# Tag Info

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No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...

5

If you're annualising your data with T it should always be the same, not changing with the length of your data. To demonstrate, annualising monthly returns, the Sharpe ratios turn out fairly similar:- Note The reason for multiplying by root 12 is that the mean return is annualised by multiplying by 12 and volatility is annualised by m = 12. 12 on the ...

5

beta_A = correlation_A_Index * (stdd_A / stdd_Index ) The difference you see is due to correlation. The correlation between A and the index is lower than B and the index, and that's why you're seeing a lower beta. The moral of the story is that risk is subjective, and in fact you need to understand how your portfolio is correlated with these stocks in ...

5

There are sufficiently different ways to calculate the Sharpe ratio that the best advice I can give is to do whatever your boss wants. Also, if it is for a paper or research document, just make clear you document your method. My approach is usually to calculate the highest frequency Sharpe ratio I can based on the data. The higher frequency choice is to get ...

4

If $$X_1, X_2, \dots, X_{12}$$ are i.i.d. (stochastic independent identical distributed) it holds $$var(\sum X_i) = \sum var(X_i) = \sum var(X_1) = 12var(X_1)$$. now take the square root to get the stated result.

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The units of returns are 'per time', while the units of variance are also 'per time', thus the units of the Sharpe ratio are 'per square root time'. See section 2.2 of the Short Sharpe Course for a discussion of units, and section 3.3.2 of the same for more information on how moments of the Sharpe are affected by the sampling rate.

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I'll try to answer according to what I've read (and I hope mostly understood). Let's assume the mean of daily returns is 1%, and the standard deviation of daily returns is 1%. Then: $$Sharpe = \sqrt{252} \frac{mean(daily\ return)}{stddev(daily\ return)} \approx \sqrt{252} \frac{1 \%}{1 \%} = \sqrt{252}$$ Now let's assume we work with monthly returns. In ...

4

Ideally you'd want to use daily returns and just annualise it, but if you only have monthly returns then calculating the weighted variance in the following way might do it: $$Var = \frac{\sum_{i=0}^{24}(R_i - \mu)^2}{24 + \frac{21}{31}} + \frac{\frac{21}{31} (R_{25}' - \mu)^2}{24 + \frac{21}{31}}$$ $$Vol = \sqrt{Var}$$ Where $R_i$ is the returns of ...

4

As indicated by @AlexC and @amdopt, the formula is exact for log returns and approximate for discrete returns. Define the factor by which a price changes as $k$ so that price tomorrow $P_{t+1}$ is the price today times $k$ : $P_{t}*k$.Then the change in the price over a business year is $$\prod_{i \in [1, 252]}{k}$$ The log of the change is by properties of ...

4

Yes, there is... BUT... it’s a ton of effort, that is very unlikely to ever make any material difference. The problem here isn’t so much that different calendar months have different numbers of calendar days. From any year to the next, different years will have a different number of trading days, depending on the accident of when weekends and public ...

3

You know that : $X \sim N(\mu,\sigma^2)$. $Z = \large\frac{X-\mu}{\sigma}$. $\text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1$. So that $Z \sim N(0,1)$. However note that the pdf evaluated for X and Z have different domains. The following figure illustrate it : $X$ is plotted in a) and $Z$ in b) Their ...

