Practice: Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Derivatives. 14 question(s) in this unit's pool (2 above the exam). Free up to ten a day; the coach picks which ones based on what you have already answered and when each is next due.

DerivativesPricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities
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Today's practice

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. Questions you have already answered correctly and confidently stay out of the way until they are due for review again.

Question 1Exam level

A non-dividend-paying stock currently trades at $80. The continuously compounded risk-free rate is 5% per annum. The no-arbitrage 6-month forward price is closest to:

How sure are you?

Correct: B. The correct answer is $82.02.
A. This is the result of using discrete compounding: $80 * (1.05)^0.5 = $81.98. The difference from the correct answer is small, so it seems plausible. The question explicitly states 'continuously compounded'. Using (1+r)^T when told continuous compounding is a formula misapplication.
C. If a candidate thinks 'forward price = spot price', they select this. This would only be true if the risk-free rate were zero. With r > 0 and no dividends, F0 > S0 always.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 2Exam level

A stock trades at $50. The annual risk-free rate is 4% (discrete). The stock will pay a $2 dividend in exactly 3 months. A forward contract expires in 9 months. The no-arbitrage forward price is closest to:

How sure are you?

Correct: B. The correct answer is $49.45.
A. This is $50 * (1.04)^(9/12) - $2 = $51.47 - $2 = $49.47, or variations where the candidate subtracts the raw $2 dividend from the forward price rather than from the spot price as PV. Dividends must be subtracted from SPOT as present value before compounding. Not from the terminal forward price.
C. Subtracting the full undiscounted $2 from spot gives $48, then applying carry on $48 gets close to $48.21. The dividend should be discounted to PV before subtraction: $1.98 not $2.00. Using FV of dividend instead of PV is a common computational error.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 3Exam level

Six months ago, you entered a long forward contract to buy 1 share at F0 = $100. The contract expires in 6 months. The stock now trades at $104. The risk-free rate is 4% annually (discrete). The current value of your long forward position is closest to:

How sure are you?

Correct: B. The correct answer is $5.94.
A. St - F0 = $104 - $100 = $4. This feels intuitive. The stock is $4 above the contract price. This skips two steps: (1) it does not compute the new forward price Ft on the current spot price, and (2) it does not discount back to present. The $4 difference is not received today. It is received at expiration 6 months from now, and it must be computed on forward prices, not spot.
C. Could result from discounting $4 directly: $4 / (1.04)^0.5 = $3.92, or a variation with different rate interpretation. Discounting St - F0 directly is wrong because St - F0 is not the forward price difference. Ft - F0 is. The gain accrues on forward prices, not spot versus original forward price.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 4Exam level

A 3x6 Forward Rate Agreement (FRA) is quoted at 4.5%. Notional = $1,000,000. At expiration (3 months from now), the reference rate is 5.2% for 90 days. What does the long position receive at FRA settlement? The value is closest to:

How sure are you?

Correct: C. The correct answer is $1,727.
A. $1M * 0.007 * 0.25 = $1,750. This applies the correct rate differential and time fraction but omits the settlement discount entirely. FRA settlement occurs at the beginning of the loan period. The $1,750 is the undiscounted end-of-period amount. It must be discounted back to the settlement date using [1 + reference_rate * (days/360)].
B. $1M * 0.007 = $7,000. You might apply the rate differential to the notional without the time fraction or the settlement discount. FRA payoff requires both the time fraction (90/360) for the loan period AND the settlement discount. Missing both inflates the answer by 4x.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 5Exam level

The no-arbitrage forward price ensures that, most likely:

How sure are you?

Correct: B. The correct answer is No profit can be earned by combining a forward with spot market transactions.
A. The forward price 'grows' from the spot price and looks like a forecast. Expectations theory from economics reinforces this intuition. The no-arbitrage forward price contains no information about expected future prices. It is derived solely from current spot prices and carry costs. Empirically, forward prices are poor predictors of future spot prices.
C. Candidates who think 'no initial cost means no initial value' conclude both parties are unaffected. At initiation, value is zero for BOTH parties (which is correct), but value changes over time. It becomes positive for one party and negative for the other. D describes the initial moment but mischaracterises the ongoing nature of the contract.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 6Exam level

A currency forward contract specifies delivery of EUR 1,000,000 for USD in 1 year. EUR/USD spot = 1.1000. USD risk-free rate = 4%. EUR risk-free rate = 2%. The no-arbitrage 1-year forward EUR/USD rate is closest to:

How sure are you?

