Curve Interpolation

Whether you build a yield curve or use it to value an instrument with cash flows on dates between the points, the choice of interpolation changes the valuation and how sensitivity is allocated for hedging.

What interpolation determines

The discount factors found in Bootstrapping a Yield Curve exist only at the maturities where a market rate is available, such as 1, 2, and 5 years. The cash flows you want to value are also paid on dates in between. Interpolation fills in the discount factor and the forward rate between the points using a stated assumption. When the number of discount factors to find is larger than the number of market instruments, the system of equations has no unique solution, and the interpolation fixes the solution by limiting the shape of the curve¹.

Consider a hypothetical example in which the following three points have been found from market rates at 1, 2, and 5 years. The zero rate is the spot rate expressed with continuous compounding, and the discount factor is $DF(t) = e^{-r(t)\,t}$. Its compounding differs from that of the annually compounded rate $r_n$ in the bootstrapping article.

The discount factor falls as maturity increases. From the differences in the log discount factor, the average forward rates over 0–1 years, 1–2 years, and 2–5 years are 3.00%, 5.00%, and 0.67% (see the formula below). The zero rates and discount factors between the points are not determined by the market rates. The interpolation determines them.

The interpolated quantity and the forward rate

Write the log discount factor as $z(t) = \ln DF(t)$. The zero rate is then $r(t) = -z(t)/t$. The instantaneous forward rate $f(t)$ is the negative of the slope of $z(t)$, and the average forward rate $f_i^d$ over the interval $[t_{i-1}, t_i]$ comes from the difference in $z$ over that interval².

$$f(t) = -\frac{dz}{dt}(t) = \frac{d\,(r(t)\,t)}{dt}, \qquad f_i^d = -\frac{z(t_i) - z(t_{i-1})}{t_i - t_{i-1}}$$

$f_i^d$ equals the average of $f(t)$ over the interval².

The discount factor, its logarithm, the zero rate, and the forward rate can be converted into one another by formulas. The quantity you interpolate is one of these, and the shape of the chosen quantity determines the shape of the others. In this example, the values at the points are fixed first, and the interpolation determines only what lies between them. In an actual construction, the values at the points can also move ("Assumptions and limits").

Methods and formulas

The papers that compare interpolations are Fabien Le Floc'h's (2013) and Jherek Healy's (2020).

Let the points be $t_0 < t_1 < \dots < t_n$, and let $t$ be a maturity inside the interval $[t_{i-1}, t_i]$.

Linear in the discount factor. Make $DF(t)$ a straight line within the interval. If you use only points where $DF$ falls as maturity increases, the forward rate $-DF'(t)/DF(t)$ stays positive. The slope is constant within the interval, but the denominator shrinks as $DF$ falls, so the forward rate rises gradually within the interval and jumps at the points.

Linear in the zero rate. Make $r(t)$ a straight line within the interval. If $r(t) = a + b\,t$, the formula above gives $f(t) = a + 2b\,t$. The forward rate moves linearly within the interval, and it jumps at the points if the slope $b$ differs from one interval to the next. If the slope $b$ is large and negative, the forward rate turns negative.

Linear in the log discount factor. Make $z(t)$ a straight line within the interval. The slope is constant, so the forward rate in the interval is constant at the average $f_i^d$. Le Floc'h's paper calls this method "raw" interpolation³.

$$z(t) = \frac{(t_i - t)\,z(t_{i-1}) + (t - t_{i-1})\,z(t_i)}{t_i - t_{i-1}}$$

Cubic spline. Join cubic polynomials interval by interval so that the value, the slope, and the curvature (second derivative) are continuous at the points. Setting the curvature at both ends to zero is called the natural boundary condition. Applied to $z(t)$, this method makes the forward rate a piecewise quadratic that joins smoothly at the points. This cubic spline is the same curve as the "area-preserving quadratic spline," which joins the forward rate with piecewise quadratics⁴. The coefficients are found by solving a system of equations over the values at all the points, so a change in the value at one point reaches distant intervals as well.

Monotonicity-preserving interpolation. Join the intervals with cubic polynomials, and choose the slope $s_i$ at each point so that $z(t)$ is monotone⁵. Applied to $z(t)$, it keeps the forward rate positive as long as the discount factor falls with maturity⁶. Ways of choosing the slope include the Hyman filter, which caps the slope, and a method that uses the harmonic mean of the slopes of the neighboring intervals.

