill market. A little more on the overnight rate: if the Bank of Canada "tightens the system," making it more expensive for jobbers and other market participants to finance their inventories, perhaps to the point where the cost of financing exceeds the yield of the securities, interest rates tend to rise as dealers attempt to shed their inventories since they would lose money by holding them. Everyone attempting this at once is like everyone trying to get out the door at the same time. The Bank of Canada can push rates lower or keep rates low by making it possible to carry inventories at a profit, of course. Past the three-month maturity range, its influence wanes as market forces of supply and demand take over.
在货币市场内部，货币市场收益率曲线（12个月及以内）时常为短期利率的近期走势提供线索，这是大家都关心的问题。加拿大银行通过隔夜市场和三个月国库券市场直接影响短期利率。关于隔夜利率略谈：如果加拿大银行“收紧系统”，使得做市商或其他市场参与者为融资持仓支付更高成本，甚至高于证券收益率，则利率往往会上升，因为交易商会试图尽快抛售持仓以避免亏损。大家同时试图逃离就像一群人同时冲出门外。当然，加拿大银行也可以通过使持仓在盈利情况下继续持有来压低或保持利率在低位。超过三个月期限后，银行的影响力则会因市场供需力量的作用而衰减。
The money market yield curve changes frequently. Remembering the general yield curve rules, when short-term money market yields (less than three months) are a lot lower than six- and twelve-month rates (a steep curve), avoid buying the long maturities, since the curve is telling you that short-term rates are too low and/or the cycle for rates is over and rates are headed higher. Frequently, the money market yield curve is flat or inverted. This is a signal to extend term: the cycle is about to turn to lower rates, and the central bank has had to push short-term rates up sharply, typically for currency defence, not for economic reasons. Since the economic fundamentals in such cases argue for lower rates, market participants look past the near term and buy six- or twelve-month bills, knowing that once the currency stabilizes, rates will move lower again, and greater returns are to be had in the longer maturities. The accompanying chart illustrates the importance of ignoring the higher yields available when a yield curve is inverted
货币市场收益率曲线频繁变化。记住一般的收益率曲线规则，当短期货币市场收益率（少于三个月）远低于六个月和十二个月的利率（形成陡峭曲线）时，应避免购买长期产品，因为这条曲线表明短期利率过低或利率周期已结束，而利率正朝更高方向发展。通常，货币市场收益率曲线会呈现平坦或倒挂状态。这时正是延长投资期限的信号：利率周期即将转向下降，中央银行不得不大幅上调短期利率（通常是为了维护货币而非经济原因）。由于此类情况下经济基本面暗示利率会下降，市场参与者会抛开短期因素，购买六个月或十二个月的票据，因为他们知道一旦货币稳定下来，利率会再次下行，而长期产品会带来更高收益。附图说明了在收益率曲线倒挂时忽略较高收益率的重要性。
On November 19, 2007, this spread was minus 40 basis points with the two-year yield at 3.64 percent and the one-year rate at 4.04 percent. One year later, the two-year yield had fallen to 1.92 percent while the one-year was at 1.87 percent for a spread of plus five basis points. For this period, the one-year earned the 4.04 percent while the two-year returned 5.44 percent. When the crunch occurred and Lehman Brothers failed, the Bank of Canada slashed the Bank Rate to 0.25 percent and the spread between the two-year and one-year soared to 107 basis points by October 2009, On November 19, 2009, the two-year yield was 1.27 percent with the one-year yield at 0,50 percent. For this one year period, the one-year treasury bill returned 1.87 percent while the two-year bond returned 2.59 percent.
2007年11月19日，二年期收益率为3.64%，一年期收益率为4.04%，两者差值为负40个基点。一年后，二年期收益率降至1.92%，而一年期收益率为1.87%，差值变为正5个基点。在这一期间，一年期票据获得了4.04%的回报，而二年期债券回报为5.44%。当危机发生、雷曼兄弟倒闭后，加拿大银行将银行利率大幅下调至0.25%，而二年期和一年期之间的息差在2009年10月猛增至107个基点；2009年11月19日，二年期收益率为1.27%，一年期收益率为0.50%。在这一年期间，一年期国库券回报为1.87%，二年期债券回报为2.59%。
Thus, buying the higher yielding one-year bill resulted in a lost opportunity:
因此，购买收益更高的一年期票据导致了机会的流失：
Investors who are well-informed can take advantage of certain anomalies that can lead to enhanced returns. These anomalies may occur if there is a very large seller of a bond of a particular maturity or if there is huge demand for another maturity. The net effect is to make a bond of one maturity less or more expensive than normally would be the case. Other factors that can produce such anomalies are a sudden surge in new-issue financings or large purchases by foreign investors. These kinds of opportunities occur at various points on the yield curve. They may alert investors that their original selected maturity date turns out to be the "expensive" part of the curve with greater yield available in a neighbouring maturity. Again, investors may return to the original maturity once the anomaly vanishes.
