The cost of carry: storage costs and yields
1. The idea
Whoever sells a forward can buy the asset today and hold it until delivery. What that costs, minus what the asset pays its holder, is the asset’s cost of carry. For a share without dividends it is the interest on the money, as in the first lesson of this level. For other assets two more things count:
- Storage costs raise it. Gold and silver must be stored and insured until delivery.
- Income lowers it. The shares in a stock index pay dividends to whoever holds them, at a yearly rate called the dividend yield.
When the storage costs and the income are yearly shares of the asset’s value, the forward price grows the spot price at the cost of carry:
\[ F_0 = S_0\, e^{cT}, \qquad c = r + u - q, \]
where \(F_0\) is the forward price, \(S_0\) the spot price, \(c\) the cost of carry, \(r\) the risk-free rate, \(u\) the storage cost and \(q\) the income as yearly shares of the value, all with continuous compounding, as decimals, and \(T\) the time to delivery in years. When storage is paid as amounts on set dates instead, add their present value \(U\) to the spot price, just as the forward price with dividends subtracts the dividends’ present value:
\[ F_0 = (S_0 + U)\, e^{rT}. \]
These formulas give the forward price of assets that many investors hold, such as gold, silver and stock indices: at any other price, a riskless profit would be possible, as the lesson on arbitrage shows. Oil and other commodities held to be used follow the next lesson.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The cost of carry on a chart
In the question, gold costs $2,000 an ounce, and storage of $1 an ounce is paid today and in 3, 6 and 9 months. The present value of those four payments is \(U = 3.9407\). The forward price grows the spot price plus that amount at the risk-free rate: \((2{,}000 + 3.9407) \times e^{0.04 \times 1} = \$2{,}085.72\).
Now silver at $25 an ounce, with storage costing 1.5% of its value a year. Move the time to delivery: the forward price rises faster than at the risk-free rate alone, and the gap between the lines comes from the storage cost.
For a stock index, income works the other way: a dividend yield makes the forward price rise more slowly than the risk-free rate alone, and when the yield is above the rate, the forward price is below the spot price.
4. One worked example, then your turn
The worked example is a stock index with a dividend yield.
Now silver with hints, one step at a time.
And gold on your own.
5. Practice
Each question has new numbers: gold with storage paid every quarter, silver with storage as a share of its value, or a stock index with a dividend yield. You master the skill when you get three right in a row, with at least one right answer for each of the three. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: