# Tag Info

7

I don't think the claim that "Lévy alpha-stable distributions are better descriptors of returns" is universally accepted. While Mandelbrot (and others before him) has correctly identified non-normality of returns in financial time series, he wasn't really equipped at the time (1963) to pursue its real nature. Appropriate models appeared only much ...

6

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 ...

6

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 ...

6

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 ...

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.

4

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 $$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ... 4 The units of returns are 'per time', while the units of variance are also 'per time', thus the units of the Sharpe ratio are 'per square root time'. See section 2.2 of the Short Sharpe Course for a discussion of units, and section 3.3.2 of the same for more information on how moments of the Sharpe are affected by the sampling rate. 4 I'll try to answer according to what I've read (and I hope mostly understood). Let's assume the mean of daily returns is 1%, and the standard deviation of daily returns is 1%. Then:$$ 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}$$The log of the change is by properties of ... 4 Yes, there is... BUT... it’s a ton of effort, that is very unlikely to ever make any material difference. The problem here isn’t so much that different calendar months have different numbers of calendar days. From any year to the next, different years will have a different number of trading days, depending on the accident of when weekends and public ... 4 As far as I understand, you're calculating the standard deviation on two different things (prices and log-returns). Assume that the values (eg. stock prices) are defined by X_t, for t=1,\ldots,T. Then the first method described above, can be formulated as: $$\bar{\sigma} = \sqrt{\frac{1}{T}\sum_{i=t}^T (X_t - \bar{X})^2},$$ ... 3 You know that : X \sim N(\mu,\sigma^2). Z = \large\frac{X-\mu}{\sigma}. \text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1. So that Z \sim N(0,1). However note that the pdf evaluated for X and Z have different domains. The following figure illustrate it : X is plotted in a) and Z in b) Their ... 3 The standard deviation (and variance) of the returns of an asset has two sources: the market beta times the market's standard deviation, and the asset's own idiosyncratic (market independent) standard deviation. Hence, an asset with high idiosyncratic standard deviation can have a high standard deviation despite a low beta. Definition of A:s beta to the ... 3 Focusing on intuition rather than theory, \beta can also be thought of as the "risk premium" of that specific asset relative to the market. In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here ... 3 Intuitively put you can say that volatility is the within variation and beta is the between variation. Within means the variation that A has within its own time-series, whereas between means between A and the index. 3 I would suggest check out the Wikipedia page first and use more stylized notations. In your update equation mean(t) = mean(t-1) + K(t) * ( price(t) - mean(t-1) ) you are basically saying that your state process is mean(t) and price(t) is a measurement of mean(t). This doesn't sound legit On the other hand, you could have a mean reverting process$$\text{...

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 ...

3

Why are you not using broker quoted implied volatilities? E.g. ICAP and TP quote all the standard expiries & tenors. If you want a reliable (i.e. executable) price you'd be better off using implied rather than historical vol. For example USD 2y5y OIS Black vol is around 40% right now (equivalent to 350bp spot premium). If that's not a concern or ...

3

Using only words and no equations: Knowing the Variance (or standard deviation) of a Brownian Motion we can calculate the uncertainty in the future position of a particle. Knowing $\sigma^2$ and assuming the particle starts at $S_0$ we can say that $S_T$ will be in $[S_0-1.96 \sigma, S_0+1.96 \sigma]$ 95% of the time. In other words 95% of the trajectories ...

3

For option pricing in the classical Black-Scholes model, you assume the underlying stock follows Geometric Brownian Motion: $$S_t = S_0 + \int_{h=0}^{h=t} S_h \mu dh + \int_{h=0}^{h=t} S_h \sigma dW_h = S_0 \exp \left( \mu t + 0.5 \sigma^2 t + \sigma W(t) \right)$$ Take the log of the solution above and you get:  \ln\left( \frac{S_t}{S_0} \right) = \mu t + ...

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

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,] # [1] 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*...

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