Let's break down the problem step by step. The answer is **4,110.6** index points, so we will work towards that and explain each step. ## **Given Data** - **Current index level \( S_0 \):** 3,933 - **Time to maturity \( T \):** 1 year - **Risk-free rate for borrowing \( r_b \):** 4.3% (annual compounding) - **Risk-free rate for lending \( r_l \):** 4.3% - 0.4% = 3.9% (annual compounding) - **Dividends received over year (reinvested):** 0.9 index points at maturity - **Bid-ask spread for stock index basket:** 4 index points - **Stock borrowing fee (if short selling):** 7 index points (paid at maturity) - **Futures round-trip commissions:** 7 index points (paid at start) - **No other transaction costs** - **Question:** Find the **highest futures price** (upper no-arbitrage bound). ## **Step 1: Understand Arbitrage Bound** The **highest futures price** is based on the **cash-and-carry arbitrage**: - **Buy** the index basket, **sell** the futures contract. - At maturity, deliver the stocks, receive the futures price. We need to include all transaction costs and yields. ## **Step 2: Calculate Net Cost and Cash Flows** ### **Buy the Basket:** - **Buy price:** Need to pay at the **ask**, so \( S_0 + \text{half spread} \). - The spread is 4 index points, so typically, buying at ask is \( S_0 + 2 \). - But we'll see that by buying, you pay the ask, so \( S_0 + 2 \). - **Commission:** 7 index points (paid at start). - **Dividend:** Receive 0.9 index points at maturity. - **Financing:** Borrow at 4.3% for 1 year. - **Sell Futures:** No short-selling of stocks (so no borrowing fee for stocks). But let's be meticulous. Since we're buying the basket and selling futures, **no short-selling fee** applies. The **borrowing rate** applies because you borrow money to buy the basket, not the lending rate. ### **Step-by-step Cash Flows at Initiation:** - **Pay for stocks:** \( S_0 + 2 \) (ask price) - **Pay commission:** 7 **Total Outflow at Start:** \[ \text{Outflow} = (S_0 + 2) + 7 = S_0 + 9 \] ### **Step 3: Financing Cost** Borrow this amount at 4.3% for 1 year (annual compounding): \[ \text{Future value at maturity} = (S_0 + 9) \times (1 + 0.043) \] ### **Step 4: Dividend Income** You receive 0.9 index points in dividends at maturity. ### **Step 5: Net Outflow at Maturity** At maturity: - **Sell stocks at bid price:** If you want to close out the position, you sell at the bid, which is \( S_0 - 2 \). - But in cash-and-carry, you **deliver the basket to close the futures**, so you don't pay the bid-ask to exit. - Assume you use the stocks to deliver against the futures contract (so no bid-ask on exit). But, since the arbitrage bounds are about **entering and exiting**, some sources say to include the half-spread, but with futures, you just deliver the basket (no bid-ask to unwind). **We use:** - **Buy at ask**: \( S_0 + 2 \) - **No sell at bid** at maturity (since you deliver stocks). ### **Step 6: Futures Price Payoff** At maturity: - **Deliver stocks (no transaction cost)** - **Receive futures price \( F \)** - **Receive dividend:** 0.9 **Net cash at maturity:** - **Pay off loan:** \( (S_0 + 9) \times 1.043 \) - **Receive:** \( F + 0.9 \) ### **Step 7: Set Arbitrage Condition** For no arbitrage, the net profit should be ≤ 0: \[ \text{Net profit} = F + 0.9 - (S_0 + 9) \times 1.043 \leq 0 \] \[ F \leq (S_0 + 9) \times 1.043 - 0.9 \] ### **Step 8: Plug in Numbers** Given \( S_0 = 3,933 \): \[ F \leq (3,933 + 9) \times 1.043 - 0.9 \] \[ F \leq 3,942 \times 1.043 - 0.9 \] \[ F \leq 4,109.406 - 0.9 = 4,108.506 \] Wait, this is a bit less than the answer you provided. Let's check if we've missed a transaction cost. #### **Check All Transaction Costs:** - **Buy at ask**: +2 - **Futures commission**: 7 - **Stock commission**: Not specified, so not included (commission is for futures) - **Bid-ask spread:** Already included in buy at ask (+2) - **Short-selling fee:** Not relevant here (we are not shorting stocks) - **Any other fees at maturity?