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Bonds for the Duration?

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Dr. Eric Gaze
Director of the QR Program
Bowdoin College

Numb: 10-year T-notes have a duration of 8.8.

Number: Fortune lists the duration of 10-year Treasury notes on January 1, 2011 as 8.8, meaning the price of these bonds, when trading on the secondary market after auction, will drop 8.8% for every 1 percentage point increase in interest rates.

Num-best: Fortune lists the duration of 10-year Treasury notes on January 1, 2011 as 8.8, meaning the price of these bonds, when trading on the secondary market after auction, will drop 8.8% for every 1 percentage point increase in interest rates. Given the current historic low rates (0.25% Federal Funds Rate) bond prices are set for a major drop.

Investing used to be so easy, stocks are risky and bonds are stodgy safe havens for our capital. Stocks give you an equity/ownership stake in a company, while bonds give you a lender/creditor stake. Stockholders are like highflying jet-setting CEO's while bondholders bring to mind our favorite banker, Mr. Banks from Mary Poppins. Lately though everyone seems to be talking about the "bond bubble" and the overheated bond market. Indeed the December 2010 issue of Fortune has an ominous looking skull graphic next to their article, The Danger in Bonds; and Money magazine tells us to stop gobbling up bonds in their January/February 2011 issue with an evil pac-man illustrating the US equity market's voracious appetite for bonds. Holy fixed-income Batman! If pac-man has joined the dark side something must truly be wrong. How can boring old bonds have gotten so sexy? Not surprisingly, the plunge in the stock market from the DOW high of just past 14,000 in October, 2007 to the bitter depths of March 2009 when it bottomed out in the 6600 range, spooked many investors from the stock market. This crash was especially frightening since investors were just getting their mojo back after the tech bubble burst in 2000 which was quickly followed by 9/11, both of which had sent stocks into a slide until 2003. Meanwhile bond funds were chugging along yielding an average of 9% a year in the super low interest rate climate. After enjoying one of the greatest bull runs in the history of the stock market from 1982 to 2000, investors had been burned twice and were swearing off stocks en masse.

A "mind-boggling" amount of money starting flowing into bond funds, $937 billion since September 2008 according to Fortune, increasing bond fund holdings by 55% while stock fund assets have decreased slightly over the same period. In order to understand the danger we need to backtrack
and cover some bond basics. As mentioned above, bonds are essentially loans to companies (corporate bonds) or governments (federal are called US Treasuries, loans to city/states are referred to as municipal bonds or munis). A bond, like a loan, is issued for a fixed period of time, maturity, with the borrower typically agreeing to make fixed interest payments at regular intervals using an interest rate, yield, on the amount borrowed, principal. This is pretty straightforward, a city might need money to build a school so will issue a 5 year bond at 4% for $20 million. This means they will make interest payments of $800,000 a year for 5 years and then will repay the full $20 million. There are only two real risks here:

  1. The borrower defaults on the loan by going bankrupt, and the lender will only recoup a fraction of the principal lent, possibly nothing. Rating agencies assess the danger of default and give companies and governments a bond rating with AAA being the best.
  2. Inflation! If inflation is at 3% a year when you lend the money at 4%, your real rate of return is only 1%. If inflation goes up to 5% you are actually losing money on this loan.

Inflation is the risk we are interested in and needs a bit more explanation. Assume you are considering buying a jacket for $100 right now, but instead decide to invest your money for 1 year at 7.5%. At the end of the year your $100 will have grown to $107.50. At the same time the price of the jacket now costs $103 due to the 3% inflation rate. Buying the jacket at $103 leaves you with $4.50, meaning you experienced a 4.5% real rate of return on your $100. The interest rates banks charge are thus linked to inflation and also to the federal funds rate. Banks can borrow money from each other at this rate which currently is 0.25%, and then lend it out at a higher rate and make money on the difference.

Federal Funds Rate 1954-2009

As an investor interested in not losing any money at all, you would seek out the "safest" AAA bonds, which will have the lowest yields. Treasuries in particular are considered to be the safest haven for your cash since the US government would have to collapse for this bond to be worthless. There are 3 types of treasuries, T-bills that mature within 1 year, T-notes that mature in 1-10 years, and T-bonds that mature in 10-30 years. T-notes and T-bonds pay interest every 6 months, while T-bills pay back the principal plus interest at their maturity date. There is no compounding of interest with a bond but we still have to differentiate between the interest rate and the effective yearly rate of return, the yield, because the price of a bond can vary as we will now discuss.

Treasury Bills

The yields in the table above come from the end of the week in which President Mubarak of Egypt finally stepped down on Friday (2/11/11), a move which calmed investors who had been anxiously trying to guess what would happen. Yields on the ten-year note had fluctuated wildly with a high of 3.74% on Tuesday. The Treasury sold $72 billion in new bonds this week (sold meaning they are borrowing this money, or issuing new debt!): $32 billion in 3-year notes on Tuesday, $24 billion in 10-year notes on Wednesday, and $16 billion in 30-year bonds on Thursday. So if 10-year notes were sold on Wednesday at 3.66% how can the yield jump all over on the other days?

