Question 1 of 30
An asset trades at USD 500. The 1-year forward price is USD 510. The risk-free rate is 5%. There are no other costs or benefits. Which arbitrage strategy generates a risk-free profit?
id: 7
model: Grok
topic: Arbitrage with Mispriced Forward (Underpriced)
Explanation
<h3>First Principles Thinking: Reverse Cash-and-Carry</h3><p><strong>A is correct.</strong> Calculate the no-arbitrage price: USD 500 × 1.05 = USD 525. The market forward is USD 510. The forward is cheap (Underpriced). Buy low, sell high. You should Buy (Long) the Forward at USD 510. To hedge this, you must sell the asset now at USD 500. This is a 'Reverse Cash-and-Carry'. You Short the spot asset (inflow USD 500), Lend that USD 500 at 5% (becoming USD 525 at maturity). At maturity, you use the Long Forward to buy the asset back for USD 510, return the asset to close the short, and pocket the difference (USD 525 - USD 510 = USD 15).</p><p>B is incorrect: This is for an overpriced forward. If you sold the forward at 510 and bought spot funded at 5%, your cost would be 525, locking in a loss.</p><p>C is incorrect: This describes the leg of a standard Cash-and-Carry but mixes terms; Reverse Cash-and-Carry always involves shorting the spot asset.</p>
Question 2 of 30
An analyst needs to price a 3-month forward contract on a stock index. The index level is currently 4,000. The continuously compounded annual risk-free rate is 3%, and the continuously compounded dividend yield on the index is 1.5%. What is the forward price?
id: 5
model: CFA Level I
topic: Equity Index Forward with Continuous Yield
Explanation
<h3>First Principles Thinking: Cost of Carry with Continuous Yields</h3><p><strong>A is correct.</strong> For indices with continuous yields, the cost of carry is the net of the risk-free rate and the dividend yield. <br> Formula: $$ F_0(T) = S_0 e^{(r - q)T} $$ <br> $S_0 = 4000$. <br> $r = 0.03$. <br> $q = 0.015$. <br> $T = 0.25$ (3 months). <br> Calculation: <br> $$ F = 4000 \times e^{(0.03 - 0.015) \times 0.25} $$ <br> $$ F = 4000 \times e^{0.015 \times 0.25} $$ <br> $$ F = 4000 \times e^{0.00375} $$ <br> $$ F = 4000 \times 1.003757 $$ <br> $$ F = 4,015.028 $$ <br> Rounded to 4,015.03.</p><p>B is incorrect. It likely assumes $T=1$ ($4000 \times e^{0.015} = 4060$). <br> C is incorrect. It likely subtracts the net rate or reverses signs ($4000 \times e^{-0.00375}$).</p>
Question 3 of 30
An asset is currently trading at USD 1,000. The annual risk-free rate is 5%. A one-year forward contract on the asset is priced at USD 1,060. Assuming no storage costs or benefits, calculate the arbitrage profit per unit at expiration (Time T).
id: 2
model: CFA Level I
topic: Arbitrage using Cash-and-Carry
Explanation
<h3>First Principles Thinking: Law of One Price</h3><p><strong>B is correct.</strong> To determine arbitrage profit, compare the market forward price to the no-arbitrage forward price. <br> No-Arbitrage Price ($F_{no-arb}$): $$ S_0 (1+r)^T = 1000 (1.05)^1 = 1050 $$ <br> Market Price ($F_{mkt}$) = 1060. <br> Since $F_{mkt} > F_{no-arb}$, the forward is overpriced. <br> <strong>Strategy (Cash-and-Carry):</strong> <br> 1. Borrow USD 1,000 at 5%. <br> 2. Buy Asset for USD 1,000. <br> 3. Sell Forward at USD 1,060. <br> <strong>At Time T:</strong> <br> 1. Deliver asset, receive USD 1,060. <br> 2. Repay Loan: $1000(1.05) = 1050$. <br> 3. Net Profit = $1060 - 1050 = 10$.</p><p>C is incorrect. It calculates the present value of the profit ($10 / 1.05 = 9.52$) instead of profit at expiration, or uses continuous compounding incorrectly.</p>
Question 4 of 30
A commodity has a spot price of USD 100. The annual risk-free rate is 4%, and storage costs are USD 2 per year. Despite this, the 1-year forward price is trading at USD 98. What implies the existence of a 'convenience yield' in this scenario?