** No. But the problem says **round-trip commissions in the futures market are 7 index points and payable at the start**. So, perhaps the 7 is only for the futures trade, not the stock. Should the 4 index points be fully included for the bid-ask spread? Let's try the other way: maybe you have to cross the full spread to buy and then sell. But in cash-and-carry, you **buy the index basket at ask** and deliver, so bid-ask is +2. But some sources add the **full spread** to be conservative. Let's try: #### **If we include full bid-ask spread (4), plus futures commission (7):** \[ F \leq (S_0 + 4 + 7) \times 1.043 - 0.9 \] \[ F \leq (3,933 + 11) \times 1.043 - 0.9 \] \[ F \leq 3,944 \times 1.043 - 0.9 \] \[ F \leq 4,112.492 - 0.9 = 4,111.592 \] Now, this is **4,111.6**, which matches more closely to the correct answer **4,110.6** (but still 1 point too high). #### **Are we missing the stock borrowing fee?** No, because we are **not shorting** stocks in cash-and-carry, so NO. #### **Is the commission double-counted?** - Futures commission: 7 index points, paid at start (so not annualized). #### **Is the dividend subtracted or added?** - Since you receive dividend, you **subtract** from the cost. #### **Are we supposed to use the lending rate for the dividend?** Sometimes, the dividend is discounted at the lending rate, but since it's given as 0.9 index points at maturity, it's already at maturity value. ## **Step 9: Final Formula and Calculation** So the correct formula is: \[ F \leq [(S_0) + (\text{bid-ask spread}) + (\text{futures commission})] \times (1 + r_b) - (\text{dividend at maturity}) \] Plug in: - \( S_0 = 3,933 \) - Bid-ask spread (full) = 4 (sometimes, because worst-case, you buy at ask and sell at bid) - Futures commission = 7 - \( r_b = 0.043 \) - Dividend = 0.9 \[ F \leq (3,933 + 4 + 7) \times 1.043 - 0.9 \] \[ F \leq 3,944 \times 1.043 - 0.9 \] \[ F \leq 4,112.492 - 0.9 = 4,111.6 \] But the answer is **4,110.6**. ## **Step 10: The One Index Point Difference** Given the answer is **4,110.6**, likely they **only add one side of the spread (2 points)** (since you just buy at ask, deliver at maturity), so: \[ F \leq (3,933 + 2 + 7) \times 1.043 - 0.9 = (3,942) \times 1.043 - 0.9 \] \[ F \leq 4,109.406 - 0.9 = 4,108.506 \] But this does **not** yield 4,110.6. But if we **exclude the commission from the financing (i.e., do not grow it at 4.3%)** (i.e., pay it at start, not borrowed): \[ F \leq (3,933 + 4) \times 1.043 - 0.9 + 7 \] \[ F \leq 3,937 \times 1.043 - 0.9 + 7 \] \[ F \leq 4,106.391 - 0.9 + 7 = 4,112.491 \] No. ### **Let's try final check: Use only half-spread and commission:** \[ F \leq (3,933 + 2) \times 1.043 - 0.9 + 7 \] \[ F \leq 3,935 \times 1.043 - 0.9 + 7 \] \[ F \leq 4,103.405 - 0.9 + 7 = 4,109.505 \] Still, not matching. ## **Final Step: Check If There Is Short Selling (Reverse Cash-and-Carry)** If you do **reverse cash-and-carry** (i.e., sell the index basket short, invest at lending rate, buy the future), then you pay the **stock borrowing fee**, the lower risk-free rate, etc. But the question asks for the **highest price** (upper bound), so **cash-and-carry** is correct. ## **Conclusion** **The correct step-by-step calculation:** \[ \boxed{ F \leq (S_0 + \text{full spread} + \text{futures commission}) \times (1 + r_b) - \text{dividend at maturity} } \] Plug in the numbers: \[ F \leq (3,933 + 4 + 7) \times 1.043 - 0.9 = 4,111.6 \] ### **Final Answer:** \[ \boxed{4,111.6} \] **But your given answer is 4,110.6**. ### **Possible Explanation for 1 Point Difference:** - Perhaps the **dividend is not exactly 0.9**, or one of the transaction costs is **not annualized**. Sometimes, the commission is paid up front and not included in the amount that accrues interest. If you don't grow the 7 point commission: \[ F \leq (3,933 + 4) \times 1.043 - 0.9 + 7 \] \[ = 3,937 \times 1.043 - 0.9 + 7 = 4,106.391 - 0.9 + 7 = 4,112.491 \] But that's higher. **Given the standard approach and the data, the calculation yielding 4,111.6 is correct.** --- ## **Summary Table** | Step | Value | |----------------------|-----------| | \(S_0\) | 3,933 | | Bid-ask spread | +4 | | Futures commission | +7 | | Sum | 3,944 | | Borrowing rate (1 yr)| × 1.043 | | Subtotal | 4,112.5 | | Dividend at maturity | -0.9 | | Final upper bound | 4,111.6 | --- ## **Final Answer (with 1 decimal place):** \[ \boxed{4,111.6} \] --- If you **must** get 4,110.6, maybe one of the costs is not included or is different by 1 point. Please double-check the data, but the above is the standard approach. --- **Summary:** - Add current index, full bid-ask spread, futures commission. - Multiply by (1 + borrowing rate). - Subtract dividend at maturity. - Round to one decimal place. **Highest non-arbitrage futures price:** \[ \boxed{4,111.6} \] (matches calculation, check if data is off by 1 point).