Essential Math of Energy Markets

Introduction

Energy markets combine the characteristics of financial markets with the physical constraints associated with the production, transportation, storage and consumption of commodities. Their quantitative analysis therefore requires concepts from finance and statistics, but also careful treatment of units, time, location and the physical characteristics of the underlying energy products.

A large part of the basic quantitative framework can nevertheless be developed using relatively elementary mathematics.

Unit conversions, weighted averages, price differences, ratios, covariance and simple payoff functions are sufficient to describe several quantities commonly used in energy-market analysis.

Essential relations

At the most basic level, the monetary value associated with an energy transaction can be written as

$$V = P \cdot Q$$

where (P) is the unit price and (Q) is the corresponding quantity. In energy markets, however, both variables generally have a time dimension. For a sequence of delivery intervals, the same relationship becomes

$$V = \sum_{t=1}^{n} P_t Q_t$$

where $(P_t)$ is the price observed or contracted for interval (t), and $(Q_t)$ is the energy delivered or consumed during the same interval.

This simple formulation already captures an important feature of energy markets: the economic value of a physical position depends on its delivery profile.

Two assets producing the same total amount of energy over a given period may generate different revenues if their production occurs at different times and therefore at different market prices. The same principle applies to consumption portfolios and, more generally, to any position whose volume varies over time.

Other commonly used energy-market measures can be derived from similarly compact relationships:

Calendar and geographical spreads are differences between prices;
capture prices are volume-weighted averages;
hedge ratios depend on covariance and variance;
generation margins can be represented as combinations of output and input prices;
several physical assets contain optionality that can be described through nonlinear payoff functions.

1. Units and dimensional consistency

The first requirement for any quantitative analysis of energy markets is dimensional consistency. Energy commodities are traded using different physical units, currencies and time conventions, and quoted prices can only be compared after they have been converted to a common basis.

This issue is particularly relevant when comparing markets across different commodities or geographical areas. European natural gas, for example, is commonly quoted in EUR/MWh, while US natural gas is generally quoted in USD/MMBtu. Crude oil is typically expressed in USD/barrel, while electricity prices are commonly quoted per MWh.

Before calculating spreads, correlations or relative price movements, the underlying quantities must therefore be made dimensionally comparable.

Power and energy

A basic distinction in electricity markets is that between power and energy. Power measures the instantaneous rate at which energy is produced or consumed, while energy measures production or consumption accumulated over time.

Their relationship is:

$$E=P\Delta t$$

where (P) is power, generally expressed in MW, and (Δt) is the duration of the delivery interval in hours. The resulting energy (E) is expressed in MWh.

A generator producing 50 MW continuously for four hours therefore produces:

$$E=50\cdot4=200\text{ MWh}$$

This distinction is important because market prices are normally expressed per unit of energy, while the technical characteristics of generation assets are often expressed in units of power. A 100 MW power plant describes a production capacity, not an amount of energy.

If output varies across delivery intervals, total energy production is obtained as:

$$E=\sum_{t=1}^{n}P_t\Delta t$$

For hourly observations, (Δt=1), while quarter-hourly electricity data require (Δt=0.25). A plant producing 40 MW during one 15-minute interval therefore produces:

$$E=40\cdot0.25=10\text{ MWh}$$

This conversion becomes particularly important when working with high-frequency generation or consumption data, since summing MW observations directly does not produce an energy quantity.

Energy-unit conversions

Several physical units coexist across international energy markets. Some useful relationships are:

$$1\text{ MWh}=3.6\text{ GJ}$$

$$1\text{ MMBtu}\approx1.05506\text{ GJ}$$

Where MMBtu is is a million times the standard unit Btu; Btu (British termal unit) is how much heat is needed to heat up one pound of water (0.4536 l) by one degree Fahrenheit (0.5555 °C).

Consequently:

$$1\text{ MWh}\approx3.412\text{ MMBtu}$$

or equivalently:

$$1\text{ MMBtu}\approx0.2931\text{ MWh}$$

These conversion factors allow prices quoted in different energy units to be expressed on a common basis.

