Taylor expansion · RiskAtoms Academy

By the end you will be able to

1Approximate a curve near a point with its tangent line: the first-order approximation.
2Add a second-order term with the second derivative to bend the line into a parabola.
3Estimate f(a + Δx) to first and second order, and compare the errors.
4Explain why duration and convexity are the first- and second-order terms for a bond price.

Why you need it

A risk manager wants to know what a 1 percentage point jump in rates would do to a portfolio of thousands of bonds, quickly. Two numbers per bond, and one line of algebra, give a very good answer. This atom is that line.

L1.4.08
Duration
Duration is the first-order term: the straight-line estimate of a price change.
L1.4.09
Convexity
Convexity is the second-order term: the correction for the curve's bend.

The idea

A line, then a bend

You know a curve well at one point: its value, its slope and how sharply it bends. Can you guess its value a little way off? Think of driving: knowing where you are and your speed, you can guess where you will be in a minute. Knowing whether you are speeding up makes the guess better. Step through three guesses for a bond price curve.

Three guesses, each better than the last

Step 1 of 3

1
Order 0: nothing changes
A flat line at today's value. Right at the dot, wrong everywhere else.

That is the whole idea of a Taylor expansion: build a simple curve that matches the real one at a point, first in level, then in slope, then in bend. Each extra term is a higher order.

The rules

To measure the bend we need one new tool, the second derivative.

Now build the approximation one order at a time.

For a bond. Put the bond price into the same formula, and two famous names drop out.

The formulas, for reference

f(a+\Delta x) \approx f(a) + f'(a)\,\Delta x + \tfrac12 f''(a)\,(\Delta x)^2

\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac12 C\,(\Delta y)^2, \qquad D = -\frac{P'}{P}, \quad C = \frac{P''}{P}

The error of a first-order estimate shrinks roughly with $(\Delta x)^2$ , and of a second-order estimate with $(\Delta x)^3$ : halve the step, and they shrink about 4 and 8 times.

Try it

Push the step Δx out in both directions and compare the two guesses with the true value (the open circle). The tangent always sits below a curve that bends upwards, so it overstates the fall and understates the rise. Then switch to the 30-year bond: more bend, so the second-order term matters more.

Try itTaylor overlay

Curve

Expansion point a5%

Step Δx+2 pt

Order 1 (tangent)Order 2 (parabola)

True value at a + Δx

50.8349

Order 1 guess49.6977

Order 1 error−1.1372

Order 2 guess50.9228

Order 2 error0.0879

the curve order 1 order 2

Halve the step and watch the errors: order 1 shrinks about 4 times, order 2 about 8 times.

Δx	order 1 error	order 2 error
2 pt	−1.13719	0.087853
1 pt	−0.294945	0.011315
0.5 pt	−0.075129	0.001436

Worked examples

1. √102 without a calculator

Step 1 of 4

1
What we know at a = 100
$f(100) = 10, \quad f'(100) = \frac{1}{2\sqrt{100}} = 0.05, \quad f''(100) = -\frac{1}{4 \times 100^{1.5}} = -0.00025$

2. A bond price after a 1 point move

Step 1 of 4

1
What we know at 5%
100 paid in 10 years, annual compounding: $P(y) = 100(1+y)^{-10}$ . At 5%: $P = 61.3913$ , $P' = -584.68$ , $P'' = 110 \times 100 \times 1.05^{-12} = 6{,}125.2$ .

3. The same, in duration and convexity

Step 1 of 3

1
The two numbers
$D = -\frac{P'}{P} = \frac{10}{1.05} = 9.5238 \qquad C = \frac{P''}{P} = \frac{110}{1.05^2} = 99.7732$

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

f(a) = 50 and f′(a) = −4. Use a first-order approximation to estimate f(a + 0.5).

Question 2 · objective 2

Same point, and now f″(a) = 6. What is the second-order estimate of f(a + 0.5)?

Question 3 · objective 3

Estimate √26 with a first-order approximation from √25 = 5. (The derivative of √x is 1 ÷ (2√x).)

Question 4 · objective 2

Which term of the expansion captures how the curve bends?

Af(a)Bf′(a)ΔxC½f″(a)(Δx)²DNone: the expansion is always a straight line

Question 5 · objective 4

In ΔP/P ≈ −DΔy + ½C(Δy)², duration D belongs to which part of the Taylor expansion?

AThe levelBThe first-order (slope) termCThe second-order (bend) termDThe error

Glossary

Second derivative: The derivative of the derivative, written f″(x). It measures how fast the slope changes: how sharply the curve bends.
Taylor expansion: An approximation of f near a point a built from f and its derivatives at a: f(a) + f′(a)Δx + ½f″(a)(Δx)² + …
Order: The highest power of the step Δx used in an approximation. First order uses Δx, second order adds (Δx)².
Curvature: How much a curve bends away from a straight line.
Parabola: The U-shaped (or upside-down U) curve of a quadratic, such as y = x².
Modified duration: D = −P′/P: the first-order sensitivity of a price to its yield, in percent per unit of yield.
Convexity: C = P″/P: the second-order term, measuring how the price-yield curve bends.