id: 5
model: ChatGPT
topic: Convenience Yield and Backwardation
Explanation
<h3>First Principles Thinking: Net Cost of Carry</h3><p><strong>A is correct.</strong> The theoretical forward price based on financial costs alone would be S0 + Cost of Carry. Here, Cost of Carry = Interest (4%) + Storage (2%). So, the Forward Price should be > USD 100 (approx USD 106). However, the market Forward Price is USD 98. The discrepancy (USD 106 - USD 98) represents the Convenience Yield. This is the non-monetary benefit of holding the physical inventory (e.g., to prevent a production shutdown). When Convenience Yield > (Interest + Storage Costs), the forward curve goes into backwardation (Futures < Spot).</p><p>B is incorrect: Forward < Spot defines Backwardation, not Contango. Contango is when Forward > Spot.</p><p>C is incorrect: Storage costs are explicitly stated as positive (USD 2). The 'negative cost' effect comes from the convenience yield, not the storage cost itself.</p>
Question 5 of 30
The spot exchange rate is 1.25 USD/EUR (1 EUR = 1.25 USD). The 1-year risk-free rate in the US is 2%, and in the Eurozone, it is 4%. According to the no-arbitrage principle (Covered Interest Parity), how should the 1-year forward rate compare to the spot rate?
id: 6
model: Gemini
topic: FX Forward Pricing (Interest Rate Parity)
Explanation
<h3>First Principles Thinking: Interest Rate Parity</h3><p><strong>A is correct.</strong> The formula for FX forwards is F = S × (1 + r_price) / (1 + r_base). Here, USD is the price currency and EUR is the base currency. So, F = 1.25 × (1.02 / 1.04). Since the numerator (USD rate) is smaller than the denominator (EUR rate), the ratio is less than 1. Therefore, the Forward Rate must be less than the Spot Rate. Conceptually, the currency with the higher interest rate (EUR) must depreciate in the forward market to offset the yield advantage; otherwise, everyone would convert to EUR, earn 4%, and hedge back for a risk-free profit.</p><p>B is incorrect: This would happen if the US rate were higher than the Eurozone rate.</p><p>C is incorrect: Interest rate differentials force forward rates to deviate from spot rates to prevent arbitrage; equality assumes r_USD = r_EUR.</p>
Question 6 of 30
Assertion (A): The no-arbitrage price of a derivative is the same as its price calculated under the assumption that all investors are risk-neutral.
Reason (R): Arbitrage arguments rely on the construction of a risk-free hedge, which renders the investors' risk preferences irrelevant to the pricing.
id: 10
model: GPT-4
topic: Arbitrage & Risk-Neutrality
Explanation
Assertion A is true: Risk-neutral pricing is a valid valuation technique. Reason R is true: It explains *why* the technique works. Since we can replicate the payoff with a risk-free portfolio (arbitrage logic), the price cannot depend on risk premiums. Therefore, we can simplify the math by assuming risk neutrality without changing the result. R is the correct explanation for A.
Question 7 of 30
An investor entered into a long forward contract on a non-dividend paying stock 6 months ago at a price of USD 50.00. The contract expires in 6 months (Total T=1 year). The current spot price of the stock is USD 55.00. The annual risk-free rate is 5% (discrete). What is the value of the forward contract to the investor today?
id: 4
model: CFA Level I
topic: Valuation of a Long Forward Contract
Explanation
<h3>First Principles Thinking: Mark-to-Market Valuation</h3><p><strong>B is correct.</strong> The value of a forward contract during its life is the difference between the current spot price and the present value of the original forward price. Alternatively, it is the PV of the difference between the current forward price and the original forward price. <br> Formula: $$ V_t = S_t - F_0 / (1+r)^{T-t} $$ <br> Where $T-t$ is time remaining = 0.5 years. <br> $S_t = 55.00$. <br> $F_0 = 50.00$. <br> $r = 5\%$. <br> Calculation: <br> $$ PV(F_0) = 50 / (1.05)^{0.5} $$ <br> $$ PV(F_0) = 50 / 1.0247 = 48.795 $$ <br> $$ V_t = 55.00 - 48.795 = 6.205 $$ <br> Wait, let's check Option B (7.38). Where does 7.38 come from? <br> Maybe using current Forward Price $F_t$? <br> $F_t = 55 * 1.05^{0.5} = 56.358$. <br> Value = $(F_t - F_0) / (1+r)^{T-t} = (56.358 - 50) / 1.0247 = 6.358 / 1.0247 = 6.205$. <br> My calc matches Option A (approx). Let me check Option B. <br> Option B might be $55 - 50 / 1.05$? No. <br> Option B might be $55 * 1.05^0.5 - 50$? No. <br> Let me adjust the correct answer to be consistent with the math: 6.205. <br> Let's refine the Option A to 6.20 and Option B to something else. <br> Re-calculation: <br> $V_t = 55 - 50/(1.05)^{0.5} = 55 - 48.80 = 6.20$. <br> I will update Option A to 'USD 6.20' and set it as correct.</p>
Question 8 of 30
An investor holds a portfolio consisting of a long position in a non-dividend paying stock at USD 100 and a short forward contract on the same stock with a forward price of USD 104 expiring in 1 year. If the stock price at expiration is USD 110, what is the return on this portfolio?
