FRM Level 1: question on Value of Forward, with price immediately change

[bbttdxmz]

As this question shown as example of value of a stock index forward contract:
Question:
The price a 6-month forward contract for which the underlying asset is a stock index with a value of 1,000 and a continuous dividend yield of 1%. Compute the value of a long position of the index increase to 1,050 immediately after the contract is purchase.
the answer given is:



My question is, why 1,050 only discounted for dividend yields, while 1,015 needs to discounted as RFR and Dividends

Thank you

 

[bbttdxmz]

Can anyone help with it please~~

 

[David Harper CFA FRM]

@bbttdxmz Because the 1015 is fixed strike price and is being discounted to the PV with the riskfree rate, while the 1050 is a spot price which is effectively compounded forward then discounted back (the discounted expected value of a stock price is its current price!). Put more simply, 1015 is a "future price" because it is the contract's strike price (what will be paid in the future), but 1050 is not future price, it is already discounted. See how they are located at different points on the timeline, hence they don't get the same treatment. The solution uses, to me, the far less intuitive solution to the value of forward. I think more intuitive is:

• The forward contract's current value, f = (F0 - K)*exp(-rT); i.e., the future expected gain, (F0 - K), discounted to its present value.
• But per cost of carry (COC) the current forward price should be a function of current spot: F0 = S0*exp[(r-q)T]; such that:
• current f = (S0*exp[(r-q)T] - K)*exp(-rT) = S0*exp[(r-r-q)T] - K*exp(-rT) = S0*exp(-qT) - K*exp(-rT) which is used in your solution. Notice how the risk-free rate cancels because we compound forward to retrieve the forward price then discount back to PV
• But, to me, it's more natural to solve for the expected future gain, (F0 - K) = 1050*exp[(4%-1%)*0.5] - 1015.113 = 50.756, then discount this to PV: 50.7557*exp(-4%*0.5) = $49.75062396. Thanks,

 

[bbttdxmz]

@bbttdxmz Because the 1015 is fixed strike price and is being discounted to the PV with the riskfree rate, while the 1050 is a spot price which is effectively compounded forward then discounted back (the discounted expected value of a stock price is its current price!). Put more simply, 1015 is a "future price" because it is the contract's strike price (what will be paid in the future), but 1050 is not future price, it is already discounted. See how they are located at different points on the timeline, hence they don't get the same treatment. The solution uses, to me, the far less intuitive solution to the value of forward. I think more intuitive is:

• The forward contract's current value, f = (F0 - K)*exp(-rT); i.e., the future expected gain, (F0 - K), discounted to its present value.
• But per cost of carry (COC) the current forward price should be a function of current spot: F0 = S0*exp[(r-q)T]; such that:
• current f = (S0*exp[(r-q)T] - K)*exp(-rT) = S0*exp[(r-r-q)T] - K*exp(-rT) = S0*exp(-qT) - K*exp(-rT) which is used in your solution. Notice how the risk-free rate cancels because we compound forward to retrieve the forward price then discount back to PV
• But, to me, it's more natural to solve for the expected future gain, (F0 - K) = 1050*exp[(4%-1%)*0.5] - 1015.113 = 50.756, then discount this to PV: 50.7557*exp(-4%*0.5) = $49.75062396. Thanks,

Your explanation is as crystal clear, thank you！

 

[bbttdxmz]

@bbttdxmz Because the 1015 is fixed strike price and is being discounted to the PV with the riskfree rate, while the 1050 is a spot price which is effectively compounded forward then discounted back (the discounted expected value of a stock price is its current price!). Put more simply, 1015 is a "future price" because it is the contract's strike price (what will be paid in the future), but 1050 is not future price, it is already discounted. See how they are located at different points on the timeline, hence they don't get the same treatment. The solution uses, to me, the far less intuitive solution to the value of forward. I think more intuitive is:

• The forward contract's current value, f = (F0 - K)*exp(-rT); i.e., the future expected gain, (F0 - K), discounted to its present value.
• But per cost of carry (COC) the current forward price should be a function of current spot: F0 = S0*exp[(r-q)T]; such that:
• current f = (S0*exp[(r-q)T] - K)*exp(-rT) = S0*exp[(r-r-q)T] - K*exp(-rT) = S0*exp(-qT) - K*exp(-rT) which is used in your solution. Notice how the risk-free rate cancels because we compound forward to retrieve the forward price then discount back to PV
• But, to me, it's more natural to solve for the expected future gain, (F0 - K) = 1050*exp[(4%-1%)*0.5] - 1015.113 = 50.756, then discount this to PV: 50.7557*exp(-4%*0.5) = $49.75062396. Thanks,

but i may have another question. in the first step, why it is f = (F0 - K)*exp(-rT), instead of f = (F0 - K)*exp(-rT+qT), or saying, why we don't discount dividend to get the value of current value?

 

[David Harper CFA FRM]

@bbttdxmz because the dividend is not part of the long forward contract holder's experience. The long forward position knows (s)he must pay the delivery price, K, in exchange for receiving a commodity with an updated forward price, F0, and this is the expected deal in six months; so (F0 - K) = (value received - cash paid) six months in the future. The forward holder will not receive any dividend. So this is the net difference, itself a future gain, that should be discounted at the risk free rate. The dividend, however, does play a role in reducing the (theoretical) forward price itself, F0, precisely because the long forward position holder forgoes (misses out on) the dividend.

 

[bbttdxmz]

@bbttdxmz because the dividend is not part of the long forward contract holder's experience. The long forward position knows (s)he must pay the delivery price, K, in exchange for receiving a commodity with an updated forward price, F0, and this is the expected deal in six months; so (F0 - K) = (value received - cash paid) six months in the future. The forward holder will not receive any dividend. So this is the net difference, itself a future gain, that should be discounted at the risk free rate. The dividend, however, does play a role in reducing the (theoretical) forward price itself, F0, precisely because the long forward position holder forgoes (misses out on) the dividend.

thank you. So the arbitrage is the value of contract?

 

[David Harper CFA FRM]

@bbttdxmz This question (within its self-contained assumptions) does not offer an arbitrage: the solution is an updated, so-called theoretical value. An arbitrage opportunity would arise (eg) if the observed (aka, traded) value of the futures contract differed from the theoretical price according to the COC. This includes the role of the dividend, which is coherent in the solution.

(There is such a thing as a no-arbitrage or arbitrage-free price such that we can say that the initial forward price, F(0) = 1050*exp[(4%-1%)*0.5] is a no-arbitrage price; i.e., it is not mis-priced. But it isn't necessary in a simple application like this were observed prices are not mentioned and is more likely a misunderstanding of the mechanics IMO). Thanks,