Why the Financing Period Matters
A forward contract sets a price today for an asset that will be exchanged at a specified future date. For an asset that pays no income before delivery, its fair forward price is closely tied to the cost of financing the asset until that date. The key is to accumulate the spot price over the actual time to delivery, using the compounding convention that matches the quoted financing rate.
The formula is compact, but applying it correctly requires care. A rate quoted as effective annually is not used in the same way as a nominal rate compounded quarterly. Likewise, the number of compounding periods must match the delivery date. These calculations assume the financing rate is available for the relevant period and that the asset generates no cash flows before delivery.
The No-Income Forward-Price Formula
Assume the asset can be bought and sold at the stated spot price, pays no income before delivery, and has no storage or other carrying costs. Assume also that investors can borrow and lend at the same known financing rate, with no transaction costs, taxes, or trading restrictions. The asset and the forward contract are treated as having no default risk. Under these conditions, the fair delivery price is the spot price accumulated to the delivery date.
The time \(T\) is measured in years in these expressions, but the rate convention determines how the time is used. With an effective annual rate, a non-integer \(T\) uses a fractional exponent. With a nominal rate convertible \(m\) times per year, count the \(mT\) conversion periods. If the dates do not fit that convention, a specific treatment of the fractional period must be given; do not silently substitute a different compounding rule.
If financing rates differ across successive periods, accumulate using the rate that applies to each period. For example, if the first year’s effective rate is \(i_1\) and the second year’s is \(i_2\), the two-year accumulation factor is \((1+i_1)(1+i_2)\). The calculation must reflect the rates and dates actually specified, rather than treating every period as if it had the same rate.
Why the Formula Holds
A short replication argument explains the result. At time 0, borrow \(S_0\) and use it to buy one unit of the asset. The purchase and borrowing offset, so the initial net investment is zero. At time \(T\), the asset is worth the delivery price under a forward sale, while the borrowing must be repaid with interest.
For the positions to have the same value at delivery, the forward delivery price \(K\) must equal the loan repayment. If \(K\) were higher, the asset purchased with borrowed funds could be sold through the forward for more than the repayment. If \(K\) were lower, this financing comparison would reverse. Thus the no-arbitrage delivery price under the stated assumptions is \(F_0(T)=S_0(1+i)^T\). This argument relies on being able to borrow, buy, and enter the forward on the stated terms; if those conditions fail, the formula need not determine a unique price.
Express the time from today to delivery in years or in the number of compounding periods required by the quoted rate.
Distinguish effective annual, nominal convertible, force-of-interest, or period-specific rates.
Multiply \(S_0\) by the accumulation factor for exactly the time to delivery.
Discount the calculated forward price back to time 0 using the same convention. It should return \(S_0\).
Worked Calculations
Worked Example: Effective Annual Financing for Three Years
A hypothetical asset has a spot price of $240. It pays no income before delivery, and the effective annual financing rate is 4%. A forward contract calls for delivery in three years. Find the fair delivery price, to the nearest cent. Assume the rate is constant and borrowing and lending are both available at 4%.
Method. Use the effective annual accumulation formula with \(S_0=240\), \(i=0.04\), and \(T=3\). The calculation uses three annual periods.
After one year, the accumulation factor is \(1.04\). After three years, it is \(1.04^3=1.04\times1.04\times1.04=1.124864\).
Rounding only at the final step gives a fair delivery price of $269.97.
Check. Discounting three years at 4% gives \(269.96736/(1.04)^3=269.96736/1.124864=240\), the original spot price.
Common mistake avoided. Using \(240(1+3(0.04))\) applies simple interest, not the stated effective annual rate. The specified compound accumulation is \(240(1.04)^3\).
Worked Example: Nominal Rate and Quarterly Periods
A hypothetical no-income asset costs $500 today. Its financing rate is 8% nominal per year, convertible quarterly. Delivery is in 15 months. Find the fair delivery price to the nearest cent. Assume the quoted financing rate applies throughout and can be used for borrowing and lending.
