The current price of the Exchange Traded Fund YHT, which does not pay dividends, is $45.25 per share. Your position, worth 27150 dollars, consists entirely of YHT shares. The effective 3-month interest rate is 0.5% 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 420 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 -17% and 24%. 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 key data and translating the scenario into a math setup that we can analyze step by step.
- Current price per YHT share: 45.25 dollars.
- Your equity position: 27,150 dollars in YHT, which implies you hold 600 shares because 27,150 / 45.25 = 600.
- Three-month non-dividend yield assumption: the 3-month effective rate is 0.5%, so the 3-month futures price F0 equals S0 multiplied by 1.005 when the futures contract is fair valued for a non-dividend-paying asset.
- Number of futures contracts sold: 420 contracts expiring in 3 months.
- Your model for the 3-month net return on YHT: R ∈ [-17%, 24%] (i.e., -0.17 to 0.24).

Next, we express the value of the combined position at the 3-month horizon. Two components contribute:
- Value of the stock position in 3 months: 600 × S_T, where S_T = S0 × (1 + R).
- Payout from the short futures position: for a short futures position opened at F0, the payoff at maturity is F0 − S_T per contract; for 420 contracts the total payoff is 420 × (F0 − S_T).

Therefore, the total value V at 3 months is:
V = 600×S_T + 420×(F0 − S_T) = (600 − 420)×S_T + 420×F0 = 180×S_T + 420×F0.
This shows the combined value is linear in S_T, and thus in R, with all else fixed.

Now plug in the known quantities. We have S0 = 45.25 and F0 = S0 × 1.005 = 45.25 × 1.005 = 45.30625 (exactly 45.30625, but we will keep the more precise product for clarity). Compute 420×F0:
- 45.25 × 420 = 19,005.0
- 19,005.0 × 1.005 = 19,100.025
So 420×F0 = 19,100.025 dollars.

Also, 180×S_T = 180 × S0 × (1 + R) = 180 × 45.25 × (1 + R) = 8,145 × (1 + R).
Thus V(R) = 8,145 × (1 + R) + 19,100.025 = 27,245.025 + 8,145 × R. (Note: 8,145 comes from 180×45.25.)

To find the lowest possible value of the combined position given the model’s range for R, we seek the smallest R, which is R = −0.17. Substituting:
- 8,145 × (−0.17) = −1,385. - precisely, 8,145 × 0.17 = 1,384.65, so the product is −1,384.65.
- Therefore, V_min = 27,245.025 − 1,384.65 = 25,860.375 dollars.

Rounding to the standard two decimals used for currency, the lowest possible value of the combined position is 25,860.38 dollars.

In summary, the lowest scenario occurs when the stock price falls by 17% (the most negative end of your model), while you hold short 420 futures that are priced at F0 = 1.005×S0 for the 3-month horizon, producing a fixed futures payoff component and a stock-dependent component that decreases with S_T. The arithmetic combines these effects to yield the 25,860.38-dollar figure.

COEC_V 371 001 002 2025W1 Lecture 8 Practice Quiz

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