Summation notation · RiskAtoms Academy

By the end you will be able to

1Read a sum written with Σ: the index, where it starts, where it ends, and the term.
2Expand a Σ into its terms, and collapse a written-out sum back into Σ.
3Pull a constant out of a sum, and split a sum of two pieces.
4Read the sums used in finance, such as a sum of discounted payments or a weighted average.

Why you need it

Σ looks frightening, but it is only an instruction: “add these up”. Once you can read it, a formula with thirty terms fits on one line.

These Core Path lessons write their formulas with Σ:

L1.4.03
Pricing a bond
A bond's price is the sum of its discounted cash flows, written with one Σ.
L1.5.05
Stock indices
An index level adds up the prices or market values of its members, often with weights.

The idea

A short way to write “add these up”

A long sum such as $C_1 + C_2 + C_3 + C_4 + C_5$ follows a pattern: the same letter, with a position number that goes up by one each time. Summation notation writes the pattern once and says which positions to use. The big symbol is sigma, the Greek S, for “sum”. Tap each part to see its job.

Take a Σ apart

Tap each part

Σ

Sigma, the Greek capital S. It means: add up everything that follows.

All together: $\sum_{t=1}^{5} C_t = C_1 + C_2 + C_3 + C_4 + C_5$

It is a recipe with three questions: what to add (the term), from where (the bottom), and to where (the top). For example, $\sum_{i=1}^{4} i^2$ means: put $i = 1, 2, 3, 4$ into $i^2$ and add. That is $1 + 4 + 9 + 16$ = 30.

The rules

First, how the notation is built. The counting letter is called the index.

Going the other way, from a written-out sum to Σ, is called collapsing it. Find what changes from term to term, call it the index, and write one typical term. Try these.

Write it with Σ

Guess, then reveal

$2 + 4 + 6 + 8 + 10$
$\frac{1}{1.05} + \frac{1}{1.05^2} + \dots + \frac{1}{1.05^{20}}$
$w_1R_1 + w_2R_2 + w_3R_3$

Three rules let you tidy sums up. They all come from the ordinary rules of adding and multiplying. A constant here means anything that does not change with the index.

The rules, for reference

\sum_{i=1}^{n} c\,x_i = c\sum_{i=1}^{n} x_i \qquad \sum_{i=1}^{n} c = n\,c \qquad \sum_{i=1}^{n} (x_i + y_i) = \sum_{i=1}^{n} x_i + \sum_{i=1}^{n} y_i

A sum from $i = m$ to $i = n$ has $n - m + 1$ terms.

Try it

Choose a term, then move the start and the end. Set c to 3 and check that pulling it out gives the same total. Try an end below the start, too.

Try itΣ expander

The term to add

Start (bottom)i = 1

End (top)4

Constant c in front1

Total

30

Number of terms4

end − start + 14 − 1 + 1 = 4

The sum in Σ notation:

\sum_{i=1}^{4} i^2

Unfolded, one term for each value of i:

i = 11
+i = 24
+i = 39
+i = 416

Worked examples

1. Adding prices with a divisor

Step 1 of 4

1
The set-up
Some stock indices add up their members' share prices and divide by a fixed number, the divisor. Three shares trade at 50, 120, 30; the divisor is 2.

2. A sum of discounted payments

Step 1 of 4

1
The set-up
Read this, then work it out: $\displaystyle\sum_{t=1}^{3} \frac{5}{1.04^t} + \frac{100}{1.04^3}$ .

3. A weighted average

Step 1 of 3

1
The set-up
A portfolio holds three funds with weights $w_i$ (the share of money in each) and returns $R_i$ over a year. Its return is $\sum_{i=1}^{3} w_i R_i$ .

Common mistakes

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

Reaction

Check what stuck

Question 1 · objective 1

How many terms does a sum have if its index runs from i = 3 to i = 10?

Question 2 · objective 2

Work out Σ i², for i from 1 to 4.

Question 3 · objective 3

Work out Σ 3, for k from 1 to 5. (The term is just the number 3.)

Question 4 · objective 3

Which expression is always equal to Σ 2xᵢ (for i from 1 to n)?

A2 + Σ xᵢB2 × Σ xᵢC(Σ xᵢ)²D2n × Σ xᵢ

Question 5 · objective 4

Work out Σ 10 ÷ 1.1ᵗ for t from 1 to 2, to two decimal places.

Glossary

Sigma (Σ): The Greek capital letter S, written Σ. In maths it means: add up the terms that follow.
Index: The counting letter in a sum, such as i or t. It runs through whole numbers from the start to the end.
Term: The expression after the Σ: what to add for each value of the index.
Lower and upper limit: The first and last values of the index, written below and above the Σ.
Constant: A number that stays the same for every term, because it does not depend on the index.
Weighted average: A sum of values each multiplied by a weight, where the weights add up to 1.