21
Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
14
I took a quick look at Matlab's Financial Toolbox and attempted to map the features to corresponding Python packages –
For asset allocation, portfolio optimization, and risk analytics:
Standard packages such as scipy provide a large number of optimizers that should suit your needs. There are also pre-canned packages that do portfolio optimizations more ...
11
Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero.
Discussion
A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ...
9
There has been a split in the community ever since Mandelbrot published his paper "On the Variation of Certain Speculative Prices."
See:
Mandelbrot, B. (1963). The variation of certain speculative prices.
The Journal of Business, 36(4):394–419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
9
The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years:
The standard model (Markowitz, 1952, 1959) originally only considered:
Constrained model where short sales are forbidden
Only risky assets considered for investment (no risk-free asset)
Scenarios that the mean-variance model ...
8
It is more complicated than that: It is not the optimization per se that leads to inferior results but the data you use.
Kritzman et al. makes a strong case in defense of optimization vs. 1/N in this popular paper:
In Defense of Optimization: The Fallacy of 1/N, Financial Analysts Journal, Vol. 66, No. 2, 2010 by Mark Kritzman, Sebastien Page and David ...
7
I am very happy with the following equivalent formulation for the risk budgeting problem (as presented in Bruder, Roncalli, 2012, Managing Risk Exposures using the Risk Budgeting Apporach):
Let $b_i$, $\Sigma_{i=1}^n b_i =1$ be the risk budgets, $y_i$ the unscaled portfolio weights and $S$ the variance covariance matrix and $c$ arbitrary.
$$ y^* = \text{...
6
Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 there is 100% certainty that he will pay you back in a year. Before the pay date, he can do whatever he wants with your $100, while you have no access to it. Even ...
6
In recent years there has been much attention given to defining indexes other than market-cap based indices. While market-cap based indices approximate the theoretical Market Portfolio enshrined in textbooks, some people believe we could do better than that. One popular idea is that "market indexes overweight the most overvalued stocks", though this is ...
5
I would create categories, and work on risk parity among the categories.
Otherwise, variance is not really a good measure of downside risk:
Change your risk measure, use a rolling window historical VaR or Expected Shortfall at some horizon that matches your investment style. downside semi-variance could do the trick too if don't want to change your algo ...
5
There are very powerful software solutions out there, so you should not reinvent the wheel.
One notable R package is PortfolioAnalytics.
You can find a very good introduction here, where your concrete constraints requirement is addressed in section 3.3, p. 6:
Benett, R.: Introduction to PortfolioAnalytics (2015)
5
Many long term investors use historical events and the market moves associated with such events to stress test their portfolios. For example, they use the dot-com bust, the latest "great recession", LTCM, Asian Crises, Black Monday, etc and any other dramatic events in history and see how their portfolios would have performed under those conditions. Of ...
5
I'll add some comments, recognizing that 1) they are highly opinionated, and 2) they don't actually offer any real solutions. Hopefully more thoughtful and useful answers will emerge.
First of all, purely from a philosophical perspective, I have to admit that I sometimes find these discussions on strategic asset allocation (SAA) "strange." ...
4
Another approach to construct a risk parity portfolio would be to use the formulation proposed by Spinu [1]: $$\begin{array}{ll}
\underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}\Sigma\mathbf{w} - \sum_{i=1}^{N}b_i\log(w_i)\\
\textsf{subject to} & \mathbf{1}^T\mathbf{w}=1.
\end{array}$$
where $\mathbf{w}$ is the vector of portfolio ...
4
This however, goes against the conventional wisdom that variance becomes smaller as you hold the portfolio longer.
Which conventional wisdom says this? If the variance decreases with time, then the likelyhood of getting a return close to the expected return increases (Cecbycev's inequality). So you are telling me, I know more about the long-time future as ...
4
In response to the original question:
Drawdown optimization is a convex problem, see our recent article:
http://ssrn.com/abstract=2430918
We do not address the issue of choosing a "good" risk model to feed the optimizer. However, even when using the history, drawdown does capture something that volatility and expected shortfall do not account for, namely ...
