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5

The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$. Thus we need a $q$ such that $$ P[L>q] = 1-\alpha, $$ where we can imagine $\alpha=99\%$ and thus we need the starting point of the $1\%$ tail. Because we have a probability of a loss of size $0$ of ...


4

If the loss distribution is normal with mean $\mu$ and variance $\sigma^2$, then the Value-at-Risk and Expexted Shortfall (or CVaR) at level $\alpha \in (0, 1)$ are \begin{align*} \mbox{VaR}_\alpha & = \mu + \sigma \Phi^{-1}(\alpha) , \\ \mbox{ES}_\alpha & = \mu + \sigma \frac{\phi\{\Phi^{-1}(\alpha)\}}{1 - \alpha} , \end{align*} where $\phi$ ...


4

First, I am quite sure that this is a typo and it should be $$ 0 < VaR_1 < VaR_0 $$ then $$ -VaR_0 < -VaR_1 $$ and the plot is correct. Second, the put strategy does not change only the expected profit but the whole distribution of the P&L. If you buy a put with strike $K_1 = -VaR_1$ then you get compensated for losses below $K_1$. But you ...


3

As a short summary and adaption of the question: You better redefine $\hat{r}_i= \frac{S_{i-1}}{S_1}-1$ and $\hat{S}_i = (1+\hat{r}_i)S_0$. The above definition of $\hat{S}_i$ yields a sample of potential values for $S$ for the future day. This approach is usually applied in historical simulation. The aim here is to use information of the past about the ...


3

The standard approach is to multiply by the square root of the number of trading days in a year. If you assume there are 250 trading days in the year, you multiply by $\sqrt{250}$. Investopedia is one source explaining this approach.


3

You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that $$ \operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\} $$ and since in your case $X = X_1+X_2$ you just need to compute $F_X$ in terms of $X_1$ and $X_2$. For the notation of partial derivatives, I denote the generic ...


3

It depends on the method by which you calculate VaR. Some models (t-distributuion, normal) lead to a form of VaR such that it is just scaled volatility: $$ VaR = c \sigma $$ with some proper $c$ (e.g. $q_{\alpha}$ in the case of normal, bit more complicated for the t-distribution). Then as $\sigma$ scales with square-root-of-time so does VaR. If VaR is ...


3

Value at risk is quoted by absolute value. This is the amount of money you can lose, so everyone knows the sign by default. For the second question, the last line explains it. Probability of at least one of the assets losing money is ~9.6%. Probability of both losing money is pretty small and is ignored. So, since 9.6% > 5%, it means that you lose on one of ...


3

If $z_\alpha$ is the so-called standard normal $z$-score of the significance level $\alpha$ such that $$ \frac 1 {\sqrt{2\pi}}\int_{-\infty}^{z_\alpha} e^{-\xi^2/2}d\xi=\alpha $$ and we assume normality, (ignoring skewness and kurtosis,) then we can estimate the $\alpha$ quantile of a distribution with cdf $\Phi$ as $$\Phi^{-1}(\alpha)=\mu + \sigma ...


3

Skewness decays with time, but the rate of that skewness decay will vary based on the instruments and how they are traded, so a simple estimator such as the square root of time rule is not appropriate. I typically recommend that to scale VaR or ES it makes more sense to lower your confidence level (raise the alpha parameter) to one that makes sense for your ...


3

the risk neutral drift is needed for pricing of derivatives. For a $100\%$ equity portfolio you can take the real world drift - sometimes a good guess is a drift of zero. For fixed-income you could do the same and might need more sophistication for the variance term. If you have short-dated bonds then you will need a special model for the pull-to-par. For ...


3

VaR is not a good measure of risk taking, in my opinion. It suffers from inherent faulty assumptions (check out VaR Wiki to start) and it omits many other important aspects of risk measurement. When I evaluate an asset's risk and return I like to start looking at the following: Historical risk and returns of an asset. This leads to the Sharpe Ratio, ...


3

You got some things wrong: You don't have to devide sd by $\sqrt{n}$, the division is already part of the definition of $sd$. The $t$ distribution has a parameter $\nu$, the degrees of freedom. The variance of a standard $t$ distributed random variable $T$ is $$ VAR(T) = \nu/(\nu-2). $$ Thus you have to define $\sigma = sd * \sqrt{(\nu-2)/\nu}$ and a ...


2

The time scaling of higher moments for ordinary (discrete) returns as per the Wingender paper is illustrated in Excel and VBA in the following spreadsheet demonstration files: Terminal-Wealth-Time-Horizon-Calcs-Normal-and-Modified-VBA and; Liqudity-VaR-With-Correct-Time-Scaling-of-Higher-Moments Available here For more on the weaknesses of the Cornish ...


