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- Omega ratio - Wikipedia
- Omega Ratio | Definition, Components, Advantages & Limitations
- Omega Ratio (Definition, Formula) | Step by Step
- Omega Ratio - PortfoliosLab
- Omega Ratio - Breaking Down Finance
- The Omega Risk Measure - Invest Excel
- Omega Ratio: Risk Metrics Series - Swan Global Investments
- Omega ratio, the ultimate risk-reward ratio? - Quantdare
- What is the Omega Ratio? (with picture) - Smart Capital Mind
- What is Omega ratio | Capital.com
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The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some threshold return target. The ratio is an alternative for the widely used Sharpe ratio and is based on information the Sharpe ratio discards.
Omega is calculated by creating a partition in the cumulative return distribution in order to create an area of losses and an area for gains relative to this threshold.
The ratio is calculated as:
Ω
(
θ
)
=
∫
θ
∞
[
1
−
F
(
r
)
]
d
r
∫
−
∞
θ
F
(
r
)
d
r
,
{\displaystyle \Omega (\theta )={\frac {\int _{\theta }^{\infty }[1-F(r)]\,dr}{\int _{-\infty }^{\theta }F(r)\,dr}},}
where
F
{\displaystyle F}
is the cumulative probability distribution function of the returns and
θ
{\displaystyle \theta }
is the target return threshold defining what is considered a gain versus a loss. A larger ratio indicates that the asset provides more gains relative to losses for some threshold
θ
{\displaystyle \theta }
and so would be preferred by an investor. When
θ
{\displaystyle \theta }
is set to zero the gain-loss-ratio by Bernardo and Ledoit arises as a special case.
Comparisons can be made with the commonly used Sharpe ratio which considers the ratio of return versus volatility. The Sharpe ratio considers only the first two moments of the return distribution whereas the Omega ratio, by construction, considers all moments.
Optimization of the Omega ratio
The standard form of the Omega ratio is a non-convex function, but it is possible to optimize a transformed version using linear programming. To begin with, Kapsos et al. show that the Omega ratio of a portfolio is:
Ω
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θ
)
=
w
T
E
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r
)
−
θ
E
[
(
θ
−
w
T
r
)
+
]
+
1
{\displaystyle \Omega (\theta )={w^{T}\operatorname {E} (r)-\theta \over {\operatorname {E} [(\theta -w^{T}r)_{+}]}}+1}
The optimization problem that maximizes the Omega ratio is given by:
max
w
w
T
E
(
r
)
−
θ
E
[
(
θ
−
w
T
r
)
+
]
,
s.t.
w
T
E
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r
)
≥
θ
,
w
T
1
=
1
,
w
≥
0
{\displaystyle \max _{w}{w^{T}\operatorname {E} (r)-\theta \over {\operatorname {E} [(\theta -w^{T}r)_{+}]}},\quad {\text{s.t. }}w^{T}\operatorname {E} (r)\geq \theta ,\;w^{T}{\bf {1}}=1,\;w\geq 0}
The objective function is non-convex, so several modifications are made. First, note that the discrete analogue of the objective function is:
w
T
E
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r
)
−
θ
∑
j
p
j
(
θ
−
w
T
r
)
+
{\displaystyle {w^{T}\operatorname {E} (r)-\theta \over {\sum _{j}p_{j}(\theta -w^{T}r)_{+}}}}
For
m
{\displaystyle m}
sampled asset class returns, let
u
j
=
(
θ
−
w
T
r
j
)
+
{\displaystyle u_{j}=(\theta -w^{T}r_{j})_{+}}
and
p
j
=
m
−
1
{\displaystyle p_{j}=m^{-1}}
. Then the discrete objective function becomes:
w
T
E
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r
)
−
θ
m
−
1
1
T
u
∝
w
T
E
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r
)
−
θ
1
T
u
{\displaystyle {w^{T}\operatorname {E} (r)-\theta \over {m^{-1}{\bf {1}}^{T}u}}\propto {w^{T}\operatorname {E} (r)-\theta \over {{\bf {1}}^{T}u}}}
Following these substitutions, the non-convex optimization problem is transformed into an instance of linear-fractional programming. Assuming that the feasible region is non-empty and bounded, it is possible to transform a linear-fractional program into a linear program. Conversion from a linear-fractional program to a linear program yields the final form of the Omega ratio optimization problem:
max
y
,
q
,
z
y
T
E
(
r
)
−
θ
z
s.t.
y
T
E
(
r
)
≥
θ
z
,
q
T
1
=
1
,
y
T
1
=
z
q
j
≥
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z
−
y
T
r
j
,
q
,
z
≥
0
,
z
L
≤
y
≤
z
U
{\displaystyle {\begin{aligned}\max _{y,q,z}{}&y^{T}\operatorname {E} (r)-\theta z\\{\text{s.t. }}&y^{T}\operatorname {E} (r)\geq \theta z,\;q^{T}{\bf {1}}=1,\;y^{T}{\bf {1}}=z\\&q_{j}\geq \theta z-y^{T}r_{j},\;q,z\geq 0,\;z{\mathcal {L}}\leq y\leq z{\mathcal {U}}\end{aligned}}}
where
L
,
U
{\displaystyle {\mathcal {L}},\;{\mathcal {U}}}
are the respective lower and upper bounds for the portfolio weights. To recover the portfolio weights, normalize the values of
y
{\displaystyle y}
so that their sum is equal to 1.
See also
Modern portfolio theory
Post-modern portfolio theory
Sharpe ratio
Sortino ratio
Upside potential ratio
References
External links
How good an investment is property?
"The Omega Measure: A better approach to measure investment efficacy" (PDF) (Press release). California: Propertini.
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Omega ratio - Wikipedia
The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the …
Omega Ratio | Definition, Components, Advantages & Limitations
Jul 4, 2023 · The Omega Ratio is a performance measurement tool used in finance and investment to evaluate the risk-return trade-off of a given investment or portfolio. It measures …
Omega Ratio (Definition, Formula) | Step by Step
The omega ratio is a ratio that assesses the risk and return of an investment at a specific expected return level. It helps us determine the likelihood of winning versus losing, with a …
Omega Ratio - PortfoliosLab
The Omega ratio is the ratio of returns above a certain target level (usually a minimum acceptable return or "MAR") to the total downside risk below that same threshold level. It can provide …
Omega Ratio - Breaking Down Finance
Omega ratio. The omega ratio is a risk-return measure, like the Sharpe ratio, that helps investors to assess the attractiveness of a hedge fund, mutual fund, or individual security. But unlike the …
The Omega Risk Measure - Invest Excel
Shadwick and Keating (2001), however, proposed a non-parametric gains-to-losses ratio called the Omega Ratio. Unlike the Sharpe Ratio, the Omega Ratio does not assume any specific …
Omega Ratio: Risk Metrics Series - Swan Global Investments
Apr 22, 2018 · Put simply, Omega is the ratio of an investment’s gains relative to its losses. It gives you an idea of whether an investment’s return will be met or exceeded. Keating and …
Omega ratio, the ultimate risk-reward ratio? - Quantdare
Jan 9, 2019 · What is the Omega ratio and why is it considered to be superior to others? Is there anything in the Omega ratio that makes it stand out from other risk-reward ratios?
What is the Omega Ratio? (with picture) - Smart Capital Mind
May 16, 2024 · The Omega ratio is a way of measuring the performance of financial assets based on the level of returns they offer in return for the risk of investing in them. It is a ratio of …
What is Omega ratio | Capital.com
Essentially, Omega is the ratio of upside returns relative to downside returns. The higher the Omega value, the greater the probability that a given return will be achieved or exceeded. The …