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5 votes

Question about adding new investment A to portfolio B

The proposition is intuitive but the proof of this is not so straight forward in my opinion. The paper Benhamou & Guez (2021), Computation of the marginal contribution of Sharpe ratio and other ...
oronimbus's user avatar
  • 1,896
5 votes
Accepted

Setup for proving equation 3.4 from Grinold

In order to derive the simplified portfolio volatility, there is also an assumption of equal variance $\sigma_i = \sigma$ for all $i$. Assume we have an equally weighted portfolio of $n$ assets with ...
Pleb's user avatar
  • 4,486
4 votes
Accepted

Terminology: "global" in "global minimum-variance portfolio"

I think it's just a common misnomer, on Google Scholar I found a 1980 article by R. Roll: "Orthogonal Portfolio's" but I doubt the term was coined there. I do think I understand why a global ...
Bob Jansen's user avatar
  • 8,562
4 votes

Question about adding new investment A to portfolio B

$SR=r/vol$ so $d(SR)=dr/vol - r/vol^2 * dvol = 0$ so that $dr/r=dvol/vol$. This is the 'indifference condition'. Let's say you add 'e~0' amount of the asset resulting in $dr=e*(r_{new}-r_p)$ and $d(...
Arshdeep's user avatar
  • 2,451
4 votes
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When optimizing a portfolio for risk parity, can any portfolio weights turn negative?

Many risk-parity implementations simply use the inverse-vol rule (i.e. weights are proportional to 1 over vol), and then all weights are (strictly) positive by construction. A number of ...
Enrico Schumann's user avatar
4 votes

Setup for proving equation 3.4 from Grinold

For sake of completeness, let me add the approach using linear algebra. Let the covariance matrix $$ \begin{align} \Sigma&=\sigma^2\begin{pmatrix} 1&\rho&\rho&\ldots&\rho \\ \rho&...
Kermittfrog's user avatar
  • 6,832
3 votes
Accepted

How to construct a delta-neutral portfolio containing stocks using correlations?

The answer depends on your definition of "neutralise". For some people it is a matter of being dollar neutral. Then the answer is straightforward. In your case it seems that you have also in ...
lehalle's user avatar
  • 12.3k
3 votes

Given a statistical model which predicts price, how to determine trading strategy?

I am going to somewhat answer your question but also give commentary as to why your question is difficult to answer. In the basic sense, you long when your model "says" the stock is ...
THATS MY QUANT MY QUANTITATIVE's user avatar
3 votes
Accepted

How to solve for the optimal portfolio weight with target variance?

Answer to updated question: The new expression for $\omega^*$ is also not correct. To see why let: $$ \begin{split} \mu &:= \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \textbf{1} \\ \Sigma &:= \...
Adam Cataldo's user avatar
3 votes
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PCA for portfolio optimization (Markowitz)

There is a rich body of knowledge about spectral decomposition of the covariance matrix. Gatheral and Cucuringu have very readable lectures on this topic. I also found a larger and more formal review ...
Adam N.'s user avatar
  • 233
3 votes
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Proof of weights maximizing sharpe of a portfolio

If the optimization problem is: $\max_{w} \frac{w'\boldsymbol{\mu}}{(w'\boldsymbol{\Sigma}w)^{\frac{1}{2}}}$ constrained to $w\textbf{1}=1$, then we just use Lagrange multipliers: $$L = \frac{w'\...
THATS MY QUANT MY QUANTITATIVE's user avatar
2 votes
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Interpretation of optimal weights in portfolio for risk-adjusted return maximization

With regards to your question about the denominator, if we expand the denominator, it becomes: \begin{equation} w_1 = \frac{\mu_1\sigma_2^2 - \mu_2\rho\sigma_1\sigma_2} {\mu_1\sigma_2^2 - \mu_1\rho\...
KaiSqDist's user avatar
  • 1,474
2 votes

If Kelly and tangent portfolios have the same weights, do they differ only empirically?

So to begin with the terms mentioned ,The Kelly portfolio is based on optimizing the growth rate of wealth over the long term, considering the probabilities of different outcomes. On the other hand, ...
Shivam Singh's user avatar
2 votes
Accepted

How to find the expression for the SDF and solve this exercise?