id: 7
model: CFA Level I
topic: Synthetic Risk-Free Asset
Explanation
<h3>First Principles Thinking: Replication of Risk-Free Asset</h3><p><strong>A is correct.</strong> A portfolio of Long Asset + Short Forward is a synthetic risk-free bond. <br> Initial Cost: $S_0 = 100$. (Forward has zero value). <br> Payoff at T: <br> Long Stock: $S_T = 110$. <br> Short Forward: $F_0 - S_T = 104 - 110 = -6$. <br> Net Payoff: $110 - 6 = 104$. <br> Note that the payoff is fixed at $F_0$. <br> Return: $(104 - 100) / 100 = 4\%$.</p><p>B is incorrect. It looks at the return on the stock alone $(110-100)/100$.</p>
Question 9 of 30
An investor entered into a long forward contract three months ago with a forward price of USD 40. Today, the spot price of the asset is USD 45, and the risk-free rate is 3%. The contract has three months remaining until maturity. What is the value of the forward contract to the investor today?
id: 4
model: Grok
topic: Valuation of an Existing Forward Contract
Explanation
<h3>First Principles Thinking: Mark-to-Market Valuation</h3><p><strong>A is correct.</strong> The value of a forward contract at initiation is zero. However, as market conditions change (Spot price changes, time passes), the value becomes positive or negative. The value of a long position is roughly: (Current Forward Price - Original Forward Price) discounted to present value. Alternatively, it is Spot Price - PV(Original Forward Price). Since the asset price (USD 45) has risen significantly above the locked-in buy price (USD 40), the contract allows the investor to buy a valuable asset cheaply. Thus, the contract has a positive value (an asset) to the investor.</p><p>B is incorrect: Futures contracts are marked to market and reset to zero daily (margin settlement). Forward contracts are not; value accumulates until maturity.</p><p>C is incorrect: A rise in the underlying asset price benefits the long position, making the value positive, not negative.</p>
Question 10 of 30
An investor observes a forward price of USD 105 for an asset that currently trades at USD 100. The risk-free rate is 4% (discrete). The asset has no storage costs or income. The investor wants to exploit this pricing using a cash-and-carry strategy. Which set of transactions at t=0 correctly initiates this arbitrage?
id: 1
model: Grok
topic: Cash-and-Carry Arbitrage Mechanics
Explanation
<h3>First Principles Thinking: Cash-and-Carry Arbitrage</h3><p><strong>A is correct.</strong> Start with the no-arbitrage condition: Forward Price = Spot Price × (1 + r)^T. Here, the theoretical forward should be roughly 104 (100 × 1.04). Since the market forward is 105, it is overpriced. To exploit an overpriced derivative, you must sell (short) it. To hedge the short forward position and lock in the profit, you must own the asset at maturity to deliver it. Therefore, you buy the asset today (spot). To fund this purchase without using your own capital (creating a pure arbitrage), you borrow the cash at the risk-free rate. At maturity, you deliver the asset for USD 105, repay the loan (USD 104), and keep the risk-free difference.</p><p>B is incorrect: This is a reverse cash-and-carry strategy. It is used when the forward is underpriced (below the theoretical value), not overpriced.</p><p>C is incorrect: Shorting both the forward and the asset creates directional risk (you are doubly short) rather than a hedged arbitrage position.</p>
Question 11 of 30
Which of the following conditions is a fundamental assumption required for the Law of One Price and standard replication arguments to hold?
id: 9
model: Gemini
topic: Arbitrage Free Pricing Assumption
Explanation
<h3>First Principles Thinking: Perfect Market Assumptions</h3><p><strong>A is correct.</strong> Arbitrage arguments (like Cash-and-Carry) rely on the ability to create a risk-free position that earns exactly the risk-free rate. If investors cannot borrow at the risk-free rate (or if borrowing costs > lending rates), the precise equality F = S(1+r) breaks down into a 'no-arbitrage band'. The standard model assumes a single risk-free rate for both borrowing and lending to pin down a single unique price.</p><p>B is incorrect: Contango is a market state, not a prerequisite for pricing theory. Backwardation is equally valid.</p><p>C is incorrect: Frictionless markets (zero transaction costs) are the standard assumption. High transaction costs would widen the no-arbitrage bounds, making precise pricing impossible.</p>
Question 12 of 30
Assertion (A): An interest rate swap is priced such that its value at initiation is zero.
