LOS 38.4 Forward Price with Cash Flows

[Sharklets]

Hello
here is the situation, personnaly I would really appreciate if you could clarify what they meant by those calcs (expecially step 2)

QUESTION
At the inception of a one-year forward contract on a stock index, the price of the index was 1,100, the interest rate was 2.6percent, and the continuous dividend was 1.2 percent. Six months later, the price of the index is 1,125.
ANSWER
The answer given is : The value of the short position is -$17.17.

EXPLANATIONS
Step 1 : At the inception of the forward contract, the delivery price would have been: 1,100e (0,026-0,012) = 1,115.51. This part I understand.

Step 2 : The value to the long position after six months is: [1,125e(-0,012)(0,5) ] - [1,115.51e (-0,026)(0,5) ] = 1,118.27 - 1,101.10 = $17.17. Therefore, the value of the short position is -$17.17.

Q1 : why do we discount the dividend back to the present? is it because we consider it accreted 50% during the first 6 months and it should accrete at the same rate for the next following 6 months?
Q2 : in the second portion of the formula, we discount the forward price @ risk free to bring it back to today. Why do we not take the dividend into account here by reducing the cost of carry?
What is the rationale behind?
Please advise
Thanks!!

 

[David Harper CFA FRM]

Hi @Sharklets

I don't either find the expression of Step 2 directly intuitive. The value of a forward contract is given by Hull 5.4: f = (F0 - K)*exp(-r*T), where f = value of contract, F is the current forward price, and K is the delivery price. Hopefully this is intuitive as the current value of a forward contract; e.g., if you are long a a contract with delivery price of (K) then, at this current point in time, you are expecting in (T) years to pay the delivery price K (which does not change during the life of the contract) and in exchange to receive the commodity that, currently at least, has expected future value of F0. So the difference (F0 - K) is the expected gain (or expected loss for the short position) in the future, so it needs to be discounted per *exp(-rT). Then we can retrieve your step (2) by distributing: f = (F0 - K)*exp(-r*T) = F0*exp(-rT) - K*exp(-rT), but F0 = S0*exp[(r-q)T] so that f = S0*exp[(r-q)T]*exp(-rT) - K*exp(-rT) = S0*exp[(r-r-q)T] - K*exp(-rT) = S0*exp[(-q)T] - K*exp(-rT), which is your Step 2 expression. It is possible to get an intuition about this, but I think f = (F0 - K)*exp(-r*T) is so much easier to grok!

Q1: The theoretical forward price, F0, is based on the cost of carry where the dividend is a tangible benefit of ownership (like storage or financing cost, r, are costs of ownership). The forward position holder forgoes (skips) these benefits/costs of ownership, so they must decrease/increase the forward price under the carry perspective. This model assumes continuous but the lump sum version has the same effect with F0 = (S - I)*exp(rT)
Q2: The delivery price was determined in the first step (six months ago, T - 0.5 years), at which time it was reduced by the expected dividend. But then the delivery price (aka, strike price) of the contract is fixed. I hope that's helpful!

 

[Sharklets]

David
Thanks for this comprehensive answer which clarifies the point for me.
Best