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We can obtain a closed-form solution for the expected return over an arbitrary holding period under some typical assumptions. Assuming geometric Brownian motion with drift $\mu$ and volatility $\sigma$, the stock price at time $t \geqslant 0$ is $$S(t) = S(0)e^{(\mu - \frac{1}{2}\sigma^2)t}e^{\sigma \sqrt{t} z},$$ where $z \sim \mathcal{N}(0,1)$, a standard ...


3

When using arithmetic returns the right way to calculate an average is via the geometric average. The reason is that there is a multiplicative relationship between the returns. Example: Let $P_t$ denote the stock price at time $t$, then the simple (arithmetic) net return is defined as: \begin{equation} r_t=\frac{P_t-P_{t-1}}{P_{t-1}}=\frac{P_t}{P_{t-1}}-1 \...


2

Solving it algebraically: As seen in the above provided reference (just above " 1) "), the general formulation for the unconstrained Markowitz portfolio optimization scheme, is given by: \begin{align} &\text{arg}\max_{w} \; w^T\mu-\frac{\delta}{2} w^T\Sigma w.\\ \end{align} In absence of any constraints, the above optimization scheme have the ...


1

Short answer [the link/URL doesn't work]. In a world without risk premia, this logic would be correct. Variance drag would cause all volatile risk assets to have negative expcted returns. Which is precisely why most financial markets apply a discount rate to risky assets that more-than compensates for these Kelly Betting problems. Even if we took the ...


1

This is a mathematical fact but imo there's not much to square with intuition, without appropriate fractional sizing (kelly) volatility is guaranteed to eat away returns over very long time horizons but people are not investing over time horizons where this effect will materialise (though volatility drag is still a thing.)


1

The number of up moves of the stock $S$ after 100 days follows binomial distribution. To calculate expected value of the stock we have to weight values by probability mass function. After 100 days we have $k$ up moves of $1+10\%$ and $100-k$ down moves of $1-10\%$ i.e. the value of the stock is $S_0*(1+10\%)^k*(1-10\%)^{(100-k)}$ with probability ${100}\...


1

"How to roughly measure how much premium investors would demand if a stock could not be sold and its investors had to stick with it permanently using just dividends not capital gain as return?" A starting point might be to look at the private/public valuation arbitrage in the sector.


1

This is a hopefully clearer explanation of what I've been saying in my comments. The background is that you have an average monthly return $\bar{r}_{m} = \frac{1}{12}\sum_{i=1}^{12} r_{i}$ and a monthly variance estimate $\sigma^2_{\bar{r}_{m}}$ where $\sigma^2_{\bar{r}_{m}} = \frac{\sigma^2_{r}}{12}$. In what follows, it is assumed that these estimates ...


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