# Tag Info

16

Basically, prices usually have a unit root, while returns can be assumed to be stationary. This is also called order of integration, a unit root means integrated of order 1, I(1), while stationary is order 0, I(0). Time series that are stationary have a lot of convenient properties for analysis. When a time series is non-stationary, then that means the ...

16

Intro: Duration-Based Asset Pricing Similar to bonds, we can define the duration of stock $i$ as $$Dur_{i,t} = \sum_{s=1}^\infty s\cdot\frac{\mathbb{E}_t[CF_{i,t+s}]e^{-s r_{i,t}}}{P_{i,t}},$$ where $P_{i,t}$ is today's stock price, $r_{i,t}$ a discount rate and $CF_{i,t}$ are cash flows. The variable $Dur_{i,t}$ tells you the weighted average of when a ...

13

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...

10

The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow. There isn't necessarily a difference between these quantities in a long-only ...

10

Looking at transaction prices, they would occur at the market bid if the active part is a seller, and at the ask if the active part is a buyer. With a random flow of sellers and buyers, the price will bounce between the bid and ask prices, creating a negative autocorrelation in returns. This penomenon is known as the bid-ask bounce, and has been discussed ...

9

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...

8

We actually managed to come up with the answer to this question ourselves but wanted to share the answer since it might be relevant to others as well. The calculation depends on what method is used to calculate the cost. There is the FIFO, LIFO and the average cost method, see: http://www.accounting-basics-for-students.com/fifo-method.html If FIFO or LIFO ...

8

An easy way to perform what you need is do it this way: if your data are daily then : > prices <- data$cl > log_returns <- diff(log(prices), lag=1) would provide you with daily log returns, if you change the$lag=1$to$lag=5$then you will get weekly moving log returns. 8 The correct answer has some intuition though it doesn't generalize to continuous time very easily: Think about the paper below like this:$Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)$The generalization is slightly hard because the dynamics of$\mu$and$\sigma^2$could be dependent for arbitrary returns. You can use a GMM estimator to derive the asymptotic ... 8 Perhaps overly simplistic and repeating the pt above, but when doing statistics, ideally we want to compare like with like. Returns can be comparable with each other. Prices on the other hand always depend on the previous price. 8 Usually the formula for the sample variance of a stock is given by: $$Var(R_{i}) = E (R_t - E(R_t))^2$$ If you are using daily data to compute the variance then the second term:$E(R_t) \approx 0$, therefore you can drop it from the computation. Which yields: $$Var(R_{i}) \approx E (R_t)^2$$ ... 6 The answer is that it depends. In addition to the Lo paper above, there are a number of excellent references that go into depth about annualizing or time scaling non-i.i.d. returns, one of which is Roger Kauffman, "Long-Term Risk Management", 2005 which can be found at http://www.rogerkaufmann.ch/all-Budapest.pdf. There are some well known cases where the ... 6 You cannot use the clt to test something, it is a theorem about convergence. You can only use a statistical test to test something which basis is in many cases the clt. In this case you could e.g. use a so called t-test. In R you would e.g. type: t.test(data.Rb,data.Ra) to test whether the difference in the means is significant. 6 Yes, you are correct on both terms - it doesn't make much sense, and there exists a well-cited solution by C. Israelsen: "A refinement to the Sharpe ratio and information ratio." Journal of Asset Management 5.6 (2005): 423-427. The adjustment he gives is to define $$SR_{adj} = \frac{r}{\sigma^{\frac{r}{abs(r)}}},$$ which solves the ranking problem during ... 6 What you describe is known as the Equity Premium Puzzle - and it really is, as the name says, a real enigma: "The equity premium puzzle (EPP) is a phenomenon that describes the anomalously higher historical real returns of stocks over government bonds." Source: https://www.investopedia.com/terms/e/epp.asp#ixzz5HlCdHS2Z A good first introduction can be ... 6 GARCH models have little to do with the economics of the data generating process of the series you model, so both returns and excess returns (and log-returns, and inflation-adjusted ones, even ones measured in yen!) are valid input. However, there is usually the conditional mean equation besides the variance equation in a GARCH set-up, and your risk-free ... 6 Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha. CAPM The CAPM is an economic model for asset pricing. It states that the equation $$E[r_i - r_f] = \beta_i E[r_m- r_f]$$ holds for any asset$i$.$r_i$denotes the return of asset$i$,$r_f$the risk-free rate of interest,$...