Deep diveThe formula that sets the slope by a harmonic mean

Let $D_{i-1}$ and $D_i$ be the average slopes of $z(t)$ over two adjacent intervals. When $D_{i-1}D_i > 0$, the slope $s_i$ at the point $t_i$ is the weighted harmonic mean given by the following formula⁷.

$$\frac{1}{s_i} = \frac{t_i - t_{i-1} + 2(t_{i+1} - t_i)}{3(t_{i+1} - t_{i-1})}\cdot\frac{1}{D_{i-1}} + \frac{2(t_i - t_{i-1}) + t_{i+1} - t_i}{3(t_{i+1} - t_{i-1})}\cdot\frac{1}{D_i}$$

When $D_{i-1}D_i \le 0$, set $s_i = 0$. This slope depends only on the values of the neighboring intervals. The slopes at the two ends are set by a three-point formula and then limited so that the curve stays monotone⁷.

Worked example: 3 points, 5 ways

Connect the 3 points above in 5 ways. For linear interpolation of the zero rate, the rate is held constant at 3.00% over 0–1 years. The cubic spline is applied to $z(t)$ with the natural boundary condition, and the monotonicity-preserving interpolation is applied to $z(t)$ with harmonic-mean slopes. Both are values found by solving the system of equations or the formula numerically. The following table compares the instantaneous forward rate (%) and the 3-year discount factor.

The rows you can follow by hand are the three linear methods. The cubic spline and harmonic-mean rows are values found by solving the system of equations or the formula numerically, and are not worked by hand here (the harmonic-mean formula is in the deep dive above). With linear interpolation of the log discount factor, the forward rate is constant at the average 5.00% over 1–2 years and at $(0.10 - 0.08)/3 = 0.67\%$ over 2–5 years. The 3-year discount factor is $e^{-(0.08 + 0.0067)} = 0.916983$. With linear interpolation of the discount factor, the 3-year value is $DF(3) = 0.923116 + (0.904837 - 0.923116)/3 = 0.917023$. With linear interpolation of the zero rate, the zero rate over 2–5 years is $r(t) = 4 - \tfrac{2}{3}(t - 2)$ (%), so the 3-year zero rate is 3.33% and the discount factor is $e^{-0.0333 \times 3} = e^{-0.1} = 0.904837$. This is the same value as the 5-year discount factor.

The present value of 100 received in 3 years is the discount factor times 100, and it ranges from 89.81 (cubic spline) to 91.70 (linear in the discount factor), depending on the method. The gap between the upper and lower values is about 1.9, while the values at the points are the same in all 5 methods. Because the interpolation affects the valuation, it is one of the model's assumptions, and the value of an instrument with cash flows on dates between the points changes with how you choose it. This magnitude is set by the shape of these 3 points and would differ for a curve of another shape.

Figure 1 plots the instantaneous forward rate between the points, in order of maturity, for two of the methods in the table.

Figure description

This is the instantaneous forward rate when the same 3 points (zero rates of 3%, 4%, and 2% at 1, 2, and 5 years) are connected by two methods. Vertical lines mark the points at 1 and 2 years. With a straight line in the log discount factor, the forward rate is constant within each interval, a staircase of 3.00% over 0–1 years, 5.00% over 1–2 years, and 0.67% over 2–5 years, and it is positive throughout. With a straight line in the zero rate, the forward rate is 3.00% over 0–1 years, then rises linearly to about 4% at 1 year and to about 6% just before 2 years. It drops to about 2.67% at 2 years and falls linearly to −1.33% at 5 years. It is 1.33% at 3 years and negative after 4 years. The vertical changes at the interval boundaries represent jumps at the points, and the lines at 1 and 2 years are an approximation drawn with a width of 0.02 years. With the same values at the points, the method changes both the shape and the sign of the forward rate between the points.

Whether the forward rate turns negative

In the table above, the forward rates under linear interpolation of the zero rate and under the cubic spline are negative at 4.5 years. The cubic spline turns negative from about 3.6 years and reaches −1.20% at 5 years. The discount factors at the points fall with maturity, and nothing in the values at the points indicates arbitrage.

