Stock Price Trends

Paychex Inc (PAYX)

Paychex, Inc. is an American provider of human resource, payroll, and benefits outsourcing services for small- to medium-sized businesses.

Stock Price Trends

Stock price trends estimated using linear regression.

Key facts

  • The primary trend is increasing.
  • The growth rate of the primary trend is 3.06% per annum.
  • PAYX price at the close of March 31, 2025 was $154.28 and was higher than the top border of the primary price channel by $20.12 (14.99%).
  • The secondary trend is increasing.
  • The growth rate of the secondary trend is 18.94% per annum.
  • PAYX price at the close of March 31, 2025 was inside the secondary price channel.

Linear Regression Model

Model equation:
Yi = α + β × Xi + εi

Top border of price channel:
Exp(Yi) = Exp(a + b × Xi + 2 × s)

Bottom border of price channel:
Exp(Yi) = Exp(a + b × Xi – 2 × s)

where:

i - observation number
Yi - natural logarithm of PAYX price
Xi - time index, 1 day interval
σ - standard deviation of εi
a - estimator of α
b - estimator of β
s - estimator of σ
Exp() - calculates the exponent of e


Primary Trend

Start date: June 28, 2021
End date: August 21, 2024

a = 4.6339

b = 0.0001

s = 0.0573

Annual growth rate:

Exp(365 × b) – 1
= Exp(365 × 0.0001) – 1
= 3.06%

Price channel spread:

Exp(4 × s) – 1
= Exp(4 × 0.0573) – 1
= 25.77%

June 28, 2021 calculations

Top border of price channel:

Exp(Y312)
= Exp(a + b × X312 + 2 × s)
= Exp(a + b × 453 + 2 × s)
= Exp(4.6339 + 0.0001 × 453 + 2 × 0.0573)
= Exp(4.7859)
= $119.81

Bottom border of price channel:

Exp(Y312)
= Exp(a + b × X312 – 2 × s)
= Exp(a + b × 453 – 2 × s)
= Exp(4.6339 + 0.0001 × 453 – 2 × 0.0573)
= Exp(4.5566)
= $95.26

August 21, 2024 calculations

Top border of price channel:

Exp(Y1,104)
= Exp(a + b × X1,104 + 2 × s)
= Exp(a + b × 1,603 + 2 × s)
= Exp(4.6339 + 0.0001 × 1,603 + 2 × 0.0573)
= Exp(4.8808)
= $131.73

Bottom border of price channel:

Exp(Y1,104)
= Exp(a + b × X1,104 – 2 × s)
= Exp(a + b × 1,603 – 2 × s)
= Exp(4.6339 + 0.0001 × 1,603 – 2 × 0.0573)
= Exp(4.6515)
= $104.74

Description

  • The primary trend is increasing.
  • The growth rate of the primary trend is 3.06% per annum.
  • PAYX price at the close of March 31, 2025 was $154.28 and was higher than the top border of the primary price channel by $20.12 (14.99%).

Secondary Trend

Start date: February 21, 2023
End date: March 31, 2025

a = 4.1106

b = 0.0005

s = 0.0425

Annual growth rate:

Exp(365 × b) – 1
= Exp(365 × 0.0005) – 1
= 18.94%

Price channel spread:

Exp(4 × s) – 1
= Exp(4 × 0.0425) – 1
= 18.55%

February 21, 2023 calculations

Top border of price channel:

Exp(Y727)
= Exp(a + b × X727 + 2 × s)
= Exp(a + b × 1,056 + 2 × s)
= Exp(4.1106 + 0.0005 × 1,056 + 2 × 0.0425)
= Exp(4.6974)
= $109.66

Bottom border of price channel:

Exp(Y727)
= Exp(a + b × X727 – 2 × s)
= Exp(a + b × 1,056 – 2 × s)
= Exp(4.1106 + 0.0005 × 1,056 – 2 × 0.0425)
= Exp(4.5273)
= $92.50

March 31, 2025 calculations

Top border of price channel:

Exp(Y1,255)
= Exp(a + b × X1,255 + 2 × s)
= Exp(a + b × 1,825 + 2 × s)
= Exp(4.1106 + 0.0005 × 1,825 + 2 × 0.0425)
= Exp(5.0628)
= $158.03

Bottom border of price channel:

Exp(Y1,255)
= Exp(a + b × X1,255 – 2 × s)
= Exp(a + b × 1,825 – 2 × s)
= Exp(4.1106 + 0.0005 × 1,825 – 2 × 0.0425)
= Exp(4.8926)
= $133.31

Description

  • The secondary trend is increasing.
  • The growth rate of the secondary trend is 18.94% per annum.
  • PAYX price at the close of March 31, 2025 was inside the secondary price channel.