Mathematics · Statistics
Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver
Rearrange the treynor systematic-risk ratio relationship and solve for portfolio return minus risk-free return.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Use a=cb with Treynor ratio=0.06666666666666667 and portfolio beta=1.2.
- portfolio return minus risk-free return=0.08.
- Substitution into c=a/b reconstructs 0.06666666666666667.
Understand Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return
One idea, three depths
Choose how deeply to explain Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return
Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return: Rearrange the treynor systematic-risk ratio relationship and solve for portfolio return minus risk-free return.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return to answer this question: rearrange the treynor systematic-risk ratio relationship and solve for portfolio return minus risk-free return? Enter Treynor ratio and portfolio beta; the calculator shows portfolio return minus risk-free return. For example: portfolio return minus risk-free return=0.08 and portfolio beta=1.2 produce Treynor ratio=0.06666666666666667. The answer tells you portfolio return minus risk-free return.
Age 15Explain it to a 15-year-oldConnect it to the formula
The Treynor ratio divides excess return by systematic beta exposure. This page isolates portfolio return minus risk-free return and verifies it in the original relationship. The rule is a=cb. Its input values are Treynor ratio, portfolio beta, and the main result is portfolio return minus risk-free return. For example: portfolio return minus risk-free return=0.08 and portfolio beta=1.2 produce Treynor ratio=0.06666666666666667.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated treynor systematic-risk ratio: solve portfolio return minus risk-free return relation over the valid real-number domain stated below. The implemented relation is a=cb, evaluated from Treynor ratio, portfolio beta to produce portfolio return minus risk-free return. The Treynor ratio divides excess return by systematic beta exposure. This page isolates portfolio return minus risk-free return and verifies it in the original relationship. It is meaningful only when beta and return use a compatible benchmark and horizon.
Inputs and valid domain
- Treynor ratio must be a finite real number.
- portfolio beta must be a finite real number.
Important boundary: It is meaningful only when beta and return use a compatible benchmark and horizon.
The formula
a=cb
How the calculator works through it
It substitutes Treynor ratio, portfolio beta into the formula and exposes every numerical step above. The main output is portfolio return minus risk-free return, accompanied by Reconstructed Treynor ratio.
Read the result correctly
The portfolio return minus risk-free return is the direct answer to “rearrange the treynor systematic-risk ratio relationship and solve for portfolio return minus risk-free return.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
portfolio return minus risk-free return=0.08 and portfolio beta=1.2 produce Treynor ratio=0.06666666666666667.
Where this model stops being reliable
It is meaningful only when beta and return use a compatible benchmark and horizon.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return uses a=cb. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Averages and representative values
Representative values help you judge what the Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return inputs summarise and what the result can legitimately describe.
Review this foundation about 5 min
Optional enrichment
- Spread and measurement variation
Variation is not always part of the Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return formula, but it helps you judge how stable a reported result may be.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Treynor ratio, portfolio beta.
- Evaluate the principal relationship: a=cb.
- Return portfolio return minus risk-free return and check the domain conditions described above.
Python
from math import *
def treynor_systematic_risk_ratio_solve_a(c, b) -> float:
return (c * b)
assert abs(treynor_systematic_risk_ratio_solve_a(0.06666666666666667, 1.2) - 0.08) < 1e-6 * max(1.0, abs(0.08))
C
#include <assert.h>
#include <math.h>
double treynor_systematic_risk_ratio_solve_a(double c, double b) {
return (c * b);
}
int main(void) {
const double expected = 0.08;
const double actual = treynor_systematic_risk_ratio_solve_a(0.06666666666666667, 1.2);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double treynor_systematic_risk_ratio_solve_a(double c, double b) {
return (c * b);
}
int main() {
constexpr double expected = 0.08;
const double actual = treynor_systematic_risk_ratio_solve_a(0.06666666666666667, 1.2);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double treynor_systematic_risk_ratio_solve_a(double c, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global treynor_systematic_risk_ratio_solve_a
section .text
treynor_systematic_risk_ratio_solve_a:
push rbp
mov rbp, rsp
sub rsp, 32
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-16]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = treynor_systematic_risk_ratio_solve_a(c, b)
result = (c * b);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[c_, b_] := (c * b);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Introductory Statistics 2e
Read the free OpenStax statistics textbookCite this book
- APA 7
- Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- MLA 9
- Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
- Chicago author-date
- Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver. MW SysArc Tools. https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver
MLA 9
MW SysArc. “Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver.
Harvard
MW SysArc (2026) ‘Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_treynor_systematic_risk_ratio_solve_a_2026,
author = {{MW SysArc}},
title = {Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Treynor Systematic-Risk Ratio portfolio return minus risk-free return Solver
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/statistics/treynor-systematic-risk-ratio-portfolio-return-minus-risk-free-return-solver
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return do?
Rearrange the treynor systematic-risk ratio relationship and solve for portfolio return minus risk-free return.
How does the Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return work?
The calculator applies a=cb. The Treynor ratio divides excess return by systematic beta exposure. This page isolates portfolio return minus risk-free return and verifies it in the original relationship.
What can I learn from the Treynor Systematic-Risk Ratio: solve portfolio return minus risk-free return?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .