Smooth Interpolation of Zero Curves Ken Adams

Smoothness is a desirable characteristic of interpolated zero curves; not only is it intuitively appealing, but there is some evidence that it provides more accurate pricing of securities. This paper outlines the mathematics necessary to understand the smooth interpolation of zero curves, and describes two useful methods: cubic-spline interpolation—which guarantees the smoothest interpolation of continuously compounded zero rates—and smoothest forward- rate interpolation—which guarantees the smoothest interpolation of the continuously compounded instantaneous forward rates. Since the theory of spline interpolation is explained in many textbooks on numerical methods, this paper focuses on a careful explanation of smoothest forward-rate interpolation.

Risk and other market professionals often show a smooth interpolation. There is a simple keen interest in the smooth interpolation of mathematical definition of smoothness, namely, a interest rates. Though smooth interpolation is smooth function is one that has a continuous intuitively appealing, there is little published differential. Thus, any zero curve that can be research on its benefits. Adams and van represented by a function with a continuous first Deventer’s (1994) investigation into whether derivative is necessarily smooth. However, smooth interpolation affected the accuracy of interpolated zero curves are not necessarily pricing swaps lends some credence to the smooth; for example, curves that are created by intuitive belief that smooth interpolation gives the well-known technique of linear interpolation more accurate results than linear interpolation. of a set of yields are not smooth, since the first The authors took swap rates for maturities of derivative is discontinuous. This article extends one, two, three, five, seven and 10 years together the simple mathematical definition of with the six-month zero rate, removed the seven- smoothness, and then describes the implications year swap rate from the data and created the of the extended definition for the smooth implied zero-rate curve from the remaining data. interpolation of zero curves. The resulting zero-rate curve was used to calculate the missing seven-year swap rate which The mathematics of zero curves was then compared to the actual seven-year swap The mathematics of zero curves is derived from rate. The authors found that swap rates the prices of discount bonds; a discount bond calculated with smoothly interpolated zero-rate being a security that pays, with certainty, a unit curves were closer to the actual seven-year swap amount at maturity. The following axioms define rate than swap rates calculated with curves that the discount bond market: were linearly interpolated. • The market trades continuously over its This paper aims to describe the mathematics and trading horizon: it extends from the current finance theory necessary for an understanding of time to some distant future time such that

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the maturities of all the instruments to be maturity of coupon-bearing bonds and yield valued fall between now and the trading curves derived from the continuously horizon. compounded yields to maturity of zero coupon (discount) bonds. This article is concerned only • The market is frictionless: no transaction with the latter, which are called zero curves. In costs or taxes are incurred in trading, there addition, the term zero yield is used to refer to are no restrictions on trade (legal or the continuously compounded yield to maturity otherwise) such as margin requirements on of a zero coupon bond. Note that all rates are short sales, and the goods in the market are continuously compounded. It is an easy matter to infinitely divisible. convert to and from periodic compounding, and the use of continuous compounding enables the • The market is competitive: every trader can expression of the mathematics of zero curves in a buy and sell as many bonds as desired particularly simple and elegant form, which without changing the market price. greatly simplifies the discussion. • The market is efficient: information is Prices and yields available to all traders simultaneously, and every trader makes use of all the available Consider a bond which is sold now, at time t, and ≤ < information. is due to mature at time x, where txx∞ . The

• The market is complete: any desired cash trading horizon, x∞ , is much greater than zero flow can be obtained from a suitable self- and is longer than the maturity of any bond. financing strategy based on a portfolio of Suppose the price of the bond is denoted by discount bonds. P()tx, . Since the bond pays a unit amount at < • There are no arbitrage opportunities: the maturity, we must have P()xx, = 1 . When tx price of a portfolio is the sum of its the bond sells at a discount and Ptx(), < 1 . Thus, constituent parts. in general, we have Ptx(), ≤ 1 .