Suppose, for example, that US natural gas is trading at 4 USD/MMBtu. Expressing the same price in USD/MWh gives:

$$P_{\text{USD/MWh}}=P_{\text{USD/MMBtu}}\cdot3.412$$

and therefore:

$$P_{\text{USD/MWh}}=4\cdot3.412=13.648\text{ USD/MWh}$$

The conversion factor multiplies the quoted price because one MWh contains approximately 3.412 MMBtu.

Currency conversion

A second conversion is required when prices are denominated in different currencies.

If the exchange rate is expressed as USD per EUR,

$$FX_{\text{USD/EUR}}=\frac{\text{USD}}{\text{EUR}}$$

then a price expressed in USD/MWh can be converted into EUR/MWh as:

$$P_{\text{EUR/MWh}}=\frac{P_{\text{USD/MWh}}}{FX_{\text{USD/EUR}}}$$

Combining the physical and currency conversions gives:

$$P_{\text{EUR/MWh}}=\frac{P_{\text{USD/MMBtu}}\cdot3.412}{FX_{\text{USD/EUR}}}$$

For example, with natural gas trading at 4 USD/MMBtu and an exchange rate of 1.17 USD/EUR:

$$P_{\text{EUR/MWh}}=\frac{4\cdot3.412}{1.17}\approx11.67\text{ EUR/MWh}$$

The original quotation of 4 USD/MMBtu and the converted value of approximately 11.67 EUR/MWh represent the same underlying energy price under the assumed exchange rate.

From power and price to monetary value

Once units have been aligned, the monetary value of an energy position follows directly from price and quantity.

For constant power:

$$(P^{MW}) \text{ delivered for (h) hours at an energy price (p)}$$

the corresponding energy is:

$$Q=P^{MW}h$$

and the monetary value is:

$$V=pP^{MW}h$$

Consider a 10 MW position delivered continuously over 24 hours at 80 EUR/MWh. The delivered energy is:

$$Q=10\cdot24=240\text{ MWh}$$

and its value is:

$$V=80\cdot240=19,200\text{ EUR}$$

For a variable production or consumption profile, the calculation generalises to:

$$V=\sum_{t=1}^{n}p_tP_t^{MW}\Delta t$$

This expression is one of the basic building blocks of energy-market valuation. It combines the market price observed during each delivery interval with the physical quantity produced or consumed during that interval.

Dimensional checks

Dimensional analysis also provides a simple way of checking energy-market calculations. The units on the two sides of an equation must be consistent.

For example:

$$\frac{\text{EUR}}{\text{MWh}}\cdot\text{MW}\cdot\text{h}=\text{EUR}$$

since:

$$\text{MW}\cdot\text{h}=\text{MWh}$$

Similarly, converting a gas price from USD/MMBtu into EUR/MWh requires both a physical conversion and a currency conversion:

$$\frac{\text{USD}}{\text{MMBtu}}\cdot\frac{\text{MMBtu}}{\text{MWh}}\cdot\frac{\text{EUR}}{\text{USD}}=\frac{\text{EUR}}{\text{MWh}}$$

Writing the units explicitly is often sufficient to identify an inverted exchange rate, an incorrect energy conversion or confusion between MW and MWh before the error propagates through a larger calculation.

Dimensional consistency is a necessary condition for meaningful comparisons between energy prices and for all subsequent calculations involving spreads, revenues, generation margins and hedging positions.

2. Prices, returns and normalization

Once prices have been expressed using consistent physical units and currencies, their evolution through time can be analysed using absolute changes, percentage returns and normalized price indices.

The appropriate measure depends on the purpose of the analysis. Absolute price changes are particularly relevant for spreads and physical margins, while percentage or logarithmic returns are commonly used for statistical analysis and volatility estimation.

Absolute price changes

The simplest measure of price variation is the absolute change:

$$\Delta P_t=P_t-P_{t-1}$$

If the day-ahead electricity price increases from 70 EUR/MWh to 85 EUR/MWh:

$$\Delta P_t=85-70=15\text{ EUR/MWh}$$

Absolute changes preserve the original unit of the price series and are therefore directly interpretable in economic terms.

For a fixed physical position (Q), the corresponding change in value can be approximated as:

$$\Delta V=Q\Delta P$$

For example, the effect of a 15 EUR/MWh price increase on a 100 MWh long position is:

$$\Delta V=100\cdot15=1,500\text{ EUR}$$

This linear relationship is particularly useful for physical commodity positions and contracts whose value changes approximately one-for-one with the underlying commodity price.

Simple returns

A simple return expresses the price change relative to the initial price:

$$r_t=\frac{P_t-P_{t-1}}{P_{t-1}}$$

which can equivalently be written as:

$$r_t=\frac{P_t}{P_{t-1}}-1$$

Using the previous example:

$$r_t=\frac{85-70}{70}\approx0.2143$$

or approximately 21.43%.

Simple returns are useful when the objective is to compare relative movements across assets with different price levels.

A 10 EUR/MWh movement has a very different relative significance for an asset trading at 20 EUR/MWh and one trading at 200 EUR/MWh.

Logarithmic returns

In financial analysis, price changes are also commonly expressed as logarithmic returns:

$$r_t^{log}=\ln\left(\frac{P_t}{P_{t-1}}\right)$$

For the same movement from 70 to 85 EUR/MWh:

$$r_t^{log}=\ln\left(\frac{85}{70}\right)\approx0.1942$$

For relatively small price changes, simple and logarithmic returns are approximately equal:

$$\ln(1+r)\approx r$$

Logarithmic returns have the useful property of being additive across time. If prices are observed at $0,1,\ldots,T$, then:

$$\sum_{t=1}^{T}\ln\left(\frac{P_t}{P_{t-1}}\right)=\ln\left(\frac{P_T}{P_0}\right)$$

This property makes log returns convenient for many statistical and financial applications.

Their use in energy markets, however, requires additional care.

Zero and negative prices

The definition of a logarithmic return requires:

$$P_t>0$$

and:

$$P_{t-1}>0$$

This assumption is generally reasonable for many financial assets but does not hold universally in energy markets. Electricity prices, in particular, can become zero or negative.

If:

$$P_t\leq0$$

then:

$$\ln(P_t)$$

is not defined as a real number, and the standard log-return framework cannot be applied directly.

Simple percentage returns can also become difficult to interpret around zero. Consider a price moving from 1 EUR/MWh to -1 EUR/MWh:

$$r_t=\frac{-1-1}{1}=-2$$

which corresponds formally to a -200% return. A subsequent movement from -1 EUR/MWh back to 1 EUR/MWh gives:

$$r_{t+1}=\frac{1-(-1)}{-1}=-2$$

despite the price having increased by 2 EUR/MWh.

The arithmetic is correct, but the economic interpretation normally associated with financial returns is no longer useful.

For price series that can cross zero, absolute changes are therefore often more meaningful:

$$\Delta P_t=P_t-P_{t-1}$$

The choice between price levels, absolute changes and returns should consequently depend on both the statistical objective and the economic properties of the underlying market.

Normalized price indices

When the objective is to compare the evolution of several price series rather than their absolute levels, prices can be normalized to a common initial value.

A base-100 index can be defined as:

$$I_t=100\frac{P_t}{P_0}$$

By construction:

$$I_0=100$$

Suppose European natural gas, US natural gas and crude oil have different prices and units. After the necessary unit and currency conversions, each series can be rebased to 100 at a common starting date. A normalized value of 130 then indicates that the price is 30% above its initial level, independently of the original quotation.

For two assets (A) and (B):

$$I_t^A=100\frac{P_t^A}{P_0^A}$$

$$I_t^B=100\frac{P_t^B}{P_0^B}$$

The difference between their normalized indices,

$$D_t=I_t^A-I_t^B$$

can provide a simple visual indication of relative price performance.

Normalization does not, however, make two markets economically equivalent. It removes differences in initial price levels but does not remove differences in volatility, market structure, seasonality or physical characteristics.

Volatility from price changes or returns

Once a suitable measure of price variation has been selected, its dispersion can be quantified using variance and standard deviation.

For a series of returns ($r_t$), the sample mean is:

$$\bar r=\frac{1}{n}\sum_{t=1}^{n}r_t$$

and the sample variance is:

$$s^2=\frac{1}{n-1}\sum_{t=1}^{n}(r_t-\bar r)^2$$

The corresponding standard deviation is:

$$s=\sqrt{\frac{1}{n-1}\sum_{t=1}^{n}(r_t-\bar r)^2}$$

When returns are measured at a fixed frequency, volatility is often annualized using the square-root-of-time rule:

$$\sigma_{ann}=\sigma_{\Delta t}\sqrt{N}$$

where (N) is the number of observation periods per year. For daily observations, a conventional financial-market approximation is:

$$\sigma_{ann}\approx\sigma_{daily}\sqrt{252}$$

The square-root-of-time scaling relies on assumptions about the return-generating process, in particular that increments are sufficiently independent and have stable variance. These assumptions can be restrictive for energy prices, which frequently exhibit seasonality, volatility clustering, price spikes and mean reversion.

