Expected Returns

In portfolio theory, we define the sample mean return vector of \(n\) assets (\(\hat{\boldsymbol{\mu}} \in \mathbb{R}^n\)) (also called the expected return vector) as the column‑wise average of each asset’s log‑returns over the sample period. As I explained in log‑returns, the term “return” refers to log‑returns. You can read Osborne’s work to see why we use log‑returns in the first place.

Since you work with log‑returns, recall

\[ r_t^{(i)} \;=\;\ln\!\Bigl(\frac{p^{(i)}_{t+1}}{p^{(i)}_{t}}\Bigr), \]

which represents the log‑return of the \(i\)‑th asset on day \(t+1\) relative to day \(t\).

Assume we observe each asset’s closing price over \(T\) consecutive trading days. Because each log‑return compares day \(t\) to day \(t+1\), this yields \(T-1\) returns per asset. You first compute the \(T-1\) log‑returns for each of the \(n\) assets, and then stack them column‑wise into the \((T-1)\times n\) return matrix

\[ \mathbf R = \begin{bmatrix} \mathbf r^{(1)} & \mathbf r^{(2)} & \cdots & \mathbf r^{(n)} \end{bmatrix}, \]

with

\[ \mathbf r^{(i)} = \begin{pmatrix} r_1^{(i)}& r_2^{(i)}& \cdots& r_{T-1}^{(i)} \end{pmatrix}^{\top}. \]

Finally, taking the mean of each column gives the mean‐return vector

\[ \hat{\boldsymbol\mu} = \frac{1}{T-1} \begin{pmatrix} \hat{\mu}^{(1)}& \hat{\mu}^{(2)}& \cdots& \hat{\mu}^{(n)} \end{pmatrix}^{\top} \]

where \(\hat{\mu}^{(i)}=\frac{1}{T-1}\sum_{t=1}^{T-1}r_t^{(i)}.\)

Another way to compute it is via the matrix‑vector product:

\[ \hat{\boldsymbol{\mu}} \;=\;\frac{1}{T-1}\,\mathbf{R}^{\top}\mathbf{1}, \]

where \(\mathbf{1}\) is a vector of ones.

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Numerical Example

To illustrate these steps in practice, consider a simple example. We have daily closing prices for six assets over a ten‑day period. First, we construct the price table and then compute the corresponding log‑returns. Finally, by averaging each asset’s log‑returns, we obtain the mean return vector shown below.

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Stock Prices Table

Date	AAPL	AMZN	GOOG	MSFT	TQQQ	TSLA
2025‑04‑28	210.14	187.70	162.42	391.16	53.82	285.88
2025‑04‑29	211.21	187.39	162.06	394.04	54.87	292.03
2025‑04‑30	212.50	184.42	160.89	395.26	54.88	282.16
2025‑05‑01	213.32	190.20	162.79	425.40	56.77	280.52
2025‑05‑02	205.35	189.98	165.81	435.28	59.43	287.21
2025‑05‑05	198.89	186.35	166.05	436.17	58.40	280.26
2025‑05‑06	198.51	185.01	165.20	433.31	56.74	275.35
2025‑05‑07	196.25	188.71	152.80	433.35	57.40	276.22
2025‑05‑08	197.49	192.08	155.75	438.17	59.11	284.82
2025‑05‑09	198.53	193.06	154.38	438.73	58.97	298.26

Log‑Returns Table

Date	AAPL	AMZN	GOOG	MSFT	TQQQ	TSLA
2025‑04‑29	0.0051	−0.0017	−0.0022	0.0073	0.0193	0.0213
2025‑04‑30	0.0061	−0.0160	−0.0072	0.0031	0.0002	−0.0344
2025‑05‑01	0.0039	0.0309	0.0117	0.0735	0.0339	−0.0058
2025‑05‑02	−0.0381	−0.0012	0.0184	0.0230	0.0458	0.0236
2025‑05‑05	−0.0320	−0.0193	0.0014	0.0020	−0.0175	−0.0245
2025‑05‑06	−0.0019	−0.0072	−0.0051	−0.0066	−0.0288	−0.0177
2025‑05‑07	−0.0115	0.0198	−0.0780	0.0001	0.0116	0.0032
2025‑05‑08	0.0063	0.0177	0.0191	0.0111	0.0294	0.0307
2025‑05‑09	0.0053	0.0051	−0.0088	0.0013	−0.0024	0.0461

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The sample mean return vector is:

\[ \hat{\boldsymbol{\mu}} \;=\;[-0.0063,\;0.0031,\;-0.0056,\;0.0128,\;0.0102,\;0.0047]^{\!\top}. \]

This means that over this 10‑day period:

• Apple (AAPL) decreased by 0.63 % on average

• Amazon (AMZN) increased by 0.31 % on average

• Google (GOOG) decreased by 0.56 % on average

• Microsoft (MSFT) increased by 1.28 % on average

• TQQQ increased by 1.02 % on average

• Tesla (TSLA) increased by 0.47 % on average