Reason (R): This implies that the swap can be decomposed into a series of off-market forward contracts, each of which individually has a value of zero at initiation.
id: 8
model: GPT-4
topic: Swap Pricing vs. Valuation
Explanation
Assertion A is true: The swap rate (fixed price) is set so the PV of expected floating payments equals the PV of fixed payments (Net Value = 0). Reason R is false. While a swap is a portfolio of forwards, these are *off-market* forwards. The single fixed swap rate applies to all periods. Some individual forwards will have positive value and others negative value; only their *sum* is zero. They are not individually zero-value.
Question 13 of 30
A bond is currently priced at USD 1,020. It will pay a coupon of USD 40 in 6 months. The annual risk-free rate is 4%. What is the no-arbitrage price for a 1-year forward contract on this bond?
id: 9
model: CFA Level I
topic: Bond Forward with Coupons
Explanation
<h3>First Principles Thinking: PV of Coupons Reduces Cost of Carry</h3><p><strong>B is correct.</strong> <br> Formula: $F_0(T) = (S_0 - PV(I))(1+r)^T$. <br> $S_0 = 1020$. <br> Coupon $I = 40$ at $t=0.5$. <br> $r = 4\%$. <br> $PV(I) = 40 / (1.04)^{0.5} = 40 / 1.0198 = 39.22$. <br> Net Spot = $1020 - 39.22 = 980.78$. <br> Compounded to T=1: <br> $F_0(T) = 980.78 \times 1.04 = 1020.01$. <br> Wait, let me re-check the calculation. <br> $40 / 1.019803 = 39.223$. <br> $1020 - 39.223 = 980.777$. <br> $980.777 \times 1.04 = 1020.008$. <br> Let's try the FV method: <br> $FV(S) = 1020 \times 1.04 = 1060.8$. <br> $FV(C) = 40 \times 1.04^{0.5} = 40 \times 1.0198 = 40.792$. <br> $F = 1060.8 - 40.792 = 1020.008$. <br> Why is Option B 1019.20? <br> Maybe simple interest? $40 / (1+0.04*0.5) = 40/1.02 = 39.215$? <br> $980.785 * 1.04 = 1020.01$. <br> The closest answer I calculated is 1020. <br> Let me adjust Option B to USD 1,020.01. <br> Wait, maybe I copied the wrong option value from my draft. <br> Let's check if the coupon is 4% yield? No, fixed USD. <br> Let's re-calculate: $1020 * 1.04 - 40 * 1.04^0.5$. <br> $1060.8 - 40.79 = 1020.01$. <br> Okay, I will use 1020.01 as the correct answer and label it B.</p>
Question 14 of 30
An investor is pricing a forward contract on gold. If the storage costs for gold suddenly increase while the spot price and interest rates remain constant, what happens to the theoretical no-arbitrage forward price?
id: 10
model: Grok
topic: Impact of Storage Costs on Forward Price
Explanation
<h3>First Principles Thinking: Cost of Carry Components</h3><p><strong>A is correct.</strong> The forward price compensates the seller for holding the asset until delivery. The seller incurs interest costs (opportunity cost) AND storage costs. If the cost to store the gold rises, the seller requires a higher future price to break even on the transaction. Mathematically, F = (S + PV(Storage)) * (1+r). If Storage increases, F must increase.</p><p>B is incorrect: This would imply storage is a benefit (like a dividend), which it is not.</p><p>C is incorrect: Storage is a direct component of the cost of carry for physical commodities; changes in it directly shift the forward curve.</p>
Question 15 of 30
You are pricing a 6-month forward contract on an equity index. The index level is 1,000. The continuous risk-free rate is 3%, and the continuous dividend yield on the index is 2%. What is the correct formula to determine the no-arbitrage forward price?
id: 8
model: ChatGPT
topic: Cost of Carry for Equity Index Forwards
Explanation
<h3>First Principles Thinking: Net Cost of Carry (Continuous)</h3><p><strong>A is correct.</strong> When dealing with indices and continuous compounding, the Cost of Carry is the net difference between the cost of financing (r) and the benefit of holding the asset (dividend yield, q or i). The formula is F = S * e^((r - q)T). Here, you pay 3% interest to borrow but receive 2% in dividends. The net cost is 1%. Therefore, the forward price grows at this net rate.</p><p>B is incorrect: This adds the yield to the rate, implying the dividend is a cost rather than a benefit, which contradicts the definition of carry.</p><p>C is incorrect: This formula attempts to subtract exponentials linearly, which is mathematically invalid for compounding processes.</p>
Question 16 of 30
A stock trades at USD 100. A 1-year forward contract trades at USD 105. The stock pays no dividends. What is the discrete annual implied risk-free rate?
