Chapter 4 Bond Mathematics
Overview • This chapter covers the fundamental ideas behind fixed income risk and risk measurement. Although it uses the Treasury market as the mechanism for examining these ideas, they form the basis for all fixed income analysis. • The specific topics covered include: • Pricing conventions • Yield to maturity, yield to call, yield to worst, current yield • Duration and convexity • Hedging of interest rate risk • Trading: spreads and butterflys
Pricing Conventions • If you pick up the Wall Street Journal or some other source of Treasury Bond prices, the prices that you find in there are the “quoted” prices (sometimes called the “clean” or “flat” price.) • If you were to buy one of those bonds, you would have to pay the quoted price plus the accrued interest. • Calculating the accrued interest is not particularly difficult, just quirky. • First, realize that Treasury Bonds/Notes pay ½ of their stated coupon every six months. Thus an 8% bond pays 4% of the principal amount ($1000) every 6 months.
Pricing Conventions • The convention in the market is that if the bond is sold between coupon payment dates, the accrued interest is calculated as equal to the percentage of the time between coupon dates that the seller held the bond. • Thus, if you are two-thirds of the way through the coupon period, the accrued interest that would have to be paid would be equal to two-thirds of the coupon payment that would be made at the next coupon date. • This percentage is calculated on an “actual/actual” basis, meaning that you take the exact number of days since the last payment and divide it by the exact number of days between coupon payments. • One effect of this is that since the number of days between payments will vary (from 178 to 184) the daily interest accrual rate changes from period to period!
Pricing Conventions • Let LC stand for the last coupon payment date, let NC stand for the next coupon payment date, and let T stand for today. C is the annual coupon on the bond. • To calculate the accrued interest, simply do the following: T ND LD
Pricing Conventions • So let’s say that we had an 8% bond, with a face value of $1000, that pays interest on February 15 and August 15 of every year. If we purchase this bond on January 22, how much accrued interest would we owe? • There are 184 days between August 15 and February 15, and 160 days between August 15 and January 22. • The accrued interest, therefore is:
Pricing Conventions • Note, however, that since normally prices are quoted as a percentage of face (par) value, the accrued interest will also be quoted that way. • This means that the 34.78 would be quoted as 3.478 if prices were quoted in terms of par. Thus if the bond were quoted as a price of 103.5, the accrued interest would be quoted as 3.478.
Pricing Conventions • It is very common in debt markets to quote bonds in terms of yield instead of price. Since the two are (generally) monotonic transformations of each other, traders use whichever is convenient. Using yield avoids confusion in quotes because of differences in par amounts, etc. • Yield for Treasury Bonds and Notes are the same – they are “bond equivalent yields”, and are quoted under the assumption that interest is paid on a semi-annual basis. • The market quotes Treasury Bills differently. They are quoted on a “discount” basis.
Pricing Conventions • Let’s say that it is January 22, 2003, and you are quoted a rate of 1.25 on a T-Bill maturing on April 15, 2003. This trade will settle on January 23, 2003. • The actual price you would pay for the bill is given by the following formula: • Where SD is the settlement date and MD is the maturity date, and D is the rate of 1.25.
Pricing Conventions • Notice that unlike the Treasury Bond, Treasury Bills pay interest on what is called the actual/360 basis. Thus, if you held a Treasury for exactly 1 calendar year, you would earn slightly more than the quoted rate! • Converting between the discount yield and the bond equivalent yield is cumbersome, and depends on how many days the bill has outstanding. • The book covers this in great detail, and you will implement this as part of the first project set.
Yield Conventions • Treasury Bonds and notes are quoted on a Yield to Maturity convention, and Treasury Bills are based on a discount rate convention. • Bonds that have a callable feature will be quoted on a yield to call basis – meaning assume the bond is called on its call date and solve for yield. • Callable bonds can also be quoted on a yield to worst basis, meaning solve for yield to maturity and yield to each call date (there may be more than one), and assume you will get the lowest of all of those yields.
Yield Conventions • Recall that the yield curve is just a plot of each bond’s yield against its maturity date for a given set of bonds. • The following are the yield curves for January 13, 2003, and July 21, 2003. • The Federal Reserve releases interest rate data daily on their web site at http://www.federalreserve.gov/releases/h15/update/ • They also have historical data available there. You will need this site to collect data for some of the projects.
Measuring Risk in Fixed Income • The most basic risk in fixed income is price risk. That is, that they price of the asset will change because of a change in interest rates. • Normally, yields and prices are inversely related. • The primary methods that finance people use to measure price risk is a concept known as duration, and its related concept of convexity. • In the next slides we will examine these concepts.
