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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 ...


14

Basically the Total Return Index assumes reinvestments compared to "regular" indices. "A total return index is an index that measures the performance of a group of components by assuming that all cash distributions are reinvested, in addition to tracking the components' price movements.1 While it is common to refer to equity based indices, there ...


13

Let $P_t$ be the price of the overall market index at the end of quarter $t$ Let $D_t$ be the dividend for the overall market in quarter $t$ Let $X_t = \frac{D_t}{P_t}$ be the dividend to price ratio. Two key concepts in time-series statistics are stationarity and ergodicity. If the dividends to price ratio is a stationary, ergodic process, then dividend ...


10

Maybe I am a little bit late to the party, but I want to give a shot. As in Campbell and Shiller, start from the identity $R_{t+1}\equiv\frac{P_{t+1}+D_{t+1}}{P_t}$ where $R_{t+1}$ is the gross return between time $t$ and $t+1$, and $P_t$ is the price at time $t$. Rearrange the relationship as $R_{t+1} =\frac{D_{t+1}}{D_t}\frac{\left(1+\frac{P_{t+1}}{D_{t+1}}...


7

There are 2 ways to do it. The good-enough way, and the complete and complex way. The Good-Enough Way Here you will convert to a situation where you can apply put-call parity. Begin by finding the strike $K$ where put and call prices are closest to each other. This might not end up being the closest-to-the-money strike, but it will do. Now run the ...


6

I'm not sure how deep of a question you are asking. The dog that did not bark is from a Sherlock Holmes murder mystery. The dog at the house did not bark at the intruder, so Holmes believed the dog knew the intruder. Therefore, the lack of evidence like barking, was itself the evidence. In the Cochrane paper, the introduction mentions that the lack of ...


6

You could compute index dividend yield from ATM options using linearized put-call parity (assuming index options are European.) The present value of the dividend payment is: $PV(div) = P - C + (S - K) + K(e^{rT} - 1)$ where $r$ is interest rate to the option expiration and $T$ is time to maturity in years. Then the implied dividend is: $d = \frac{PV(div)}{...


6

Theoretically, this is a more difficult problem than it looks like at first glance. Unfortunately, existing literature taking into account a proper dividend consideration is rare (at least from a practical viewpoint). There are several options: 1) Use what is called "De-Americanization": In this case, based on your input dividends (maybe based on other ...


5

Vanguard S&P 500 index fund tracks the index and not the total return because it pays dividends out to the owners of the fund... some investors reinvest the dividends, some investors spend their dividends, etc., so, because they cannot control the reinvestment and distribute the dividends, they benchmark against the S&P 500 index and not the total ...


5

If you assume that dividends are discrete but proportional to the pre dividend date stock price then the BS formula is exact provided you correctly compute the expiry date stock forward price, hence the continuous dividend yield case calibrated to the correct forward will give the correct result (this is because with proportional dividends, discrete or ...


5

Diviends or not, the put-call parity (with European options) always hold: $ C(S,K) - P(S,K) = F - K*DF $ In the RHS, dividends will impact the forward $F$ (higher dividends imply lower forward). So the LHS should be lower as well: the Call costs less and the Put costs more. The proof is straightforward, you just notice that at maturity $T$ you have: $S_T - K ...


5

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

If you assume the same tax rate $\alpha$ for all shareholders, then out of a dividend $D$ the amount $\alpha D$ goes to the government and the amount $(1-\alpha) D$ goes to the shareholders. In a theoretical pure no arbitrage environment, and assuming no interest rate discounting for the sake of simplicity, this would imply that the stock price would go down ...


4

The paper is generally correct, but it is not a general statement, as in a general truth of options hedging in a theoretical context, rather a statement regarding how the structured derivs market is typically set up: retail and institutional investors buy a large number of products that at their core entail the dealer buying (from the investor) long-dated (...


4

It is indeed no rounding error, but follows from the way Yahoo computes the adjusted price: it does not reflect the actual returns of the investor. Just look at August 17 and 20. The actual close prices were 10.75 and 9.95. On August 20 the company went ex-dividend for an amount 0.4508. The return on that day is $\frac{P_t+D_t}{P_{t-1}} -1 = \frac{9.95+0....