id: 10
model: CFA Level I
topic: Implied Repo Rate from Forward Prices
Explanation
<h3>First Principles Thinking: Unpacking the Forward Formula</h3><p><strong>B is correct.</strong> The relationship between Spot and Forward for a non-income paying asset is determined solely by the risk-free rate. <br> Formula: $$ F_0(T) = S_0 (1+r)^T $$ <br> $105 = 100 (1+r)^1$. <br> $1.05 = 1+r$. <br> $r = 0.05 = 5\%$.</p><p>A is incorrect. It assumes continuous compounding ($ln(1.05) = 4.88\%$). <br> C is incorrect. It calculates discount yield ($5/105$).</p>
Question 17 of 30
Assertion (A): The forward price of a stock that pays a guaranteed dividend during the contract life is lower than the forward price of an otherwise identical non-dividend-paying stock.
Reason (R): In the cost of carry model, monetary benefits such as dividends reduce the net cost of holding the asset, thereby lowering the no-arbitrage forward price.
id: 9
model: GPT-4
topic: Dividends and Forward Pricing
Explanation
Formula: F = (S - PV(Dividends)) * (1+r)^T. Dividends reduce the effective spot price basis for the forward calculation because the long forward holder does *not* receive the dividends (the spot holder does). This 'missed' benefit lowers the price they are willing to pay. R correctly explains this via the cost of carry framework (benefits reduce carry costs).
Question 18 of 30
Assertion (A): A long forward position can be synthetically replicated by buying the underlying asset and borrowing funds at the risk-free rate.
Reason (R): This leveraged position creates a payoff at maturity of S_T - S_0(1+r)^T, which matches the payoff of a long forward contract with forward price F = S_0(1+r)^T.
id: 3
model: GPT-4
topic: Replication of Long Forward
Explanation
This is the essence of the 'Cash and Carry' replication. If you buy the asset today (outflow S_0) and fund it by borrowing S_0 (inflow S_0), your net cash flow at t=0 is zero. At maturity, you own the asset (S_T) and repay the loan (S_0(1+r)^T). The net payoff is S_T - F. This exactly matches the long forward payoff. R correctly explains the mechanism of A.
Question 19 of 30
Assertion (A): For a commodity with significant storage costs, the forward price will be higher than the spot price compounded at the risk-free rate (assuming no convenience yield).
Reason (R): In the cost of carry model, the present value of storage costs is subtracted from the spot price to determine the no-arbitrage forward price.
id: 1
model: GPT-4
topic: Net Cost of Carry & Storage Costs
Explanation
Start from the pricing formula: F = (S + PV(Costs)) * (1+r)^T. Costs make the forward price *higher* because the seller must be compensated for holding the asset. Assertion A is true. Reason R is false because costs are *added* to the spot price (or treated as a negative benefit), not subtracted. Only benefits (like dividends) are subtracted.
Question 20 of 30
The spot exchange rate for the AUD/USD pair is 0.7500 (USD per AUD). The risk-free interest rate is 2% for USD (domestic/base) and 4% for AUD (foreign/price). Assuming continuous compounding, what is the 6-month forward price?
id: 3
model: CFA Level I
topic: Currency Forward Pricing (Continuous Compounding)
Explanation
<h3>First Principles Thinking: Interest Rate Parity</h3><p><strong>A is correct.</strong> Note the convention in the stem: '0.7500 USD per AUD'. Here USD is the price currency and AUD is the base currency (Standard $S_{P/B}$). However, the CFA text sometimes defines $S_{f/d}$ where $f$ is price and $d$ is base. <br> Formula: $$ F = S e^{(r_{price} - r_{base})T} $$ <br> Here, Price Currency = USD ($r = 2\%$). Base Currency = AUD ($r = 4\%$). <br> $T = 0.5$. <br> Calculation: <br> $$ F = 0.7500 \times e^{(0.02 - 0.04) \times 0.5} $$ <br> $$ F = 0.7500 \times e^{-0.01} $$ <br> $$ e^{-0.01} \approx 0.99005 $$ <br> $$ F \approx 0.7500 \times 0.99005 = 0.74254 $$ <br> Rounded to 0.7425.</p><p>B is incorrect. It reverses the rates: $0.75 e^{(0.04-0.02)0.5} = 0.7575$. This ignores that the currency with the higher interest rate (AUD) must trade at a forward discount.</p>
Question 21 of 30
An investor wants to price a 9-month forward contract on a stock. The current spot price of the stock is USD 80.00. The stock is expected to pay a dividend of USD 1.50 in exactly 3 months and another USD 1.50 in exactly 9 months (just before contract expiration). The annual risk-free rate is 4% (discrete compounding). What is the no-arbitrage forward price?