3

Assume the weights of the two assets are $w$,$1-w$ respectively;the expected returns and standard deviations are denoted by $\mu$,$\sigma$ with subscripts 1,2,p(for portfolio),i.e,we have $\mu_1$,$\mu_2$,$\mu_p$,$\sigma_1$,$\sigma_2$,$\sigma_p$.The correlation coefficent is $\rho$ Then $$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ... 3 Focusing on intuition rather than theory, \beta can also be thought of as the "risk premium" of that specific asset relative to the market. In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here is what we would see: States:... 3 Intuitively put you can say that volatility is the within variation and beta is the between variation. Within means the variation that A has within its own time-series, whereas between means between A and the index. 3 The simplest and most common method for finding the Tracking Error of Fund X versus a Benchmark B is to compute the standard deviation of the differences in monthly returns of the Fund and the Benchmark:$$TE=\text{STDEV}(r_{X,i}-r_{B,i})$$A slightly more complicated method involves performing a regression of Fund X returns on the benchmark returns. Then ... 2 Here is an example calculation according to the formula by William F. Sharpe, 1994. The OP's method of annualising the variance (as used below), is also specified by the Committee of European Securities Regulators in this document, page 5, box 1. For this example, taking 24 months of returns of risk-free proxy (US 4-week T-bills) and an example stock, (and ... 2 It seems to me that you want to use the series of option prices to estimate the Sharpe ratio given the option prices in your sample. If so, the idea is to realise that for each option price you have at different times t_1, t_2, ... you could actually close the position and realise the profit or loss. So, basically if you have the option prices you just ... 2 Let me give you an example to show how this can happen. Suppose you invest 0.50 in a coin flip that will pay 1 on heads and 0 on tails a month later. The monthly variance will be .5*(1-.5)^2+.5*(0-.5)^2=.5 so the standard deviation will be .25. This is significantly higher standard deviation than a market index or almost all stocks. So by one measure this is ... 2 PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf 2 The standard deviation (and variance) of the returns of an asset has two sources: the market beta times the market's standard deviation, and the asset's own idiosyncratic (market independent) standard deviation. Hence, an asset with high idiosyncratic standard deviation can have a high standard deviation despite a low beta. Definition of A:s beta to the ... 2 TLDR: Beta = systematic risk Standard deviation = total risk Long Answer: There are two types of risk, systematic and unsystematic risk. Systematic risk affects the entire stock market. The recession of '08 is a good example of systematic risk. It affected all stocks. On the other hand, unsystematic risk is risk that only affects a particular security. ... 2 A pithy way to put it is "implied volatility is the wrong number to put in the wrong formula to get the right price." That is, implied volatility is by definition the parameter \sigma to plug into the Black-Scholes option pricing formula to get the market price of a vanilla option. This is called "volatility," but in reality it isn't the same as the ... 2 Another way to skin cat: # risk-free = 0 require(quantmod) require( PerformanceAnalytics) getSymbols('DJIA', src='yahoo', from = '2009-01-01', to ='2014-12-31') price <- Cl(DJIA) simple.ret <- price/lag(price)-1 table.AnnualizedReturns(simple.ret,Rf=0)[3,] #  0.7267 log.ret <- na.omit(ROC(price)) SD <- sd(log.ret)*sqrt(252) R <-... 2 This is how people usually approach calculating SR with logreturns: library(quantmod) getSymbols('DJIA', src='yahoo', from = '2009-01-01') price <- Cl(DJIA) log_ret <- log(price/lag(price,1)) mean_log_ret <- mean(log_ret, na.rm=T) sd_log_ret <- sd(log_ret, na.rm=T) rf <- 0.0025 # benchmark SR <- (252 * mean_log_ret - log(1+rf))/(sd_log_ret*... 2 What is risk? If one defines risk heuristically as deviation from expectation, then (assuming returns have finite variance) standard deviation can be considered a first approximation for risk. For most distributions the mean and variance do not fully parameterize the distribution. Some standard measures of risk for general distributions include Value at ... 2 You can calculate variance of a portfolio/basket by taking direct weighed averages of the components and then adding the relevant correlation terms * weights for each pair. Can take sqrt of the expression obtained to have Standard deviation. Exact formula for calculation goes like this : (source: benetzkorn.com) 2 Rt in your notation is "filtered" variance R(t|t). The prediction of variance R(t+1|t) adds another term which is not guaranteed to be decreasing overtime. I think another critical assumption is Ve in your equation. How do you define Ve? For price series Ve as a proxy for volatility makes sense to be time-varying, and probably exhibit some auto-correlation.... 2 I would suggest check out the Wikipedia page first and use more stylized notations. In your update equation mean(t) = mean(t-1) + K(t) * ( price(t) - mean(t-1) ) you are basically saying that your state process is mean(t) and price(t) is a measurement of mean(t). This doesn't sound legit On the other hand, you could have a mean reverting process$$\text{...

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This is only correct if the expected returns are normally distributed. Remember that z-score is in essence the quantile function or the VaR of the normal distribution. If you try to apply this to any other distribution, you are going to be sorry. Take a lognormal distributed variable$\sim \text{logN}(\mu,\sigma^2)$, the VaR in that case is $$\text{VaR}_{95\... 2 Your estimator \hat{s_i} for stock i is an unbiased estimator of its latent standard deviation \sigma_i (which is constant for your model). When applying your "window rolling" for calculating \hat{s_i}, you get a time-series ts_{\sigma_i} for each stock i. With an intercept-only OLS-regression for each time-series ts_{\sigma_i},$$\hat{s_{it}}...

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