信息灵通的投资者能够利用某些异常现象来获取更高回报。这些异常可能发生在某一到期的债券出现大卖盘，或者另一到期的债券需求极大时。其总体效果是使某个到期日的债券定价比通常情况更低或更高。其他可能导致这种异常的因素还有新发债券融资的突然激增或外国投资者的大量买入。这类机会会在收益率曲线的不同位置出现。它们可能提醒投资者，他们原先选择的到期日实际上是收益率曲线中“昂贵”的部分，而邻近到期日却有更高的收益率。一旦这种异常现象消失，投资者可以重新回到原先选择的到期日。
SUMMARY
总结
The yield curve is a better forecaster of yields than any economist. For example, consider that one-year treasury bills yield 2 percent and two-year Canadas yield 3 percent. If you buy the two-year at 3 percent and then consider selling it after one year, what must the one-year rate be so that you would earn 3 percent? If the yield curve was unchanged, you would have earned 3 percent for the first year plus the 1 percent gain from the fact that the one-year rate was 2 percent. With the yield curve unchanged, you could buy the two-year again and earn another 4 percent. This shifting maturity strategy is a proven strategy. The only potential drags on performance are transaction costs (so make sure you are getting competitive offerings) and the price risk of longer term securities. In addition, another rule of thumb used by some portfolio managers is to extend term when you are able to pick up an extra 20 basis points per annum.
总的来说，收益率曲线比任何经济学家都更能预测收益率。例如，假设一年期国库券收益率为2%，而两年期加拿大债券收益率为3%。如果你以3%的价格购买了两年期债券，然后在一年后考虑出售，那么为了使你的回报达到3%，一年期的收益率必须是多少？如果收益率曲线保持不变，你在第一年就获得了3%的收益，加上一年期票据由2%涨至3%所带来的1%的增幅。若收益率曲线不变，你可以买入另一只两年期债券再获得4%的回报。这种到期转变策略已被证明是有效的。对表现的唯一潜在拖累是交易成本（所以务必确保你获得竞争力的报价）以及长期证券的价格风险。此外，一些投资组合经理还遵循另一个经验法则：如果每年能额外获得20个基点，便应延长到期时间。
It seems to me that this approach can enhance returns and it makes the maintenance of a yield curve very important. I encourage you to keep Your own yield curve up-to-date and watch for opportunities.
在我看来，这种方法可以提高回报，并使维护收益率曲线变得非常重要。我鼓励你保持自己的收益率曲线最新，并密切关注其中出现的机会。
MONETARY POLICY
货币政策
It is very important to be aware of monetary policy, as the actions of the Bank of Canada have a bearing on the economy, profits, currency, and the bond market. I will introduce here a simple and practical way to follow monetary policy. Market professionals chart the spread between the two-year Government of Canada benchmark yield and the Bank Rate as well as the spread between the same two-year bond and the ten-year Canada benchmark. At this stage, I would like to define the Bank Rate: it is the lender of last resort rate. In other words, the Bank of Canada stands ready to supply credit at the Bank Rate to money market jobbers who would have to supply acceptable collateral. In practice, the Bank Rate is used to indicate what the overnight lending rate should be among market participants. Typically, the overnight rate is 25 basis points above the Bank Rate and it is the single rate used by the Bank of Canada to indicate to the market where it wants short-term interest rates to be. This rate is announced after each of the eight predetermined dates during the year. The target for the overnight rate is set by the Bank of Canada to "achieve a rate of monetary expansion consistent with the inflation-control target. The transmission mechanism is complex and involves long and variable lags — the impact on inflation from changes in policy rates is usually spread over six to eight quarters." (Bank of Canada Monetary Policy Report, April 2008.)