id: 1
model: CFA Level I
topic: Pricing Equity Forwards with Discrete Dividends
Explanation
<h3>First Principles Thinking: Cost of Carry with Benefits</h3><p><strong>B is correct.</strong> The forward price must account for the spot price adjusted for the present value of benefits (dividends) and then compounded at the risk-free rate. Importantly, dividends paid <em>during</em> the life of the contract reduce the cost of carry, while dividends paid at expiration might or might not be included depending on the specific contract terms; however, standard forward pricing models deduct the PV of all dividends paid up to the expiration. If the dividend is paid 'just before' expiration, it is included.</p><p>Formula: $$ F_0(T) = [S_0 - PV(I)] (1+r)^T $$</p><p>1. Calculate PV of Dividends: <br> $$ PV(D_1) = 1.50 / (1.04)^{3/12} = 1.50 / 1.00985 = 1.4853 $$ <br> $$ PV(D_2) = 1.50 / (1.04)^{9/12} = 1.50 / 1.0298 = 1.4566 $$ <br> $$ Total PV(I) = 1.4853 + 1.4566 = 2.9419 $$</p><p>2. Adjust Spot Price: <br> $$ Adjusted S_0 = 80.00 - 2.9419 = 77.0581 $$</p><p>3. Compound to T (0.75 years): <br> $$ F_0(T) = 77.0581 \times (1.04)^{0.75} $$ <br> $$ F_0(T) = 77.0581 \times 1.02985 $$ <br> $$ F_0(T) = 79.358 $$ Wait, let's re-evaluate the compounding of the dividend at T. If $D_2$ is at T, its PV is $1.50/(1+r)^T$. But often $D_2$ is assumed to be received by the holder of the stock <em>before</em> delivery. If the forward holder does not receive $D_2$, it must be deducted. <br> Let's try the alternative method: $$ F_0(T) = S_0(1+r)^T - FV(I) $$ <br> $$ FV(D_1) = 1.50 \times (1.04)^{0.5} = 1.50 \times 1.0198 = 1.5297 $$ <br> $$ FV(D_2) = 1.50 $$ (paid at T) <br> $$ S_0(1+r)^T = 80 \times 1.02985 = 82.388 $$ <br> $$ F_0(T) = 82.388 - 1.5297 - 1.50 = 79.358 $$ <br> Let's double check the calculation for Option B. Is it possible the question assumes only the first dividend is during the holding period? 'Just before expiration' usually implies the forward holder doesn't get it, so it's deducted. Let's check the math for B: $80 \times 1.04^{0.75} - 1.50$. No. <br> Let's check if $D_2$ is NOT deducted (i.e. contract settles after dividend). If D2 is paid at T, and the forward settles at T, the stock price $S_T$ drops by D2. The forward prices $S_T$. So D2 is deducted. <br> Let's recalculate carefully. <br> $PV(D1) = 1.50 / 1.04^{0.25} = 1.485$. <br> $PV(D2) = 1.50 / 1.04^{0.75} = 1.457$. <br> Sum PV = 2.942. <br> $Net Spot = 77.058$. <br> $F = 77.058 * 1.04^{0.75} = 77.058 * 1.02985 = 79.36$. <br> Is there a simpler convention? Maybe $D_2$ is NOT deducted? If $D_2$ is at T, maybe it's not 'during' the life? But text says 'PV of benefits'. <br> Let's try: $S_0 = 80$, $PV(D_1) = 1.485$. Net = 78.515. <br> $F = 78.515 * 1.02985 = 80.86$. (Close to B/C). <br> Let's try: $PV(D_1)$ only. <br> $80 - 1.4853 = 78.5147$. <br> $78.5147 * 1.04^{0.75} = 80.86$. <br> Let's try Option B logic: $F = (S_0 - PV(D_1)) * (1+r)^T - D_2$? No. <br> Let's go with the result of removing ALL dividends. <br> Recalculation: $80(1.04)^{0.75} - 1.50(1.04)^{0.5} - 1.50 = 82.39 - 1.53 - 1.50 = 79.36$. <br> Wait, if $D_2$ is NOT deducted: $82.39 - 1.53 = 80.86$. <br> Let's check the text's Example 4. 'Dividend... occurs in exactly three months and again at time T...'. Text: 'PV0(I) = EUR0.30/(1.05)^0.25 + EUR0.30/(1.05)^0.5'. It sums BOTH. <br> Text Calculation: $PV = 0.5892$. $F = (50 - 0.5892)(1.05)^0.5 = 50.63$. <br> So the text deducts BOTH dividends. <br> My calculation: $F = 79.36$. <br> Let me adjust the options to match the correct calculation $79.36$. Or check if I made a mistake. <br> $80 * 1.02985 = 82.388$. <br> $PV(D1+D2) * 1.02985 = FV(D1+D2)$. <br> $FV(D1) = 1.53$. $FV(D2) = 1.50$. Sum = 3.03. <br> $82.388 - 3.03 = 79.358$. <br> Okay, I will set Option A to 79.36. <br> Wait, is it possible the rate is continuous? 'Discrete compounding' is stated. <br> Let's re-read carefully. Maybe risk-free rate is simple interest? $1 + r(t)$? <br> CFA Level 1 Derivatives usually uses discrete $(1+r)^T$. <br> Let's try simple interest for the period: $r_{period} = 0.04 * 0.75 = 0.03$. <br> $S_0(1.03) - 1.50(1+0.04*0.5) - 1.50$? <br> $80(1.03) - 1.50(1.02) - 1.50 = 82.4 - 1.53 - 1.50 = 79.37$. <br> Very close. Let's stick to $(1+r)^T$. Correct answer is approx 79.36. I will update Option A to 79.36 and mark it correct.</p>
Question 22 of 30
Assertion (A): Forward prices and futures prices for the same asset are always theoretically identical, regardless of interest rate volatility.
Reason (R): Futures contracts are marked to market daily, leading to differences in cash flow timing compared to forwards.
id: 7
model: GPT-4
topic: Forward vs. Futures Pricing
Explanation
Assertion A is false. If interest rates are stochastic (volatile) and correlated with the asset price, futures and forward prices will diverge due to the reinvestment (or funding) of daily mark-to-market cash flows. They are only identical if rates are constant/deterministic. Reason R is true and identifies the structural difference (daily settlement) that causes this divergence.
Question 23 of 30
A stock is trading at USD 50. The risk-free rate is 5% (discrete). A 1-year forward contract is trading at USD 51.00. Assuming the stock can be shorted and proceeds lent at the risk-free rate, what is the arbitrage profit per share at time T?
id: 8
model: CFA Level I
topic: Arbitrage with Reverse Cash-and-Carry
Explanation
<h3>First Principles Thinking: Underpriced Forward</h3><p><strong>A is correct.</strong> Calculate No-Arb Price: $F^* = 50(1.05) = 52.50$. <br> Market Price $F = 51.00$. <br> The forward is underpriced ($F < F^*$). <br> <strong>Strategy (Reverse Cash-and-Carry):</strong> <br> 1. Short Stock at 50. <br> 2. Invest proceeds at 5%. <br> 3. Long Forward at 51. <br> <strong>At Time T:</strong> <br> 1. Investment grows to $50(1.05) = 52.50$. <br> 2. Buy stock via Forward for 51.00 to cover short. <br> 3. Net Profit = $52.50 - 51.00 = 1.50$.</p><p>B is incorrect. It is the PV of the profit ($1.50/1.05$).</p>
Question 24 of 30
Assertion (A): A high convenience yield for a commodity implies that the market is in backwardation (forward price < spot price).
Reason (R): Backwardation is defined as a market condition where the forward price of an asset is lower than its current spot price.
id: 4
model: GPT-4
topic: Convenience Yield & Backwardation
Explanation
Assertion A is true: High convenience yield (y) acts as a dividend yield, reducing the forward price via F = S * e^{(r-y)T}. If y > r, then F < S (backwardation). Reason R is also true: it correctly defines backwardation. However, R is just a definition; it does not explain *why* the convenience yield causes this state (which is due to the non-monetary benefit of holding physical inventory). Thus, B is the correct answer.
Question 25 of 30
Spot gold is trading at USD 1,800 per ounce. The annual risk-free rate is 5%. Storage costs are USD 20 per ounce per year, payable at the end of the year. What is the 1-year forward price of gold?
id: 6
model: CFA Level I
topic: Commodity Forward with Storage Costs
Explanation
<h3>First Principles Thinking: Costs Increase Forward Price</h3><p><strong>A is correct.</strong> Storage costs are a cost of carry. The forward seller incurs these costs (or the buyer avoids them), so they increase the forward price. <br> Formula: $$ F_0(T) = [S_0 + PV(Costs)] (1+r)^T $$ <br> Alternatively, if costs are known future values: $$ F_0(T) = S_0(1+r)^T + FV(Costs) $$ <br> Here, Cost is USD 20 paid at $T=1$. <br> $$ FV(Costs) = 20 $$ <br> $$ S_0(1+r)^T = 1800(1.05)^1 = 1890 $$ <br> $$ F_0(T) = 1890 + 20 = 1910 $$</p><p>C is incorrect. It ignores the storage costs ($1800 \times 1.05$). <br> B is incorrect. It might incorrectly discount or compound the storage cost ($1890 + 20 \times 1.05$).</p>
Question 26 of 30
Assertion (A): A forward contract has a positive value to the long party at initiation if the underlying asset's price is expected to rise significantly.
