Hi,
The book says FP of an equity contract is spot - present value of Future dividends (1+Risk free)^t
My question is aren’t the present value of future dividends already factored into the spot price?
If yes, why do we need to deduct it.
Thanks.
They’re not factored in because if you buy the stock you get the all of the dividends in the future – hence, today you get the present value of all of the dividends – but if you enter into a long position in a forward contract you don’t get the dividends paid before the contract expires – hence, today you don’t get the present value of those dividends. Thus, for the forward price you must remove the value of those dividends.
to help in the understanding, let’s say you call up a bank and ask to buy 1 share of a $100 stock in 3 months that pays a dividend of $1 in 2 months.
risk free rate is 5%.
the bank will say ok and quote you a price…
lets say the bank doesn’t first take into account the present value of the dividend and instead quotes you just the future value of the spot price at 5% (which would be the no-arbitrage price), assuming the stock pays no dividend.
The forward price is: 100 * (1+5%)^(3/12) = $101.2272
is there an arbitrage for the bank?
the answer is of course… yes there is. here is what the bank does…
at initiation…
-
the bank borrows $100 at 5% for 3 months.
-
the bank buys 1 share of the company at spot for $100.
-
the bank agrees to sell you the 1 share in 3 months at $101.2272
at maturity…
-
bank takes the share and sells it to you for the agreed price of $101.2272
-
bank repays loan plus interest of $101.2272 with the proceeds from the sale
-
bank keeps dividend (because it was holding the stock) for the three months.
The bank can make a risk free gain of the dividend (technically future value of the dividend) over the 3 months.
Instead, let’s say the bank first subtracts the present value of the dividend.
So the new forward price is ($100 - $1/(1.05)^(2/12)) * (1.05)^(3/12) = $100.2231
Is there an arbitrage opportunity for the bank?
Lets do the same steps…
at initiation…
-
the bank borrows $100 at 5% for 3 months.
-
the bank buys 1 share of the company
-
the bank agrees to sell you the share in 3 months at $100.2231
in 2 months…
- bank receives $1 of dividend and reinvests it at 5% for 1 month
at maturity…
-
bank takes the share and sells it to you for the agreed price of $100.2231
-
bank redeems the dividend investment for $1 * (1.05)^(1/12) = $1.004074
(bank now has cash of $101.2231 + $1.004074 = $101.2272)
- bank repays loan plus interest of $101.2272 with the proceeds from the sale and the future value of the dividend.
You see the bank has exactly the cash they need to repay the loan. so there is no arbitrage opportunity here.
awesome. thx
Hi,
someone pls explain how the following formulas are equivalent in FP calculation.
-
FP = FV (Spot - Carry benefits), when we you use annual compounding.
-
FP =Spot*e(continuously compounded risk free rate - continuously compounded dividend yield)
Why the continuously compounded dividend yield is subtracted from the risk free rate?
Thanks.
It’s math:
[e(-yt)] e(rt) = e(r-y)t
Hi,
Can you pls explain how the continuously compounded dividend yield is taken as e(-yt)?
Thanks.
Why are we dividing the FV of spot with the FV of carry benefits? We should be subtracting it ryt?
The continuously compounded dividend yield is _ not _ e−yt; the continuously compounded dividend yield is y.
If y were the annual, compounded dividend yield, then you would add dividends to the spot price by multiplying by (1 + y)t; the continuous analogue to that is multiplying the spot price by e_yt_.
Because you don’t receive dividends when you own only the futures contract, you have to subtract the dividends from the spot price. If y were the annual, compounded dividend yield, then you would subtract dividends from the spot price by dividing by (1 + y)t; the continuous analogue to that is dividing the spot price by e_yt. And dividing by eyt_ is the same as _ multiplying _ by e−yt.
Many Thanks 
You’re welcome.