• All traders in the market act to maximize Now, define the zero yield in terms of the bond their profits: they are rational and prefer price. The zero yield, as seen at time t, of a bond more to less. that matures at time x, t ≤ xx< ∞ , is denoted by ytx(), and is defined, for tx< , by the relationship These axioms are necessary for the development of the mathematics of zero curves. However, they Ptx(, ) = exp()–()xt– ytx(), (1) may not apply to real markets. For example, one conclusion that can be drawn from the axioms is Equation 1 states that the price of the bond at that the market prices equal the intrinsic value of time t is equal to its discounted value. Note that the bonds; that is, there is no “noise” in the this relationship does not define ytt(), since market prices. Further, the market completeness (), axiom implies that all points on the zero curve PTT = 1 for all T; so, for a discount bond are known. In fact, the zero curve is not defined maturing at tT= , both sides of the equation are by an infinite set of values, but rather by a equal to unity, irrespective of the value of ytt(), . discrete, finite set. The zero yield in terms of the price of a bond is obtained by rearranging Equation 1: Discount bonds are often called zero coupon bonds, in contrast to coupon bonds that make –ln()Ptx(), ytx(,) = ------(2) more than one cash payment to their owner. It is xt– usual to distinguish between yield curves derived from the periodically compounded yields to provided tx< .

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Forward rates where

Suppose that at time t we enter into a forward ∂ y ()tx, = ytx(), x ∂ contract to deliver at time x1 a bond that will x mature at time x . Let the forward price of the 2 To derive an equation for the instantaneous , bond be denoted by Pt() x1,x2 . At the same time, forward rates in terms of the bond prices, Equation 2 can be rearranged to obtain a bond that matures at time x1 is purchased; the (), –ln()Ptx(), = ()xt– yt(), x (8) price of this bond is Ptx1 . Further, again at time t, a bond that matures at time x2 is bought; Differentiating Equation 8 with respect to x gives (), the price of this bond is Ptx2 . Note that the , –Px()tx ------= yt(), x + ()xt– y ()tx, (9) complete-market axiom guarantees that these Pt(), x x bonds exist. In addition, the axiom specifying that there are no arbitrage opportunities implies Finally, by direct comparison of Equation 7 and that the price of the bond maturing at time x2 Equation 9 must be equal to the product of the price of the –Px()tx, ftx(), = ------(10) bond maturing at time x1 and the forward price: Ptx(),

It is now possible to define yt(), t , which, as noted, Ptx(), 2 = Ptx(), 1 Ptx(), 1,x2 (3) is not defined by Equation 2. First, note that Let the implied forward rate, as seen at time t, for Equation 7 implies ftt(), = ytt(), . Then, noting (),, , the period x1 to x2 be ftx1 x2 , defined by: that Pt() t = 1 , Equation 10 can be used to obtain ()()() Ptx(), 1,x2 = exp – x2 – x1 ftx, 1,x2 (4) –Px()tx, ytt(), ==lim------lim – Px()tx, Note the similarity between the definition of the xt→ Ptx(), xt→ implied forward rate (as defined in Equation 4) Defining the zero curve and the zero rates (as defined in Equation 1). On substituting Equation 1 and Equation 4 into Assume that the prices of all bonds in the market Equation 3 we obtain are known; the implication being that the value (), ()≤≤ of ytx for txx∞ is known. Then, the exp()–()x – t ytx(), = exp()–() x – t ytx(), 2 2 1 i (5) current zero curve (i.e., the one seen at time t) × ()() exp – x2 – x1 ftx(), 1,x2 comprises the zero yields, as seen at time t, of the zero coupon bonds, which mature between t and Rearranging Equation 5 gives x∞ , inclusive; that is, the current zero curve () () defined by ytx(), for ()txx≤≤ . x2 – t ytx(,2 )– x1 – t ytx(,1 ) ∞ ftx(),,x = ------(6) 1 2 x – x 2 1 Though the zero curve is defined in terms of the zero yields, it can be defined in terms of the The forward rate ftx(),,x , defined in 1 2 instantaneous forward rates. In this case, the zero Equation 6, is the period forward rate. However, curve is defined by the instantaneous forward the instantaneous forward rate is of much greater rates ftx(, ) for txx≤≤∞ . The zero yield in terms of importance in the theory of the term structure. the instantaneous forward rate is obtained by The instantaneous forward rate for time x, as integrating Equation 7: seen at time t, is denoted by ftx(, ) and is the continuously compounded rate defined by x 1 ytx(), = ------òft(),u du (11) , () x – t ftx(), ==limftxx(), , + h y() t x + xt– yx()tx, (7) h0→ t