了解货币政策非常重要，因为加拿大银行的行为会影响经济、利润、货币和债券市场。接下来我将介绍一个简单实用的方法来跟踪货币政策。市场专业人士通常会绘制两项利差的图表：一是两年期加拿大政府基准债券收益率与银行利率之间的利差，二是同一两年期债券与十年期加拿大债券之间的利差。在此，我想先定义一下银行利率：它是贷款人的最后利率。换句话说，加拿大银行随时准备以银行利率向需要提供合格抵押品的货币市场参与者提供信贷。实际上，银行利率用于指示市场上隔夜贷款利率应处于何位。通常，隔夜利率比银行利率高25个基点，而且加拿大银行用这一单一利率告诉市场它希望短期利率处于何水平。这个利率每年在八个预定日期后公布一次。加拿大银行设定隔夜利率目标以“实现与控制通胀目标相符合的货币扩张率”。其传导机制复杂，涉及漫长且不确定的滞后期——政策利率变化对通胀的影响通常分布在六到八个季度。（《加拿大银行货币政策报告》，2008年4月）
Similar to most other central banks in the developed world, the Bank of Canada has agreed on targeting a rate for Canadas consumer price inflation (Consumer Price Index or CPI). In November 2006, in agreement with the federal government, the Bank of Canada extended the acceptable annual range of 1 to 3 percent to the year 2011.
与其他大多数发达国家的中央银行类似，加拿大银行也同意将加拿大消费者物价指数（CPI）的目标保持在1%至3%的年化范围内，该安排在2006年11月与联邦政府达成一致，并延续至2011年。
Since the Bank of Canada adopted its regular policy and Bank Rate setting meetings, the two-year Canada benchmark has become a good proxy for what market participants believe to be the trend in monetary policy, Should the market believe that the Bank of Canada is planning to lower the Bank Rate and ease monetary policy, the market will move the yield on the two-year bond below the Bank Rate with the converse being true should the market anticipate a higher Bank Rate.
自从加拿大银行开始定期召开政策会议并设定银行利率以来，两年期加拿大基准债券已成为市场参与者对货币政策趋势判断的良好代理指标。如果市场认为加拿大银行计划降低银行利率并放宽货币政策，市场就会使得两年期债券收益率低于银行利率；反之亦然，如果市场预期银行利率上调。
The ten-year benchmark trades independently of the Bank Rate and its yield reflects the collective will of all bond market participants. At a time of extreme monetary ease and an expanding economy, the yield spread from the two-year benchmark to the ten-year benchmark will be very wide (or steep). When monetary policy becomes less stimulating, this spread will narrow, leading to a flattening of the spread. This graph shows the recent trend in this spread:
十年期基准债券独立于银行利率交易，其收益率反映了所有债券市场参与者的共同意愿。在极度宽松和经济扩张时期，两年期债券与十年期债券之间的收益率利差通常非常宽（或称陡峭）；而当货币政策收紧时，这一利差会缩小，导致曲线趋平。下图显示了这一利差近期的趋势：
Believe it or not, this spread was negative up July 2007 after a period when the yield curve was flat. We know what happened next; the credit crisis hit with full force and the Bank of Canada began to ease monetary conditions significantly, pushing the Bank Rate down to a mere 0.25 percent in the process. The spread from two to ten years moved up sharply, reaching an extreme level of 223 basis points in June 2010. The Bank of Canada began to back away from its monetary ease but the current positive spread of 145 basis points indicates that monetary conditions are still very stimulative.
信不信由你，直到2007年7月，这一利差还是负值，此前收益率曲线处于平坦状态。我们都知道随后发生了什么；信贷危机全面爆发，加拿大银行大幅放宽货币条件，将银行利率降至仅0.25%，同时两年期与十年期之间的息差在2010年6月急剧上升至223个基点。之后加拿大银行开始逐步撤出宽松政策，但目前145个基点的正息差表明，货币条件仍然非常宽松。
Above all else, make it a point to maintain your own yield curve and keep it current so that you maintain a good perspective on what is happening in the money market and with monetary policy.
最重要的是，务必保持你的收益率曲线并使其与时俱进，这样你才能对货币市场和货币政策的动态有一个清晰的认识。
SUMMARY
总结
So, you made it through the math chapter! The most important point to remember is the present value concept. It shows you that a bond is merely the sum of the present value of all its parts. Now you know something that not more than 5 percent of the investing population knows: Which way do bond prices go when interest rates go up? You know that higher interest rates at different maturities reduce the present value of the parts of a bond and thus the total price or value of the bond moves lower. You also know about duration, which is a vital term as it allows you to judge the relevant volatility or risk of different bonds. (Perhaps go back to the teeter-totter. Someone who is chicken will not sit at the end of the board and thus will have less risk.) You know what basis points are, along with the yield curve, compound interest, and the important topic of reinvestment risk and bond price dynamics.
