Finance-Theory · Unit 11 · Video 5 · Interactive Practice

A Synthetic Index Fund Built from S&P 500 Futures

IKey Formulas

Formula	Name	What it needs
$\text{Notional} = 250 \times I$	Contract notional	The index level $I$ ; one contract is $250 per index point, so at $I = 1{,}000$ it carries $250,000 of exposure
$N = \dfrac{W}{250 \times I_0}$	Contracts to buy	The dollars to invest $W$ and today's index $I_0$ : $1,000,000 ÷ $250,000 = 4 contracts
$G_t = N \times 250 \times (F_t - F_{t-1})$	Daily settlement	Credited to (or debited from) the account every day; four contracts pass along $1,000 per index point
$V_t = W + 250N\,(F_t - F_0) = W \cdot \dfrac{I_t}{I_0}$	Futures portfolio	Futures price taken equal to the index and $250 N I_0 = W$ ; ignoring interest and dividends it equals the cash portfolio

Key Insight: Entering the contracts costs nothing, so the whole deposit stays in the account as collateral and earns interest, while the daily settlements add the index's gains and losses. Account plus futures gains and losses tracks the cash portfolio dollar for dollar, yet you own no stock: you hold a pure bet on the index, guaranteed by the clearing corporation.

IIHow Many Contracts?

Matching the exposure means dividing the dollars to invest by one contract's notional, $250 times the index.

Cash to invest $1,000,000 S&P 500 level 1,000

IIIMarked to Market, Day by Day

Settled daily, four contracts pass along $1,000 per index point, so the account tracks the cash portfolio.

Trading day 0 Contracts held

💡 Days 0, 5 and 10 use the video's levels (1,000, 900, 1,100); the days between are our illustration. Like the video's table, the portfolio values ignore the interest the account earns and the dividends the stocks pay.

IVDegree Days: A Contract on the Weather

Each day adds its shortfall below 65°F to the heating index, or its excess above to the cooling index.

💡 CME Group lists heating and cooling degree-day futures and options on 13 U.S. cities plus international locations (as of 2026), at $20 times the respective degree-day index; the seven-day strip and its one-week contract are our illustration.

VQuiz Questions

Problem 1 · Sizing the Position

Given: you want $3,500,000 of S&P 500 exposure, the index stands at 1,400, and one contract is $250 times the index. Find the number of contracts to buy.

✅ Correct! Each contract carries $250 × 1,400 = $350,000 of exposure, and $3,500,000 ÷ $350,000 = 10.

❌ That uses the video's notional. $250,000 per contract assumed the index at 1,000; at 1,400 each contract carries $250 × 1,400 = $350,000.

❌ The multiplier is missing. Dividing by the index level alone treats a contract as one index unit; it is $250 per index point, so divide by $250 × 1,400.

❌ Four belonged to the video's numbers. It came from $1,000,000 at an index of 1,000; here both the amount and the index differ.

❌ Not quite. Divide the dollars to invest by one contract's notional, $250 × index.

Show solution

Notional of one contract: $250 × 1,400 = $350,000.

N = \frac{3{,}500{,}000}{250 \times 1{,}400} = \frac{3{,}500{,}000}{350{,}000} = 10

Check: 10 × $350,000 = $3,500,000, so every index point moves the position by 10 × $250 = $2,500, exactly what a $3,500,000 cash portfolio gains or loses per point: $3,500,000 ÷ 1,400 = $2,500.

Problem 2 · What Is the Futures Portfolio Worth?

Given (illustrative figures): you deposit $1,500,000 in a futures account and buy 5 S&P 500 contracts ($250 × index) with the index at 1,200. Over the following week the index rises to 1,236, settled daily. Take the futures price equal to the index and ignore interest and dividends. Find the value of your futures portfolio.

✅ Correct! The settlements add 5 × $250 × 36 = $45,000 to the $1,500,000 in the account, the same 3% gain the cash portfolio earns.

❌ That counts one contract. $250 × 36 = $9,000 is one contract's gain; you hold five.

❌ The contracts are not an extra asset. Their notional, 5 × $250 × 1,236 = $1,545,000, is exposure, not money you own on top of the deposit; they cost $0 to enter and add only their daily gains and losses.

❌ That is the video's multiplier. $1,000 per index point was four contracts × $250; five contracts pass along $1,250 per point.