6

It depends. For example, if you're doing option pricing in the log normal world returns are completely described by the mean and standard deviation. If you add jumps, you would also need to parametrize the underlying Poisson process which is fully described by one parameter and the jump size. In other words, if you have a (log)normal distribution and the ...

6

Hi: Even if returns were stationary ( which is probably dependent on the time series one is considering ), cumulative returns, where $n$ is not fixed ( as it in say a rolling sum with a fixed window size or a non-overlapping sum with a fixed window size ) definitely can't be stationary. Consider a pure noise process. $logret_t = log(P_{t}) - log(P_{t-1}) = \... 5 You are calculating the geometric mean as if these are arithmetic returns. If you let $$L_{t}\equiv \frac{P_{t}}{P_{t-1}}-1$$ and $$C_{t}\equiv log(P_{t})-log(P_{t-1})$$ then $$L_{t}=exp\left(C_{t}\right)-1$$ Thus, to calculate the geometric return on log returns, you would recognize that $$\prod\left(1+L_{t}\right)=exp\left(\sum C_{t}\right)$$ The ... 5 You're compounding correctly but the discrepancy is not just because of rounding. SMB and HML are formed as averages of 6 and 4 different portfolios, respectively. As French's website explains, this results from cutting all stocks into 2x3 SizexBook portfolios. French compounds each of these portfolios to the proper horizon (eg monthly) and then averages ... 5 In Python, simple geometric returns: import numpy as np import pandas as pd sp500 = pd.io.data.DataReader('^GSPC', 'yahoo')['Close'] simple_ret = sp500.pct_change() (1+simple_ret).cumprod()[-1] -1 0.74751768460019963 Log-returns: log_ret = np.log(1+simple_ret) np.exp(log_ret.cumsum()[-1]) -1 0.74751768460020074 In ... 5 For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference. 5 I have written an entire paper on this approach at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744 As to your specifics 1) "Volatility" as defined by variance does not exist, which is why it is changing. The first moment is undefined so the second cannot exist. See the paper as to why. Your fitted pdf will treat the outcomes as having a ... 5 Consider these two simple portfolios: Portfolio 1 returns -10% in month 1 and 10% in month 2. Average arithmetic return is zero, and cumulative return is$(1-10\%)(1+10\%)=0.99$. Portfolio 2 returns -50% in month 2 and 50% in month 2. Average arithmetic return is still zero, but cumulative return is$(1-50\%)(1+50\%)=0.75$, a much lower terminal value! In ... 5 Let x represent the percent change-e.g. 2%, let k represent the number of decreases, and z the number of increases. Something like this? We want to find z such that:$\left(1-x\right)^k\left(1+x\right)^z=1$Rearrange,$\left(1+x\right)^z=\frac{1}{\left(1-x\right)^k}$And take log:$z \ln \left(1+x\right)=-k \ln \left(1-x\right)$and solve for z:$z =-...

5

You're right but a GBM doesn't assume that percentage returns are normally distributed. It's about log-returns. If the log-return $r_t=\ln\left(\frac{S_{t+dt}}{S_t}\right)$ is normally distributed (GBM assumption), then $r_t$ can indeed be any arbitrarily large (positive or negative) number with positive probability. This also implies that stock prices are ...

5

Yes, you can just do IGE - SPY if you assume the short finances the long. The Sharpe ratio will be the same whether or not you divide by 2.

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The term structure of returns refers to returns on assets with the same underlying cash flows, where the return is measured over the same holding period, but for different maturities. The price of a stock or equity index $S_t$ is given by the discounted value of its dividends $D_t$: P_t = \sum^\infty_{n=1} E_t(M_{t:t+n}D_{t+n}) = \sum^T_{n=1} E_t(M_{t:t+n}...

4

The correct answer is "arithmetic mean, because Bill Sharpe says so". He invented the thing, and he's pretty clear on which one he was looking at. If you use the geometric mean, which is lower the higher the volatility in the returns, and then you divide by standard deviation, you have essentially discounted your result TWICE for volatility.

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