## Stock Price Trends

Stock price trends estimated using linear regression.

### Summary

#### Key facts

- The primary trend is increasing.
- The growth rate of the primary trend is 31.91% per annum.
- TJX price at the close of September 10, 2024 was $117.83 and was inside the primary price channel.
- The secondary trend is increasing.
- The growth rate of the secondary trend is 79.49% per annum.
- TJX price at the close of September 10, 2024 was inside the secondary price channel.

### Linear Regression Model

Model equation:

Y_{i} = α + β × X_{i} + ε_{i}

Top border of price channel:

Exp(Y_{i}) = Exp(a + b × X_{i} + 2 × s)

Bottom border of price channel:

Exp(Y_{i}) = Exp(a + b × X_{i} – 2 × s)

where:

i - observation number

Y_{i} - natural logarithm of TJX price

X_{i} - time index, 1 day interval

σ - standard deviation of ε_{i}

a - estimator of α

b - estimator of β

s - estimator of σ

Exp() - calculates the exponent of e

### Primary Trend

Start date: March 4, 2022

End date: September 10, 2024

a = 3.3450

b = 0.0008

s = 0.0505

Annual growth rate:

Exp(365 × b) – 1

= Exp(365 × 0.0008) – 1

= 31.91%

Price channel spread:

Exp(4 × s) – 1

= Exp(4 × 0.0505) – 1

= 22.39%

#### March 4, 2022 calculations

Top border of price channel:

Exp(Y_{625})

= Exp(a + b × X_{625} + 2 × s)

= Exp(a + b × 905 + 2 × s)

= Exp(3.3450 + 0.0008 × 905 + 2 × 0.0505)

= Exp(4.1326)

= $62.34

Bottom border of price channel:

Exp(Y_{625})

= Exp(a + b × X_{625} – 2 × s)

= Exp(a + b × 905 – 2 × s)

= Exp(3.3450 + 0.0008 × 905 – 2 × 0.0505)

= Exp(3.9306)

= $50.94

#### September 10, 2024 calculations

Top border of price channel:

Exp(Y_{1,257})

= Exp(a + b × X_{1,257} + 2 × s)

= Exp(a + b × 1,826 + 2 × s)

= Exp(3.3450 + 0.0008 × 1,826 + 2 × 0.0505)

= Exp(4.8314)

= $125.39

Bottom border of price channel:

Exp(Y_{1,257})

= Exp(a + b × X_{1,257} – 2 × s)

= Exp(a + b × 1,826 – 2 × s)

= Exp(3.3450 + 0.0008 × 1,826 – 2 × 0.0505)

= Exp(4.6294)

= $102.45

#### Description

- The primary trend is increasing.
- The growth rate of the primary trend is 31.91% per annum.
- TJX price at the close of September 10, 2024 was $117.83 and was inside the primary price channel.

### Secondary Trend

Start date: April 4, 2024

End date: September 10, 2024

a = 1.8650

b = 0.0016

s = 0.0275

Annual growth rate:

Exp(365 × b) – 1

= Exp(365 × 0.0016) – 1

= 79.49%

Price channel spread:

Exp(4 × s) – 1

= Exp(4 × 0.0275) – 1

= 11.62%

#### April 4, 2024 calculations

Top border of price channel:

Exp(Y_{1,148})

= Exp(a + b × X_{1,148} + 2 × s)

= Exp(a + b × 1,667 + 2 × s)

= Exp(1.8650 + 0.0016 × 1,667 + 2 × 0.0275)

= Exp(4.5915)

= $98.64

Bottom border of price channel:

Exp(Y_{1,148})

= Exp(a + b × X_{1,148} – 2 × s)

= Exp(a + b × 1,667 – 2 × s)

= Exp(1.8650 + 0.0016 × 1,667 – 2 × 0.0275)

= Exp(4.4815)

= $88.37

#### September 10, 2024 calculations

Top border of price channel:

Exp(Y_{1,257})

= Exp(a + b × X_{1,257} + 2 × s)

= Exp(a + b × 1,826 + 2 × s)

= Exp(1.8650 + 0.0016 × 1,826 + 2 × 0.0275)

= Exp(4.8463)

= $127.27

Bottom border of price channel:

Exp(Y_{1,257})

= Exp(a + b × X_{1,257} – 2 × s)

= Exp(a + b × 1,826 – 2 × s)

= Exp(1.8650 + 0.0016 × 1,826 – 2 × 0.0275)

= Exp(4.7363)

= $114.01

#### Description

- The secondary trend is increasing.
- The growth rate of the secondary trend is 79.49% per annum.
- TJX price at the close of September 10, 2024 was inside the secondary price channel.

- Discover a new approach to trend analysis.
- Check the power of mathematical statistics.
- Price channels have never been so accurately estimated.

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