Interpolation and splines for curves

Drawing a smooth line through a few points

Markets quote a yield curve at a few maturities, such as 2, 3 and 5 years, but you need the value at any maturity, such as 2.5 years. Interpolation fills the gaps between known points. The simplest way is linear: draw a straight line between two neighbours with np.interp or interpolate.interp1d. A cubic spline, interpolate.CubicSpline, joins the points with pieces of cubic curve so that the slope and the curvature also match at every point, and the result looks smooth. A spline can overshoot between points. PchipInterpolator avoids that by never going above or below its neighbours. Inside the known range any of these is fine, but outside it you are extrapolating, which is a guess, and a cubic can then run away.

Which quantity you interpolate matters. Linear interpolation of yields is easy, but finance often interpolates the logarithm of the discount factor instead. With continuous compounding the discount factor is $D(t)=e^{-y(t)\,t}$ , so $\ln D(t)=-y(t)\,t$ . Interpolating this linearly keeps forward rates constant between knots. The forward rate between times $t_1$ and $t_2$ is the rate you can lock in today for that future period: $f=\dfrac{y(t_2)t_2-y(t_1)t_1}{t_2-t_1}$ . A smooth, sensible forward curve is the real test of an interpolation.

D(t)=e^{-y(t)\,t},\qquad \ln D(t)\ \text{linear between knots},\qquad f(t_1,t_2)=\frac{y(t_2)t_2-y(t_1)t_1}{t_2-t_1}

(1)

The 2.5-year yieldThe yield is

4.0\%

2

years and

4.5\%

3

years. Linear interpolation gives

4.25\%

2.5

years. Interpolating

\ln D=-yt

, which is

-0.080

2

years and

-0.135

3

years, gives

-0.1075

2.5

years, so the yield is

0.1075/2.5=4.30\%

. The forward rate between

2

and

3

years is

(0.135-0.080)/1=5.5\%

. A natural cubic spline through

(0,0)

(1,1)

and

(2,0)

S(x)=-0.5x^{3}+1.5x

[0,1]

, so

S(0.5)=0.6875

, while a straight line gives

0.5

. On real data, the Treasury curve of 9 October 2026 has

5.24\%

10

years and

5.65\%

20

years. At

15

years, linear interpolation gives

5.445\%

, a cubic spline through all ten points gives

5.450\%

and PCHIP gives

5.493\%

. Beyond the data it goes wrong: the same spline at

40

years says

4.26\%

, although the

30

-year yield is

5.60\%

Interpolate inside, be careful outside: linear, cubic spline and PCHIP agree inside the data and part ways outside it.
Choose what to interpolate: the log of the discount factor keeps forward rates steady between knots.
Check the forward curve: a good interpolation gives sensible forward rates, not just a pretty line through the points.

Lab · what's graded

A linear interpolation10 pts
Interpolating the discount factor15 pts
A forward rate15 pts
A spline value10 pts
The right tool15 pts
Building a curve10 pts
Why not extrapolate a cubic spline25 pts

Drawing a smooth line through a few points

D(t)=e^{-y(t)\,t},\qquad \ln D(t)\ \text{linear between knots},\qquad f(t_1,t_2)=\frac{y(t_2)t_2-y(t_1)t_1}{t_2-t_1}

(1)

The 2.5-year yieldThe yield is

4.0\%

2

years and

4.5\%

3

years. Linear interpolation gives

4.25\%

2.5

years. Interpolating

\ln D=-yt

, which is

-0.080

2

years and

-0.135

3

years, gives

-0.1075

2.5

years, so the yield is

0.1075/2.5=4.30\%

. The forward rate between

2

and

3

years is

(0.135-0.080)/1=5.5\%

. A natural cubic spline through

(0,0)

(1,1)

and

(2,0)

S(x)=-0.5x^{3}+1.5x

[0,1]

, so

S(0.5)=0.6875

, while a straight line gives

0.5

. On real data, the Treasury curve of 9 October 2026 has

5.24\%

10

years and

5.65\%

20

years. At

15

years, linear interpolation gives

5.445\%

, a cubic spline through all ten points gives

5.450\%

and PCHIP gives

5.493\%

. Beyond the data it goes wrong: the same spline at

40

years says

4.26\%

, although the

30

-year yield is

5.60\%

Interpolate inside, be careful outside: linear, cubic spline and PCHIP agree inside the data and part ways outside it.
Choose what to interpolate: the log of the discount factor keeps forward rates steady between knots.
Check the forward curve: a good interpolation gives sensible forward rates, not just a pretty line through the points.

Lab · what's graded

A linear interpolation10 pts
Interpolating the discount factor15 pts
A forward rate15 pts
A spline value10 pts
The right tool15 pts
Building a curve10 pts
Why not extrapolate a cubic spline25 pts

Interpolation and splines for curves

Drawing a smooth line through a few points

Lab · what's graded

Log in to run your code.

Interpolation and splines for curves

Drawing a smooth line through a few points

Lab · what's graded

Log in to run your code.