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This completes the essential mathematical methods, here, we consider those methods that theory of zero curves. In the following sections, require knowledge of the data points only. This the relevance of this theory to the interpolation excludes, for example, Hermitian interpolation, of zero curves is shown, with particular emphasis which requires knowledge of the derivative on smoothest forward-rate interpolation. values as well as the values at the data points. In addition, the use of algebraic polynomials, such The smooth interpolation of zero curves as Lagrange polynomials, are excluded because the order of the interpolating polynomial must, in To construct zero curves from market data, general, be n1– , which implies that there could assume that the n data values are be as many as n2– maxima and minima, and this {}… is not a desirable property of a zero curve. (,)x1 y1 ,(,)x2 y2 , ,(,)xn yn Consequently, we consider only piecewise spline ≤ … where 0x1 <

In developing the mathematical theory of zero Analogously, the zero curve is defined by the n {}… curves, it is assumed that the value of ytx(), for data points (,)x1 y1 ,(,)x2 y2 , ,(,)xn yn , where each ()txx≤≤∞ is known. In reality, the current zero one of the data values represents a knot point, that is a point at which the (as yet unspecified) curve is not defined by this infinite set of values spline segments join. If the spline segment implied by the complete market axiom, but, ≤ < {} function for the interval xi x xi1+ is denoted rather, by a set of discrete data values (,)xi yi , by S , then the zero curve can be defined by the each value comprising a time to maturity and a i … zero rate. If we wish to use the mathematics of set of functions {,S1 S2, ,Sn1– } . If it is necessary zero curves derived above, the discrete set of to extrapolate beyond the end values, (,)x1 y1 and values must be extended to an infinite set. This is (,)xn yn , two further spline functions, S0 and Sn , achieved by defining the current zero curve by a will be needed; the former being used for the combination of the set of discrete data values and range at the left-hand end, 0x≤ < x , and the a method for interpolating those values. Given 1 these, the value of yx() for any value of x in the latter for the range at the right-hand end, ≤ < ()≤ xn xx∞ . Though nothing has been said about range 0x< x∞ can be found. Note that one the form of the spline functions, polynomial consequence of this definition of the zero curve is splines are sufficient for our purpose. that changing the interpolation method changes the zero curve. The simplest type of interpolation algorithm is the two-point algorithm, where interpolation of Interpolation ≤ < values in the interval xi x xi1+ depends only Interpolation methods provide a means to on the two points (,)xi yi and (,)xi1+ yi1+ . In calculate values of yx() for times that do not xi contrast, the multi-point algorithm requires coincide with the given times to maturity, knowledge beyond the adjacent knot points. The … x1,x2, ,xn . Though there are many interpolation well-known technique of linear interpolation is a

