Portfolio Optimization Objectives and Practical Constraints
Summary
The document introduces mean-variance portfolio optimization, including a minimum-variance portfolio and a variance-minimizing portfolio constrained to a target expected return. It then describes practical variations: long-only holdings, caps on individual weights, limits on country, issuer, sector, or currency exposure, turnover controls, and transaction-cost bounds. For fixed-income portfolios, objectives may include yield or option-adjusted spread while controlling rate and spread duration; other objectives can include minimizing capital or risk-weighted asset measures.
A separate formulation maximizes expected return minus a variance penalty, with a risk-aversion parameter that ranges from return-seeking toward minimum variance. Linear constraints can be added to such quadratic problems. The responses characterize many applications as linear or quadratic optimization and note that analytic solutions are often unavailable, although numerical methods can solve many cases efficiently. The discussion is illustrative rather than a catalog of formulations or solution recipes, and it does not address estimation error, parameter stability, or how to choose constraints and objectives for a particular portfolio.
Key ideas
- Mean-variance optimization can minimize variance subject to a target expected return, while minimum variance drops that target constraint.
- Portfolio rules such as long-only weights, position caps, exposure limits, and turnover limits can be expressed as constraints.
- A return-minus-risk objective uses a risk-aversion parameter to balance expected return and variance.
- Fixed-income portfolio objectives may include yield, duration, transaction cost, or capital requirements.
- Many practical formulations require numerical linear or quadratic optimization rather than a simple analytic solution.
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# Optimal Portfolios
# Optimal Portfolios
In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving
$$\underset{\mathbf{w}}{\mbox{min}\,\,}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}$$
subject to $\mathbf{w}^{T}\mathbf{1}=1$ and $\mathbf{w}^{T}\boldsymbol{\mu}=\eta$.
Another example that I've seen in lectures is the Minimum Variance Portfolio which is the same as above except the condition $\mathbf{w}^{T}\boldsymbol{\mu}=\eta$ is dropped.
I was wondering, there are surely lots of other similar sorts of optimisation problems similar to these. For example,
- imposing each entry of $\mathbf{w}$ is >0 -- to avoid short shelling
- imposing each entry of $\mathbf{w}$ is < $\alpha$ to avoid putting too much weight into one stock
My question is as follows: is there a convenient list of these sorts of optimisation problems, and their solutions?
## Answer by ash (score 8, accepted)
https://quant.stackexchange.com/a/15440
As a practitioner, I have worked on the following
- Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There are other constraints such as No short selling Max amount you can buy is X% of Max outstanding amount in market Maximum exposure to a perticular country , issuer, Sector , currency etc is constrained Maximum portfolio turnover is within a certain limit. Transaction Cost (Defined as function of DV01 Bid-Offer Spread) is within a range
- Instead of the objective function being Yeild/OAS or any other measure of return we can also try minimize functions such as RWA(Risk Weighted Assets) , Basel 3 Capital required etc. These problems have similar set of constraints as the previous one.
- I am attempting to solve a dynamic optimization exercise where we would have re-balancing based on a simulated environment of rates , inflation , fx etc.
Most of these are not purely Markowitch type and I end up using Linear / Quadratic programming based on the use case.
Hope this helps you in some small way.
## Answer by Ulysses (score 1)
https://quant.stackexchange.com/a/15510
One more kind of problem in your basket: $$ \max_w \left(w^T \mu -q \cdot w^T \Sigma w\right) $$ where $q\geq 0$ is a risk-aversion parameter. In case $q\to\infty$ you are extremely risk-averse, and you minimize the variance without caring about the mean. If $q =0 $ you are risk-neutral, and you're only interested in maximizing the mean. You can put all possible linear constraints on top: $Aw \leq b$, $A'w = b'$ etc. and they will all fall into the class of quadratic optimization problems, that are very-well studied in math - in particular, most of them won't have nice analytic formulas for $w$, but can compute it numerically rather fast.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.