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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 $$\beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}}$$ The idiosyncratic risk is the portion of risk unexplained ...

10

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

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

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

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 ...

6

I am a risk taker and I can say with confidence that you will never convince those individuals, you cited in your question, that they incur too much risk, because there will always be certain traders who prefer lottery tickets over longevity with the same firm (running high risk books unfortunately in the current environment runs equal to a free option; blow ...

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

Values of VaR are just the inverses of the cumulative distributions. CVaR is not a very commonly used term, its more frequently used synonym is Expected Shortfall. See http://www.maths.manchester.ac.uk/~saralees/chap17.pdf for the list of Expected Shortfall values for more than 20 distributions.

6

What happened was totally unexpected end of peg against the euro @ 1.2CHF regime that Swiss central bank aborted. See some articles about it. As far as I know nobody in the markets knew, there was no indication whatsoever.. In terms of management, I'm afraid lots of people got heavy losses, particularly banks (Austria, Poland, Hungary) - lots of Swiss loans ...

6

Different portfolio risk decompositions answer different questions. Before discussing what method to use, first ask why you want a decomposition and what definition of risk are you using. Is the point to examine how portfolio return volatility is affected by a tiny change in portfolio weights? On the other hand, if the point is to make a statement like, "30%...

5

Via Liquidity Horizons $LH$ (which have to be taken into consideration anyway when modelling for $Basel_3$) as function of the specific concentrations $c$'s. Increasing the effective maturity of the contract, $M_0+LH_0$, by a quantity proportional to its concentrations with respect to different slicings magnifies the credit risk. $M_0$ is the maturity ...

5

The following paper (Identifying Small Mean Reverting) is not directly related to portfolio risk minimization but it provides a method to build tradable mean reverting portfolios based on a multivariate co-integration approach. It has the advantages of providing a theoretical framework along with two algorithms. It also takes into account financial strict ...

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