Continuous Income Yields And Forward Pricing

Why Income Yield Changes a Forward Price

For an asset with no income, the fair forward price reflects the cost of financing the spot purchase until delivery. If the asset produces income while it is held, that income offsets part of the financing cost. A continuous income yield captures this effect as a rate proportional to the asset's value. The resulting formula is compact, but its rates must be expressed on compatible conventions.

This tutorial considers a forward contract agreed at time 0 for delivery at time $T$. The forward price is the delivery amount that makes the contract fair at initiation. The model assumes a known, constant financing rate and income yield over the term. All assets and financing are assumed divisible, income can be reinvested in the asset, and there are no taxes, transaction costs, or restrictions on borrowing or trading.

Definition: A continuous income yield $q$ is a constant rate at which an asset produces income proportional to its value. Under the model, income is received continuously and, when reinvested in the asset, increases the number of asset units held at rate $q$. It is not a fixed cash dividend or a known discrete distribution.

Let $S_0$ be the asset's spot price. Let $\delta$ be the constant force of interest on financing, measured per year, and let $T$ be the delivery time in years. The income yield $q$ is also an annual continuously compounded rate. The result below gives the fair delivery price, denoted $F_{0,T}$, in this constant-rate model.

Theorem: For an asset with spot price $S_0$, constant financing force $\delta$, and constant continuous income yield $q$, the no-arbitrage forward price for delivery at time $T$ is

F_{0,T}=S_0 e^{(\delta-q)T}.

Why the Formula Holds

The key is to identify the cost today of obtaining one asset at the delivery date without paying the forward price. First consider holding the asset and reinvesting its continuous income in additional units of the same asset. If $N(t)$ is the number of units held at time $t$, income reinvestment gives $dN(t)/dt=qN(t)$. Solving this growth equation gives $N(T)=N(0)e^{qT}$. Thus, starting with $e^{-qT}$ units at time 0 and reinvesting all income produces exactly one unit at time $T$.

The initial cost of those $e^{-qT}$ units is $S_0e^{-qT}$. This is the prepaid forward cost: the amount paid today to receive one asset at time $T$, with no payment due at delivery. Financing that amount to time $T$ multiplies it by $e^{\delta T}$. Therefore, the delivery-date cost associated with the prepaid forward is $S_0e^{-qT}e^{\delta T}$, which must equal the fair forward price.

F_{0,T} = \left(S_0e^{-qT}\right)e^{\delta T} = S_0e^{(\delta-q)T}.

The yield reduces the forward price relative to the no-income case: for positive $q$, $F_{0,T}<S_0e^{\delta T}$. The formula does not treat a fixed cash payment on a particular date as a continuous yield. A known discrete payment must instead be handled as a discrete distribution.

Converting Financing Rates to a Force

The formula uses a force of interest $\delta$, so a financing rate stated as an effective annual rate or as a nominal rate convertible several times a year must first be converted. These are equivalent ways of describing the same annual accumulation under their respective conventions.

Rate conversion: If the effective annual financing rate is $i$, then $1+i=e^\delta$, so $\delta=\ln(1+i)$. If the nominal annual rate is $i^{(m)}$, convertible $m$ times per year, then the annual accumulation is $\left(1+\frac{i^{(m)}}{m}\right)^m$, so

\delta = \ln\left[\left(1+\frac{i^{(m)}}{m}\right)^m\right] = m\ln\left(1+\frac{i^{(m)}}{m}\right).

Conversely, a force $\delta$ corresponds to effective annual rate $e^\delta-1$, or nominal rate convertible $m$ times per year equal to $m(e^{\delta/m}-1)$. These conversions matter because the nominal quoted rate is not itself the force. The continuous yield $q$, by contrast, is already a continuously compounded annual rate in this model.

Put time in years.
Check that $T$, the financing rate, and the income yield use the same annual time basis.

Convert the financing rate.
Use $\ln(1+i)$ for an effective annual rate or $m\ln(1+i^{(m)}/m)$ for a nominal rate convertible $m$ times per year.

Apply the yield adjustment.
Subtract $q$ from the financing force, multiply by $T$, and calculate $S_0e^{(\delta-q)T}$.

Worked Applications

Worked Example: Effective Annual Financing Rate

A hypothetical asset has a spot price of $1,000 and a continuous income yield of 1.0% per year. The financing rate is an effective annual rate of 4.0%. Find the fair forward price for delivery in 3 years. Assume constant rates and the no-friction model stated above.

Method. Convert the effective annual financing rate to a force, then use the continuous-yield forward formula.

The financing force is

\delta=\ln(1.04)=0.0392207.

The net force after accounting for income yield is $0.0392207-0.0100000=0.0292207$. Over 3 years, the exponent is $0.0292207(3)=0.0876621$. Therefore,

F_{0,3} = 1000e^{(0.0392207-0.0100000)3} = 1000e^{0.0876621} \approx \$1{,}091.62.

