Finance-Theory · Unit 11 · Video 4 · Interactive Practice

Cost of Carry Across Asset Classes: Gold, Gasoline, Stock Index Futures and the 1987 Crash

IKey Formulas

Formula	Case	What it needs
$F_{t,T} \approx H_{t,T} = (1+r_f)^{T-t} S_t + \text{FV}_T(\text{net storage costs})$	Any date $t$	Today's spot $S_t$ , the rate $r_f$ , the remaining time $T - t$ , and the net cost of holding the asset
$F_{t,T} \approx H_{t,T} = (1+r_f)^{T-t} S_t$	Gold	No storage cost, no dividends, no convenience yield: spot grown at $r_f$
$F_{t,T} \approx H_{t,T} = (1+r_f+c-y)^{T-t} S_t$	Gasoline	Storage cost $c$ added, convenience yield $y$ subtracted (both % per period)
$F_{t,T} \approx H_{t,T} = (1+r_f-d)^{T-t} S_t$	Stock index	Annualized dividend yield $d$ subtracted; the contract is cash settled

Key Insight: Discounted, the relation reads $\dfrac{F_{t,T}}{(1+r_f)^{T-t}} \approx S_t + \text{PV}_t(\text{net storage costs})$ : what you pay at settlement, in today's money, equals the asset's value now plus the net cost of holding it until $T$ . A quote that breaks it is either an arbitrage or a sign that the market is dislocated.

IIOne Relation, Three Carry Rates

Each asset adds to or subtracts from $r_f$ ; the sign of the net carry decides whether futures sit above spot.

r_f 3.5% storage c 0.0% convenience y 14.2% dividend d 0.3% T − t 6 mo

💡 The video's gasoline figure of about 14.18% is a convenience yield net of storage, which is why $c$ starts at zero here: the quotes cannot separate the two.

IIIRunning It Backwards: The Implied Carry

A spot price and one futures quote pin down the growth rate between them, and so the implied carry.

Quotes of 2 Aug 2001

IVRich or Cheap: Which Trade?

When the quote leaves the carry value, buy the cheaper route to owning the asset at $T$ and sell the dearer one.

Asset Quote − carry value 0

💡 After Hurricane Katrina destroyed Gulf Coast storage in 2005, the gasoline relation stayed violated for a period; the video reads that gap as a signal that building your own storage had become worthwhile, if you could.

VQuiz Questions

Problem 1 · Gold Futures From Spot

Given: spot gold is $1,250.00 an ounce, $r_f = 3\%$ a year, and the futures contract settles in 3 months. Gold has no storage cost, dividend or convenience yield. Find the no-arbitrage futures price $F_{t,T}$ , to the nearest cent.

✅ Correct! Gold's futures price is just spot grown at

r_f

over the remaining time:

(1.03)^{3/12} \approx 1.007417

, so the futures price ≈ $1,259.27.

❌ Not quite. That grows spot for a whole year. The exponent is the remaining time

T - t = 3/12

of a year, not 1.

❌ Not quite. The exponent counts years, because

r_f

is an annual rate: 3 months is

T - t = 3/12

, not 3.

❌ Not quite. Dividing by

(1.03)^{3/12}

gives a present value. The futures price is paid at

T

, so spot is grown forward, not discounted.

❌ Not quite. Use

F_{t,T} \approx (1+r_f)^{T-t} S_t

with

T - t = 3/12

Show solution

With no storage, dividend or convenience term, the net storage term is zero and the relation collapses to

F_{t,T} \approx H_{t,T} = (1+r_f)^{T-t} S_t, \qquad T - t = \tfrac{3}{12}.

The growth factor is $(1.03)^{3/12} \approx 1.007417$ (rounded to six places), so the futures price is 1.007417 × $1,250.00 ≈ $1,259.27.

The other options use the wrong exponent (1 year gives $1,287.50; 3 "years" gives $1,365.91) or discount instead of compounding ($1,240.80).

Problem 2 · Gasoline: Which Way Do c and y Go?

Given: spot gasoline is $3.20 a gallon, the futures contract settles in 8 months, $r_f = 5\%$ , storage cost $c = 2\%$ and convenience yield $y = 12\%$ , all per year. Find the no-arbitrage futures price, to the nearest cent.

✅ Correct! The net carry is

0.05 + 0.02 - 0.12 = -0.05

, so the futures price sits below spot: the benefit of having gasoline on hand outweighs interest plus storage.

❌ Not quite. The convenience yield is a benefit of holding the asset, so it is subtracted: the factor is

1 + r_f + c - y

, not

1 + r_f + c + y

❌ Not quite. That includes interest and storage but leaves out the convenience yield

y = 12\%

, which the futures holder does not get.

