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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 ...


7

Not sure about all of the complicated math and programming above, but I can tell you that, if you want to calculate for 1 Standard Deviation from the current stock price X days away, the following calculation will give you a +/- value from the current stock price. 1 StdDev Move = (Stock Price X Implied Volatility X the Square Root of 'how many days') all ...


6

I guess what they are trying to say here is that, assume you have two time series $X$ and $Y$ which are exactly the same i.e. $X=Y$, the correlation is : $$\rho_{X,Y}= \frac{Cov(X,Y)}{\sigma_X \sigma_Y}\overset{X=Y}{=}\frac{Cov(X,X)}{\sigma_X \sigma_X}=\frac{\sigma_X^2}{\sigma_X^2}=1$$ Now assume a time series $Z=2 \cdot X$, you have: $$\sigma_Z=2 \...


4

I did not look at the data, but recall that beta is a parameter in the following equation: $$ r_A = \alpha + \beta r_B + \epsilon $$ relating two returns (random variables, samples) $r_A$ and $r_B$. To calculate beta you peform $$ \beta = \frac{cov(r_A,r_B)}{var(r_B)}. $$ Thus if assets $A$ and $B$ exchange roles, then only the denominator changes. In your ...


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.


4

You can use refined methodologies but if you just need a rough estimation of liquidity, you can simply use an average of daily volume over N days. In practice, for equities, people tend to use N = 20 or 30. Once you have the average daily volume (say 100,000 shares), you compare it to your holding (say 50,000 shares) to determine the the size of your ...


4

Use the call put parity : $$C(t,F_{t,T},T,\sigma,K)-P(t,F_{t,T},T,\sigma,K)=F_{t,T}-K$$ where $F_{t,T}$ is the forward rate(underlying), $K$ is the strike, $t$ the valuation date, $\sigma$ the model volatility, $T$ is the maturity. Differentiate the equation with respect to $\sigma$, and you will get the result wanted.


4

"How do they choose the forex price number's precision?" By convention. And/or by vendor, e.g. BBG has 4dp for some pairs where Oanda has 5. And of course convention for forwards differs from spot, even for the same pair. There is some colour on the move to 'fractional pips' in the book "Why Aren't They Shouting?". EDIT: perhaps also ...


3

It depends on the ratio you are looking at. Most of them are scaled by $\sqrt{12}$, but the Treynor index is a bit different and is scaled by $12$. Sharpe and Information ratios are both ratios of average returns to standard deviations. They are annualized by assuming that the monthly returns are IID. Hence, average monthly return is scaled up by 12 and ...


3

Due to the leap year 366 days need to be used here to match UST conventions (which is ACT/ACT). In this case it doesn't matter whether your interest period extends to only 1 day after the 29th of February or, e.g., 200. In fact if you look at the daycount description of the bill it says: "the day count basis for price and yield calculations is 365 ...


2

The only correct way is using log returns. To keep everything consistent, take a arithmetic mean of log returns. Then calculate it net of the risk free (how do you subtract properly using geometric returns?). Then divide (how do you do this properly using geometric returns?) by the standard deviation (how would you calculate this properly with geometric ...


2

In the dot.com era the Internet was considered a-winner-takes-it-all market, new tech start-ups (like Netscape, Amazon.com and the famous Pets.com) was measured by how much the capital they where able to chew through, the logic being that the more they spend the more aggressive they were (at least in the investors' eyes), conquering this new market known as ...


2

I would consider Amihud (2002) as a good first approximation with that level of data.


2

I am trying to fill in what Richard left for the second part: \begin{align*} \exp(-r(T-t))E\, N'(d_2) &= \frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\, \exp\left(-\frac{1}{2}d_2^2\right) \\ &=\frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\, \exp\left(-\frac{1}{2}\big(d_1-\sigma\sqrt{T-t}\,\big)^2\right) \\ &=\frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\\ &\qquad\qquad \...


2

Assume we start at $t=0$ with $P_0$, there are $t=1...N$ subsequent periods, and at each period-end $t$ an (entirely arbitrary) portion $c$ of our portfolio $P_t$ is churned and $(1-c)$ remains untouched. $P$ grows over each period by a factor $(1+g)$: $P_t = P_{t-1}(1+g)$. We can partition $P_t$ into sub-portfolios, each with its own churn history, as in: ...


2

You wrote: $$d[5] = (DJIR[5] - \mu) * Covariance$$ but you left out half of it (the inverse and the transposed vector on the right side). The correct formula is $$d[5] = (DJIR[5] - \mu)^2 / Var[DJIR]$$ The covariance "matrix" becomes the variance in a 1-dimensional case (in other words $x_i$ and $y_i$ are both equal to DJIR[i] in this case) and the "matrix ...