ALGO RESEARCH QUARTERLY 14 MARCH/JUNE 2001 Interpolation of zero curves two-point algorithm, and cubic-spline Though the theory of strain energy is not interpolation is a multi-point algorithm. discussed, note that the smaller the quantity, the less the strain energy or, intuitively, the smaller Smoothness the “bending” of the elastic spline. This quantity Recall that a smooth function is one that has a is taken as the measurement of smoothness for continuous differential. Thus, any zero curve that the splines: the smaller this measure of can be represented by a continuous function is smoothness, the smoother the interpolating smooth. However, the same is not necessarily curve. The measure of smoothness shall be used true for interpolated zero curves; for example, to determine the “best” interpolation methods curves that are created by linear interpolation are for zero curves. not smooth. This observation applies to all two- A second property of elastic splines is useful. point interpolation formulae, since, in general, Since the only constraints are that the splines there is a discontinuity in the first derivative at must pass through the given points, the parts of the knot points. For similar reasons, it also the splines that project beyond the curve defined applies to any method that does not use the full by the specified points are not subject to any set of data points in the construction of the spline constraints. This implies that the portions of the curve. In general, there must be at least one knot splines beyond the ends of the specified points point at which the derivatives are not are linear (since this minimizes the strain energy continuous. Consequently, only those in those parts of the splines). Thus, the second interpolation methods that use all of the data derivatives of the splines beyond the given data points to construct the spline curve are points are zero. considered. The determination of the smoothest possible The mathematical definition of smoothness does interpolation method depends on whether we not help to distinguish between different spline want to find the smoothest zero curve or the functions; in particular, it does not provide a smoothest forward-rate curve (specifically, the measure of smoothness. To define a measure of smoothest continuously compounded forward- smoothness, we begin with a simple idea that rate curve). Thus, two interpolation methods are makes intuitive sense and then give it a precise considered. Cubic-spline interpolation meaning. Intuitively, interpolation functions with guarantees the smoothest zero curve, and the smallest number of maxima and minima have smoothest forward-rate interpolation guarantees the fewest possible “bends,” that is, they are as the smoothest continuously compounded close as possible to a straight line. Recall that forward-rate curve. Since cubic-spline straight lines can not be used to join the knot interpolation is a standard technique dealt with points, because when data sets contain more in many books on numerical methods, it is than two points, linear interpolation is not reviewed only briefly here. smooth.

Elastic splines are a mechanical equivalent of our Smoothest zero-rate interpolation intuitive idea to draw a curve with the fewest possible bends, as well as a means of defining a The cubic-spline interpolation method produces precise measure of smoothness. The strain energy smooth zero curves. Moreover, it can be shown depends on the shape, gx() , assumed by the spline that (see, e.g., Burden and Faires (1997)), for the measure of smoothness defined by the strain (gx() is used to avoid confusion with fx() , which is energy, no interpolating function passing through used to denote forward rates). It can be shown (see, for example, Schwarz (1989)) that the the given data values is smoother than the cubic spline passing through the same points. The most strain energy over the interval [,]ab is related to commonly used cubic spline is the natural cubic the quantity spline, which is constructed so that the second b 2 derivatives at both end points are zero. This is æö∂ 2 ç÷gx() dx analogous to allowing the ends of the splines to ò 2 èø∂x a be unconstrained so that the free ends are linear.

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Values beyond the two end points are calculated Smoothest forward-rate interpolation using linear extrapolation. However, if the gradient of this line is too steep, the extrapolated Though smooth zero curves are desirable, values may be unacceptably high (positive practitioners often state a preference for zero gradient) or negative (negative gradient). If our curves that have the smoothest forward rates. aim is to create a zero curve with a better shape, The interpolating function that guarantees the smoothest continuously compounded it is possible to constrain the cubic spline so that instantaneous forward-rate curve is a quartic the gradient at the right-hand end is zero; the spline. Recall that the data consist of zero rates, constrained spline behaves like zero curves that not forward rates. Therefore, the quartic spline tend to flatten at longer maturities. Then linear that is constructed does not pass through the extrapolation gives a smooth curve, albeit a data points. In this section, the equations of the horizontal one for maturities longer than the smoothest forward-rate interpolating function maturities in the data set. are given, and the linear equations to be solved to find the coefficients defining that function are A financial cubic spline denotes a cubic spline specified. that is constrained so that its derivative at its right-hand end is zero, and its second derivative We wish to construct a quartic spline for each of at the left-hand end is also zero. These additional the n1– sections between the knot points. The constraints mean that it will have a slightly ith spline segment can be expressed as different shape than the natural cubic spline 2 3 4 S x abxcx dx ex (12) passing through the same set of points, so it will i()= ++ + + not be the smoothest curve to interpolate those where S ()x represents the function fx() over the points (the natural cubic spline is). However, no i range x ≤ xx< . other interpolation function that is subject to the i i1+ same constraints as the financial cubic spline, Each spline segment is a quartic polynomial and and which fits the given data, is smoother than there are n1– segments; 5n()– 1 = 5n– 5 the financial cubic spline that interpolates that unknown coefficients must be found. data. The system of linear equations is defined by the A further property of cubic-spline interpolation is following constraints: worthy of mention. The general equation for a 2 3 • the n original zero rates must be recoverable cubic is yabxcx= ++ +dx , suggesting the from the zero curve need for four coefficients (a, b, c and d) for each section of the spline curve. However, it is also • there must be continuity: possible to define a cubic spline in terms of the given x and y values and the second derivatives • at the n2– interior knot points at the knot points. Thus, to create a cubic-spline • of the first derivatives at the n2– zero curve, it is necessary to find only the second interior knot points derivatives at the knot points, which leads to a set of tri-diagonal linear equations. A standard • of the second derivatives at the n2– algorithm for solving sets of linear equations is interior knot points LU decomposition and back-substitution. When • of the third derivatives at the n2– the set of equations is tri-diagonal, the algorithm interior knot points. takes a particularly efficient form so that the Thus far, n4n2+5n8()– = – conditions have implementation of cubic-spline interpolation is been defined. The additional three conditions computationally efficient. These algorithms are imposed are f′()x =f0 , ″()x =f0 and ″()x = 0 . described in standard texts on numerical n 1 n ′ methods (see, e.g., Burden and Faires (1997)). The first condition, f ()xn = 0 , ensures that the