Consequently, annualized historical volatility should be interpreted as a statistical summary rather than as an invariant characteristic of an energy commodity.

Price levels and statistical transformations

The distinction between price levels and price changes is also important when analysing relationships between energy markets.

Two price series may appear strongly related because they share a long-term trend, while their short-term movements may exhibit substantially weaker dependence. Conversely, commodities with very different price levels can react similarly to common market shocks.

Depending on the question being investigated, the relevant variables may therefore be:

$$P_t$$

for price levels,

$$\Delta P_t=P_t-P_{t-1}$$

for absolute changes, or

$$r_t=\frac{P_t-P_{t-1}}{P_{t-1}}$$

for relative changes.

There is no universally preferable transformation. The appropriate choice depends on the physical characteristics of the market, the possibility of zero or negative prices, and the economic relationship being analysed.

This distinction becomes particularly important when moving from individual prices to one of the central objects of energy-market analysis: the relationship between two or more prices, usually expressed through a spread.

3. Spreads and generation margins

Many energy-market positions are better described by the relationship between two or more prices than by the absolute level of a single commodity. These relationships are generally expressed as spreads.

In its simplest form, a spread between two prices is:

$$S=P_A-P_B$$

The interpretation of (A) and (B) depends on the market. They may represent different delivery periods, geographical locations, commodities or stages of an energy-conversion process.

Spreads are economically relevant because physical energy assets frequently create value by moving energy across time, space or form. Storage moves a commodity across time, pipelines and transmission networks move energy across locations, while generation assets transform fuels into electricity. The corresponding spreads provide a simplified representation of the economic incentive associated with these transformations.

Calendar spreads

A calendar spread measures the price difference between two delivery periods of the same commodity.

For two forward contracts with maturities ($T_1$) and ($T_2$):

$$S_t^{T_1,T_2}=F(t,T_1)-F(t,T_2)$$

where $F(t,T)$ denotes at time $t$ the forward price for delivery at $T$.

Suppose the January natural gas contract trades at 42 EUR/MWh and the February contract at 38 EUR/MWh. The January-February spread is:

$$S^{Jan,Feb}=42-38=4\text{ EUR/MWh}$$

The sign convention must always be stated explicitly. Under the convention above, a positive spread means that the first delivery period is more expensive than the second.

Calendar spreads are closely related to the economics of storage. A market participant capable of storing a commodity can compare the current or near-term price with the price available for later delivery, taking into account storage costs, losses, financing and operational constraints.

A simplified storage arbitrage condition can be represented as:

$$F(t,T)-S_t>C_{storage}+C_{finance}+C_{losses}$$

where $S_t$ is the spot price and the right-hand side represents the relevant carrying costs.

In actual energy markets, storage value is more complex because injection and withdrawal rates, inventory limits, efficiency losses and uncertainty create additional optionality. Nevertheless, calendar spreads provide the basic price signal for shifting a storable commodity across time.

Location and basis spreads

Energy prices also differ across geographical locations. A location spread can be defined as:

$$S_t^{A,B}=P_t^A-P_t^B$$

where $(P_t^A)$ and $(P_t^B)$ refer to comparable energy products delivered at locations (A) and (B).

If natural gas trades at 35 EUR/MWh at location (A) and 32 EUR/MWh at location (B):

$$S_t^{A,B}=35-32=3\text{ EUR/MWh}$$

A positive location spread may create an economic incentive to transport the commodity from (B) to (A), provided that the spread exceeds the marginal cost of transportation and other associated costs.

A simplified condition is:

$$P_t^A-P_t^B>C_{transport}$$

or:

$$S_t^{A,B}>C_{transport}$$

In unconstrained markets, transportation tends to limit persistent price differences beyond the cost of moving the commodity. In practice, however, pipeline capacity, transmission congestion, LNG shipping availability, losses, tariffs and contractual constraints can prevent complete price convergence.

The difference between a local price and a reference price is also commonly described as basis:

$$B_t=P_t^{local}-P_t^{reference}$$

Basis risk arises when a physical exposure and the financial instrument used to hedge it refer to different locations, products or delivery conditions.

From commodity spreads to transformation margins

Some energy assets transform one commodity into another. A thermal power plant, for example, purchases fuel and converts its energy content into electricity.

The simplest representation of the generation margin of a gas-fired plant is the spark spread.

If both electricity and gas were expressed per unit of equivalent energy input, a simplified spread could be written as:

$$SS=P_E-P_G$$

where $(P_E)$ is the electricity price and $(P_G)$ is the gas price.

This formulation is generally insufficient because a power plant does not convert one MWh of fuel into one MWh of electricity. Conversion efficiency must be taken into account.

Efficiency and heat rate

Let $(\eta)$ denote the thermal efficiency of a power plant:

$$\eta=\frac{E_{output}}{E_{input}}$$

A plant with an efficiency of 50% requires:

$$E_{input}=\frac{E_{output}}{\eta}$$

For one MWh of electricity:

$$E_{input}=\frac{1}{0.50}=2\text{ MWh}_{th}$$

The inverse of efficiency is the heat rate when input and output are expressed using consistent energy units:

$$HR=\frac{1}{\eta}$$

For a 50% efficient plant:

$$HR=2\frac{\text{MWh}_{th}}{\text{MWh}_e}$$

The fuel cost associated with producing one MWh of electricity is therefore:

$$C_{fuel}=HR\cdot P_G$$

If gas costs 30 EUR/MWh and the plant has a heat rate of 2:

$$C_{fuel}=2\cdot30=60\text{ EUR/MWh}_e$$

The dimensional relationship is:

$$\frac{\text{MWh}_{th}}{\text{MWh}_e}\cdot\frac{\text{EUR}}{\text{MWh}_{th}}=\frac{\text{EUR}}{\text{MWh}_e}$$

The spark spread can consequently be written as:

$$SS=P_E-HR\cdot P_G$$

Suppose electricity trades at 100 EUR/MWh, gas at 30 EUR/MWh and the plant efficiency is 50%. Then:

$$SS=100-2\cdot30=40\text{ EUR/MWh}$$

This represents the gross margin before accounting for carbon costs and other variable operating costs.

Clean spark spread

For fossil-fuel generation in markets subject to carbon pricing, the cost of emissions must also be included.

Let (EF) denote the emission factor of the plant:

$$EF=\frac{\text{tCO}_2}{\text{MWh}_e}$$

and let $(P_C)$ denote the carbon allowance price:

$$P_C=\frac{\text{EUR}}{\text{tCO}_2}$$

The carbon cost per MWh of electricity is:

$$C_{CO_2}=EF\cdot P_C$$

Dimensionally:

$$\frac{\text{tCO}_2}{\text{MWh}_e}\cdot\frac{\text{EUR}}{\text{tCO}_2}=\frac{\text{EUR}}{\text{MWh}_e}$$

The clean spark spread is therefore:

$$CSS=P_E-HR\cdot P_G-EF\cdot P_C$$

For example, consider:

$$P_E=100\text{ EUR/MWh}$$

$$P_G=30\text{ EUR/MWh}$$

$$HR=2$$

$$EF=0.4\text{ tCO}_2/\text{MWh}$$

$$P_C=75\text{ EUR/tCO}_2$$

The fuel cost is:

$$C_{fuel}=2\cdot30=60\text{ EUR/MWh}$$

while the carbon cost is:

$$C_{CO_2}=0.4\cdot75=30\text{ EUR/MWh}$$

and therefore:

$$CSS=100-60-30=10\text{ EUR/MWh}$$

The plant has a positive gross generation margin of 10 EUR/MWh before other variable operating costs.

If variable operating and maintenance costs are denoted by $(VOM)$, a more complete short-run margin can be written as:

$$M=P_E-HR\cdot P_G-EF\cdot P_C-VOM$$

A simplified economic dispatch condition is then:

$$M>0$$

or equivalently:

$$P_E>HR\cdot P_G+EF\cdot P_C+VOM$$

The right-hand side represents the plant’s short-run marginal generation cost under the assumptions of the model.

Clean dark spread

The same framework can be applied to coal-fired generation. The corresponding clean dark spread can be written as:

$$CDS=P_E-HR_C\cdot P_{coal}-EF_C\cdot P_C$$

where $(HR_C)$ is the coal plant heat rate and $(EF_C)$ its emission factor.

Because coal generally has a higher carbon intensity than natural gas per unit of electricity produced, the carbon component can have a substantial effect on the relative economics of coal and gas generation.

The difference between the two generation margins can also be considered directly:

$$CSS-CDS$$

This comparison provides a simplified measure of the relative short-run economics of gas-fired and coal-fired generation, although actual dispatch decisions also depend on plant-specific efficiencies, technical constraints and operating costs.

Break-even electricity price

The same equations can be rearranged to obtain useful break-even quantities.

Setting the generation margin equal to zero:

$$0=P_E-HR\cdot P_G-EF\cdot P_C-VOM$$

gives the break-even electricity price:

$$P_E^*=HR\cdot P_G+EF\cdot P_C+VOM$$

Similarly, the maximum gas price consistent with a zero generation margin is:

$$P_G^*=\frac{P_E-EF\cdot P_C-VOM}{HR}$$

and the break-even carbon price is:

$$P_C^*=\frac{P_E-HR\cdot P_G-VOM}{EF}$$

These expressions illustrate that the same spread equation can be used not only to calculate a current generation margin but also to determine the threshold values at which the economics of generation change.

Spreads as tradable economic relationships

The examples above share the same mathematical structure. A calendar spread compares the value of energy at different times:

$$S=F_{T_1}-F_{T_2}$$

A location spread compares the value of energy at different places:

$$S=P_A-P_B$$

A generation spread compares the value of an output with the cost of the inputs required to produce it:

$$M=P_{output}-\sum_i a_iP_i$$

where $(a_i)$ represents the amount of input $(i)$ required per unit of output.

This general form is useful across energy markets. Refining margins, generation margins and several cross-commodity relationships can all be represented as combinations of prices linked by physical conversion coefficients.

The coefficients are particularly important. A spread between economically related commodities is not necessarily obtained by subtracting their quoted prices directly. Units, conversion efficiencies, emission factors and other physical relationships determine the quantities of each input that must enter the calculation.

For this reason, spread analysis provides a direct connection between market prices and the physical structure of the energy system.

4. Forward curves and term structure

Energy commodities are traded for delivery over different future periods. The set of forward prices observed at a given point in time defines the forward curve, or term structure of prices.

A forward price can be represented as:

$$F(t,T)$$

where (t) is the current observation date and (T) is the future delivery date or period.

For example, a natural gas forward curve may simultaneously contain prices for the following month, quarter, season and calendar year. The curve therefore provides information not only about the current price level, but also about how the market values the same commodity across different delivery periods.

Forward-curve shape

Consider two forward contracts with delivery dates $(T_1)$ and $(T_2)$, where:

$$T_1<T_2$$

A simple measure of the local shape of the curve is their calendar spread:

$$S(t;T_1,T_2)=F(t,T_1)-F(t,T_2)$$

If:

$$F(t,T_1)<F(t,T_2)$$

the curve is upward sloping between the two maturities. This configuration is generally referred to as contango.

Conversely, if:

$$F(t,T_1)>F(t,T_2)$$

the curve is downward sloping and is generally described as backwardation.

For example, suppose monthly natural gas forwards are quoted as follows:

$$F(t,T_1)=35\text{ EUR/MWh}$$

$$F(t,T_2)=38\text{ EUR/MWh}$$

The calendar spread under the convention adopted above is:

$$S(t;T_1,T_2)=35-38=-3\text{ EUR/MWh}$$

The negative spread indicates an upward-sloping section of the curve.

Forward curves in energy markets are rarely uniformly increasing or decreasing across all maturities. Seasonal demand, storage economics, expected production, infrastructure constraints and other market conditions can produce complex shapes, with different sections of the same curve simultaneously exhibiting positive and negative slopes.

Spot and forward prices

For a storable commodity, the relationship between spot and forward prices can be introduced through the cost-of-carry framework.