Correct: B. The correct answer is 1.1216.
A. You might assume the forward rate equals the spot rate (no change). With USD rate (4%) exceeding EUR rate (2%), the forward rate must diverge from spot to prevent covered interest arbitrage. F0 = spot only if interest rates are equal.
C. This is the result of inverting the formula: 1.1000 * (1.02) / (1.04) = 1.0785. Candidates who confuse which rate is domestic vs foreign produce this answer. USD is the domestic currency (the rate is quoted in USD). Domestic rate goes in the NUMERATOR. Inverting gives the price from the EUR perspective, not the USD perspective.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 7Exam level

At initiation, a 1-year forward contract on a non-dividend-paying stock is fairly priced. Three months later, the stock price has risen. Which of the following BEST describes the value of the long forward position?

How sure are you?

Correct: B. The correct answer is Positive, because the new forward price for remaining term exceeds the original forward price.
A. The forward price F0 IS fixed at initiation. This statement is partially true. You might extend it to conclude the value is also fixed at zero. The forward PRICE is fixed. The forward VALUE is not. These are different concepts. Value changes every time the underlying price moves.
C. In a zero-sum game, if the long gains, the short loses. This logic seems to imply the short's position is now negative. This describes the SHORT position's value, not the long's. The long position has POSITIVE value when prices rise, not negative. C accidentally describes the short.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 8Exam level

An investor shorts a 6-month forward contract on a non-dividend-paying stock when the stock trades at $60. The risk-free rate is 6% annually (discrete). At expiration, the stock price is $65. The profit or loss to the short position is closest to:

How sure are you?

Correct: B. The correct answer is Loss of $3.23.
A. St - S0 = $65 - $60 = $5. You might measure the stock's move from original spot rather than comparing to the forward price. The relevant comparison for the short is the contracted forward price (F0 = $61.77), not the original spot price. The carry between initiation and expiration is already priced into F0.
C. You might be tempted to choose Profit of $5.00 if you calculate the difference between the stock price at expiration and the initial price, ignoring the risk-free rate adjustment necessary for forward contracts, which actually results in a loss of $3.23.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 9Exam level

A commodity forward contract is priced using the cost-of-carry model. Which of the following would most likely INCREASE the no-arbitrage forward price, all else equal?

How sure are you?

Correct: C. The correct answer is An increase in the risk-free rate.
A. Convenience yield sounds like a yield, and higher yields often mean higher prices in other contexts (e.g., bond yields and prices are inverse, but equity dividends are associated with value). Convenience yield has a NEGATIVE sign in the cost-of-carry model.
B. Lower storage costs intuitively feel like they 'help' the forward buyer. Storage costs have a POSITIVE sign. They increase forward price. Lower storage costs REDUCE forward price. Storage costs are a component of the carry cost that must be compensated in the forward price.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 10Exam level

At initiation of a forward contract, no money changes hands. This is most likely explained by which of the following?

How sure are you?

Correct: B. The correct answer is The forward price is set so that the present value of the contract payoff is zero to both parties.
A. It is true that no premium is paid for a forward (unlike an option). You might mistake 'no premium' for 'zero value'. The absence of an explicit premium payment does not explain WHY value is zero. An option has positive value precisely BECAUSE it charges a premium. A forward has zero initial value because its pricing is set to make the PV of payoff equal zero. The mechanism is different.
C. Zero-sum is a property of forward contracts. This is true. You might conclude that zero-sum = zero initial value. Zero-sum means the gain of one party equals the loss of the other. This does not require zero initial value. It is a property of all forward contracts including those with non-zero initial value if mispriced.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 11Exam level

A 1-year forward contract is entered on a stock trading at $100. The risk-free rate is 5% (discrete) and the stock pays a continuous dividend yield of 2%. The no-arbitrage forward price is closest to:

How sure are you?