❌ Not quite. Futures portfolio = deposit + (contracts × $250 × index change).

Show solution

The position is sized to match: 5 × $250 × 1,200 = $1,500,000 of exposure.

Settlements: the index rises 1,236 − 1,200 = 36 points, and the daily credits over the week sum to 5 × $250 × 36 = $45,000.

Futures portfolio: $1,500,000 + $45,000 = $1,545,000.

V = W \cdot \frac{I_t}{I_0} = 1{,}500{,}000 \times \frac{1{,}236}{1{,}200} = 1{,}545{,}000

The cash portfolio is worth the same $1,545,000: a 3% rise in the index is a 3% rise in both.

Problem 3 · When the Contracts Don't Divide Evenly

Given (illustrative figures): you want $2,000,000 of S&P 500 exposure with the index at 1,250, contracts are $250 × index and trade only in whole numbers, and you buy the whole number of contracts whose exposure is closest to $2,000,000. The index then rises to 1,300. Take the futures price equal to the index and ignore interest and dividends.

How many contracts do you buy?

How much do the contracts gain in total?

✅ Correct! Six contracts carry $1,875,000 of exposure, so the 4% rise earns them $75,000, $5,000 less than the $80,000 a $2,000,000 cash portfolio gains.

❌ That uses the video's notional. $2,000,000 ÷ $250,000 = 8 assumes the index at 1,000; at 1,250 one contract is $312,500.

❌ Rounding up overshoots more. 7 contracts carry $2,187,500, $187,500 too much; 6 carry $1,875,000, only $125,000 short.

❌ Contracts come whole. 6.4 is the exact ratio, but you can't buy 0.4 of a contract; pick the whole number that lands closest.

❌ Check the count. Divide $2,000,000 by $250 × 1,250, then round to the nearest whole contract.

❌ That is the cash portfolio's gain. Six contracts don't match $2,000,000 exactly; they pass along 6 × $250 = $1,500 per point, not $1,600.

❌ That is seven contracts' gain. 7 × $250 × 50 = $87,500, but seven overshoot $2,000,000 by more than six fall short; use the six you bought.

❌ Check the gain. Gain = contracts × $250 × (1,300 − 1,250), using the count from the first part.

Show solution

Step 1 — Exact ratio: one contract is $250 × 1,250 = $312,500, so

N = \frac{2{,}000{,}000}{250 \times 1{,}250} = \frac{2{,}000{,}000}{312{,}500} = 6.4

Step 2 — Whole contracts: 6 contracts carry $1,875,000 ($125,000 short); 7 carry $2,187,500 ($187,500 over). Six is closer.

Step 3 — Gain: 6 × $250 × (1,300 − 1,250) = 6 × $250 × 50 = $75,000.

Step 4 — Compare: the cash portfolio gains $2,000,000 × 50 ÷ 1,250 = $80,000. The $5,000 shortfall is the uncovered $125,000 of exposure times the 4% rise.

Problem 4 · Pricing a Week of Heating Degree Days

Given (illustrative figures): a city's daily average temperatures over one week are 58, 61, 66, 52, 65, 70 and 55 °F. Suppose a contract on this week's heating-degree-day (HDD) index, base 65°F, is worth $20 times the index. Find the contract's value.

✅ Correct! The four days below 65°F contribute 7 + 4 + 13 + 10 = 34 heating degree days, and $20 × 34 = $680.

❌ Warm days don't cancel cold ones. Subtracting the 6 cooling degree days gives 28; a day above 65°F adds 0 to the HDD index, it does not reduce it.

❌ That adds the cooling degree days too. The 66°F and 70°F days count toward the CDD index (1 + 5 = 6), not the HDD index.

❌ That is the cooling index. $20 × 6 prices the days above 65°F; heating degree days count the days below.

❌ Not quite. Each day adds 65 − T when T is below 65°F, and 0 otherwise.

Show solution

Each day contributes $\text{HDD} = \max(0,\ 65 - T)$ :

58 → 7, 61 → 4, 66 → 0, 52 → 13, 65 → 0, 70 → 0, 55 → 10.

\text{HDD} = 7 + 4 + 0 + 13 + 0 + 0 + 10 = 34

Contract value: $20 × 34 = $680.

The warm days feed the separate cooling index, $\text{CDD} = \max(0,\ T - 65)$ : 1 + 5 = 6, worth $120 on a CDD contract.

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