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< right-hand end of the curve is flat. The last two In the second case, where 0x1 , the spot rate at of these conditions constrain the second time zero is unknown. Again, from Equation 11 derivatives of the forward curve at both the left- x hand and right-hand ends to be zero. Together, 1 these ()5n– 5 conditions ensure that the ò fu()du = x1y1 (16) interpolating spline has the smoothest 0 instantaneous forward rates. In addition, the functional form of f(x) must be ′′ Each of these constraints is considered in specified. Since S1 ()x1 = 0 , an obvious choice is ≤≤ constructing the linear equations that will be to use linear extrapolation for the range 0xx1 ; solved to find the coefficients. this corresponds to the straight line that an Recovering the yields unconstrained elastic spline would take. Thus, the equation for the left-hand extrapolation is: Recall from Equation 11 that: () fx()= f1 + mx– x1 (17) x ()xt– ytx(), = òft(), udu (13) where t 2 3 4 f = a ++++b x c x d x e x (18) Then, since t0= , Equation 13 takes the form 1 1 1 1 1 1 1 1 1 1 2 3 x mb= 1 +++2c1x1 3d1x1 4e1x1 (19) xy() x = òfu()du The values of m and f are derived directly from 0 1 Equation 12. Substituting Equation 17 into This must apply at each knot point. Thus, at the Equation 16, we obtain knot point (,)xi yi x x x x ()() i i1+ i1+ xy= ò f1 + mu– x1 du xi1+ yi1+ ==òfu()du + ò fu()du xiyi + ò fu()du (14) 0 0 x x i i Hence, the extrapolated zero rate is ≤≤ This is true for 1in1– . Substituting mx yf= + ------– mx (20) Equation 12 for fx() in Equation 14 over this 1 2 1 range and integrating gives The validity of Equation 20 can be checked by () () noting that y0 ==f1 – mx1 f0 , as expected. b 2 2 x y – x y = a x – x + ---ix –x (15) i1+ i1+ i i i i1+ i 2 i1+ i Therefore,

3 ci 3 3 di 4 4 mx1 b1x1 d1x1 4 ++--- ()x – x ----()x – x y0()==y f – ------=a + ------– ------– e x (21) 3 x1+ i 4 i1+ i 0 1 2 1 2 2 1 1

e i()5 5 Equation 21 is also valid for x1 = 0 ; so it suffices +--- xi1+ – xi 5 whether or not the spot rate at time zero is We consider two cases. In the first case, the spot known. rate at time zero is known; in the second case, it First, note that the validity of the extrapolation is unknown. backward from x1 to time zero depends on the

Equation 15 can be used to match all yields value of x1 . Here, x1 is the shortest maturity; so except the first. If the spot rate at t0= is known the larger its value, the greater the time to () maturity over which extrapolation is linear. Thus, (x1 =fx0 ), then ()1 ==f0 y1 and the smaller x1 , the better; if possible, the y1 ===S1()x1 S1()0 a1 overnight rate should be used.