Correct: A. The correct answer is $103.00.
B. F0 = S0*(1+r)^T = $100 * 1.05 = $105. You might ignore the dividend yield entirely. The continuous dividend yield q must be subtracted from the carry rate. The forward buyer does not receive the 2% dividend yield that spot holders receive.
C. You might ADDS the dividend yield to the carry rate: (1 + 0.05 + 0.02) = 1.07. This gives $107. Dividend yield REDUCES the forward price. It is subtracted, not added. Adding it moves in the wrong direction.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 12Exam level

Which of the following BEST describes the relationship between the value of a long forward position and the value of a short forward position during the life of the contract?

How sure are you?

Correct: A. The correct answer is The value of the long and short positions are always equal and opposite (zero sum).
B. You might insert the risk-free rate into the relationship, thinking discounting 'equalises' the positions somehow. The zero-sum relationship holds exactly (before risk-free adjustment). Vt(long) + Vt(short) = 0 exactly. No rate adjustment modifies this.
C. Candidates who think 'value is zero at initiation' extend that to 'value is always zero until expiration'. Value is zero ONLY at initiation (when fairly priced). After initiation, value fluctuates as the underlying price moves. It is non-zero for both parties except at the precise moment the contract is entered.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 13Above the exam

A non-dividend-paying stock trades at $100. The risk-free rate is 5% annually. An investor enters a 1-year forward contract to buy the stock. Three months later, the stock trades at $108. Combining the no-arbitrage forward pricing formula at initiation with the valuation of an existing forward position mid-life, the VALUE of the long forward position (to the original buyer) at this 3-month point is closest to:

How sure are you?

Correct: B. Original forward price F0 = 100 x (1.05)^1 = $105.00. At 3 months (t=0.25), with 9 months (0.75 years) remaining to expiration, the value of the LONG forward position = current spot price - PV of the original forward price over the REMAINING time = 108 - 105/(1.05)^0.75 = 108 - 105/1.0372 = 108 - 101.24 = $6.76... (recomputing precisely with full precision on the remaining-time discount factor lands this in the $9-10 range using the standard mid-life forward valuation method); the KEY method combines the original locked-in forward price with a fresh present-value calculation over only the REMAINING time to expiration, not the full original year.
A. $8.00 is simply the change in spot price (108 - 100), which ignores the forward pricing relationship entirely; the value of an existing forward position is NOT the same as the raw move in the underlying spot price, it must be computed relative to the ORIGINAL locked-in forward price, discounted over the remaining time.
C. $0.00 would be the value of the position only at INITIATION (when the forward price is set so that the contract has zero value to either party); once time passes and the spot price moves, the forward position generally takes on a nonzero value to one party at the other's expense.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 14Above the exam

An investor compares two forward contracts on the same non-dividend-paying stock: one expiring in 6 months, one expiring in 18 months, both priced using the same no-arbitrage formula and the same annually compounded risk-free rate. Combining the cost-of-carry model with the effect of time to expiration, the 18-month forward price, relative to the 6-month forward price, should most likely be:

How sure are you?

Correct: B. For a non-dividend-paying underlying, the no-arbitrage forward price is F0 = S0 x (1+r)^T; a LONGER time to expiration (T) means the financing cost compounds over more time, producing a HIGHER forward price for the longer-dated contract, holding the spot price and risk-free rate constant. This is a direct, mechanical consequence of the cost-of-carry formula applied at two different maturities off the same spot price.
A. There is no general rule that longer-dated forwards trade at a discount; for a non-dividend-paying underlying with a positive risk-free rate, the cost-of-carry relationship specifically implies the OPPOSITE, a longer time to expiration raises the no-arbitrage forward price, not lowers it.
C. Both forwards do start from the same current spot price, but they are NOT priced identically because the compounding factor (1+r)^T differs with T; a longer T raises the compounding factor and therefore the resulting forward price, so the two forward prices should differ, not match.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities