Pricing and Valuation of Futures Contracts _1

Formula: F = S * [(1 + rdomestic) / (1 + rforeign)]. If rdomestic < rforeign, the fraction is less than 1. Therefore, F < S. The forward/futures rate will trade at a discount to spot. This logic often trips students up who confuse the currency quotation convention (DC/FC vs FC/DC).

First Principles Thinking: Inverting the Formula

B is correct. The cash-and-carry relationship implies:
$$ F = Se^{rT} $$

Substitute the values:
$$ 105 = 100e^{r} $$
$$ 1.05 = e^r $$
$$ r = \ln(1.05) \approx 0.04879 = 4.88\% $$

A is incorrect because it is the discrete return \( 5/100 = 5\% \).

C is incorrect because it is the discount yield \( 5/105 \).

This is a classic 'trick' question regarding futures. Unlike forwards, which accumulate value over time, futures are marked to market (settled) daily. Any gain or loss (in this case, a gain for the short position because the price dropped) is credited to the margin account. Immediately after this settlement, the contract is reset to the new market price, returning its value to zero.

First Principles Thinking: Interest Rate Parity

B is correct. For a currency futures or forward contract with continuous compounding:
$$ F_0 = S_0 e^{(r_{dc} - r_{fc})T} $$

Here, \( S_0 = 1.3000 \), \( r_{dc} = 0.02 \), \( r_{fc} = 0.04 \), and \( T=1 \):
$$ F_0 = 1.3000e^{(0.02 - 0.04)} = 1.3000e^{-0.02} \approx 1.2743 $$

A is incorrect because it swaps the domestic and foreign rates.

C is incorrect because it uses discrete compounding rather than the continuous compounding specified in the stem.

It is crucial to distinguish between the 'price' of the contract (the quote, for example, 100) and the 'value' (PV of gain/loss). At initiation, no money changes hands (except margin, which is collateral, not payment). The market 'price' is the specific number that sets the current net present value (NPV) of entering the contract to zero for both parties.

First Principles Thinking: Cumulative Realized Gain

A is correct. While the current value of a futures contract resets to zero daily, the question asks for the cumulative MTM gain/loss generated.
Short Position Payoff:
$$ Gain = Entry Price - Current Price $$
$$ Gain = 500 - 450 = USD 50 $$
The trader has received USD 50 in their margin account over the holding period.

C is incorrect. This would be the value of the contract for a new entrant or post-settlement.
B is incorrect. This attempts to discount the gain, but futures gains are realized immediately in cash, so no discounting applies to the realized amount.

First Principles Thinking: Futures Margin Mechanics

A is correct.

Price change:
$$ 1{,}770 - 1{,}800 = -30 $$

Total loss:
$$ 30 \times 100 \times 10 = \text{USD } 30{,}000 $$

Initial margin balance is \( \text{USD } 100{,}000 \). After the loss, the balance is \( \text{USD } 70{,}000 \). Maintenance margin is \( 7{,}500 \times 10 = \text{USD } 75{,}000 \), so a margin call is triggered. The account must be restored to the initial margin level, so the required deposit is:
$$ 100{,}000 - 70{,}000 = \text{USD } 30{,}000 $$

B is incorrect because it only restores the account to maintenance margin.

C is incorrect because it does not restore the account to initial margin.

First Principles Thinking: Net Cost of Carry

A is correct. The cost-of-carry relationship is driven by financing cost minus income. Here, the financing cost is approximately the market yield, \( 4\% \), while the coupon income is \( 6\% \). Because income exceeds financing cost, the net carry is negative.

With negative carry:
$$ F_0 = S_0 + (\text{Financing Cost} - \text{Income}) < S_0 $$

So the futures price trades at a discount to spot. The basis is \( S_0 - f_0 \), which is positive when \( f_0 < S_0 \). As expiration approaches, the cost/income differential shrinks and the futures price converges to the spot price, so the basis narrows to zero.

B is incorrect because it describes positive carry.

C is incorrect because convergence requires the basis to disappear by expiration.