With linear interpolation of the zero rate, $r(t)\,t$ is a quadratic function of maturity. $r(t)\,t$ can rise at both ends of an interval and still fall in the middle. In this example $r(t)\,t$ peaks at 10.67% at 4 years and falls to 10.00% at 5 years. $f(t)$ is the slope of $r(t)\,t$, so it is negative after 4 years.

A negative forward rate means the discount factor rises with maturity. Le Floc'h's paper lists a positive forward rate as one condition for an arbitrage-free curve³. Meanwhile, Healy's paper says that since negative rates appeared, many practitioners have removed the modifications that keep the forward rate positive⁸. To tell whether a negative forward rate comes from the market rates or only from the interpolation, change the interpolation and see if it persists. In this example, the cubic spline and the straight line in the zero rate are negative because of the shape of the interpolation, even though the values at the points are the same. The straight line in the log discount factor and the monotonicity-preserving interpolation stay positive at the same points. The monotonicity-preserving interpolation is nearly 0 at the 5-year end.

Interpolation and hedging

Interpolation also changes how sensitivity to interest rate changes is allocated. One way to find the key rate sensitivity is to move one market rate, rebuild the curve, and revalue the position⁹. If the interpolation carries a change at one point to distant maturities, the value of cash flows at distant maturities moves, and the sensitivity is allocated to rates at distant maturities as well. One of Le Floc'h's criteria for hedging is whether most of the sensitivity is allocated to hedging instruments with nearby maturities³.

The following table shows the change in the zero rate (bp) when the 2-year zero rate among the 3 points above is raised by 1bp. In practice you would move a market rate and rebuild the curve, but here the zero rate at the point is moved directly, as a simplification. All 5 rows of the table are found numerically.

With the three linear methods, moving the 2-year point does not move anything over 0–1 years. The cubic spline also moves the zero rate at 0.5 years, before the 1-year point, and the change has the opposite sign. The monotonicity-preserving interpolation also moves the 0.5-year zero rate by −0.39bp, but the effect of moving the 2-year point appears mainly near the 1-year and 2-year points, and it is 0.02bp at 4.5 years. The cubic spline still moves 0.21bp at 4.5 years, a wider spread than the other methods.

Because interpolation changes the hedge allocation, the same position is allocated to the instruments at each maturity differently depending on the method. Which way that difference moves the actual hedging result depends on how the market moves, and this example does not settle it.

How to choose an interpolation

Compare interpolations with the following criteria used in Le Floc'h's paper. Criteria a to e are described as those of the 2006 paper by Hagan and West, and f is a criterion added by Le Floc'h³. Le Floc'h's paper states the criteria as questions rather than names, and the names in the left column of the table are the short labels used here for those questions.

This page does not cover the contents of Hagan and West's monotone convex interpolation, only the assessment that Le Floc'h's paper gives. In Le Floc'h's assessment, linear interpolation of the log discount factor satisfies the criteria except for the continuity of the forward rate (criterion b)³. The cubic spline fails criteria a and e, and layering the Hyman filter on top still does not satisfy criterion e³. Le Floc'h's paper concludes that the monotonicity-preserving interpolation, which sets the slope by a harmonic mean, is local, stable, and monotone, and it also says Hagan and West's monotone convex interpolation can fail to produce a stable curve when the points are moved one at a time¹⁰.

Every method has trade-offs. For example, a straight line in the log discount factor keeps jumps in the forward rate, but it is local and positive. The cubic spline is smooth, but it produces negative forward rates and a wide spread of changes. As for market trends, interpolating the discount factor is the traditional approach. After Hagan and West's monotone convex interpolation, interpolating the forward rate became more widespread, according to Healy's paper (2020)¹¹. The choice depends on which criteria you give priority to, and there is no single answer.

Assumptions and limits

This example fixes the values at the points first and then interpolates, so bootstrapping and interpolation are separated. In practice, the parameters of the interpolation formula are solved numerically to match the market rates, so changing the interpolation can move not only the values at maturities with no market rate but also the values at the points¹. For example, the bootstrapping article has an example separate from this one, with fixed rates (1.00%, 1.50%, 3.00%) at 1, 2, and 5 years. In that example, the 3-year par rate splits into 2.00% and 2.34% depending on whether you connect the fixed rates with a straight line or connect the log discount factor with a straight line.

