9

There are a few related reasons: The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with discrete holdings is quite hard; For small retail investors fixed trading costs will swamp the rounding in your example but also for larger amounts (at least ...


6

Yes. It comes from a core theorem of statics, Stein's Lemma. It shook the foundations of the field of statistics when it came out. It blew up an entire way of viewing mathematical statistics. Although it followed from critical work by Robbins in Bayesian estimation, Stein's work is really what is now remembered. There are three primary schools of ...


6

Defining asset classes from a quantitative perspective is an interesting question that is not really addressed "officially" as far as I know. Let's try to write some requirements you want strategic decisions to make sense: each asset class should have at least one or two different "economic drivers" than the others you want tactical ...


5

This was too long for a comment, so I'm writing it as an answer. I have provided some interesting literature that will give you insight into the common pitfalls of backtesting algorithmic trading strategies. Marcos Lopéz de Prado on backtesting: Marcos Lopéz de Prado provides some very good slides giving you a quick introduction to the goal of backtesting, ...


4

As @stans already said in the comments to your question, the existence of the market portfolio hinges on the existence of a risk free rate $r_f$, where risk free, in this context, means that its value can be perfectly contracted for the relevant return horizon, e.g. you will with probability one get that rate for 1 month or 1 year. In theory, we must also be ...


4

I can list a couple of things that are very reasonable to start off with. As written in the above comment, you will not be able to find any "secret sauce" in books and journals. You will, however, be able to find some good ideas that are commonly shared in the industry. "Free" applied course: Portfolio optimization with R There exists a ...


4

Factor Portfolios are by-products of the cross-sectional regression techniques used to estimate style factor returns - the estimates of the return of a set of styles for a particular period. Style Factor Returns are estimated by regression of the returns for all assets in a single period on a matrix containing the styles for each asset in that period. So, ...


4

The first article on this was Fisher and Lorie "Some studies of variability of returns on investments in common stocks" JB April 1970. https://www.jstor.org/stable/2352105 The Statman article you quote "How many stocks make a diversified portfolio" is from 1987, it is still referenced and broadly agrees with the other. A more recent ...


3

Too long for a comment - so I add this here as an answer. Not sure what delta hedging arbitrage is but I think you define delta as a difference in price? While cross listing is not uncommon, I think AAPL is actually Nasdaq only as opposed to say IBM which is cross-listed. This is verified by Reuters when looking at IBM vs Apple If you ever were to see ...


3

In addition to classical texts by Grinold and Kahn, and the sources cited in the previous answer, I can't help mentioning my book. It has a chapter on Portfolio Construction which includes alphas at different horizons, multi-factor risk, slippage and impact costs, and Kelly bet sizing.


3

The asset returns are \begin{align*} X_1 &= \mu_1 + \sigma_1 \varepsilon \\ X_2 &= \mu_2 - \sigma_2 \varepsilon \end{align*} where $\varepsilon \sim N(0,1)$. The returns of the portfolio are then \begin{align*} X &= w_1 X_1 + w_2 X_2 \\ &= w_1 \mu_1 + w_2 \mu_2 + (w_1 \sigma_1 - w_2 \sigma_2) \varepsilon \end{align*} and the expected ...


3

The conditional notation is indeed a bit confusing for those who do not spend a lot of time with mathematics. Downside beta is computed simply by taking only those data rows for which we see underperformance, and then doing the regular beta calculation. So, for example, if you have percent returns like Asset Benchmark 5 4 -2 2 8 -1 4 6 3 1 3 5 Then ...


3

If the link is insufficient, does this work?


3

As correctly mentioned in an earlier answer, portfolio optimization is something used for books in hedge funds and other institutions. As a smooth utility function changes slowly near its maximum, perturbation by rounding is inconsequential. If you would like to adopt a utility approach to personal investing, the following algorithm could work: (1) solve ...


2

You can find a lot of good papers by just typing keywords like "deep reinforcement learning finance" in the arXiv or Google Scholar or looking at top researchers websites which provides an overflow of applications and research directions to engage with. Anyway, here are a few off the top of my head: If you are looking for a more introductory level ...


2

The problem is, what do you define as the market? You can easily construct the efficient frontier using the covariance matrix. You can easily show the efficient frontier contains the solution, maybe with or without leverage. Without the definition of the market (e.g. the cap-weighted index, tangency portfolio of the 2 assets), the question is however not too ...