Duration • Duration is a measure of how much the price of a bond or other fixed-income asset will change when the discount rate changes. • What duration measures is the instantaneous rate of change in price with respect to yield (i.e. the discount rate.) • In other words, what we want to measure is the rate at which the bond price changes when yield changes. • This means we want to know the slope of the price curve. • Note that technically, the price of a bond is a mathematical function of interest rates.
Duration • Recall that in general the price of a fixed income asset is given by the following formula: • Note that we are denoting price as a function of r: P(r).
Duration • For our purposes, it perhaps more convenient to write this as a product instead of as a quotient. • A couple of rules from differential calculus are also useful to remember:
Duration • First, the derivative of a sum is equal to the sum of the derivatives. This means that we can treat each term of our summation independently. • Second, when dealing with an equation of the form:
Duration • In this context, g(x) is (1+r/m). So that means our derivative will be: • Notice that the 1/m term come from the fact that we have to take the derivative of (1+r/m), which is simply 1/m.
Duration • Reassembling this into a perhaps more conventional form:
Duration • Thus, the first derivative of price with respect to r is: • The first derivative tells us the instantaneous rate at which P is changing – that is, it is the rate at which P is changing given a specific value of r. • The derivative of a specific bond calculated at two different values of r will be different. • Let’s work a couple of examples to see exactly how this is calculated.
Duration • Let’s start with the simple example of a Treasury Bond that matures in exactly 1 year. Let’s assume a coupon rate of 6%, and that the current yield is 4%. • This bond will pay $30 in 6 months: • $1000 * .06/2 = $30 • And $1030 in 1 year. • The price of the bond, therefore is:
Duration • The first derivative of the bond with respect to price, therefore is given by: • or
Duration • The first derivative of price with respect to r is frequently referred to in Finance as “dollar duration”. • By convention the negative sign is usually omitted, so that dollar duration is quoted as a positive number. • The reason that it is referred to as “dollar duration” is that you can use it to predict the dollar change in price for a given change in interest rate. • To do this, you simply multiply the dollar duration by the change in rate (but you must keep in mind the sign of the change and dollar duration!).
Duration • Mathematically this means: • Or, in this specific case • So for a 10 drop in rate, you would expect the price of the bond to rise approximately $0.985
Duration • In reality if rates fell from 4% to 3.9%, the bond’s price will rise from 1019.41 to 1020.40, a change of .98573. • The reason that this is not exact, of course, is because duration uses a linear approximation to the curved price function – we make a tradeoff between ease of calculation and accuracy.
Duration • To demonstrate this, let us use another example, one using a longer-maturing treasury bond. • In particular let us use a 30 year Treasury bond with a coupon of 8%. • If the yield on this bond is 8%, then the bond is worth $1000, but at 10% it is worth $810.71. • The following graph shows the price for all interest rates between 1% and 20%.
Duration • Since there are 60 cash flows associated with a 30 year treasury bond, it is probably easiest to work with the present value of an annuity formula to get this price. • Which in this case would work out to be (at a 10% yield): You may wish to verify for yourself that if r=10%, the price is $1000.
Duration • We could use the same formula for the derivative that we did in the original equation, but, with 60 cash flows, it is cumbersome to do so. • Instead we can use a variation of that formula that is based on the present value of annuity formula we just used. That formula is:
Duration • So, at 10%, the first derivative of the bond with respect to yield would be given by: • Or:
Duration • At a yield of 8%, the first derivative of the treasury bond is: –11311.7 • Recall, that the first derivative tells us the slope of the curve for an instantaneous change in rate. The next slide presents these slopes:
Duration • From the graph we can see that as the interest rate increases, the curve becomes less steep – indicating that as price of the bond is less sensitive to interest rate changes. • By looking at the graph you can see that the rate at which price changes in not constant. • What we want to do is develop a measure of the rate of change given a specific yield. • Dollar duration provides this measure, but it does have some drawbacks.
Duration • One drawback in particular is that it is difficult to compare the relative risk of two bonds that have different face amounts. • A bond with a $5000 face amount will have a derivative that is 5 times larger than one with a $1000 face amount. • It would be nice if we could have a somewhat more standardized way of measuring the risk.
Duration • There are several other variants of duration other than dollar duration. They include: • DV01 • Modified Duration • Macaulay’s duration • Usually finance textbooks will provide you with either Modified or Macaulay’s duration (which is why the number so far may have seemed a little odd-looking to those of you that have seen duration in other courses.) • Let us examine each of these in detail.
Duration DV01 • Since dollar duration numbers tend to be large in absolute terms, it is more convenient to scale them. One way of scaling them is to multiply them by a small yield amount. One choice is to use 1 basis point. This will tell you the approximate change in an instrument’s price for a 1 basis point change in yield.