2

There is a formula for calculating ES from a normal distribution. There is also a formula for ES of arbitrary distributions using a Cornish-Fisher expansions (easy for univariate processes but frustrating for multivariate). However, the most common approach is a scenario representation of the distribution. This could include using the historical distribution ...


2

I don't know what you did when you tried pulling out $1-\alpha$, the correct expression would be $\lim_{\alpha \to 1} \frac{\mu(1-\alpha) + \sigma {\phi^{-1}(\alpha)}}{(1-\alpha)(\mu + \sigma \phi^{-1}(\alpha))}$. Anyhow, you can try using the substitution $\Phi^{-1}(\alpha) = x$, $x \to \infty$ and $\alpha = \Phi(x)$. Then the expression becomes ...


2

It is correct! You can also see it this way: $$ \text{CVaR}_\alpha(X)=\mathbb{E}(X|X\leq \text{VaR}_\alpha(X)) = \frac{\int_{\mathbb{R}} x\cdot 1_{X\leq \text{VaR}_\alpha(X)}dF(x)}{\int_\mathbb{R}1_{X\leq \text{VaR}_\alpha(X)}dF(x)} = \frac{1}{\alpha} \int_{-\infty}^{\text{VaR}_\alpha(X)}xdF(x) $$ The sign problem still remains (in both versions). If you ...


2

These are identical definitions of ES. It's just a matter of expressing losses as negatives or positives. First definition Notice the integral bounds are $a$ and $1$: losses are positive; this is so-called Loss(+)/Profit(-). Here alpha might be 95%, as in 95% confidence VaR or ES. Second definition Losses are negative, and the corresponding quantile is ...


2

Note that \begin{align*} \mathbb{E}\big(L \mid L\geq q_\alpha(L)\big) &= \frac{\mathbb{E}\big(\pmb{1}_{\{L\geq q_\alpha(L)\}} L\big)}{\mathbb{P}\big(L\geq q_\alpha(L) \big)}. \end{align*} The formula follows immediately.


2

The best solution is to matrix-price these bonds first. For each bond, either find a comparable bond or use your own judgment to determine the appropriate spread to a benchmark curve (e.g., OAS to LIBOR), then use the daily LIBOR curve and the corresponding OAS to obtain the daily prices.


2

Better to compute it by yourself either using Historical simmulation, Monte Carlo, or simple parametric method such as variance-covariance. Alternatively subscribe toBloomberg Risk Analytics, populate the ISIN(s) for your ETF(s) and get the relevant metrics.


2

I won't base my answer on your example as i couldn't understand what you mean. Firstly, when you ask a question "what is better?" you should address this question to the model and not the output values. Secondly, model is good only when it as accurately as possible explains the reality (with a degree of confidence). The model is useless if it ...


2

I think you are right. What he calls the approximation is the correct amount, the other is an approximation.


2

As you and @Malick noted, VaR only gives a certain threshold given a certain confidence but says nothing about what happens beyond that point (tail risk). For loss distributions with long tails, this would underestimate the risk. Regarding VaR having a problem with diversification - VaR is technically not a coherent risk measure. In simple terms, we would ...


1

After your remarks: So you have 3 lines of business and calculate VaR's for them: $$ VaR_{99.9\%}(L_1) ,VaR_{99.5\%}(L_2) \text{ and } VaR_{99\%}(L_3), $$ so if we speak in terms of events you model at different events - once an event of $0.1\%$ probability and so on. Thus mathematically in my mind it does not make sense to add these VaRs up and see it as ...


1

Not too sure about the second part of your questions but as far as VaR, R has some pretty neat functions. First I took you subset A and converted it to discrete returns since using actual prices for VaR may be a bit harder to interpret. # Load PerformanceAnalytics for VaR & Calculating Returns library("PerformanceAnalytics") # Calculate Returns a ...


1

The most commonly used approach is multiplication by the square-root of T, 19.1 in this case. This assumes no autocorrelation, however (Markov process). Interest rates tend to show a mean reversion, so the number would be smaller than 19.1. Other cases could show the oppoite effect if there are positive feedbacks. In both of these cases, a simple time ...


1

It might help to think of the two as special cases of $$S_{i+1}-S_i = \sigma (c+S_i)^\beta \epsilon$$ which looks like a Constant Elasticity of Variance extension. Taking squares of both sides and then logs will (nearly) linearise it, allowing you to carry some basic estimation using OLS. The parameter $c$ will control the lower bound and can impose some ...


1

When you backtest VaR, you are essentially backtesting your model. You are essentially saying: "If I had this model back in 2007, what would be its calculated VaR?" The model is tested given all available information up to that time. A good model will adapt given new data and the VaR will change after an extreme event. Even one data point should make a ...



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