First Principles Thinking: Cost of Carry (Continuous)

A is correct. Equity indices usually use continuous compounding conventions in CFA Level I (Module 6).
$$ f_0(T) = S_0 e^{(r - q)T} $$
$$ S_0 = 4,000 $$
$$ Net Rate = 0.03 - 0.01 = 0.02 $$
$$ T = 0.5 $$
$$ f_0(T) = 4,000 \times e^{0.02 \times 0.5} $$
$$ f_0(T) = 4,000 \times e^{0.01} $$
$$ f_0(T) \approx 4,000 \times 1.01005 $$
$$ f_0(T) = 4,040.20 $$

B is incorrect. It ignores the dividend yield (\( 4000 e^{0.03 \times 0.5} \)).
C is incorrect. It uses discrete compounding: 4000(1.02)^{0.5} = \( 4039.8 \), which is close but different, or perhaps \( 4000(1+0.02/2) = 4040 \). The exponential term is specific.

First Principles Thinking: Future Value of Costs

A is correct. We must sum the Future Value of the spot price and the Future Value of the storage costs.
1. FV of Spot:
$$ 25.00 \times (1.04)^{3/12} = 25.00 \times 1.00985 = 25.246 $$
2. FV of Storage Costs:
Cost 1 (t=1m): USD 0.10 \times (1.04)^{2/12} = 0.10 \times 1.0065 = 0.1007$
Cost 2 (t=2m): USD 0.10 \times (1.04)^{1/12} = 0.10 \times 1.0033 = 0.1003$
Cost 3 (t=3m): USD 0.10$
Total FV Costs = USD 0.3010$
Total Futures Price:
$$ 25.246 + 0.301 = 25.547 $$
Rounded to USD 25.55.

B is incorrect. It likely assumes costs are paid at \( t=0 \) and compounded (\( 0.30 \times 1.04^{0.25} \)).
C is incorrect. It ignores the time value of money on the spot price or costs.

First Principles Thinking: Tail Risk and Convexity

A is correct. This demonstrates the 'Convexity Bias'.
Futures (Short): Payoff is linear. $Gain = \text{Notional} \times \Delta Rate \times Time$. The cash is received daily (no discounting).
FRA (Long): Payoff is $ \frac{\text{Notional} \times \Delta Rate \times Time}{1 + (Rate \times Time)} $.
Mechanism: As rates ($R$) rise, the denominator in the FRA formula increases, dampening the payoff. The Futures contract does not have this denominator effect. Therefore, for a large rate increase, the Futures payoff (undiscounted) > FRA payoff (discounted).

B is incorrect. The FRA payoff is concave (dampened) relative to rates.
C is incorrect due to the daily settlement vs term settlement difference.

First Principles Thinking: Daily Reset

A is correct. Futures contracts are marked to market each day. The price move from \( 70 \) to \( 72 \) creates a realized gain of \( 2 \), which is credited to the trader's margin account. After settlement, the contract is reset to the new settlement price, so the value of the futures contract itself is \( V_t = 0 \).

B is incorrect because USD 2.00 is the realized gain, not the remaining contract value.

C is incorrect because USD 72.00 is the futures price, not the contract value.

First Principles Thinking: Futures Invoice Price

A is correct. The invoice price is the cash amount the buyer pays the seller when a specific bond is delivered against a futures contract. The formula is:
$$ \text{Invoice Price} = (\text{Futures Price} \times \text{Conversion Factor}) + \text{Accrued Interest} $$

Convert the quoted price: \( 98 ext{-}16 = 98 + 16/32 = 98.50 \). Then:
$$ \text{Invoice Price} = 98.50 \times 1.10 + 0.50 = 108.85 $$

B is incorrect because it ignores accrued interest.

C is incorrect because it ignores the conversion factor.

The entire premise of the cost-of-carry model is 'no-arbitrage'. We replicate the position using the asset and risk-free borrowing. Because the payoff is certain (hedged), the discount rate used must be the risk-free rate. Risk premiums (systematic risk) are irrelevant for finding the theoretical no-arbitrage price, although they do determine the expected future spot price.

