What is the relationship between dividends and no arbitrage?

[courtrigrad]

Hello all

1. If a company makes a 3-for-1 stock split would the share price decrease by a factor of $$\frac{1}{4}$$? In other words if we have a stock valued at $500, with a stock split would we have 4 stocks valed at $125?

2. A company whose stock price is cirrently $$S$$ pays out a dividend $$D$$, where $$0\leq D\leq 1$$. What is the price of the stock just after the dividend date? Would it just be $$S - DS$$?

3. A particular forward contract costs nothing to enter inato at time t and obligates the holder to buy the asset for an amount $$F$$ at expiry $$T$$. The asset pays a dividend $$DS$$ at time $$t_{d}$$, where $$0\leq D\leq 1$$ and $$t\leq t_{d}\leq T$$. Use an arbitrage argument to find the forward price $$F(t)$$
Hint: Consider the point of view of the writer of the contract when the dividend is re-invested immediately in the asset Would $$F = S(t)e^{-r(T-t}?$$

Thanks

 

[dextercioby]

I'm no expert in economy,my gf tends to be,but,just outta curiosity,how did u get that 1/4 for the first problem??

Daniel.

 

[courtrigrad]

Well consider a stock split such as: A company with a stock price of $900 announces a 2-for-1 stock split. This means that instead of 1 stock with a price of $900, we have 3 stocks valued each at $300. So the price of one share decreases by a factor of $$\frac{1}{3}$$

 

[dextercioby]

Hmmm,that's weird...My logics would tell me:
2 of 450$ for 1 of 900$...

Daniel.

P.S.There's something fishy about economy problems,i should warn my gf...

 

[courtrigrad]

Heh. Anybody have any advice for #2 or #3?

 

[Curious3141]

Well consider a stock split such as: A company with a stock price of $900 announces a 2-for-1 stock split. This means that instead of 1 stock with a price of $900, we have 3 stocks valued each at $300. So the price of one share decreases by a factor of $$\frac{1}{3}$$

Daniel's intuition is correct, you are wrong.

http://www.investopedia.com/terms/s/stocksplit.asp

Economics *is* weird, but so is the tendency to overthink it.

I'm just getting interested in this stuff. But I'm an amateur trying to learn the ropes, and am more interested in derivatives than equity because of the leverage provided.

For number two, why should the stock price change at all ? The dividends (as I understand them) are declared on the earnings of the company not directly off the stock price.

I don't think there is an easy way to predict this. Let's consider a simplistic model with a multiple (P/E) of k.

If a per-share dividend of D is declared, then the two simplest possibilities as I see it are as follows :

The first is if the stock price remains the same, in which case the multiple changes to $$\frac{P}{E - D} = \frac{kP}{P - D}$$, or rather, that it increases by a proportion of $$\frac{P}{P - D}$$

The second is if the multiple remains constant, meaning the new stock price has to decrement by an absolute value of $$kD$$.

I think these are both extreme and highly theoretical possibilities. In reality, I would expect the market behaves anomalously before a dividend declaration, because the psychology of existing stockholders is to hold on to the equity until the dividends are received, so there probably is going to be a decrease of trading. Plus if the dividends declared are higher than usual, sentiment is going to be bullish on the stock, so the stock is going to be driven up.

But I could be talking out my arse, I'm a rank amateur in all this.

 

[Curious3141]

I was wrong. Apparently, the stock price will "definitely" fall after a dividend declaration, at least accoding to investopedia.com, a resource I trust.

Quoting,

...So the price of the stock will drop approximately by the amount of the dividend on the ex-dividend date. The word "approximately" is crucial here. Due to tax considerations and other happenings in the market, the actual drop in price may be slightly different. In any case, the point is that you can't make free profits on the ex-dividend date.

They didn't give a formula, but they imply that the stock will readjust to $$P - D$$, in which case, of course, the multiple will decrease, but in any case, it's more complicated than all this because of the other factors at play (as the site mentioned).

You should read the whole article (link : http://www.investopedia.com/articles/02/110802.asp) , and in fact, try to read the whole site, it's a goldmine.

And if you think this is tough, wait till you see the mathematics of option pricing.

 

[courtrigrad]

Thanks a lot guys. How about #3? IS this a trick question?

Thanks

 

[courtrigrad]

I think it would still be
$$F = S(t)e^{-r(T-t}$$ if we invest in a special portfollio.