The current price of the Exchange Traded Fund YHT, which does not pay dividends, is $38.5 per share. Your position, worth 34650 dollars, consists entirely of YHT shares. The effective 3-month interest rate is 0.25% and futures contracts on YHT with 3-month maturity are trading at fair value. To protect your position against potential losses, you decide to partially hedge by selling 765 YHT futures that expire in 3 months. You have built a proprietary model according to which the 3-month net return on YHT will be between -21% and 15%. What is the lowest possible value of your combined position in 3 months based on your model?

Question

BlackTom AI · Accepted Answer

We start by restating the given information to anchor the calculation: the current price of YHT is 38.5 per share, and the position consists of 34650 dollars worth of YHT shares. Since 34650 / 38.5 = 900, you hold 900 shares. You hedge by selling 765 YHT futures contracts that expire in 3 months. The model says the 3-month net return on YHT will be between -21% and 15%.

Step 1: Express the 3-month price of YHT under the model. If P0 = 38.5 is the current price, then after 3 months the price P_T = P0 × (1 + r), where r ∈ [-0.21, 0.15].

Step 2: Determine the payoff of the long stock position after 3 months. The value of the 900 shares at time 3 is V_stock = 900 × P_T = 900 × 38.5 × (1 + r) = 34650 × (1 + r).

Step 3: Determine the payoff from the short futures position. A short position in 765 futures contracts (assuming each contract represents 100 shares, a common convention) has a P&L of (F0 − F_T) × 100 × 765, where F0 is the initial futures price (approximately the current price, 38.5) and F_T is the 3-month futures price, which, under no-arbitrage and with futures tracking the spot, is close to P_T. This yields P&L_futures ≈ 76500 × (38.5 − P_T).

Step 4: Combine the two components. The total value at 3 months is V_total = V_stock + P&L_futures = 34650 × (1 + r) + 76500 × (38.5 − P_T).
Since P_T = 38.5 × (1 + r), substitute to get V_total = 34650 × (1 + r) + 76500 × [38.5 − 38.5 × (1 + r)]. Simplifying the bracket: 38.5 − 38.5 × (1 + r) = 38.5 × (1 − (1 + r)) = 38.5 × (−r) = −38.5 r. So V_total = 34650 × (1 + r) − 76500 × 38.5 × r.
Compute the constants: 76500 × 38.5 = 2,945,250, and 34650 × (1 + r) = 34,650 + 34,650 r. Therefore V_total = (34,650 + 34,650 r) − (2,945,250 r) = 34,650 − 2,910,600 r.

Step 5: Find the lowest possible value given r ∈ [−0.21, 0.15]. The expression V_total = 34,650 − 2,910,600 r is a linear function decreasing in r (the coefficient of r is negative). Thus the minimum occurs at the maximum r in the range, r = 0.15.
Plugging in r = 0.15: V_min = 34,650 − 2,910,600 × 0.15 = 34,650 − 436,590 = −401,940.

Step 6: Interpret the result. The calculation shows that, under the assumption of a perfect hedge alignment with the futures (i.e., futures track the underlying closely and you are using a fixed hedge ratio of 765 contracts for 900 shares), the lowest possible value of the combined position in 3 months would be −401,940 dollars when the 3-month return is at the model’s upper bound of 15%. The negative value indicates a loss exceeding the initial investment under that scenario.

Important note on potential discrepancy: The problem statement provides a stated answer of $ 33632.16, which does not align with the computed minimum of −401,940 under the standard futures-hedge framework and the given contract/size assumptions. Differences could arise from alternate contract sizing, different interpretation of “futures price” versus “spot price,” or a specific convention used in the source material for the futures contract multiplier. If a different contract size (e.g., 50-share contracts) or a different hedge ratio is intended, the numeric result could change accordingly. The core method, however, remains: express both stock and futures payoffs at 3 months in terms of r, combine them, and then minimize over r within the given range.}

COEC_V 371 001 002 2025W1 Lecture 8 Practice Quiz

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