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Continuity constraints are numbered 12,,… ,n1– to correspond to the spline segments S ,,,S … S , then the rows in Continuity at the interior knot points is 1 2 n1– equivalent to requiring that block i (1in1≤ < – ) are, in order, ≤≤ Si()xi1+ =0in2Si1+ ()xi1+ for – , so that, on •Match yi1+ substituting the equations for Si and Si1+ • Ensure continuity at internal knot points:

2 3 4 Si()xi1+ = Si1+ ()xi1+ ai ++++bixi1+ cixi1+ dixi1+ eixi1+ 2 3 4 = ai1+ ++++bi1+ xi1+ ci1+ xi1+ di1+ xi1+ ei1+ xi1+ • Ensure continuity of first derivative at internal knot points: ′ ′ Continuity of the first derivatives at the interior S i()xi1+ = S i1+ ()xi1+ knot points is equivalent to ′ ′ ≤≤ S i()xi1+ =1in1S i1+ ()xi1+ for – , so that • Ensure continuity of second derivative at internal knot points: ′′ ′′ 2 3 S i()xi1+ = S i1+ ()xi1+ bi +++2cixi1+ 3dixi1+ 4eixi1+ 2 3 • Ensure continuity of third derivative at = bi1+ +++2ci1+ xi1+ 3di1+ xi1+ 4ei1+ xi1+ internal knot points: ′′′ ′′′ Continuity of the second derivatives at the S i()xi1+ = S i1+ ()xi1+ interior knot points is equivalent to The last block, in1= – , contains the rows ′′ ′′ ≤≤ S ()xi1+ = S i1+ ()xi1+ for 1 i n1– , so that •Match yn 2 2 c ++3d x 6e x = c ++3d x 6e x i i i1+ i i1+ i1+ i1+ i1+ i1+ i1+ ′ •Set S n1– ()xn = 0 Continuity of the third derivative at the interior S′′ ()x = 0 knot points is equivalent to the constraint •Set n1– n ′′′ ′′′ ≤≤ S i()xi1+ =1in1S i1+ ()xi1+ for – , so that •Match y1 d + 4e x = d + 4e x i i i1+ i1+ i1+ i1+ ′′ •Set S 1()x1 = 0 Additional constraints Note that the second, third and fifth lines ′ The first additional constraint, f ()xn = 0 , implies contain the additional constraints.

2 3 Unlike cubic splines, the system of linear b +++2c x 3d x 4e x = 0 n1– n1– n n1– n n1– n equations describing quartic splines cannot be The second additional constraint, f″()x = 0 , represented by a tri-diagonal matrix; standard n diagonal decomposition and back-substitution implies must be used to solve the system of equations.

2 (For a description of this standard method, see c d x e x 0 n1– ++n1– n n1– n = Burden and Faires (1997)). ″′ The third additional constraint, f ()x1 = 0 , Once the coefficients are known, it is possible to implies interpolate and extrapolate values from the zero curve, such that these values correspond to the 2 smoothest forward-rate curve. c1 ++3d1x1 6e1x1 = 0

Solving the system of equations Interpolating values from the zero curve The linear equations defined above are arranged In order to interpolate the value of fx() for ≤ into blocks of five equations each. If the blocks x1

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x ≤ x < x . Then, from Equation 11 i i1+ value fn where

2 3 4 x x x æöi æö fn = an1– ++++bn1– xn cn1– xn dn1– xn en1– xn 1ç÷1ç÷ y ==-- fu()du + fu()du -- y x + fu()du xç÷ò ò xç÷i i ò èøèøThus, the integral of the forward rate is 0 xi xi