Let me write this problem without the vector notation since it is easier: $$ \max E[v(c_0,c_1)] $$ s.t. $$ \text { subject to } c_0+\sum_{i=1}^n \theta_i p_i=w_0 \text { and }\tilde{c}_1=Y+\sum_{i=1}^...
phdstudent's user avatar
  • 8,431
2 votes

Portfolio Optimization with ETFs and Futures

I did the same experiment using yahoo finance to get charts. I took the SPY, the VIX and third ETF for comparison. First have a look at the volatility of the 3 vehicles: You can see that the VIX has ...
lehalle's user avatar
  • 12.3k
2 votes

Volatility Tax/Variance Drag and Drawdowns/Breakevens

Assume the standard deviation of daily returns is $\sigma$. If the market is up the typical amount one day and down the typical amount next day, the 2 day return is $(1+\sigma)(1-\sigma)-1= -\sigma^2$....
nbbo2's user avatar
  • 11.5k
2 votes

Application of Leverage in Different Interest Rate Environments to an Efficient Portfolio

The composition of the tangency portfolio in standard mean-variance analysis does depend on the risk-free rate. The degree of that dependence depends on whether or not we hold the expected returns of ...
RRL's user avatar
  • 3,700
2 votes

Optimal weights in portfolio after rebalancing

It depends what your optimization problem is. The simplest would be return maximization: $$\max_{w \geq 0} w^\top x \text{ subject to } \mathbf{1}^\top w$$ This is a standard linear program, and the ...
msantama's user avatar
  • 151
2 votes

Bonds in a zero interest rate environment

In addition to @dm63's answers, it's worth noting that there won't necessarily be a free, zero-risk alternative. Bank accounts are only insured up to a certain balance (depends on where the bank is ...
Rylan's user avatar
  • 635
2 votes

Bonds in a zero interest rate environment

Couple of possibilities A) at the time, many believed that rates would indeed stay low for a long time and/or would actually go negative. Seems ridiculous now but that is with hindsight. B) many ...
dm63's user avatar
  • 17.2k
2 votes

Excess Return Covariance Matrix is Singular - Cash return and risk free rate are the same

I believe this answers your question? "Adding" risk-free asset to covariance matrix after the fact but the answer is replacing cash with the riskless asset. The key message is either - ...
KaiSqDist's user avatar
  • 1,474
2 votes

Improving Portfolio Optimization on a Mean-Variance Basis

Yes there is a big benefit in doing research on improving MVO. After all, the tangency portfolio is the best portfolio under several not super crazy assumptions. There has been a lot of work and ...
phdstudent's user avatar
  • 8,431
2 votes

Asset rate (elasticities ?) of substitution

Your language needs to be more clear. What do you mean by a shock to asset 2? A decrease in price of asset 2, should not change your allocation and have no spillovers (since the expected return of ...
phdstudent's user avatar
  • 8,431
1 vote

Reverse Optimization: finding the returns that satisfy specific weights given one known return

OP takes Black-Litterman reverse optimization expected returns, overrides one of the values with a different expected return, and finds that the resulting portfolio didn't match the original one. This ...
John's user avatar
  • 5,421
1 vote

Reverse optimization: How to generate the expected portfolio returns given the weights and a series of constraints on those weights?

This is not yet an answer, but too long for a comment. Let's start without the box constraints and solely impose $\sum_iw_i=1, i.e. w^T\mathbf{1}=1$: $$ \begin{align} \max_w\quad & w^T\mathbf{\mu}-...
Kermittfrog's user avatar
  • 6,832
1 vote
Accepted

If an option is undervalued, how does shorting a portfolio generate profit?

Question 1: Why do we short one call option? Why do we not long a call or short a put? You could do the other combinations, but then you would have to: Short Put > Short Stock Long Call > Short ...
KaiSqDist's user avatar
  • 1,474
1 vote

Question about marginal risk contribution / portfolio volatility decomposition

The numerator in that formula is the covariance between asset $i$ and portfolio $p$ where the latter is written as a weighted sum of assets. The fomula also employs the following facts: $\text{Cov}(...
Richard Hardy's user avatar
1 vote

Constructing a Corporate Bond portfolio?

The best thing you should do is to download returns of a corporate bond portfolio, and then create a diversified bond portfolio whose beta against the benchmark is such that it yields an expected ...
phdstudent's user avatar
  • 8,431
1 vote
Accepted

Why not inequality constraint in mean-variance portfolio optimization?

I'm not sure if I understand your question correctly. I'll try to answer, but you ay want to clarify what you're asking. I'll review portfolio optimization and constraints. Typically, you have a ...
Dimitri Vulis's user avatar
1 vote

Calculate minimum variance hedge ratio for foreign-denominated asset hedged to domestic currency

From what I understand about your problem, you are a EUR investor looking to hedge the downside risk of USD depreciating against EUR such that returns earned in a USD ETF are worth less in your ...
KaiSqDist's user avatar
  • 1,474

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