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17

Nick Higham's specialty is algorithms to find the nearest correlation matrix. His older work involved increased performance (in order-of-convergence terms) of techniques that successively projected a nearly-positive-semi-definite matrix onto the positive semidefinite space. Perhaps even more interesting, from the practitioner point of view, is his ...


15

I can think of three reasons. First, and simplest, is that people care about variance. Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing ...


15

I guess it depends on what they're referring to... The traditional swap curve (LIBOR-based) is certainly not risk free, as evidenced by the experience of the financial crisis and the resulting migration to OIS discounting. The OIS curve (which is a kind of swap curve...) is now the standard risk-free curve. The Treasury yield curve is not favored, because ...


12

I'm just providing a global answer to the question, as I think it can be interesting for some beginners in quant finance. The properties given by TheBridge: Normalize $\rho (\emptyset)=0$ This means you have no risk in taking no position. Sub-addiitivity $\rho(A_1+A_2) \leq \rho(A_1)+\rho(A_2)$ Having a position in two different can only decrease the ...


12

E.g. on Monday you get forced to buy some Friday expiry OTM puts, say 95% strike S&P weeklies. Of course, you go and buy some delta against them to "hedge" yourself. Next thing you know, the the market tanks. Unfortunately, by Friday it's only down 3.5%, so it's does not fall far enough to reach the strike. So, on Friday expiration, you are out your ...


12

The majority of the movement in currencies is in the spot rates, rather than in the term structure. A 3-month rolling hedge would always be protecting against movements in the spot rates, no matter when they happen. Using your example, if the current EUR/USD rate is 1.3333, you might be able to get a 3-month forward at 1.3339. (Forgive me if I have the ...


11

I'm not sure about the "CAPM formula" that you are referring to. I assume you are referring to the estimated coefficient of a regression of a security on a market portfolio. That is to say \begin{equation} \beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}} \end{equation} The idiosyncratic risk is the portion of risk unexplained ...


9

The risk-netural measure has a massively important property which is worth making very clear: The price of any trade is equal to the expectation of the trade’s winnings and losses under the risk-neutral measure. This property gives us a scheme for pricing derivatives: take a collection of prices of trades that exist in the market (eg swap rates, bond ...


9

Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to ...


9

I would use the identity and three step process that: $$\textrm{Total Variance} = \textrm{Systematic Variance} + \textrm{Unsystematic Variance}$$ You can calculate systematic variance via: $$\textrm{Systematic Risk} = \beta \cdot \sigma_\textrm{market} \Rightarrow \; \textrm{Systematic Variance} = (\textrm{Systematic Risk})^2$$ then you can rearrange ...


9

There is no definitive answer to this question and there are infinite papers out there. I personally think they are better explained as mispricings. Several points: 1) Persistence of HML does not imply it has to be a risk factor. If there are idiosyncratic mispricings in individual stocks, then by construction, the ones that look cheap are going to be ...


8

A market is said to be complete if any contingent claim can be replicated by an admissible (i.e. with value process bounded from below) self-financing (i.e. all gains and losses exactly offset each other) trading strategy, a so-called replicating strategy. This strategy being constructed from primary securities - the market prices of which are unique - it ...


8

Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution. Here's another approach by Kuhn. These are all one-period ...


8

The "Component ES" section of ?ES says: For the decomposition of Gaussian ES, the estimated mean and covariance matrix are needed. For the decomposition of modified ES, also estimates of the coskewness and cokurtosis matrices are needed. The estimate of the coskewness and cokurtosis matrices are what take such a long time. You can calculate them ...


8

Great question. We would expect 3rd party risk providers to have specialized expertise (robust regression techniques, factor research, data cleansing etc.). We might grant them these advantages but still find weakness in the product design. Let's start off with the different uses of risk models and the procedure or metric which is maximized to solve for ...


8

In Oracle Crystal Ball, we use an old algorithm, that works pretty well and converges fast. It is from Iman-Conovar. Here is the reference: Iman, R.L., Conover, W.J. 1982. A distribution-free approach to inducing rank correlation among input variables. Commun. Statist.-Simula. Computa. 11, 311-334. That said, Prof. Higham's method based on optimization ...


8

The underlying problem: your ACTR constraints aren't convex The $i$th constraint on your risk contribution can be written: $$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$ And this isn't a convex constraint because of the $w_j w_i$ terms (a function $g(x,y)=xy$ isn't convex in $x$ and $y$). They're not convex constraints, so you won't be able to write them as ...


7

The short answer is that I don't know, but your question gives some hints about how to find out. The key thing for me is that you want a minimum variance portfolio. I don't think you should be thinking about some abstract mathematical operation that is "best", but rather look over a few mathematical operations and see which seems to work best for your ...


7

If you have the mathematical sophistication, you should review the original papers referenced on the Equity Premium Puzzle page, particularly Mehra and Prescott (1985). Note, however, that contrary to other opinions on this page, the puzzle is NOT that there is an equity risk premium. On the contrary, the puzzle is that the premium had been so high, at ...


7

What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at ...


7

You are absolutely right, I would say that how the interview question was posed and the example given is very misleading, if not outright incorrect. Here is why: Hedging does not increase your risk in this particular example: You take on delta exposure by buying the short dated option outright. Thus buying/selling underlying (put/call) in any case will ...


7

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B-\alpha)}\\ &=\sqrt{...


6

There are all sorts of financial and non-financial risks. I define financial risk as all risks defined from events in the financial markets that affect all participants. Non-financial risks are all other forms of risk (including risks that a particular firm may face). Financial: Market value risk (interest rate risk, exchange prices, equity prices, ...


6

The Historical Method, which I would call Historical Simulation requires that you have a reasonably clean and accurate time series of data for the underlying asset. Essentially, you are using the past performance of the asset to model its likely behaviour over a time frame of typically 1 to 10 days. Choosing and updating your time series data set needs to ...


6

This is one index I find to quite credible (Kansas City Fed Financial Stress Indicator): http://www.kansascityfed.org/research/indicatorsdata/kcfsi/


6

Take a look at historical short-term risk-free rate proxies such as Fed Funds, LIBOR, short Treasuries, and you will find plenty of periods where rates have been significantly above or below inflation (as measured by any CPI series) in the same period. In fact, controlling this difference, known as the real interest rate, is the primary tool of modern ...


6

I know you're really looking for some empirical work on this topic, but I think the following theoretical paper puts your question into proper perspective.* Risk-Based Asset Allocation: A New Answer to an Old Question by Wai Lee, JPM 2011. Overall, he finds that supposedly risk-based approaches to portfolio construction are really making implicit ...


6

Meucci covers this example precisely in his paper "Fully Flexible Views: Theory & Practice". You can find his code here for three examples related to the paper. The Butterfly Trading example covers the CVAR scenario.


6

This is the equity premium puzzle. (See that article for references.) My thoughts are that individual investors are rational to be risk-averse and demand a premium for bearing a type of market risk that cannot be diversified away. This risk is actually worse and more insidious than it appears, because "personal" circumstances tend to correlate in ...


6

Actually, Ralph Vince's Leverage Space Trading Model does utilise draw down. A short introductory pdf is available here, and the R-forge package is here. Briefly, a genetic algorithm is used to model the maximum expected portfolio return based on a joint probability distribution of the portfolio component returns, subject to an overall maximum draw down ...


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