Portfolio Moments:

An Introduction to Mean-Variance
Optimization

Vincent Milhau
vincent.milhau@edhec.edu

Academic year 2024-2025

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Objectives
▶ Calculate the expected return and the variance of a portfolio,
which will be used in mean-variance optimization (next
chapter).

▶ Calculate the covariance between two portfolios.

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Outline

Reminder On Expectation, Variance and Covariance

Portfolio Expected Return

Portfolio Variance

Covariance Between Two Portfolios

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Outline

Reminder On Expectation, Variance and Covariance

Portfolio Expected Return

Portfolio Variance

Covariance Between Two Portfolios

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Expectation of a Random Variable
Definition
The expectation (mean) of a random variable is the mean of
outcomes weighted by probabilities.

▶ Consider a security with 3 possible returns and the associated

probabilities:
11% Up
1/6
The expected return is
3/6
0% Middle 1 × 11% + 3 × 0% − 2 × 5%
2/ ≈ 0.17%
6 6

−5% Down
▶ Mathematical notation for the expectation of a random variable
𝑟, e.g. expected return:
𝔼[𝑟]
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Variance of a Random Variable
Definition
The variance of a random variable 𝑟 is the expected squared
difference between 𝑟 and its mean:

𝕍[𝑟] = 𝔼 [[𝑟 − 𝔼[𝑟]]2 ]

▶ Example:

[11% − 𝔼[𝑟]]2 The variance is

1 /6 1 × [11% − 𝔼[𝑟]]2
𝕍[𝑟] =
3/6 6
[0% − 𝔼[𝑟]]2 3 × [0% − 𝔼[𝑟]]2
2/ +
6 6
2 × [−5% − 𝔼[𝑟]]2
+ ≈ 0.28%
[−5% − 𝔼[𝑟]]2 6
▶ Variance is a measure of risk.
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Covariance of Two Random Variables
Definition
The covariance between two random variables 𝑟1 and 𝑟2 is

ℂov [𝑟1 , 𝑟2 ] = 𝔼 [[𝑟1 − 𝔼[𝑟1 ]] [𝑟2 − 𝔼[𝑟2 ]]]

▶ Covariance is a measure of the intensity of the comovement

between two random variables.

Properties
▶ Covariance is symmetric with respect to the inputs:

ℂov [𝑟1 , 𝑟2 ] = ℂov [𝑟2 , 𝑟1 ]

▶ The covariance of a random variable with itself equals its

variance:
ℂov[𝑟, 𝑟] = 𝕍[𝑟]

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Outline

Reminder On Expectation, Variance and Covariance

Portfolio Expected Return

Portfolio Variance

Covariance Between Two Portfolios

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Portfolio Return
▶ In mean-variance optimization, we consider a single
investment period ranging from time 0 to time 1 with
buy-and-hold portfolios.
▶ The return on a portfolio 𝑝 is given by

𝑟0,1,𝑝 = 𝑤0,1 𝑟0,1,1 + 𝑤0,2 𝑟0,1,2 + ⋯ + 𝑤0,𝑁 𝑟0,1,𝑁

▶ We drop the time subscripts 0 and 1 for notational simplicity.

Therefore,
𝑟𝑝 = 𝑤1 𝑟1 + 𝑤2 𝑟2 + ⋯ + 𝑤𝑁 𝑟𝑁

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Portfolio Expected Return
▶ Denote the constituents’ expected returns with 𝜇1 , 𝜇2 , ..., 𝜇𝑁 and
the portfolio’s expected return with 𝜇𝑝 .

▶ Introduce the column vector of weights and the column vector

of constituents’ expected returns:

𝑤1 𝜇1
𝐰 = [ ⋯ ], 𝛍 = [⋯]
𝑤𝑁 𝜇𝑁

Property
If a portfolio is buy-and-hold over the investment period, its
expected return is the inner product of the vector of weights and the
vector of constituents’ expected returns:

𝜇𝑝 = 𝐰 ′ 𝛍

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Mathematical Proof
▶ The expectation operator 𝔼 is linear, so

𝔼[𝑟𝑝 ] = 𝔼[𝑤1 𝑟1 ] + 𝔼[𝑤2 𝑟2 ] + ⋯ + 𝔼[𝑤𝑁 𝑟𝑁 ]

▶ Moreover, weights are known at period start, so they are

non-random. Hence,

𝔼[𝑟𝑝 ] =𝑤1 𝔼[𝑟1 ] + 𝑤2 𝔼[𝑟2 ] + ⋯ + 𝑤𝑁 𝔼[𝑟𝑁 ]

=𝑤1 𝜇1 + 𝑤2 𝜇2 + ⋯ + 𝑤𝑁 𝜇𝑁

which is the inner product of 𝐰 and 𝛍.