Equation: F = S * e^{(r + u - y)T}, where r=rate, u=storage, y=convenience yield. For F < S (backwardation), the exponent term (r + u - y) must be negative. This implies that y > (r + u). In other words, the benefit of holding the physical inventory (convenience yield) is so high that it outweighs the costs of financing and storage.

First Principles Thinking: Inverse Pricing and BPV

A is correct. Interest rate futures use the convention:
$$ \text{Price} = 100 - (100 \times \text{Rate}) $$

Thus:
$$ \text{Rate} = 100 - 98.50 = 1.50\% $$

For a 90-day contract, BPV is:
$$ \Delta Value = 1{,}000{,}000 \times 0.0001 \times (90/360) = \text{USD } 25.00 $$

B is incorrect because it ignores the time fraction or the basis point definition.

C is incorrect because the implied rate is 1.50%, not 98.50%.

First Principles Thinking: Valuation of an Off-Market Forward

B is correct. The value of a forward contract at time \( t \) is the current spot price minus the present value of the original forward price:
$$ V_t = S_t - F_0 e^{-r(T-t)} $$

Here, \( S_t = 52 \), \( F_0 = 50 \), \( r = 0.05 \), and \( T-t = 0.5 \).
$$ V_t = 52 - 50e^{-0.05(0.5)} $$
$$ V_t = 52 - 48.765 \approx 3.23 $$

A is incorrect because it ignores the time value of money.

C is incorrect because it uses the new forward price rather than the current value of the existing forward contract.

Cost of Carry model: Futures Price = (Spot Price + PV of Storage Costs) (1+r)^t. Storage costs are a negative cash flow for the holder of the physical asset. To be indifferent between holding physical (paying storage) and holding a future (avoiding storage), the futures price must be higher to compensate the seller for incurring those storage costs. Therefore, F > S (1+r)^t (ignoring convenience yield for this specific comparison of storage cost impact).

This tests the convexity adjustment for interest rate futures. For Eurodollar-style contracts, the quoted futures price is inversely related to the implied rate: \( ext{Price} = 100 - ext{Rate} \). The standard convexity adjustment implies that the futures rate is greater than the corresponding forward rate. Equivalently, \( ext{Forward Rate} = ext{Futures Rate} - ext{Convexity Adjustment} \). Therefore, the futures rate is higher than the forward rate.

First Principles Thinking: Mark-to-Market and Margin Balance

B is correct. The question asks for the margin balance per contract and does not specify a contract multiplier, so the price changes are treated as per-contract gains and losses.

Day 1:
Loss = \( 1{,}800 - 1{,}750 = 50 \).
Balance = \( 6{,}000 - 50 = 5{,}950 \).
Since \( 5{,}950 > 4{,}500 \), no margin call occurs.

Day 2:
Gain = \( 1{,}790 - 1{,}750 = 40 \).
Ending balance = \( 5{,}950 + 40 = 5{,}990 \).

A is incorrect because the balance does not fall to maintenance margin.

C is incorrect because the Day 1 loss and Day 2 gain do not exactly offset.

First Principles Thinking: Cash-and-Carry Arbitrage / Hedging

B is correct. A long stock position plus a short futures position synthetically locks in the futures price. At maturity:
$$ \text{Long stock value} = S_T $$
$$ \text{Short futures payoff} = F_0 - S_T $$
$$ \text{Net value} = S_T + (F_0 - S_T) = F_0 $$

Because \( F_0 = S_0(1+r)^T \) under cost of carry, the combined position earns the risk-free rate.

A is incorrect because the hedge locks in more than principal when the futures price embeds the risk-free carry.

C is incorrect because it is the unhedged stock return.

T-Bond futures allow the delivery of various bonds. To make them comparable, the exchange defines a 'standard' bond (usually 6% coupon). The conversion factor is a multiplier that adjusts the price of the actual bond delivered to what it would be if it yielded 6%. It essentially makes all deliverable bonds roughly equal in value per unit of futures contract.

First Principles Thinking: Contract Definition

A is correct. The June contract prices the 3-month rate starting in June.
$$ Rate_{June} = 100 - Price_{June} $$
$$ Rate_{June} = 100 - 96.00 = 4.00\% $$
This rate applies to the period June to September.
Note: The September contract price (95.50) implies the rate starting in September (4.50%), which is irrelevant to the specific question asked (rate starting in June).