4
This problem is from the exercise for Chapter 2 of Kerry Back's Asset Pricing Book. The setup of the problem is rather simple. You want to
\begin{equation*}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \phi'\mu + \frac{1}{2} \alpha \phi' \Sigma \phi\\
& \text{subject to}
& & 1'\phi = w_0
\end{aligned}
\end{equation*}
The ...
4
As a start: I am sure some asset managers don't know too much mathematical finance and do a good job. They exist.
On the other hand as a mathematician I see mathematics (and classicial mathematical finance - math.fin.) still in various areas and sometimes it is not that easy to draw the line. Let's look at examples:
If you think of stocks then mathematical ...
4
Many pension funds use projected asset class returns (capital market assumptions or CMAs) and backward-looking estimates of volatilities and correlations to set the strategic asset allocation. A 10-year period for the return projection is typical. The determination of actual weights is more or less an exercise in constrained mean-variance optimization.
...
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 take it that 5 out of 10 assets is just an example, because in this case all combinations could easily be checked.)
Here would be an example how to do it in R with an algorithm called Threshold Accepting.
library("neighbours") ## https://github.com/enricoschumann/neighbours
library("NMOF") ## https://github.com/enricoschumann/...
3
The skewness and kurtosis values you obtain appear to be of realistic magnitude.
In general higher frequencies are more non-normal, i.e. have higher skewness and kurtosis. If non-normal returns are aggregated the central limit theorem starts working and the return distribution coverges to a normal. Convergence can be quite slow under fat tails.
You can try ...
3
In this case it is important to differentiate between a liability-driven investment strategy (LDI) and a (the classical) benchmark-driven investment strategy. The first one is what you need in this case.
LDI was first established by Martin Leibowitz in 1986 ("Liability returns: A new perspective on asset allocation"). So googling that might help you already....
3
Portfolios for some kind of investors effectively balance asset investments with liabilities incurred. Think about a pension account, where the future liability of the pension payment represents the liability and the currently invested monies are the assets. I am sure you can think of other similar situations but I will illustrate regarding pensions below.
...
3
The risk free rate is important and the reason for the inclusion and consideration of the risk free rate is that investors do not get compensated for not taking on risk. Now, we can argue whether the risk free rate truly provides risk free returns (we all should know that it does not, but ...) but it is important in the context of pricing risky assets that ...
3
This question has already been answered on Stack Overflow. As it is important to Quant Finance, so I have added R code here. Others users may add code of other programming software to simulate ARMA(1,0)-GARCH(1,1) model.
sim.GARCH <- function(
horizon=5, N=1e4,
h0 = 2e-4,
mu = 0, omega=0,
alpha1 = 0.027,
beta1 = 0.963
){
ret <- zt &...
3
I believe the question to be too vague to be a good interview question. If you want to do Mean Variance Optimization (MVO) it's hard to see the point of Monte Carlo simulation. One of the good thing of MVO is its analytic tractability. Clearly, the topic is not widely discussed as this Google Search has this question as the first result (I was in incognito ...
3
The basic approach is as follows:
When you estimate the HMM you estimate three things:
When you are in which state
The drifts of your assets
The covariance matrices of your assets
You would then take 2. and 3. for each state (1.) and feed it into your favourite allocation optimizer to estimate your optimal portfolio for each state.
Voila!
3
I would say that an important part of the trading cost is ultimately shaped by the balance of informed (those with private information) and non-informed (liquidy traders) market participants. Imagine a market maker in a market with only non-informed traders that randomly sell att the bid and buy at the ask. The cost for the market maker would be small and ...
3
Markowitz' method for mean-variance optimization was originally intended to demonstrate the "free lunch" of diversification. In that regard, it was and is still very successful. The method gained wider notoriety due its articulation within the Capital Asset Pricing Model (CAPM) (a-la the Sharpe-Black-Lintner Model). The CAPM was really the first application ...
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