The interesting thing about bonds is that you do not have to hold them until maturity, but can sell them for a price. Thus bonds are liquid like stocks. When you buy a stock your stockbroker will charge you a commission (fee) for conducting the transaction, online brokers like TD Ameritrade charge $9.95 per transaction on most trades. Bond traders however, make their money by buying bonds at one price (the bid) and selling them at another price (the ask), the difference between the two prices (the spread) is their profit or loss. Thus bond trading is more risky than brokering stocks, bond traders are playing with their own money so to speak, while stockbrokers are simply trying to spend their client's money. Michael Lewis' first book, Liar's Poker, gives an excellent insider account of the high testosterone (in the 1980's almost exclusively male) world of bond traders; these are the high rollers of finance moving hundreds of millions of dollars of packaged debt daily, with proportionate profits and losses. A stockbroker could never make or lose a billion dollars in a day but bond traders can and have. Fraternities of men who enjoy an elite status seem to naturally develop jargon which furthers the aura that what they do is mysterious and complicated. Cowboys in the wild west playing Texas Hold-em Poker speak of the cards being turned over as the river and the turn, and say blinds instead of ante; while bond traders refer to the interest rate as the coupon, with the issue price of the bond being the face value and all subsequent prices paid being a % of par which is 100% of the face value.

The fact that bonds can be bought and sold after they are first issued is what causes the yields to jump around. The original interest rate (coupon) is fixed when the bond is sold, but as investors begin trading the bond and paying different amounts for it, the yield or effective return on your investment will change. For simplicity (please!) let us say you buy a $100 10-year note for $100 (at par) with a 5% coupon. So the US government is guaranteeing the owner of this bond $5 a year for 10 years and then will pay back the $100 in full. Now suppose there is a crisis in the Middle East the day after you buy the bond, and people want to buy your bond from you because they think the stock market will tank and the Treasury is not selling more bonds until next month. You sell your bond for $110. The new owner will now receive the $5 a year for ten years, but $5 is a smaller percentage of the $110 they paid: $5/$110 = 4.55%. Thus the price of the bond rising to $110 has dropped the yield on this bond from 5% to 4.55% and explains why bond prices and yields are inversely related, rising bond prices force the yields lower and vice versa. Rising interest rates will drive bond prices lower which brings us to the ratio du jour: the duration!

The duration helps to quantify the risk associated to bonds due to rising interest rates. If interest rates go up bond prices will go down, which can affect even those who are planning to hold bonds until maturity. If you have the choice between a 10-year note at 3% now or at 5% next year, you would be better off waiting until next year. Managers of bond funds are faced with buying more bonds as more money flows into their fund, and with selling bonds as money flows out which makes bond funds especially subject to interest rate risk. The yields in the table above are at historic lows, with the 50 year average yield on 10-year notes being 6.76% according to Fortune. Everyone expects interest rates to begin rising soon (relook at the Fed rate graphic), and when the rates begin their regression to the mean bond prices will drop. How much you ask? The duration is the ratio of percentage change in the bond price to the total change in interest rates. Now this is a fun ratio :) We are comparing a percentage drop in the price of a bond to a percentage point change in interest rates. Fortune lists a duration of 8.8 for 10-year T-notes on January 1, 2011. This means that the price of the 10-year T-note will drop 8.8% for every 1 percentage point increase in rates. So a 3 point increase in rates would result in about a 25% drop in prices for these T-notes, and similar price drops across the board for bonds causing bond funds to tank, especially as investors rush for the exits at the first whiff of inflation.


The duration is a subtle ratio. First it is listed as a single positive number in Fortune, when in fact it is negative. Second there is no mention of units. The relative change in bond price is unitless since it is comparing a drop in bond price to the original value: -$8.80 per $100 of original value, and the units ($) cancel. Thus the duration has units of the denominator, pp-1, which is unusual. Lastly Money lists the duration as 8.8 years because there are two types of duration: Macaulay and Modified. Our table is listing the Modified duration (where ΔV = change in value of bond price and Δr = change in rate):

Equation 1

Yes that is a partial derivative from Calculus! If you know some Calculus (and enjoy reading finance textbooks) you can show that the term on the right is proportional to the weighted average of time until a bond pays back the amount invested, which is the Macaulay duration and has units of years. The constant of proportionality between the two types of duration is very close to one:

Equation 2

So for a 5% bond with two payments a year the Macaulay duration is 1.025 times the Modified duration (ignoring units!). This column has taken me farther afield than I had anticipated, and I have violated the basic publishing rule that you lose 100% of your readership for every equation you include; but being able to remind everyone that the hallowed derivative from Calculus is nothing more than a ratio has made it all worthwhile.

I got interested in this topic by reading Michael Lewis' most recent book, The Big Short. He gives an incredibly lucid description of what led up to the recent housing market/mortgage industry collapse and traces the seed of this calamity all the way back to his bond trading days at Salomon Brothers in the 1980's. I then read his first book, Liar's Poker, published in 1990 and was amazed to see how the mess got started. Basically if you are in the business of trading debt (which is what bond traders do) then mortgage debt is the proverbial mother load. The problem with mortgages is that pesky homeowners can pay them off early, ruining the basic principle of maturity dates. In a hard to believe story filled with larger than life characters, Lewis details how they figured out a way to package mortgage debt and make it attractive to fixed income investors. This cleverness seems to have first resulted in the Savings and Loan collapse around 1990, next in the creation of the sub-prime market and finally in the near collapse of the entire banking industry. To try to get a better handle on how any of this could have been allowed to happen I read Irrational Exuberance by Peter Shiller (which appeared right before the tech bubble burst in 2000 and the second edition in 2005 right before the sub-prime implosion, so he's due for a third edition right about now which includes bonds). In the end all I can say is that I am somewhat hesitant to buy gold at current prices.

At any rate (pun intended), hopefully this column has shed some light on the following sentence: "T-bills are zero coupon bonds which sell at auction below face and then pull to par." I have often found it is discourse specific vocabulary and notation to be the biggest obstacle to my students' comprehension.

Sapere Aude!