B is incorrect. This is the rate starting in September.
C is incorrect. This is the spread between the two.

Calculate the loss: (1780 - 1800) 100 oz 10 contracts = -20 1000 = -20,000 USD total loss. The initial equity was 100,000 USD (10k 10). The new equity is 80,000 USD. The maintenance margin requirement is 75,000 USD (7.5k 10). Since the remaining equity (80,000 USD) is still above* the maintenance level (75,000 USD), there is NO margin call. The trader lost money, but not enough to trigger a call.

First Principles Thinking: Commodity Cost of Carry

A is correct. Use the continuous cost-of-carry formula:
$$ F_0 = S_0 e^{(r+c-y)T} $$

Substitute the values:
$$ 82 = 80e^{(0.05+0.02-y)} $$
$$ 1.025 = e^{0.07-y} $$
$$ \ln(1.025) = 0.07-y $$
$$ y = 0.07 - \ln(1.025) \approx 0.0453 = 4.5\% $$

B is incorrect because it ignores storage or subtracts the price ratio incorrectly.

C is incorrect because a negative convenience yield would increase the futures price above the no-yield cost-of-carry price.

First Principles Thinking: Convexity Bias in Rates

C is correct. A short interest rate futures position and a receive-floating, pay-fixed FRA are both positioned for rising rates. If rates fall, both positions lose, but the futures loss is linear in the rate move, while the FRA settlement is discounted by the lower market rate. The lower discount rate magnifies the present value of the FRA loss, so the short futures position performs better, or loses less, than the FRA.

A is incorrect because daily settlement is not the key comparison in this falling-rate scenario.

B is incorrect because the pay-fixed FRA is effectively short convexity relative to the linear futures position.

First Principles Thinking: Futures vs. Forward Pricing with Correlation

B is correct. Futures are settled daily, while forwards settle at maturity. If the index level and interest rates are positively correlated, a long futures position receives variation margin when rates are high and pays variation margin when rates are low. That reinvestment advantage makes the futures contract more valuable, so its initial price must be higher than the forward price: \( f_0 > F_0 \).

A is incorrect because it states the wrong inequality.

C is incorrect because dividend yield affects both prices through cost of carry; it is not the driver of the futures-forward difference here.

First Principles Thinking: Mark-to-Market Reinvestment

C is correct. This is a theoretical nuance with numerical implications.
Mechanism:
With positive correlation:
- When Gold rises (Profit), Rates rise. The investor receives cash (Variation Margin) and reinvests it at a higher rate.
- When Gold falls (Loss), Rates fall. The investor finances the loss at a lower rate.
This 'positive convexity' makes the futures contract more valuable than the forward (where gains/losses are trapped until maturity). Since the futures contract is more desirable, it trades at a higher price.
$$ f_0 > F_0 $$

A is incorrect. This would be true if the correlation were negative.
B is incorrect. This assumes constant rates or zero correlation.

First Principles Thinking: Tailing the Hedge

A is correct. Futures are settled daily. If the hedge generates variation margin gains, those cash flows can earn interest before the hedge horizon. This interest income provides additional hedging power, so fewer futures contracts are needed than the unadjusted forward-style hedge ratio suggests.

The adjustment factor is:
$$ e^{-rT} $$

Adjusted contracts:
$$ N = \left(\beta \frac{P}{F}\right)e^{-rT} $$

Since \( e^{-rT} < 1 \), the manager uses fewer contracts. This is called tailing the hedge.

B is incorrect because liquidity affects execution costs, not the tailing adjustment.

C is incorrect because dividend differences affect futures pricing, not the margin cash-flow adjustment.

This is 'Cash and Carry' arbitrage. If Future > Theoretical Limit (overpriced), you sell the expensive thing (Short Future) and buy the cheaper replication (Buy Spot + Store). The question specifies the price is strictly greater than the bounds including costs, meaning a risk-free profit is possible after paying those costs.