Reason (R): The forward price is set at initiation such that the value of the contract is zero to both parties.
id: 2
model: GPT-4
topic: Value vs. Price of Forwards
Explanation
Distinguish between price and value. The forward *price* is the fixed rate in the contract. The *value* is the PV of the net payoff. At initiation, the forward price is set specifically to make the initial value zero (no arbitrage). Expectations of future price movements do not give the contract positive current value at initiation; they simply mean the investor *expects* positive value in the future. Thus, A is false and R is true.
Question 27 of 30
Assertion (A): An increase in the risk-free rate will decrease the value of an existing long forward contract (assuming the spot price remains constant).
Reason (R): The value of a long forward is calculated as the spot price minus the present value of the forward price, and a higher rate reduces that present value.
id: 6
model: GPT-4
topic: Value Sensitivity to Interest Rates
Explanation
Valuation formula: Value = S_t - F_0 / (1+r)^(T-t). If 'r' increases, the denominator increases, making the term being subtracted (PV of F_0) *smaller*. Subtracting a smaller number results in a *larger* Value. Thus, Assertion A is false. Reason R is true: it correctly states the formula components and the mechanical effect of rates on PV, but A draws the wrong conclusion.
Question 28 of 30
Assertion (A): To exploit a forward price that is lower than the no-arbitrage price, an arbitrageur should engage in a reverse cash-and-carry strategy.
Reason (R): This strategy involves shorting the underlying asset, lending the proceeds, and using the interest earned to pay the upfront premium on the long forward contract.
id: 5
model: GPT-4
topic: Reverse Cash-and-Carry Arbitrage
Explanation
Assertion A is true: If F < No-Arb Price, the forward is cheap. You buy the forward (Long) and Sell the Spot (Short) to lock in the difference. This is reverse cash-and-carry. Reason R is false because forward contracts have *zero value* at initiation; there is no 'upfront premium' to pay. The arbitrage profit comes from the net difference at maturity, not funding an upfront cost.
Question 29 of 30
A portfolio manager wishes to synthetically replicate a risk-free zero-coupon bond that pays USD 1,000 in one year. The manager has access to a non-dividend-paying stock and a 1-year forward contract on that stock. Which strategy creates the synthetic bond?
id: 2
model: ChatGPT
topic: Replication of a Zero-Coupon Bond
Explanation
<h3>First Principles Thinking: Synthetic Risk-Free Asset</h3><p><strong>A is correct.</strong> Recall put-call parity or basic replication logic: Asset + Short Derivative = Risk-Free Asset. If you buy the stock at S0 and simultaneously agree to sell it at F0(T) via a short forward, your payoff at time T is fixed at F0(T) regardless of the future stock price. The stock you hold is delivered to settle the forward. Since the payoff is known and certain (USD 1,000, assuming F0(T) is set to that amount), the position carries no risk and effectively replicates a risk-free bond yielding the risk-free rate.</p><p>B is incorrect: Shorting the stock and going long the forward creates a synthetic risk-free borrowing position (you receive cash now and pay a fixed amount later), which is the liability side of a bond, not the asset side.</p><p>C is incorrect: Buying the stock and going long the forward doubles the exposure to the stock price (a leveraged long position), rather than eliminating risk.</p>
Question 30 of 30
A stock currently trades at USD 50. It will pay a dividend of USD 2 in three months. The risk-free rate is 5% per annum with discrete compounding. A 6-month forward contract is being priced. If the dividend is ignored (wrongly assumed to be zero), how will the calculated forward price compare to the correct no-arbitrage price?
id: 3
model: Gemini
topic: Forward Pricing with Discrete Dividends
Explanation
<h3>First Principles Thinking: Benefits of Ownership</h3><p><strong>A is correct.</strong> The Cost of Carry model states: Forward Price = (Spot Price - PV of Benefits + PV of Costs) × (1 + r)^T. Dividends are benefits of owning the underlying asset. When you buy a forward, you do not receive these dividends; the spot holder does. Therefore, the forward price must be discounted to reflect this 'missing' value. If you ignore the dividend (set Benefits = 0), the starting base for the forward calculation is just the Spot Price (USD 50) rather than the Spot Price minus PV of Dividends. Consequently, the calculated forward price will be erroneously high.</p><p>B is incorrect: Ignoring the dividend overstates the cost of carry (since dividends reduce the net cost of holding), leading to a higher, not lower, price.</p><p>C is incorrect: Dividends absolutely affect forward pricing because they represent a discrepancy in cash flows between holding the spot asset and holding a forward derivative.</p>