Duration DV01 • This measure is known as the Dollar Value of an 01 – or simply as DV01. • It is used primarily to compare the magnitude of dollar changes across assets. • Unfortunately, it does not take into account the scale of the underlying asset. That is, an asset with a face amount of $100,000 would have a DV01 100 times greater than an identical asset with face amount of $100,000.
Duration DV01 • In the example presented earlier the DV01 for the bond would be: • DV01 = (dp/dr) * .0001 = -7877.63* -.0001 = $.07877 • Note that this is expressed in Dollars.
Duration Modified Duration • Modified duration is a way of taking into account the scale of the asset being measured. • Essentially it is dollar duration divided by price. • When a trader – or most data sources – refer to “duration” they normally mean modified duration. • This is also the variant of duration that can be viewed as a true “time measure”
Duration Modified Duration • If you multiply modified duration by a change in interest rates, it gives you the approximate percentage change in price for the asset. • Using modified duration to measure the interest rate risk in an asset lets one avoid the scaling difficulty of the DV01 measure. • In our previous example, modified duration would be: • Mod. Duration = dp/dr * 1/p = 7877.63* 1/810.71 = 9.7169. • Remember that, in general, the larger the duration number, the greater the interest rate risk.
Duration Macualay’s Duration • The first derivation of duration was made in the 1930’s by an economist named Macaulay. He was not thinking of it as a risk measure per se, but rather as the price elasticity of a bond with respect to interest rates. As such his measure is given by: • One interesting fact is that for any bond with only one cash flow the Macaulay’s duration of that bond will exactly equal its maturity!
Duration • So there are actually at least four measures of duration: • Dollar Duration: (dp/dr) • DV01: (dp/dr * .0001) • Modified Duration: (dp/dr * 1/p) • Macaulay’s Duration (dp/dr * (1+r/m)/p). • Note that many books refer to duration as a time measure. It is possible to construe it that way, but I think it is much more useful to think of it as a rate of change. • Also, recall that it is really a negative number (in most cases), it is just the convention in finance that we omit the negative sign.
Duration Complications with Duration: • The example we have worked with so far considers a case where the cash flows from the bond are certain. What if they are not? • If the cash flows do not vary with interest rates, then you would calculate duration as normal – just realize there may be risks which duration is not capturing. • For example, some companies issue bonds that have contract rates which depend upon the price of some factor of production – some ski resorts have issued bonds where the interest rate is a function of the amount of snow they get. • You can still calculate duration as normal – just realize that interest rate risk is not the only risk in the bond.
Duration Complications with Duration: • Of course some assets, like mortgages, have cash flows that do vary with interest rates. • This means that the simple derivative formula does not work – cash flow itself must be treated as a function of r, and so one must, at a minimum, use the chain rule to extend the derivative. • Frankly, this is not commonly done. The reason is that most good prepayment models are so complex that they do not have easily computed derivatives. • Analysts can do one of two things, therefore. They can either: • Ignore that cash flows are a function of r: • Approximate duration
Duration Complications with Duration: • Both ways are fairly common, although if you ignore the fact that cash flow is a function of interest rates, you will misstate duration. If you feel the misstatement is small enough, you may choose to do this. • You approximate duration by approximating the derivative. To do this you calculate the price of the asset at two points on either side of the current rate: • For example, if the discount rate were at 10%, you would determine the price at 9.9% and 10.10%, and then divided the difference in prices by the 20 basis point difference in yield. This approximates the slope and hence the derivative. • Example: in our previous example, the price of the bond at 9.9% is $818.65 and at 10.10% is 802.90.
Duration Complications with Duration: • We can approximate duration as follows: • Clearly this yields an approximate duration which is very close to the true duration. • This numerical approximation for duration is commonly used in financial modeling and financial modeling software packages.
Duration and Taylor’s Theorem • Fundamentally duration is an application of Taylor’s Theorem from mathematics. Taylor’s theorem says simply that if you know the value of a function and all of its derivatives at a given point (x), then you can calculate its value at any other point (x+h). The exact formula is
Duration and Taylor’s Theorem • What we do when we use duration is we simply use the first two terms of this formula and drop the rest (although we frequently will add the second term – it is called convexity). • That is, for a bond price (r), we use:
Duration and Taylor’s Theorem • What this says is that if we have an asset with a known price at a given interest rate, (p(r)), then if we change r by dr, the price at that new rate p(r+dr), will be approximately equal to the old price plus the change in rate times the first derivative of the pricing function (which we call dollar duration!). • The reason our value is not an exact match is because we drop those higher order terms. • This can be extremely useful if are told the price of the bond and want to determine its yield.