# Tag Info

6

I personally use the simple Garch(1,1) for volatility filtering in the risk management area. In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...

3

Using https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Solution $$X^i_t = (X^i_0 + \int_0^t\sigma_i e^{a_i u} dB^i_u)e^{-a_it}$$ and $$X^i_t-\mathbb{E}[X^i_t] = e^{-a_it} \int_0^t\sigma_i e^{a_i u} dB^i_u$$ and thus : $$\text{Cov}(X^1_t,X^2_t)=\mathbb{E}\left[e^{-a_1t} \int_0^t\sigma_1 e^{a_1 u} dB^1_u e^{-a_2t} \int_0^t\sigma_2 e^{... 3 \sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b} with \rho_{ab}=-1 the term under the square root simplifies to (\omega_a \sigma_a-(1-\omega_a) \sigma_b)^2 which is equivalent to (-\omega_a \sigma_a+(1-\omega_a) \sigma_b)^2 therefore \sigma_p=\omega_a \sigma_a-(1-\omega_a) \... 3  My "answer" below is not a really an answer for I have completely misinterpreted your original question. I thought you asked about the covariance of 2 processes over a given time horizon (i.e. for a fixed \omega) and not the covariance of two random variables (fixed t). Also note that \text{cov}(x,y)=0 does not mean that x and y are ... 2 Interesting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical Simulation (FHS), etc.). 1 - EGARCH, merely because GARCH models do not take into account so-called leverage effects, which is crucial to me for skewed and ... 1 Hint: By Integration, we have$$x_t=x_{0}+\int_{0}^{t} \alpha_1(x_s,s)ds+\int_{0}^{t} \beta_1(x_s,s)dW_1(s)y_t=y_{0}+\int_{0}^{t} \alpha_2(y_s,s)ds+\int_{0}^{t} \beta_2(y_s,s)dW_2(s)$$then$$E[x_t]=x_0+E\left[\int_{0}^{t} \alpha_1(x_s,s)ds\right]E[y_t]=y_0+E\left[\int_{0}^{t} \alpha_2(y_s,s)ds\right]$$Now we apply Ito's lemma$$d(x_ty_t)=...

1

In case of 2 securities, each and every combination of portfolio lies on efficient frontier. In your question, you have given to achieve expected return of exactly 10%. So, we have $$E(R_p)=w_1E(K_1) + w_2E(K_2)=0.10 \tag{1}$$ subject to: $$w_1 + w_2=1 \tag{2}$$ Solve your equation 1 and 2 to get $w_1$ and $w_2$. Resulting weights would lead to minimum ...

1

You most probably don't want to estimate the covariance of prices but rather the covariance of returns. Thus for equities you can take the return of the traded price. For bonds: if the maturity is long enough (say bigger than 2 years), then you can take the returns of traded prices. The pull to par should not be too relevant here. if the maturity is short ...

1

The first objective is to minimize the variance by choosing a proper control variate. First note that an expectation value is just a constant, so the covariance between an expectation value and a random variable is zero: $$\text{Cov}\left(\mathbb{E}[Y], X\right) = 0$$ Similarly for the variance of an expectation value, $\text{Var}(\mathbb{E}[Y])=0$. The ...

1

I agree on Richard. the simpler you choose, the better it is so as to get reliable estimates. What's your data frequency? purpose? For model construction as far as I am concerned, daily data from 2010 is enough. Otherwise, you could use a proxy asset for asset D depending on its nature. To clarify, if D is an ETF let's say CAC 40 ETF, concatenate its return ...

1

You can consider old prices for Stocks B, C and D to be "missing data" and apply techniques used by Statisticians to deal with such missing data. One approach, the EM algorithm, suggests you estimate the covariance for the common period, use that covariance matrix and the available data to generate pseudo data for the back period for the third stock and re-...

1

The short answer: Take all time series starting from 2010 (at most). The covarianc-matrix tells you something about the assets for a certain amount of time. E.g. if I estiamte the covaraince matrix of those 4 assets taking into account data from the last year (!) then I can expect that this matrix remains valid for the coming 1-3 months - if the markets don'...

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Are your 407 stocks all different? No A and B listings contained that are strongly if not perfectly correlated? The observation that the daily covariance matrix is singular makes me wonder. You can try the package corpcor for another shrinkage estimator.

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This is the challenge for below-mean semivariance in optimization. Since the mean becomes a moving target, the observations that impact the min function change. Estrada proposed a heuristic method for optimization and Beach(2011) discusses the decomposition and semi covariances. Below target semivariance assumes investors do not change their target return,...

1

Correlation is not an asset allocation measure and thus should have nothing to do with the weights in the portfolio. What you want to do is to figure out the correlation between the two portfolios using: p = cov(x,y) / stdev(x) * stdev(y) and depending on the results, you can then run a solver function to find out the weights in each sub portfolio that ...

1

One of the more prominent proponents has a current paper on the reasons - and why this anomaly "will persist": Why the Low Volatility Anomaly Will Persistby Eric G. Falkenstein, March 2015 Abstract Common explanations of the low volatility anomaly involve biases or frictions that cause investors to overpay for high volatility assets, giving them a ...

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