Method. A quarter is three months, so 15 months contains \(15/3=5\) quarters. The nominal annual rate per quarter is \(0.08/4=0.02\), or 2%.
The accumulation factor is \(1.02^5=1.02\times1.02\times1.02\times1.02\times1.02=1.1040808032\).
Rounded to the nearest cent, the fair delivery price is $552.04.
Check. The result divided by the five-quarter accumulation factor returns \(552.0404016/1.1040808032=500\).
Common mistake avoided. Raising \(1.02\) to \(15\) treats the 15 months as 15 quarterly periods. There are five quarterly periods in 15 months, not 15.
Worked Example: Different Financing Rates by Year
A hypothetical asset has a spot price of $1,250 and pays no income before delivery in two years. The effective financing rate is 3% during the first year and 6% during the second year. Assume both rates are known at time 0 and borrowing and lending are available at those rates. Calculate the fair delivery price to the nearest cent.
Method. Accumulate over each year using that year’s rate. The first year multiplies the spot price by \(1.03\); the second year multiplies the resulting amount by \(1.06\).
After the first year, the accumulated amount is \(1{,}250(1.03)=1{,}287.50\).
After the second year, it is \(1{,}287.50(1.06)=1{,}364.75\). Equivalently, the two-year accumulation factor is \(1.03(1.06)=1.0918\).
The fair delivery price is $1,364.75.
Check. The total accumulation factor \(1.0918\) is greater than 1, so the forward price exceeds the spot price. Dividing \(1{,}364.75\) by \(1.0918\) returns $1,250.
Common mistake avoided. Applying the second-year rate to both years would give \(1{,}250(1.06)^2\), which does not use the stated first-year financing rate. Each interval must use its applicable rate.
What the Formula Does Not Say
The forward price is the delivery price that makes a newly entered contract fair at inception under the model assumptions. It is not a forecast of the future spot price, and it is not a promise that the asset’s eventual market price will equal the delivery price. The future spot price may be higher or lower. The formula instead relates today’s spot price to the financing cost through the delivery date.
This tutorial concerns assets with no income before delivery. If the asset pays dividends, coupons, or other cash distributions, the simple accumulation formula generally needs adjustment. Similarly, storage costs, other carrying costs, different borrowing and lending rates, transaction costs, or trading restrictions can change the pricing analysis. The assumptions must therefore be checked before applying the formula.
Also distinguish the fair delivery price of a new contract from the value of a forward contract that already exists. A contract’s delivery price is fixed in its terms; a new fair forward price may change as spot prices, rates, or time to delivery change. Calculating an existing contract’s value is a separate question.
- Use the right time unit. Convert months or days to years or periods consistently with the rate convention.
- Do not mix compounding conventions. A nominal rate must first be converted to a rate per conversion period.
- Use only the period to delivery. Do not accumulate for the contract’s full stated term if delivery occurs earlier.
- Check the assumptions. The basic result requires no income or carrying costs and financing terms consistent with the model.
Check Your Understanding
For each problem, assume the asset pays no income before delivery and the stated financing terms are available for borrowing and lending. Round money to the nearest cent.
- A hypothetical asset has a spot price of $640, an effective annual financing rate of 3.5%, and delivery in two years. Find its fair forward price. (Answer: $685.58)
- A no-income asset costs $1,800. The financing rate is 6% nominal per year, convertible semiannually, and delivery is in three years. Find the fair delivery price. (Answer: $2,149.29)
- A hypothetical asset costs $950. Financing is at a constant force of interest of 4% per year, and delivery is in 1.5 years. Find the fair forward price. (Answer: $1,008.74)
- A no-income asset has a spot price of $3,000. The effective financing rate is 2% for the first year and 4.5% for the second year. Delivery is in two years. Find the fair price. (Answer: $3,197.70)
- A hypothetical asset costs $1,600. Financing is at 7% nominal per year, convertible quarterly, and delivery is in 18 months. Find the fair delivery price and state the number of quarterly periods used. (Answer: 6 periods; $1,775.52)