2

Note that with $H(\cdot)$ the Heaviside function $$\frac{d}{ds} H(s-K) = \delta(s-K)$$ but $$\frac{d}{dK} H(s-K) = \color{red}{-}\delta(s-K)$$ You can also use the Leibniz integral rule to write that $$ \frac{d}{dK} \int_K^\infty \phi_{S_T}(T,s) ds = -\phi(S_T,K) $$


2

In addition to @user42108's great answer: Maybe you already had a look on the 'usual' sources. Wiki states that the convention is still 4 to 6 digits, and Bank of Canada states the same, i.e. up to six digits. There may be currency pairs that are quoted with less digits, this seems to be the case when the exchange rate is above 80 or so. Depending on your ...


2

I had to deal with discrete FX quotes a long time ago. My answer may be badly out of date, sorry. For each currency pair, there is a "PIP", which stands for "point in percentage" or "price interest point" or "percentage in point", and used to be the smallest amount by which the quote can change. It is usually 1 basis ...


2

This question is unclear to me, what is your goal? It's in BTC Both BTC You can increase your BTC position and decrease your USD position by buying the pair and decrease your BTC position and increase your USD position by selling the pair. You can just multiply by the number of decimals to get ints?


1

Just the chain rule; $\frac{d}{dK} H \left (S-K\right)=\delta \left (S-K\right) \frac{d}{dK} \left (S-K\right)=-\delta \left (S-K\right) $


1

Yes, there are sound ways to address this problem. And, depending on the level of realism required and your goals, you will need to think a lot more to devise an acceptable strategy. Bird's eye view Let us first make the assumption that each asset indeed has exactly the same growth, each period. Even in this most simple case you can follow different ...


1

No, if you want to calculate the annualized sharpe ratio you should 1) make sure that your risk free rate is in monthly terms (so if it's 3% annual you need to put .03/12 in cell KX32) 2) only multiply the result by the square root of 12 (not 36). To calculate the annualized sharpe ratio, you multiply the monthly ratio by the square root of the ...


1

RETURN Firstly, return is based upon the amount gained over a period of time. So your calculation for a percentage return should actually be be: (Sale price - Cost basis)/Cost Basis. TOTAL RETURN A "total return" price series or index is a transformation of the original traded price timeseries to a timeseries that can be used to estimate/calculate a ...


1

I think they messed up with the dataset. The dates are weird and the rolling volatilities do not match. They suddenly take 2 year history instead of 5. May I please ask why did you not take full columns for 1Y and 3Y stress volatilities? (the percentile() starts somewhere in middle of the column) Thank you. EDIT: You should use full dataset to calculate ...


1

At the end of each year you have wealth $X_t$ in an investment account which grows at the rate $r$. At the beginning of each year you withdraw the amount $W$ and keep it in cash to pay for your expenses through the coming year. For your investment to last $n$ years you need to start with $X=W+W\frac{1}{1+r}+\cdots+W\frac{1}{(1+r)^n}$. In this way you will ...


1

The numerator is $$ S N'(d_1) = S \frac{1}{\sqrt{2 \pi}} \exp(-1/2 d_1^2) = \\ S \frac{1}{\sqrt{2 \pi}} \exp\left(- \frac12 \left(\log(S/E)+ (r + \frac12 \sigma^2(T-t)) \right)^2 / \sigma^2 (T-t) \right) $$ the denominator is: $$ \exp(-r (T-t)) E N'(d_2) = \\ E \frac{1}{\sqrt{2 \pi}} \exp\left(- \frac12 \left(\log(S/E)+(r- \frac12 \sigma^2(T-t)) \...


1

I think one of the main liquidity measures is the one from Pastor and Stambaugh (2003). You can use it for both individual stocks or indexes. Just run the following intra-month regression with daily data: $r^e_{i,d+1,t} = \theta_{i,t}+\phi_{i,t}r_{i,d,t}+\gamma_{i,t}sign(r^e_{i,d,t}) \times v_{i,d,t}+\epsilon_{i,d+1,t}$. Where $r^e_{i,d+1,t}$ is the excess ...


1

I think you are confusing the goal with the means. The calculation of the PE is not the goal, the true goal is assessing whether a particular stock is an interesting investment opportunity (cheap) under an investment thesis (set of hypotheses). Therefore, there is an infinite number of ways to calculate PE ratios, as a results of a set of different ...


1

I strongly recommend reading an undergraduate finance textbook like Investments by Bodie, Kane, and Marcus. Your methodology may be limited by your data. For example, using forward P/E requires next fiscal year's EPS estimates. NTM (next twelve months) requires quarterly EPS estimates. If you do not have estimates, the best method is TTM (trailing twelve ...


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