Convenience Yield And Commodity Forward Curves

Why Inventory Can Affect a Forward Price

A commodity may be valuable not only because it can eventually be sold, but also because having it available now can help a business keep operating. A manufacturer with an input in stock may avoid a production interruption; a fuel distributor may be able to meet an unexpected delivery. These benefits are not necessarily cash payments received by the inventory holder. In forward pricing, their effect is represented by the convenience yield.

For a storable commodity, financing and storage tend to increase the cost of carrying inventory through time. The benefit of having the commodity available tends to reduce the net cost of holding it. The forward price therefore depends on all three effects. The formula in this tutorial uses the same continuous-rate framework as the cost-of-carry model, now including convenience yield.

Definition: The convenience yield is the rate used in a commodity carry model to represent the noncash benefit of holding physical inventory. It is not a cash dividend or an amount that the holder necessarily receives as money. In the standard model, it reduces the effective cost of carrying the commodity.

Forward Pricing with Convenience Yield

Assume the spot commodity can be bought and stored, and that the model uses deterministic, continuously compounded rates over the period to delivery. Let $S_0$ be the current spot price, $F_{0,T}$ the forward price for delivery at time $T$, $r$ the continuously compounded financing rate, $u$ the proportional storage-cost rate, and $y$ the convenience-yield rate. When these rates are constant over the period, the forward price is

F_{0,T}=S_0 e^{(r+u-y)T}.

The exponent contains the net carrying rate, $r+u-y$. Financing and storage add to carrying cost; convenience yield subtracts from it. Rates and $T$ must be expressed on compatible time bases. For example, if $T$ is measured in years, the rates in the exponent are annual rates.

This version assumes that storage costs are proportional to the commodity value at the rate $u$, and that the convenience yield can be represented as a proportional rate $y$. If instead storage costs are specified as particular fixed cash amounts at particular dates, or if inventory benefits cannot reasonably be represented by a proportional yield, this formula does not directly apply without modifying the cash-flow model.

Why the Net Carrying Rate Appears

The formula follows from the cost-of-carry relationship when the benefit of holding inventory is treated as a yield. Financing at rate $r$ and proportional storage at rate $u$ increase the cost of maintaining the position over time. The convenience yield $y$ offsets those costs. With constant rates, the net accumulation over $T$ years is exponential at rate $r+u-y$.

Start with the spot position.
A unit of the commodity costs $S_0$ at time 0.

Account for carrying costs and inventory benefit.
Over time, financing and proportional storage accumulate at combined rate $r+u$, while the convenience yield reduces the net rate by $y$.

Accumulate to delivery.
Under the stated deterministic continuous-rate assumptions, the carried spot position corresponds to $S_0 e^{(r+u-y)T}$ at time $T$. This is the model’s fair forward price.

The same argument allows rates to vary deterministically with time. If $r(t)$, $u(t)$, and $y(t)$ are the respective rates at time $t$, then the accumulated net carrying rate is their integral:

F_{0,T}=S_0\exp\left(\int_0^T [r(t)+u(t)-y(t)]\,dt\right).

In this version, the integral is the total net carry over the delivery period. A maturity-specific average net carrying rate can be obtained by dividing that integral by $T$, for $T>0$. This is useful when comparing delivery prices across maturities, but the average need not equal the net carry rate for any particular shorter interval.

Worked Example: Pricing with a Stated Convenience Yield

A hypothetical commodity has spot price $S_0=\$100$. The annual continuously compounded financing rate is $r=6\%$, the proportional storage-cost rate is $u=2.5\%$, and the convenience yield is $y=4.5\%$. Find the two-year forward price. Also quantify the effect of incorrectly omitting convenience yield.

Method. Use the constant-rate cost-of-carry formula. First calculate the net carrying rate:

$$ r+u-y=0.060+0.025-0.045=0.040. $$

For $T=2$ years, the exponent is $0.040(2)=0.080$. Thus

F_{0,2}=100e^{0.080} =100(1.0832871) =\$108.3287 \approx \$108.33.

Check. Since net carry is positive, the model gives a forward price above the spot price: $\$108.33>\$100$. The result is consistent with the sign of the net rate.

Common mistake avoided. Omitting convenience yield would give a net rate of $0.060+0.025=0.085$, and the incorrect calculation would be $100e^{0.085(2)}=100e^{0.170}\approx \$118.53$. That overstates the model price because it ignores the inventory benefit. Convenience yield enters with a minus sign, not a plus sign.

Interpreting Forward Curves

A commodity forward curve plots forward prices against delivery maturities. Under constant rates and a constant convenience yield, the ratio of the forward price to spot is governed by the net carrying rate. A positive net carrying rate gives $F_{0,T}>S_0$ for $T>0$; a negative net carrying rate gives $F_{0,T}<S_0$. These patterns are commonly described as contango and backwardation, respectively, when comparing forward prices with spot.

The labels describe price relationships, not the cause of those relationships. A forward price below spot is consistent with a sufficiently large convenience yield relative to financing and storage costs, but the curve’s shape alone does not identify the yield unless the other inputs and model assumptions are specified. With rates or yields that vary by maturity, compare the cumulative net carry over each delivery period rather than relying on one constant-rate calculation.

Worked Example: Inferring an Implied Convenience Yield

A hypothetical commodity has spot price $S_0=\$120$, and its one-year forward price is $F_{0,1}=\$124.20$. The annual continuously compounded financing rate is $5\%$, and the proportional storage-cost rate is $2\%$. Under the constant-rate model, find the implied annual convenience yield.

