Evaluating a forward contract

[Hend Abuenein]

Hi David, I hope you're feeling well today

Please consider this question and the way it was answered:

At the inception of a one-year forward contract on a stock index, the value of the index was $1,100, the interest rate was 2.6 percent, and the continuous dividend was 1.2 percent. Six months later, the value of the index is $1,125. What is the value of the forward contract (long or short position):

Answer :
At the inception of the forward contract, the delivery price would have been:
1,100e(0.026 - 0.012) = $1,115.51.

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.

Would you please explain what happened in the red?

Thank you

 

[David Harper CFA FRM]

Hi Hend,

That's a tough question but a good training one (source, if you mind?)

It's using Hull 5.7 value of forward (f) = S(0)*exp(-qT) - K * exp(-rT). But i doubt a candidate can be expected to know this version of the forward value (it's not assigned)

i suppose you could use f = S(0) - K*exp(-rT) and just remember the "rule" that dividends reduce the asset price

Although we could start with the key, intuitive value of the forward: f = (F(0) - K)*exp(-rT)
i.e., in the long position, at maturity, my expected gain is F(0) - K, so value of (f) is that expected gain discounted to today. F(0) is volatile, but K is fixed.

f = (F(0) - K)*exp(-rT), and per the carry model F(0) = S(0)*exp(r - q), so:
f = [S(0)*exp(r-q)T - K]*exp(-rT)
f = [S(0)*exp(r-q)T*exp(-rT) - K*exp(-rT); here's why the final frankly escapes my direct intuition, because the riskfree rate gets canceled
f = S(0)*exp(-qT) - K*exp(-rT)

I hope that helps, I hope studies are going well! Thanks, David

 

[Hend Abuenein]

But i doubt a candidate can be expected to know this version of the forward value (it's not assigned)

Thank you, that's a relief...wondered how come I didn't know it!

It's from SchweserPro QBank. A "recycled CFA" question maybe..you think? Why else would they include it if not assigned?

Just so I get this right:

f = [S(0)*exp(r-q)T - K]*exp(-rT)
f = [S(0)*exp(r-q)T*exp(-rT) - K*exp(-rT)

The blue T here is 1 year, full term of the contract to compound spot price by cost of carry to forward price,
but red T is 0.5 year, to discount expected value of contract from delivery back to half term value.

Am I right?

studies are going well Thanks

 

[Hend Abuenein]

But wait...I don't have K in this question!
I can only calculate f(T) and assume it's the price of the contract.
So how can I solve for the expected gain at T then discount it back six months?

We seem to have drifted apart on what the question asks for here.
My understanding is that it's asking for the forward contracts worthiness mid-term, when actual price was found to be 1125.

 

[David Harper CFA FRM]

Hi Hend,

Not quite, your red & blue (T) are always the same (T); To reduce this: exp(r-q)T*exp(-rT) ... to this: exp(-qT) requires same (T).
So, at T = 0.5, you can see the solution is using 0.5 throughout
And the formula works at T = 1.0, although it's maybe overkill/circular but makes the point:
At T = 1, f = S(0)*exp(-1.2%*1) - K*exp(-2.6%*1) = 1100*exp(-1.2%*1 year) - 1115*exp(-2.6%*1 year) = 0.

So the (T) are all keyed off the same time to maturity, which assumes the time to delivery. I didn't really think about the before, but the (T) is sort of "baked into" the (K). The T = 0.5 is pricing because the delivery price (K) refers to + 0.5. I hope that helps, David

append: I answering #3 as you posted #4. You are correct, the question doesn't give you the delivery price (K).

But the first step is to infer the (K) based on the assumption the contract value is zero at initiation. We do assume that when the contracts starts, it's fair price to both counterparties such that f = 0 and F(0) = K (at inception is the only time when F(0) = K, then the K remains fixed as the delivery while the F(0) goes on to undulate per changing conditions.

 

[Hend Abuenein]

I'm more confused about it now
But I can't give it anymore time than this, since you say it's not assigned to begin with.

Thanks a lot though

 

[David Harper CFA FRM]

Okay, I will try to make a concise note in "David's notes" about it

I just meant the formula isn't assigned (and i just said "this version" of the formula please) .

GARP doesn't 100% honor the AIMs, this kind of question is "in bounds". And it's a good question, IMO, the CONCEPTS are definitely testable; in particular, the difference exploited in this question by fixed delivery K, volatile F(0) and varying forward value (f)

Thanks David

 

[Hend Abuenein]

Thanks for clarifying about the formula
I'll go over it again

Thank you