1æ bi 2 2 = -- x y + a ()xx– + --- ()x – x x xè i i i i 2 i x n x () () c d e òfu()du = ò fudu + ò fudu i()3 3 i()4 4 i()5 5 ö + --- x – xi ++---- x – xi --- x – xi ø 3 4 5 0 0 xn x (22) = x y + f du The value xx= 1 has been excluded to avoid n n ò n x problems with division by zero when x = 0 . n 1 () = xnyn + xx– n fn Extrapolating values from the zero curve From Equation 22, the zero rate is ≤≤ It has already been shown that, if 0xx1 , the x y + ()xx– f forward rates can be extrapolated using y = ------n n n n Equation 17. Now, the value of the zero rate at x x0= is given by ≤ for xn x . y0()==f0() f – mx 1 1 Comparing interpolation methods where f and m are defined by Equations 18 and 1 Most practitioners judge the quality of a zero 19. curve not by the quality of the underlying ≤≤ mathematics, but by the quality of the curve Two cases must be considered, 0xx1 and itself. In the absence of an objective measure of x < x . n quality, they rely on subjective observation of the The integral of the forward rate is candidate curves. To demonstrate the nature of the curves produced by the methods discussed, x x zero curves constructed from the same initial æöæöx fu()du ==()f + mu()– x du xf + m -- – x data using different interpolation methods are ò ò 1 1 èø1 èø2 1 0 0 presented and compared. The data is presented in Table 1. ≤ < where 0xx1 . Therefore, Maturity Zero rate æöx (years) yf= + m -- – x 1 èø2 1 0.5 0.0552

Finally, consider extrapolating zero rates for 1 0.0600 values of x such that x < x . Recall that the n 2 0.0682 additional constraints stipulate that the first and second derivatives of the right-hand end of the 4 0.0801 forward-rate curve be zero, that is 5 0.0843

′ 10 0.0931 S n1– ()xn = 0 ′′ 15 0.0912 S n1– ()xn = 0 20 0.0857 Thus, it is possible to extrapolate the right-hand end of the forward-rate curve with constant Ta b l e 1 : Zero-rate data

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The following three graphs show the zero- and extrapolated rates, but distorts the forward-rate forward-rate curves interpolated from this data curve. Though the zero curve is smooth, there is using the smoothest forward rate, natural cubic- an abrupt transition at the maximum maturity spline and financial cubic-spline interpolation (20 years) where the forward-rate curve ceases to methods. Note that though the maximum be smooth. maturity in the data is 20 years, the graphs have been extended to 30 years to show the differences between the extrapolated values.

Figure 1 shows the curves constructed from the smoothest forward-rate interpolation method. Note that the gradient of the forward-rate curve is zero at the maximum maturity of 20 years and the gradient is zero when extrapolated. Beyond the maximum maturity, the zero curve tends towards the forward-rate curve with increasing maturity. Both the zero and the forward-rate curves are smooth.

Figure 2: Natural cubic-spline interpolation

Figure 1: Smoothest forward-rate interpolation

Figure 2 shows the curves constructed from the natural cubic-spline interpolation method. Note Figure 3: Financial cubic-spline interpolation that the extrapolated zero curve is downward The zero curves are very similar in the maturity sloping, whereas most zero curves tend to zero range of the given data, but differ substantially in gradient at the right-hand end. Also, note that the extrapolated sections. As illustrated in the extrapolated zero curve is linear and leads Figure 4, the zero curves are similar up to about inevitably to negative zero and forward rates. 11 years when the zero rates attain their Both the zero and the forward-rate curves are maximum value. Beyond that point, they show smooth. considerable differences and only the smoothest forward-rate curve has a shape consistent with Figure 3 shows the curves constructed from the that of the majority of zero curves. financial cubic-spline interpolation method. This method corrects the problem with the