Method. Rearrange the forward-pricing formula to solve for $y$:

y=r+u-\frac{1}{T}\ln\left(\frac{F_{0,T}}{S_0}\right).

Here $T=1$, and the price ratio is

\frac{F_{0,1}}{S_0}=\frac{124.20}{120}=1.035.

Taking the natural logarithm gives $\ln(1.035)\approx 0.0344014$. Therefore,

$$ y=0.050+0.020-0.0344014 =0.0355986. $$

The implied annual convenience yield is $0.0355986$, or approximately 3.56% when rounded to two decimal places in percentage terms.

Check. Substituting the unrounded yield back into the formula gives $120e^{(0.050+0.020-0.0355986)(1)}=120e^{0.0344014}\approx \$124.20$, which agrees with the quoted forward price.

Common mistake avoided. Do not treat the forward-to-spot price ratio as a simple interest rate and subtract it directly from $r+u$. The model is continuously compounded, so the correct annualized carry is the natural logarithm of the ratio, divided by $T$.

Worked Example: Reading Two Points on a Commodity Forward Curve

A hypothetical commodity has spot price $S_0=\$75$. The annual continuously compounded financing rate is $r=4\%$, and the proportional storage-cost rate is $u=3\%$ throughout. For delivery in six months, the convenience yield is $1\%$ per year. For delivery in 18 months, the average convenience yield over the full period is $8\%$ per year. Calculate both forward prices and describe their relationship to spot.

Method. Apply the constant-rate formula separately to each maturity, using the stated average yield for the full delivery period.

For six-month delivery, $T=0.5$ and the net carrying rate is

$$ 0.040+0.030-0.010=0.060. $$

The exponent is $0.060(0.5)=0.030$, so

F_{0,0.5}=75e^{0.030} =75(1.0304545) =\$77.2841 \approx \$77.28.

For 18-month delivery, $T=1.5$ and the net carrying rate based on the stated average yield is

$$ 0.040+0.030-0.080=-0.010. $$

The exponent is $-0.010(1.5)=-0.015$, so

F_{0,1.5}=75e^{-0.015} =75(0.9851119) =\$73.8834 \approx \$73.88.

Interpretation and check. The six-month forward price is above spot because its net carry is positive. The 18-month forward price is below spot because its average net carry is negative. The longer-dated price is also below the six-month forward price. These relationships are consistent with the inputs: the stated longer-period convenience yield more than offsets financing and storage costs.

Common mistake avoided. Do not use the 18-month maturity as $18$ in the exponent. The rates are annual, so the correct time is $1.5$ years. Also, do not infer that every maturity must be above spot merely because financing and storage rates are positive; convenience yield can change the sign of net carry.

What an Implied Yield Does—and Does Not—Tell You

A convenience yield inferred from observed spot and forward prices is a model-implied quantity. It summarizes the residual relationship after the specified financing and storage costs have been included. It is not automatically a directly observed cash return, and it should not be interpreted as a guaranteed profit for an investor. The physical benefit of inventory depends on circumstances such as availability, location, and the ability to use or deliver the commodity.

The standard formula is most useful when the assumptions match the problem: deterministic rates, proportional storage costs, and a yield representation for the benefit of holding inventory. If a question supplies actual dated storage payments instead, those cash flows need to be handled in the specified pricing model rather than silently converted into a proportional rate. Likewise, rates stated as effective annual rates cannot be inserted as though they were continuously compounded; convert them to compatible rates before using this formula.

Common Exam Traps

Giving convenience yield the wrong sign. The exponent is $r+u-y$, because the modeled inventory benefit offsets carrying costs.
Mixing time units. An annual rate must be multiplied by time in years; six months is $0.5$, not $6$.
Using an arithmetic price ratio in a continuous-rate model. To infer a continuous net rate, use $\ln(F_{0,T}/S_0)/T$.
Calling convenience yield a cash payment. It represents a benefit of holding inventory, not necessarily money received by the holder.
Reading a curve as proof of a cause. A forward price below spot indicates negative cumulative net carry under this model; it does not by itself reveal the convenience yield without assumptions about financing and storage.

Key takeaway: Under the stated constant-rate assumptions, commodity forward price equals spot accumulated at the net carrying rate $r+u-y$. Convenience yield lowers the forward price relative to a calculation that includes only financing and storage, and its effect can vary across delivery maturities.

Check Your Understanding

For each problem, use the stated continuous-rate model and keep time in years. Monetary answers are rounded to the nearest cent unless the problem asks for a rate.

A hypothetical commodity has spot price $\$50$, annual financing rate $4\%$, storage-cost rate $1\%$, and convenience yield $2\%$. Find the two-year forward price. (Answer: $\$53.09$)
A commodity has spot price $\$100$, a one-year forward price of $100e^{0.04}$, financing rate $3\%$, and storage-cost rate $2\%$. Find the implied annual convenience yield. (Answer: $1.00\%$)
A commodity has spot price $\$90$, one-year forward price $\$87$, financing rate $4\%$, and storage-cost rate $1.5\%$. Find the implied annual convenience yield, rounded to two decimal places in percentage terms. (Answer: $8.89\%$)
A commodity has spot price $\$60$. The net carrying rate is $2\%$ for the first year and $-1\%$ for the second year. Find the two-year forward price using the integrated net carry. (Answer: $\$60.60$)
A commodity has spot price $\$80$, two-year forward price $\$80e^{-0.06}$, annual financing rate $5\%$, and storage-cost rate $2\%$. Under the constant-rate model, find the implied annual convenience yield. (Answer: $10.00\%$)

What You'll Learn