Slopes and rates of change · RiskAtoms Academy

By the end you will be able to

1Find the slope of a straight line as rise over run.
2Work out the average rate of change of a function between two points.
3Explain the slope of a curve at a point as the limit of chord slopes.
4State a slope with its units, such as dollars per year or price per percentage point.

Why you need it

“If yields rise by one percentage point, how much does the price fall?” That is a question about a slope: how fast one number changes when another moves.

These Core Path lessons use slopes:

L1.4.06
The price-yield relationship
The price-yield curve is steeper in some places than others; slopes say how much.
L1.4.08
Duration
Duration is built from the slope of the price-yield curve at today's yield.

This atom is also the stepping stone to M-B01 Derivatives, which gives quick rules for the slope at a point.

The idea

How steep, and in which direction

A road sign that says “10% gradient” means: for every 100 metres along, the road climbs 10. That is a slope: how far up for each step across. Mathematicians call the up part the rise and the across part the run. Pick a line and compare.

Rise over run

Pick a line

slope = 6 ÷ 2 = 3

Bigger than 1: the line climbs faster than it moves across.

In finance the “road” is usually a graph, and the slope is a rate of change: dollars per year, or price per percentage point of yield. A straight line has the same slope everywhere. A curve does not: it can be steep in one place and almost flat in another. So for a curve we ask two questions. What is the average slope between two points? And what is the slope at one exact point?

The rules

First, the slope of a straight line, with a new symbol, Δ.

Average rate of change. On a curve, rise over run between two points gives the slope of the chord joining them: the average steepness over that stretch.

The slope at a point. To get the steepness at one spot, make the stretch tiny. As the step shrinks, the chord turns into the tangent.

The three ideas, for reference

m = \frac{\Delta y}{\Delta x} \qquad \frac{f(a+h)-f(a)}{h} \qquad f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

Slope of a line · average rate from a to a + h · slope of the curve at a.

Try it

Press “Shrink h step by step” and watch the chord swing onto the dashed tangent. The chord slope settles on the slope at A. Then move A: on the price curve, the slope is steeper at low yields than at high ones.

Try itFrom chord to tangent

Open the numerical calculus calculator →

Curve

Point A: x1.5

Step h2

Show the tangent at A

Chord slope, h = 2

5

Rise f(a + h) − f(a)10

Run h2

Slope at A (the limit)3

Gap2

the curve chord through A and B tangent at A

Chord slopes at this A, as h shrinks:

h = 14

h = 0.13.1

h = 0.013.01

h = 0.0013.001

Worked examples

1. A straight line: steady saving

Step 1 of 3

1
The set-up
A savings pot in Sydney grows in a straight line from A$2,000 to A$2,600 over 4 years, through equal deposits and no interest. What is its rate of change?

2. An average rate on a curve

Step 1 of 4

1
The set-up
The price of a bond paying 100 in 10 years is $P(y) = 100/(1 + y/100)^{10}$ , with $y$ the yield in percent. What is the average rate of change from 4% to 5%?

3. The slope of x² at x = 3

Step 1 of 3

1
The set-up
Use shrinking chords to find the slope of $f(x) = x^2$ at $x = 3$ .

Common mistakes

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

Reaction

Check what stuck

Question 1 · objective 1

What is the slope of the line through (2, 5) and (6, 13)?

Question 2 · objective 2

f(x) = x². What is its average rate of change from x = 1 to x = 4?

Question 3 · objective 4

A loan balance falls from 10,000 to 8,500 over 3 years. What is its average rate of change, per year?

Question 4 · objective 3

f(x) = x². What is the slope of the chord from x = 5 to x = 5.1? (Give it to one decimal place.)

Question 5 · objective 3

As the step h shrinks towards zero, the slope of the chord from a to a + h gets closer to…

AzeroBthe slope of the tangent at aCthe average rate of change over the whole graphDf(a)

Glossary

Slope: How steep a line is: the change in output for each 1 unit of change in input. Rise over run.
Rise and run: The change up (in the output) and the change across (in the input) between two points.
Δ (delta): The Greek capital letter Δ. Placed before a letter, it means 'the change in', so Δx is the change in x.
Rate of change: How fast one quantity changes as another changes, measured as change in output over change in input.
Chord: A straight line joining two points on a curve.
Tangent: A straight line that just touches a curve at one point and runs in the same direction as the curve there.
Limit: The value an expression gets closer and closer to as something, such as h, shrinks towards zero.