4

Well, consider using $S_t$ as the numeraire and let the asset be the reinvested stock $S_te^{qt}$. Then this ratio equals $e^{qt}$ so can never be a martingale.


3

I believe the exact answer to the question of what the S&P 500 price number assumes you do with the dividends is that you do NOT receive them at all. They are not included in the calculation AFAIK. So, yes, the price of one of the 500 companies drops a bit with a dividend payment (actually on the ex-dividend date), and the index drops a tiny bit because ...


3

You get nothing, by this logic you could accumulate risk-free money all day by buying/selling on the ex-date as long as the dividend is larger than the spread.


3

The right formula: $$ S_{t+\Delta t} = S_t+(r-q)S_t \Delta t+\sigma\,S_t \sqrt{\Delta t}\,Z+\frac{1}{2}\sigma^2\Delta t(Z^2-1)*S_t $$ We can extract the formula from the Brownian motion equation (Wiener process): $$ dS_t = (r-q) S_t dt + \sigma S_t dW_t $$ where $ W_t $ is Wienere process. Applying Itô's lemma (r, q and $\sigma$ are constants) with $ f(S_t) ...


3

What do you mean by annotation date, there is a declaration(announcement) date, ex-date, record date but I've never heard of an annotation date. Dividends are not decided always at the fiscal year end, in some countries they are approved by the shareholders general meeting which can happen at any time during the year, some companies pay quarterly, others ...


3

I would have put this in a comment, but it was too long. I wouldn't really classify it as an answer though. You are correct that the company paying out \$1 in dividends drops the value of the company by \$1. You are also correct that it is more complicated than this. Here are some things to consider: The dividend yields of stocks also drive demand for ...


3

A long equity forward position initiated at $t=0$ for delivery at $T$ can be replicated by borrowing cash to purchase the stock at $t=0$, carrying that stock up to $T$ and paying the interests on the cash borrowed (cash & carry). This shows that the forward price is basically the cost of funding the equity purchase. Now if the stock pays dividends the ...


3

To add to the above on a more practical note: In general, SP desks make money on the individual product when the underlying declines. Dividends make the underlying decline, hence they are naturally long dividends. Take an auto-callable product which is exercised if the spot is above a pre-determined strike each year and say the SP desk sells this ...


3

Yahoo Finance For example BASF (listed in Germany): https://finance.yahoo.com/quote/BAS.DE/history?period1=796867200&period2=1589932800&interval=div%7Csplit&filter=div&frequency=1d Roche (listed in Switzerland): https://finance.yahoo.com/quote/ROG.SW/history?period1=796867200&period2=1589932800&interval=div%7Csplit&filter=div&...


3

There is no real "risk-free" rate. Now to answer your question, $r$ is time-dependent and should correspond to the repo rate corresponding to the maturity of your forward. In $I$, dividends should be "discounted" using the same time-dependent repo rate. Contrary to what others have suggested here, the use of an OIS rate or some other rate is not ...


3

The advantage is that you get to keep the option premium. The obvious drawback is that your option can be exercised. You’re effectively capping your maximum gains on stock price increase.


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In your example, I believe it's assumed that the exercise date is after the dividend date. If the dividend date is after the exercise date, nothing happens. The value would decrease, consider the following timeline: $t=0$: You have a call option worth $4.73$ and a stock worth $S_0$ $t=1$: The increase in dividend is announced but dividends are not paid out ...


3

When the dividend is paid, the stock price on your tree should drop by the same amount. Ie if the dividend is 10 and the value of stock is 100 before the dividend at a node, you should change it to 90 and then continue building the tree from there. A cursory google brings quite a few results, eg this around slide 46 seems to explain it well


2

"S&P Dow Jones Indices calculates a total return index for the S&P 500 that includes the impact of investing dividends back into the index itself. In the calculation, dividends are invested in the entire index, not just in the stock that paid the dividend. The invested dividends then grow (or fall) as the overall index grows (or falls), rather ...


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