因此，你终于完成了这章关于数学的内容！最重要的点是现值的概念。它表明一只债券仅仅是其所有组成部分现值的总和。现在你知道了一个不超过5%投资者了解的秘密：当利率上升时，债券价格如何变化？你知道，不同到期日上较高的利率会降低债券各部分的现值，从而使债券的总价格或价值下降。你还了解了久期这一关键概念，它能让你判断不同债券的相关波动性或风险。（或许你可以回忆一下跷跷板的比喻——胆小的人不会坐在跷跷板的边缘，因此风险较低。）你知道基点是什么，也了解了收益率曲线、复利，以及再投资风险和债券价格动态等重要话题。
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https://www.bankofcanada.ca/rates/interest-rates/提供了很多有用的信息。
Policy interest rate: https://www.bankofcanada.ca/core-functions/monetary-policy/key-interest-rate/
Policy interest rate，政策利率，也就是上文中的隔夜利率（overnight rate），是加拿大金融机构之间隔夜贷款的利率。这个利率也就是加拿大央行每次8次发布利率公告中的利率。在页面上提供了链接来解释政策利率，涉及到了隔夜利率、一夜期银行利率、一夜期存款利率、利率操作区间。一夜期存款利率也就是隔夜利率，利率操作区间一般是0.25%，一夜期银行利率是一夜期存款利率加上利率操作区间。这个叫做底部系统（floor system）。一夜期看起来与公众没有关系，但是政策利率是设定所有其他利率的起点。比如政策利率改变，房屋贷款、信用额度（line of credit）的最优惠利率（prime rate）一般都会调整，存款、GIC、定期存款的利率都会调整。加拿大央行的授权是设置的政策利率应尽量保证就业同时保证通货膨胀率在2%左右。
Money market yields: https://www.bankofcanada.ca/rates/interest-rates/money-market-yields/
Selected treasury bill yields: https://www.bankofcanada.ca/rates/interest-rates/t-bill-yields/
两者都是在一年期内的短期债务工具。货币市场包含了国库券，但是也包含了更广泛的短期融资工具，比如商业票据、银行承兑汇票、回购协议、短期存单等。国库券作为由政府发行的短期债务工具，是货币市场中流动性极高且风险非常低的一部分。
Bond yields: https://www.bankofcanada.ca/rates/interest-rates/canadian-bonds/
这是与本文关系最密切的一部分。
Marketable bond average yields，可流通债券平均收益率，是对一系列在市场上交易的可流通债券所计算出的收益率平均值。
Current benchmark bond yields，当前基准债券收益率，是当前市场上广泛公认并使用的基准债券的收益率。这一收益率基于政府债券的中间市场收盘收益率。什么是中间市场收盘收益率？中间市场价格就是买入价（bid）和卖出价（ask）两个价格之间的中间值，用以代表市场上买卖双方意见的折衷结果。类似地，中间市场收盘收益率就是基于中间价格计算出的收益率。收盘收益率指的是在交易日结束时，根据市场最后交易价格或报价计算得出的收益率。
Selected benchmark bond yield，选定基准债券收益率。哪些债券被选定了？可以看当前基准债券收益率下面的列表，比如，目前选定的基准债券是
- 2 year - 2026.11.01, 3.25% (2024.10.11);
- 3 year - 2027.09.01, 2.75% (2025.01.16);
- 5 year - 2029.09.01, 3.50% (2024.09.12);
- 7 year - 2031.12.01, 1.50% (2025.01.16);
- 10 year - 2034.12.01, 3.25% (2024.12.13);
- Long - 2055.12.01, 2.75% (2024.05.30);
- RRB - 2050.12.01, 0.50% (2020.06.01)
选定的债券在金融市场中成为基准债券后，一般会在最后一次拍卖之后进行更新或变更，从而构成当前基准债券。这一收益率基于政府债券的中间市场收盘收益率，这些债券的到期时间大致符合指定的期限。注意，“选定”意味着所使用的具体债券并不一定是剩余期限最接近所指示期限的那一只。也就是说，虽然到期日相对接近，但可能与其他来源所选的债券不同。
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