Beyond the last point, you are extrapolating rather than interpolating. Extrapolating with a constant forward rate corresponds to extending $z(t)$ along a straight line¹².

When a cubic spline is fitted to the log discount factor, the change in the underlying values is smoothed by the multiplication by maturity. A clear oscillation needs a change in the zero rate far from reality¹³. This example, in which the zero rate moves from 4% to 2%, uses values chosen to produce a negative forward rate. A market curve will not necessarily take this shape, and whether the forward rate turns negative for an actual curve depends on the values at the points.

The interpolation here is written on the assumption of a single curve for both discounting and projection. Some environments build separate curves by index and tenor, and Healy's paper introduces a method that fits the cubic spline directly to discrete forward rates over a fixed period¹⁴. The structure is covered on The Multi-Curve Framework page. Besides the 5 methods above, Le Floc'h's paper discusses tension splines, and Healy's paper discusses smoothing splines¹⁰¹⁵.

Notes

1. Healy's paper (arXiv:2005.13890, 2020), Section 2 (page 3). The number of instruments is smaller than the number of discount factors, and the interpolation parameters are solved with a nonlinear solver. ↩ ↩²

2. Healy's paper, Section 2 (page 2), equations (2) to (7). The definitions of $f(t)$ and $f_i^d$, and the fact that $f_i^d$ is the average of $f(t)$ over the interval. Healy's $z(t) = \ln P(t)$ corresponds to $\ln DF(t)$ in the text. ↩ ↩²

3. Le Floc'h's paper (version 1.3, July 2013; the author's draft), Section 1.3 (page 2). Criteria a to e are those used by Hagan and West (2006), and f is added by Le Floc'h. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

4. Healy's paper, Sections 4 and 5 (pages 4 to 5). It states that the curve obtained by fitting a natural cubic spline to the log discount factor is the same as the area-preserving quadratic spline shown by Hagan. ↩

5. Le Floc'h's paper, Sections 3.1 and 3.2 (pages 6 to 8). Cubic Hermite interpolation, which chooses the slope $s_i$ at each point, and the Hyman filter. ↩

6. Le Floc'h's paper, Section 3.1 (page 7). It states that if the interpolation is applied to the log discount factor and preserves monotonicity, the forward rate is positive as long as the discount factor is falling. ↩

7. Le Floc'h's paper, Section 3.4 (pages 9 to 10), equations (27) to (32). The text uses the same symbols $D_i$ and $s_i$. ↩ ↩²

8. Healy's paper, Section 3 (page 3). It states that since negative rates appeared, many practitioners have removed the modifications that preserve monotonicity and positivity, and that the modifications can make hedging unstable. ↩

9. Le Floc'h's paper, Section 1.2 (page 2). It describes the method of moving a market rate by 1bp, rebuilding the curve, and revaluing the position as the Par-Point method. ↩

10. Le Floc'h's paper, Section 5 (page 14) and the abstract (page 1). Monotone convex interpolation and tension splines can be unstable, and the Hermite spline with harmonic-mean slopes is local, stable, and monotone. ↩ ↩²

11. Healy's paper, Section 1 (page 1). It states that discount factor interpolation was the traditional approach, and that after Hagan and West's monotone convex interpolation, constructions based on interpolating discrete forward rates became more widespread. ↩

12. Healy's paper, end of Section 4 (page 5). It states that extrapolation with a constant forward rate corresponds to linear extrapolation of the log discount factor. ↩

13. Healy's paper, end of Section 5 (page 5). It states that a cubic spline can oscillate for monotone or convex input, but since the input is the log discount factor (the zero rate times maturity), a clear oscillation needs an unrealistically large change in the zero rate. ↩

14. Healy's paper, Sections 1 and 6 (pages 1 and 6). For environments that build multiple curves, with a curve for each index and tenor, it describes a method of fitting the cubic spline directly to discrete forward rates over a fixed period. ↩

15. Healy's paper, last paragraph of Section 8 (page 11). It states that if a smooth curve is the goal, a smoothing spline (such as a P-spline) can be considered. ↩