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A 3-Asset Example
▶ Assume the following annual expected returns:
▶ 10-year bond, 3.5%;
▶ Cash, 0.8%;
▶ Stock, 10.4%.

Bond

30 %
▶ Consider the following
portfolio: 50 % Stock

20 %

Cash

▶ Exercise: Calculate the expected return on that portfolio in the

workbook Portfolio_moments.xlsx. (Think of the function
SUMPRODUCT.)
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Outline

Reminder On Expectation, Variance and Covariance

Portfolio Expected Return

Portfolio Variance

Covariance Between Two Portfolios

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Portfolio Variance and Constituents’ Covariances
▶ Reminder: the return on a buy-and-hold portfolio is the
weighted sum of constituents’ returns:

𝑟𝑝 = 𝑤1 𝑟1 + 𝑤2 𝑟2 + ⋯ + 𝑤𝑁 𝑟𝑁

▶ The portfolio variance depends not only on the constituents’

variances, but also on their covariances (interactions).
▶ Standard notation for the portfolio variance: 𝜎𝑝2 , i.e.,

𝜎𝑝2 = 𝕍 [𝑟𝑝 ]

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The 2-Asset Case
▶ The variance of a 2-asset portfolio is

2 2
𝜎𝑝2 = 𝑤12 × 𝜎⏟1 +𝑤22 × 𝜎⏟2
variance of asset 1 variance of asset 2

+ 2𝑤1 𝑤2 × 𝜎12
⏟
covariance of assets 1 and 2

▶ A negative covariance reduces the portfolio variance.

▶ A positive covariance increases the portfolio variance.

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Portfolio Variance Calculation for 𝑁 = 2
▶ We have

𝕍[𝑤1 𝑟1 + 𝑤2 𝑟2 ] = ℂov[𝑤1 𝑟1 + 𝑤2 𝑟2 ,
𝑤1 𝑟1 + 𝑤2 𝑟2 ] (variance equals self covariance)

= 𝑤1 𝑤1 ℂov[𝑟1 , 𝑟1 ]
+ 𝑤1 𝑤2 ℂov[𝑟1 , 𝑟2 ]
+ 𝑤2 𝑤1 ℂov[𝑟2 , 𝑟1 ] (covariance is linear in both arguments)
+ 𝑤2 𝑤2 ℂov[𝑟2 , 𝑟2 ]

= 𝑤12 𝜎12 + 𝑤22 𝜎22

+ 𝑤1 𝑤2 𝜎12 + 𝑤2 𝑤1 𝜎21 (weights are non-random)

▶ Finally, because 𝜎12 = 𝜎21 , we can group terms and obtain

𝜎𝑝2 = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝜎12

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The 3-Asset Case
▶ The variance of a 3-asset portfolio contains 3 variance terms and
3 covariance terms:

𝜎𝑝2 = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 𝑤32 𝜎32

+ 2𝑤1 𝑤2 𝜎12 + 2𝑤1 𝑤3 𝜎13 + 2𝑤2 𝑤3 𝜎23

▶ The number of terms grows quickly as the number of

constituents increases.
▶ A more compact expression for the variance is obtained by
introducing the covariance matrix of the constituents.

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Covariance Matrix
Definition
The covariance matrix of a set of assets is the matrix of the pairwise
covariances between returns.

▶ Standard notation: 𝚺.
▶ Standard notation for the covariance between the returns on
assets 𝑖 and 𝑗: 𝜎𝑖𝑗 .

▶ Covariance matrix of 𝑁 = 3 assets:

𝜎11 𝜎12 𝜎13

⎡
⎢ ⎤
⎥
𝚺= ⎢
⎢
⎢
𝜎21 𝜎22 𝜎23 ⎥
⎥
⎥
𝜎31 𝜎32 𝜎33
⎣ ⎦

▶ Diagonal elements are variances.