Check. With no income yield, the forward price would be $1000(1.04)^3=\$1{,}124.86$. The positive yield lowers the price, as expected. The common mistake to avoid is inserting 0.04 directly for $\delta$; 4.0% is an effective annual rate, not a force.

Worked Example: Nominal Rate Convertible Monthly

A hypothetical asset has a spot price of $180 and a continuous income yield of 2.4% per year. Financing is available at a nominal annual rate of 6.0% convertible monthly. Find the fair forward price for delivery in 1.5 years. Assume constant rates and no market frictions.

Method. Convert the nominal rate to a force using $m=12$, then apply the formula with $T=1.5$.

The monthly rate is $0.06/12=0.005$, and the financing force is

\delta = 12\ln(1+0.005) = 12\ln(1.005) \approx 0.0598505.

Subtract the yield: $0.0598505-0.024=0.0358505$. Multiply by the term: $0.0358505(1.5)=0.0537757$. Thus,

F_{0,1.5} = 180e^{0.0537757} \approx \$189.94.

Check. The financing force exceeds the yield, so the forward price is above the spot price of $180. The answer uses the monthly nominal rate only after converting it to a force; treating 6.0% as a force would overstate the financing accumulation.

Worked Example: Converting a Monthly Nominal Quote Carefully

A hypothetical asset has a spot price of $500, a continuous income yield of 2.0% per year, and a financing rate of 5.0% nominal convertible monthly. Find its fair forward price for delivery in 2 years. Assume constant rates and no market frictions.

Method. Convert the financing quote to a force before subtracting the yield. Here $m=12$, so

\delta = 12\ln\left(1+\frac{0.05}{12}\right) \approx 0.0498961.

The net force is $0.0498961-0.0200000=0.0298961$. Over 2 years, the exponent is $0.0298961(2)=0.0597922$. The forward price is

F_{0,2} = 500e^{0.0597922} \approx \$530.81.

Check. Since the financing force is greater than the yield, the forward price should exceed $500, which it does. The common mistake is to use the nominal quote 0.05 as if it were the force. That incorrect calculation gives $500e^{(0.05-0.02)2}=500e^{0.06}\approx \$530.92$, rather than $530.81. The difference arises because monthly nominal conversion produces a force of about 0.0498961, not 0.05.

Interpreting the Result and Avoiding Traps

The sign of $\delta-q$ provides a quick reasonableness check. If financing is more expensive than the income yield, then $\delta-q>0$, and the forward price exceeds the spot price. If the yield exceeds the financing force, then $\delta-q<0$, and the forward price is below spot. If they are equal, the formula gives $F_{0,T}=S_0$. These comparisons are useful checks, not a substitute for ensuring that both rates are stated on compatible annual continuous bases.

Do not confuse a yield with a fixed payment. A yield is proportional to asset value over time. A fixed cash distribution on a specified date is not represented by simply subtracting that cash amount as $q$.
Do not confuse a nominal rate with a force. Convert $i^{(m)}$ using $m\ln(1+i^{(m)}/m)$ before using the exponential formula.
Keep the time unit consistent. If $T$ is stated in months, convert it to years when rates are annual; for example, 18 months is $1.5$ years.
Subtract yield once. The exponent is $(\delta-q)T$, not $\delta T-q$, and the yield is not subtracted from the spot price.

This relationship extends the no-income forward pricing result by incorporating the benefit of income earned while the asset is held. Its practical use depends on correctly identifying whether the stated income is genuinely a continuous proportional yield and matching the financing-rate convention before calculating.

Check Your Understanding

For each hypothetical asset, assume constant rates and no market frictions. Convert financing rates when needed, use years for the term, and give the fair forward price to the nearest cent.

An asset has spot price $200, a continuous income yield of 1.0%, and a continuous financing force of 3.0%. Find the forward price for delivery in 4 years. (Answer: $216.66)
An asset has spot price $750, a continuous yield of 0.8%, and an effective annual financing rate of 3.0%. Find the forward price for delivery in 2 years. (Answer: $783.05)
An asset has spot price $320, a continuous yield of 1.2%, and a 4.0% nominal annual financing rate convertible quarterly. Find the forward price for delivery in 3 years. (Answer: $347.83)
An asset has spot price $1,200, a continuous yield of 2.0%, and a 6.0% nominal annual financing rate convertible semiannually. Find the forward price for delivery in 18 months. (Answer: $1,272.52)
A hypothetical asset has spot price $500, a continuous yield of 2.0%, and a 5.0% nominal annual financing rate convertible monthly. Find the forward price for delivery in 2 years. (Answer: $530.81)

What You'll Learn