The current price of the Exchange Traded Fund YHT, which does not pay dividends, is $33.5 per share. Your position, worth 23450 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 525 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 -18% and 22%. 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 setup so the calculation is clear.
- Current price per YHT share: 33.50
- Position size: 23,450 USD in YHT stock, so number of shares = 23,450 / 33.50 = 700 shares
- Three-month interest rate (risk-free): 0.25%
- You sell 525 YHT futures contracts expiring in 3 months
- Your proprietary model bounds the 3-month net return on YHT to [-18%, +22%]
- We are asked for the lowest possible value of the combined position in 3 months based on the model

Step 1: Determine the stock value in 3 months under the model
- Let R be the 3-month net return on YHT per the model, with R ∈ [-0.18, +0.22]
- The ending price per share P_T = P_0 × (1 + R) = 33.50 × (1 + R)
- The value of the 700 shares in 3 months is V_stock = 700 × P_T = 700 × 33.50 × (1 + R) = 23,450 × (1 + R)
- With the worst-case for the stock from the model, R = -0.18, we would have V_stock = 23,450 × 0.82 = 19,249 (approximately 19.25k). With the opposite extreme R = +0.22, V_stock = 23,450 × 1.22 ≈ 28,609 (approximately 28.61k). The question asks for the lowest possible combined value, so we will focus on the lower end of R.

Step 2: Understand the futures hedge payoff
- You are short 525 futures contracts. Each futures contract on a single share passes through the same price change as the underlying stock (ignoring minor basis effects). If the stock price moves from P_0 to P_T, the futures price also moves by approximately ΔP = P_T − P_0.
- With a short futures position, the payoff is: Futures_profit = −(ΔP) × (number of underlying shares per contract) × (number of contracts).
- Assuming standard contract size of 100 shares, the total notional moved by the futures hedge is 525 contracts × 100 shares/contract = 52,500 shares. Therefore, Futures_profit = −(P_T − P_0) × 52,500.
- Substitute P_0 = 33.50 and P_T = 33.50 × (1 + R): ΔP = 33.50 × (1 + R) − 33.50 = 33.50 × R. Then Futures_profit = −(33.50 × R) × 52,500 = −1,760,? × R (in dollars), but we keep it symbolic for now as Futures_profit = −52,500 × 33.50 × R = −1,757,? × R.

Step 3: Compute combined value in 3 months as a function of R
- Combined_value(R) = V_stock + Futures_profit
- V_stock = 23,450 × (1 + R)
- Futures_profit = −52,500 × 33.50 × R = −1,757,? × R (we can keep exact product for a precise number later)
- For exact arithmetic: 52,500 × 33.50 = 1,757,? 33.50 × 52,500 = 1,758,750. So Futures_profit = −1,758,750 × (R/33.50) × (33.50/33.50)? A cleaner approach is to express Futures_profit directly in terms of R using the relationship ΔP = 33.50 × R, giving Futures_profit = −52,500 × (33.50 × R) = −1,757,? × R; numerically 52,500 × 33.50 = 1,758,750, so Futures_profit = −1,758,750 × R / 33.50? This simplification is messy in text, so we keep the standard chain: Futures_profit = −52,500 × (P_T − P_0) with P_T = 33.50 × (1 + R).
- Substituting: Futures_profit = −52,500 × [33.50 × (1 + R) − 33.50] = −52,500 × 33.50 × R = −1,758,750 × R / 1 (since 33.50 × 52,500 = 1,758,750).
- Therefore, Combined_value(R) = 23,450 × (1 + R) − 1,758,750 × R
- Expand: Combined_value(R) = 23,450 + 23,450R − 1,758,750R = 23,450 + (23,450 − 1,758,750)R = 23,450 − 1,735,300R

Step 4: Find the lowest possible value over R ∈ {−0.18, +0.22} (the model bounds)
- The coefficient of R in Combined_value(R) is −1,735,300, which is negative. This means Combined_value is decreasing in R: as R increases, the combined value decreases.
- Therefore, the lowest combined value occurs at the highest R in the model, R = 0.22 (i.e., +22% return).
- Compute: Combined_value_min = 23,450 − 1,735,300 × 0.22
- 1,735,300 × 0.22 = 381,766
- Finally, 23,450 − 381,766 = −358,316 (approximate).

Step 5: Compare with the provided answer (and interpret the result)
- The calculation above yields a notably negative value in the extreme high-return scenario because the futures hedge is highly levered relative to the stock position when contract sizes are assumed as 100 shares each.
- If an answer option exists, this would imply a very large potential downside relative to the starting value, given the size mismatch between the long stock position (700 shares) and the short futures position (52,500 shares equivalent).
- The explicitly provided answer in your prompt is $2,2438.72, which does not align with the above arithmetic under the standard contract-size assumption. This discrepancy suggests either a different contract multiplier, a different hedging ratio, or an alternative interpretation of how the futures payoff is defined in this particular question (e.g., nonstandard contract size, discrete rounding, or a model that directly ties the futures payoff to percentage moves rather than price moves).

Note on missing options: The question you provided lists a single numerical answer as the correct option but supplies an empty set of answer choices. If you share the full list of answer options (including any that correspond to the value around 22,438.72 or nearby figures), I can map the reasoning to the exact option and verify which one matches the model’s lowest-possible value under the given hedging setup.

COEC_V 371 001 002 2025W1 Lecture 8 Practice Quiz

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