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Portfolio Variance Formula
Property
The variance of a portfolio invested in assets with covariance matrix
𝚺 with the percentage weights 𝐰 is

𝜎𝑝2 = 𝐰 ′ 𝚺𝐰

▶ 𝚺𝐰 is a vector, equal to the matrix product of the matrix 𝚺 and

the vector 𝐰.
▶ The variance is the inner product of the vectors 𝐰 and 𝚺𝐰.
▶ Useful Excel functions:
▶ MMULT: matrix product;
▶ SUMPRODUCT: inner product.

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Proof of Variance Formula
▶ The 𝑖th element of 𝚺𝐰 is
𝑁
[𝚺𝐰]𝑖 = ∑ 𝜎𝑖𝑗 𝑤𝑗
𝑗=1

▶ Therefore, the inner product of 𝐰 and 𝚺𝐰 is

𝑁
𝐰 ′ 𝚺𝐰 = ∑ 𝑤𝑖 [𝚺𝐰]𝑖
𝑖=1
𝑁 𝑁
= ∑ 𝑤𝑖 ∑ 𝜎𝑖𝑗 𝑤𝑗
𝑖=1 𝑗=1

𝑁 𝑁
= ∑ ∑ 𝑤𝑖 𝜎𝑖𝑗 𝑤𝑗
𝑖=1 𝑗=1

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Proof of Variance Formula (Con’t)
▶ The portfolio return can be written as
𝑁
𝑟𝑝 = 𝐰 ′ 𝐫 = ∑ 𝑤𝑖 𝑟𝑖
𝑖=1

▶ Therefore, the portfolio variance is

𝑁
𝕍 [𝑟𝑝 ] = 𝕍 [∑ 𝑤𝑖 𝑟𝑖 ]
𝑖=1

𝑁 𝑁
= ℂov [∑ 𝑤𝑖 𝑟𝑖 , ∑ 𝑤𝑖 𝑟𝑖 ] (variance equals self covariance)
𝑖=1 𝑖=1

𝑁 𝑁
= ℂov [∑ 𝑤𝑖 𝑟𝑖 , ∑ 𝑤𝑗 𝑟𝑗 ] (change summation index from 𝑖 to 𝑗)
𝑖=1 𝑖=1
(1)
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Proof of Variance Formula (Con’t)
▶ Continuation of calculation:
𝑁 𝑁
𝕍 [𝑟𝑝 ] = ∑ ∑ ℂov [𝑤𝑖 𝑟𝑖 , 𝑤𝑗 𝑟𝑗 ] (covariance is linear in (2)
𝑖=1 𝑗=1

both arguments)

𝑁 𝑁
= ∑ ∑ 𝑤𝑖 𝑤𝑗 ℂov [𝑟𝑖 , 𝑟𝑗 ] (weights are non-random)
𝑖=1 𝑗=1

▶ Rewrite the covariance between assets 𝑖 and 𝑗 as 𝜎𝑖𝑗 . Then, the

portfolio variance is
𝑁 𝑁
𝕍 [𝑟𝑝 ] = ∑ ∑ 𝑤𝑖 𝑤𝑗 𝜎𝑖𝑗
𝑖=1 𝑗=1
′
▶ It is the same as 𝐰 𝚺𝐰, hence the portfolio variance is

𝕍 [𝑟𝑝 ] = 𝐰 ′ 𝚺𝐰
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A 3-Asset Example
▶ Consider 3 assets with the following covariance matrix:
Bond Cash Stock
0.003721 0.00108885 0.0005673 Bond
𝐑 = [0.00108885 0.000441 0 ] Cash
0.0005673 0 0.034596 Stock

▶ The corresponding volatilities (square roots of variances) are

▶ 10-year bond, 6.1%;
▶ Cash, 2.1%;
▶ Stock, 18.6%. Bond

30 %

▶ Portfolio composition: 50 % Stock

20 %

Cash
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Variance Calculation in Excel
▶ Calculate the portfolio variance and the volatility (square root
of variance) in the workbook Portfolio_moments.xlsx.
▶ Use function MMULT to calculate the first matrix product, 𝚺𝐰.
▶ Use SUMPRODUCT to calculate the inner product, 𝐰′ 𝚺𝐰.

▶ If you have an “old” version of Excel, you may need to enter an

array formula to calculate matrix products (see next slide); else
you can use regular formulas.

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Array Formulas in Excel
▶ Array formulas are only needed for “old” versions of Excel
(released before Sep. 2018).
▶ Some Excel functions return a matrix:
▶ MMULT: matrix product;
▶ TRANSPOSE: transposition;
▶ MINVERSE: matrix inverse.