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all produce smooth zero curves. Only smoothest forward-rate and natural cubic-spline interpolation methods produce smooth (instantaneous) forward-rate curves. In addition, all three methods guarantee the smoothest interpolation of a curve, though the smoothest curve depends on the chosen method. The natural cubic-spline interpolation method guarantees the smoothest zero curve and a smooth, but not necessarily the smoothest, (instantaneous) forward-rate curve. Similarly, the financial cubic-spline interpolation method guarantees the smoothest zero curve whose right- Figure 4: Comparing zero curves hand end is constrained to have a zero gradient; it does not guarantee a smooth forward-rate The differences in the resulting forward-rate curve. Finally, the smoothest forward-rate curves are more notable, as illustrated in interpolation guarantees, as its name implies, the Figure 5. Though the data was chosen to smoothest (instantaneous) forward curve and a illustrate the differences among the interpolation smooth, but not necessarily the smoothest, zero methods, financial cubic-spline interpolation curve. always causes similar distortions of the forward- rate curve. Though natural cubic-spline There is some evidence that using smooth zero interpolation does not distort the forward-rate curves results in more accurate pricing, though curve, the right-hand end of the curve does not there is insufficient evidence to show that one flatten, and the method is less suitable for interpolation method is always the best. Hence, calculating rates for maturities close to the practitioners have to make a choice, and that will maximum maturity. depend on the nature of the market data from which the zero curve is constructed. Swap traders, in particular, desire smooth forward rates and one would expect them to prefer smoothest forward-rate interpolation to cubic-spline interpolation. Other practitioners may regard the degree of smoothness of the zero curve to be paramount. In that case, they will have to choose between the two cubic-spline interpolation methods, choosing the one that best suits their needs. Both cubic-spline interpolation methods give better results for short- and medium-term maturities, whereas, the smoothest forward-rate interpolated curve can be used over the whole range of maturities. As smoothest interpolation methods are more widely adopted, the benefits and trade-offs of the various methods will be Figure 5: Comparing forward curves better defined and better understood. Discussion Conclusions Given that there are three ways to interpolate the zero curve, it is natural to ask what This discussion has extended the mathematical differences arise from the use of these methods. concept that a smooth function is one that has a First, it is important to note that all three continuous first derivative. To do this, a measure methods interpolate the same set of points, and of smoothness used by engineers when fitting

ALGO RESEARCH QUARTERLY 21 MARCH/JUNE 2001 Interpolation of zero curves smooth curves to a finite set of points was of how to set up the system of linear equations to adopted. This measure of smoothness implies solve for the coefficients of these quartic splines. that there in no smoother interpolating function In addition, it is shown how to interpolate values than the set of cubic splines interpolating the and how to deal with rates beyond the ends of given points. the zero curve.

In addition, we develop a theory of the There is anecdotal evidence that finance mathematics of zero curves that enables the practitioners consider maximum smoothness to definition of a real zero curve in terms of a set of be intuitively important. The little research that market data points and an interpolation method. has been done indicates there is some basis for Without this dual identity (data points and this intuitive judgement. A better understanding interpolation method) the assumed bond market of the interpolation of zero curves depends on is incomplete and the theory does not apply. the dual nature of the definition of a zero curve Many practitioners have been in the habit of in terms of both the market data points and the extracting a curve by bootstrapping, which interpolation method. This understanding— typically implies the use of a two-point together with the knowledge of which method interpolation formula, and then using a different guarantees which smoothest curve, and interpolation formula when using the curve to experience in the use of those methods—will value cash flows. The dual nature of interpolated allow risk practitioners to make informed choices zero curves implies that more care should be about the appropriate interpolation method to taken in defining the way in which the rates are interpolated. use in different situations.

Finally, by combining the mathematical theories References of zero curves and smoothness, we show that the smoothest interpolation method depends on Adams, K. and D. van Deventer, 1994, “Fitting whether the smoothest zero rates or the yield curves and forward rate curves with smoothest, continuously compounded forward maximum smoothness,” Journal of Fixed rates are desired. The well-known cubic-spline Income, June, 4(1):52–62. interpolation method ensures that the smoothest zero rates approach is well established as an Burden, R. and J. Douglas Faires, 1997, interpolation method in financial software Numerical Analysis, New York, NY: Brooks/ systems. The smoothest forward-rate Cole Publishing Co. interpolation method, which ensures the smoothest continuously compounded forward Schwarz, H., 1989, Numerical Analysis: A rates, is as well known. This method uses quartic Comprehensive Introduction, Stanford, CT: splines and it was possible to give a full account Wiley & Sons.

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