2

This discussion might be of use: Asset Allocation with near zero rates. I have heard (at conferences) large, institutional investors say they are reconsidering their allocation to govys and/or IG because of ZIRP and NIRP. I don't know what criteria they use to make their strategic asset allocation decisions or whether they've implemented any changes.


2

In the meantime, Gianluca de Nard has published Oops! I Shrunk the Sample Covariance Matrix Again: Blockbuster Meets Shrinkage, which works out the idea eluded to in the comments explicitly.


2

Using some of @noob2 notation, if: $x_i$: the initial dollar amount of asset i $t_i$: the transacted dollar amount of asset i $\theta$: the fee fraction (0.2%). $w_i$: the desired post balancing weights Then you have the minimisation problem: $$ \min_{t}{f(t)} = \sum_i \left ( \frac{x_i + t_i -\theta|t_i|}{\sum_j x_j +t_j -\theta|t_j|} - w_i\right )^2 $$ ...


2

I do not think there is a closed form solution. I have applied a simple iterative method to your example problem. See below. Let $N=4$ be the number of assets, indexed by $i$ ranging from 1 to $N$. Let $x_i$ be the dollar allocations before rebalancing (in your example they are called "value (P*Q)"). Let $w_i$ be the desired post-rebalancing ...


2

I'm not sure if this truly belongs in quantitative finance, but as an actuary, I can't resist responding. The answer to your question literally fills thousands of pages of regulations, research papers, best-practice articles, and study materials. Pension funds are VERY exotic options. They're not just puts. They're puts tied to mortality, employee behavior, ...


2

You would typically find such information in a prospectus. For example the prospectus of the Vanguard World Equity fund (VGHEX) doesn't have any wording putting hard limits on industry allocation or any kind (to be honest I didn't read every page). Vanguard is only bound by what they describe in their prospectus. For other funds which are set out to ...


2

So, in short, you've told BL that Exxon sucks, but you haven't specified the same for Chevron or Conoco! So the model sells XOM, but maintains the longs in the others given the more attractive pairs-trades given their correlations ;-) Mathematically, it makes perfect sense... the question is how much (and how) to attribute to "oil stocks" versus to ...


2

Risk-free assets refer to assets with a definite rate of return and no risk of default.   From the perspective of mathematical statistics, risk-free assets refer to assets with zero variance or standard deviation of investment returns. Of course, the covariance and correlation coefficient between the rate of return of risk-free assets and the rate of return ...


2

There are two primary asset allocation decisions: Strategic asset allocation: This tends to be a long-term, passive portfolio mix that an institution would hold if there are no active views. In principle, measuring success/failure is easy – does the SAA process generate the necessary return profile to support an institution's needs/missions? For example, ...


2

To answer the first question recall that $$r_i = E[r_i] + \epsilon_i + m,$$ where $E[\epsilon_i] = E[m] = 0$ by assumption. With this, we have: $$E[r_i] = E[E[r_i] + \epsilon_i + m] = E[r_i]$$ as one would expect. To find the variance: $$Var(r_i) = E[(r_i - E[r_i])^2] = E[(\epsilon_i + m)^2] = E[\epsilon_i^2 + 2m\epsilon + m^2].$$ From this we get $$Var(r_i) ...


2

Why would you need to hedge the full notional? Your currency exposure is via the margin (if you post in GBP) and the PNL, neither of which are anywhere near as much as the notional.


2

Paraphrasing some quote: "they are different but same but still different" In reality the number of correct bets $N_c$ is the number of times the analyst was correct predicting the direction of a stock (either up or down), therefore the first formula gives information only about how the analyst performed w.r.t the direction, while the second ...


2

Below, I describe three cases: The standard $$\beta=Cov(r_p,r_m)/Var(r_m)$$ The case of a (up-)sided beta with arbitrary market return threshold $\theta$, $$\beta^+_m+(\theta)= Cov(r_p,r_m|r_m>\theta)/Var(r_m|r_m>\theta)$$ The case where we condition on your portfolio instead of the market, $$\beta^+_p(\theta)=Cov(r_p,r_m|r_p>\theta)/Var(r_m|r_p>...


2

A quant technique that could be used to (partially) address this problem is the Mean Variance Spanning Test of Huberman and Kandel (1987). Abstract This is a statistical test of whether adding K new assets to an existing set of N assets improves the Efficient Frontier or not. Roughly speaking the test involves checking whether the new assets "add ...


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