Jam 6

[Agent Orange]

Can someone please tell me how to construct the integration on page C-14 of the Jam outline?

[OT]

0.05 gains off of 500.

Dividends get paid continuously. The continuously paid dividends earn 0.03 from time, t, to time 1 which is a span of time, 1-t. Hence e ^ [ 0.03 * (1-t) ].

I am not sure about the, e ^ [ (0.03 - 0.05) * t) ], which is what you are probably inquiring about. My best guess or description is that as the stock dividend gets paid, it decreases at the rate the risk-free rate falls short of the dividend rate.

<font size=-1>[ This Message was edited by: ot on 2002-02-12 14:21 ]</font>

[secada]

Mr. wwsituation there is a private message for you.

[Agent Orange]

Anyone?

Anyone?

[arwen]

give a bit of context (i.e., state the question!) for those who've done this exam already, and we may be able to help.

[Dr T Non-Fan]

If you've done this exam, then surely you have a JAM nearby? The page number shouldn't be so different.

[OT]

Assume the current MMI is 500, the risk-free rate is 3% continuous, and the continuous dividend yield is 5%.

Futures price one year from now should be, F = 490.10.

If actual futures price is 510, then could do the following to make an arbitrage-free profit.

Sell a futures contract.

Borrow at the risk-free rate to purchase the current index for 500.

At expiration of future contract, must pay off amount borrowed which equals 515.23.

During the contract, your receive dividends on the stocks owned equal to 25.13 = integral from zero to one of 500 * { e ^ [ (0.03 - 0.05) * t) ] } * 0.05 * { e ^ [ 0.03 * (1-t) ] }.

Earn dividends at a %5 rate on a declining stock index (3% - 5%); the dividends are accumulated at 3%.

At expiration of the futures contract you sell the index stocks you own for the agred price of 510.

Net Cash flow at time one is 510 + 25.13 – 515.23 = 19.90.

This profit at the end does not depend on the ending index value; also, it required no initial investment (at time one).

Note that this profit equals the difference between the actual futures price and the theoretical futures price, 510 – 490.10 = 19.90.

(I was kind of ripping along while typing, so please indicate any errors.)

[arwen]

no JAM nearby (it's buried somewhere in the closet, not here at work).

But as I suspected, this is the same question I asked someone about last year. His answer follows:


F = S (1+r) - D

F=future now expiring in one year
S= stock now
r= risk free rate
D = future value of the cumulative div earned on the index

If this equation isn't true, you have an arbitrage opportunity by "longing" the smaller side and "shorting" the bigger side. I always set these problems up in terms of cash outlay/inflow at the beg vs cash outlay/inflow at the end and seeing if a "free lunch" is earned.

Ok. Intuitively it makes sense even if one doesn't rely on the formula. After all, if you long a future, you'll pay less if div are being paid on the underlying as you miss out on these div by waiting to pay for the stock. Assuming D=0, the formula still makes intuitive sense as the person longing the future should pay more for the future because of the time value of money (the money he's saving by paying for it later earns interest). It's starting to come back to me.

Ok, back to your question. It's true that a stock's (and presumably an index's) price are adjusted downward by the amount of a div. Then, the price starts creeping up again until the point of the next ex-div date.

In his calc of div, it does appear that the div is being paid on the value of the index that is continuously adjusted downwards. However, the future value of the div grows since it gets interest. I have problems with all the integration stuff that you guys probably remember better than a do. But, even though his wording is confusing, it seems to makes sense. The div is a function of the stock value that does go down by the div amount. But, in valuing a future, you need to take into account the interest earned from the time the div is paid to the expiration of the future. He seems to be netting these two effects. I don't really like his wording by implying the stock index declines by (3%-5%). The future value of the cumulative div that affects the price of the future is the net of these two effects.