❌ Not quite. Storage is a cost of holding the asset, so it is added to

r_f

, not subtracted: the factor is

1 + r_f + c - y

❌ Not quite. Use

F_{t,T} \approx (1+r_f+c-y)^{T-t} S_t

with

T - t = 8/12

Show solution

F_{t,T} \approx H_{t,T} = (1 + r_f + c - y)^{T-t} S_t, \qquad 1 + 0.05 + 0.02 - 0.12 = 0.95, \quad T - t = \tfrac{8}{12}.

The growth factor is $(0.95)^{8/12} \approx 0.966383$ (six places), so the futures price is 0.966383 × $3.20 ≈ $3.09.

The mistakes behind the distractors: adding $y$ gives $(1.19)^{8/12}$ and $3.59; dropping $y$ gives $(1.07)^{8/12}$ and $3.35; subtracting $c$ gives $(0.91)^{8/12}$ and $3.00.

Problem 3 · Implied Dividend Yield

Given: a stock index stands at 4,000.00 and its futures contract settling in 6 months at 4,030.00, with $r_f = 4\%$ a year. Find the annualized dividend yield $d$ the quotes imply, using $F_{t,T} \approx (1+r_f-d)^{T-t} S_t$ .

✅ Correct!

d = 1.04 - (4030/4000)^2 = 1.04 - 1.01505625 = 0.02494375 \approx 2.49\%

. The futures price is above spot because

r_f > d

❌ Not quite. With

T - t = 6/12

, the ratio

F/S = 1.0075

equals

(1.04 - d)^{1/2}

; you have to square it before subtracting from 1.04.

❌ Not quite.

(F/S)^2 - 1 \approx 1.51\%

is the net carry

r_f - d

, not

d

itself. Subtract it from

r_f

❌ Not quite. The dividend yield enters with a minus sign, so

d = 1 + r_f - (F/S)^2

; adding the net carry to

r_f

flips the sign.

❌ Not quite. Divide by

S_t

, square both sides, then solve for

d

Show solution

Write the relation with $T - t = 6/12$ :

4030 = 4000\,(1 + 0.04 - d)^{6/12}.

Divide by the spot price and square both sides:

d = 1.04 - \left(\frac{4030}{4000}\right)^{2} = 1.04 - 1.01505625 = 0.02494375 \approx 2.49\%.

Check the direction: $F > S$ goes with $r_f > d$ , and indeed $4\% > 2.49\%$ .

Problem 4 · Index Futures Out of Line

Given: a stock index is at 2,000 points, $r_f = 5\%$ and $d = 2\%$ a year, and the futures contract settles in exactly one year. The futures price is quoted at 2,040.

Which trade locks in a riskless profit?

Profit per index unit, received at settlement?

✅ Correct! The carry value is

1.03 \times 2000 = 2060

, so the futures are 20 points cheap: buy them and short the stocks. That is the same direction as the arbitrage on the morning of 19 October 1987, when index futures were cheap relative to the stocks.

❌ Not quite. Selling futures and carrying the stocks is the trade when futures are rich. Compare 2,040 with the carry value

(1 + 0.05 - 0.02)^1 \times 2000

❌ Not quite. The relation predicts a specific price, not just a direction:

F

should be

1.03 \times 2000 = 2060

, and 2,040 is below that.

❌ Not quite. Two long positions add exposure to the index instead of cancelling it; an arbitrage buys one route and sells the other.

❌ Not quite. First find the carry value

(1 + r_f - d)^{T-t} S_t

and compare it with the quote.

❌ Not quite. 60 = 2,060 − 2,000 is the carry itself; the profit is the gap between the carry value and the quote.

❌ Not quite. 40 = 2,040 − 2,000 compares the quote with today's spot; the profit is the gap between the carry value 2,060 and the quote.

❌ Not quite. 19.42 ≈ 20 / 1.03 is the value today; the question asks for the amount received at settlement.

❌ Not quite. The profit at

T

is the carry value minus the quoted futures price.

Show solution

Step 1: carry value. With $T - t = 1$ ,

F^{*} = (1 + 0.05 - 0.02)^{1} \times 2000 = 1.03 \times 2000 = 2060.

Step 2: compare. The quote 2,040 is below 2,060, so the futures are cheap relative to the stocks.

Step 3: trade. Buy the futures at 2,040. Short the index stocks for 2,000 and invest the proceeds at 5%, which grows to 2,100; as the short seller you owe the dividends, $0.02 \times 2000 = 40$ , leaving $2100 - 40 = 2060$ . At settlement the index stands at some level $S_T$ : the cash-settled futures pay $S_T - 2040$ , and buying the stocks back to close the short costs $S_T$ , so the index level cancels.

Step 4: profit. $2060 - 2040 = 20$ points per index unit at settlement, whatever the index does.

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