▶ To display the complete matrix, you need to:

▶ Highlight the entire range of cells before typing the formula;
▶ Press Shift + Enter after typing the formula.

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Outline

Reminder On Expectation, Variance and Covariance

Portfolio Expected Return

Portfolio Variance

Covariance Between Two Portfolios

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Covariance Formula
▶ Consider two portfolios with the weight vectors 𝐰1 and 𝐰2 and
the returns 𝑟𝑝,1 and 𝑟𝑝,2 .
▶ We have the following result:

Property
Consider 𝑁 assets with covariance matrix 𝚺 and 2 portfolios invested
in these assets with the weight vectors 𝐰1 and 𝐰2 . The covariance
between the two portfolio returns is

ℂov [𝑟𝑝,1 , 𝑟𝑝,2 ] = 𝐰1′ 𝚺𝐰2

▶ Consider a portfolio 𝐰 and fix an asset 𝑖, let 𝐰2 = 𝐰 and let 𝐰1

represent a portfolio 100% invested in asset 𝑖. Then, the
covariance between asset 𝑖 and the portfolio 𝐰 is the 𝑖th
element of the vector 𝚺𝐰.
▶ In other words, 𝚺𝐰 is the vector of covariances between the
portfolio 𝐰 and the constituents.
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Proof for 𝑁 = 2
▶ Consider 2 assets with returns 𝑟𝐴 and 𝑟𝐵 , and 2 portfolios
invested in these assets with the percentage weight vectors

𝑤 𝑤𝐴,2
𝐰1 = [ 𝐴,1 ] , 𝐰2 = [ ]
𝑤𝐵,1 𝑤𝐵,2

▶ Portfolio returns:

𝑟𝑝,1 = 𝑤𝐴,1 𝑟𝐴 + 𝑤𝐵,1 𝑟𝐵

𝑟𝑝,2 = 𝑤𝐴,2 𝑟𝐴 + 𝑤𝐵,2 𝑟𝐵

▶ By using the bilinearity of covariance, we have that the

covariance between the two portfolios is

ℂov [𝑟𝑝,1 , 𝑟𝑝,2 ] = 𝑤𝐴,1 𝑤𝐴,2 ℂov [𝑟𝐴 , 𝑟𝐴 ] + 𝑤𝐴,1 𝑤𝐵,2 ℂov [𝑟𝐴 , 𝑟𝐵 ]
+ 𝑤𝐵,1 𝑤𝐴,2 ℂov [𝑟𝐵 , 𝑟𝐴 ] + 𝑤𝐵,1 𝑤𝐵,2 ℂov [𝑟𝐵 , 𝑟𝐵 ]

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Proof for 𝑁 = 2 (Con’t)
▶ Rewrite with the usual notation 𝜎𝑖𝑗 for the covariance between 𝑖
and 𝑗:

ℂov [𝑟𝑝,1 , 𝑟𝑝,2 ] = 𝑤𝐴,1 𝑤𝐴,2 𝜎𝐴𝐴 + 𝑤𝐴,1 𝑤𝐵,2 𝜎𝐴𝐵

+ 𝑤𝐵,1 𝑤𝐴,2 𝜎𝐵𝐴 + 𝑤𝐵,1 𝑤𝐵,2 𝜎𝐵𝐵
▶ The covariance matrix of the two assets is
𝜎 𝜎𝐴𝐵
𝚺 = [ 𝐴𝐴 ]
𝜎𝐵𝐴 𝜎𝐵𝐵
▶ Therefore,
𝜎 𝑤 + 𝜎𝐴𝐵 𝑤𝐵,2
𝚺𝐰2 = [ 𝐴𝐴 𝐴,2 ]
𝜎𝐵𝐴 𝑤𝐴,2 + 𝜎𝐵𝐵 𝑤𝐵,2
▶ Therefore,

𝐰1′ 𝚺𝐰2 = 𝑤𝐴,1 𝜎𝐴𝐴 𝑤𝐴,2 + 𝑤𝐴,1 𝜎𝐴𝐵 𝑤𝐵,2

+ 𝑤𝐵,1 𝜎𝐵𝐴 𝑤𝐴,2 + 𝑤𝐵,2 𝜎𝐵𝐵 𝑤𝐵,2

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Implementation in Excel
▶ Do the calculation in the workbook Portfolio_moments.xlsx.

▶ We consider 2 portfolios invested in bonds, cash and stocks.

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