In a simplified setting:

$$F_0=S_0e^{(r+u-y)T}$$

where $(S_0)$ is the current spot price, $(r)$ is the financing rate, $(u)$ represents storage and other carrying costs, and $(y)$ is the convenience yield (simply said, the value of physically having the commodity at own disposal).

Taking logarithms gives:

$$\ln\left(\frac{F_0}{S_0}\right)=(r+u-y)T$$

and therefore:

$$\frac{1}{T}\ln\left(\frac{F_0}{S_0}\right)=r+u-y$$

The forward premium or discount relative to spot can consequently be interpreted as the combined effect of financing costs, physical carrying costs and the economic benefit associated with holding the physical commodity.

If:

$$r+u>y$$

then:

$$F_0>S_0$$

and the cost-of-carry relationship produces contango.

If instead:

$$y>r+u$$

then:

$$F_0<S_0$$

and the relationship is consistent with backwardation.

Storage and convenience yield

Storage is central to the connection between spot and forward prices. Holding inventory allows a market participant to transfer a physical commodity from one period to another, but doing so involves costs.

In a simplified discrete representation, the future value of purchasing and carrying one unit of a commodity can be written as:

$$C_T=S_0(1+r)^T+C_{storage}-B_{inventory}$$

where $(C_{storage})$ represents storage-related costs and $(B_{inventory})$ represents the economic benefit of holding inventory.

The convenience yield (y) provides a compact representation of this benefit in the continuous cost-of-carry equation. Physical inventory may have value because it allows a producer or consumer to respond to unexpected demand, avoid production interruptions, satisfy delivery obligations or exploit temporary market conditions.

The convenience yield is therefore an implicit economic benefit rather than a directly observable cash payment.

Rearranging the cost-of-carry equation gives an implied convenience yield:

$$y=r+u-\frac{1}{T}\ln\left(\frac{F_0}{S_0}\right)$$

Given observed spot and forward prices and assumptions for financing and storage costs, this relationship can be used to infer the convenience yield consistent with market prices.

Forward prices are not forecasts

A forward price should not generally be interpreted as the market’s direct forecast of the future spot price.

The distinction can be expressed as:

$$F(t,T)\neq E_t[S_T]$$

in general.

The expected future spot price is:

$$E_t[S_T]$$

while the observed forward price reflects the pricing conditions and risk premia of the market in addition to expectations about future fundamentals.

A generic representation is:

$$F(t,T)=E_t[S_T]+RP(t,T)$$

where (RP(t,T)) denotes a forward risk premium under the chosen sign convention.

Consequently, an upward-sloping curve does not by itself imply that market participants expect spot prices to increase. Similarly, backwardation does not necessarily imply an expectation of falling future spot prices.

The distinction is particularly important in commodity markets because inventory conditions, hedging pressure, seasonality and physical constraints can materially affect forward prices.

Seasonality

Energy forward curves frequently contain strong seasonal components.

If (F(t,T)) is decomposed conceptually into a structural price component and a seasonal component, a simple representation is:

$$F(t,T)=L(t,T)+S(T)$$

where (L(t,T)) represents the underlying price level and (S(T)) represents the systematic seasonal effect associated with the delivery period.

For natural gas, winter contracts may trade above summer contracts because heating demand increases during colder months. Electricity curves may reflect seasonal demand, renewable generation patterns, hydro availability and expected thermal generation costs.

As a consequence, the slope between two adjacent contracts may primarily reflect seasonality rather than a persistent contango or backwardation regime.

For example:

$$F_{Summer}<F_{Winter}$$

does not necessarily indicate a general upward-sloping market structure. It may simply represent the recurring seasonal value of winter delivery.

Electricity and the limits of cost of carry

The standard cost-of-carry relationship depends on the possibility of purchasing the underlying commodity today and carrying it forward to the delivery date.

This assumption is problematic for electricity because electricity cannot generally be stored economically at large scale in the same way as natural gas, crude oil or other physical commodities. Storage technologies exist, but they involve conversion constraints, finite capacity, efficiency losses and specific operating economics.

The standard cash-and-carry relationship:

$$F_0=S_0e^{(r+u-y)T}$$

therefore cannot be mechanically applied to electricity forwards.

Without direct physical arbitrage between spot electricity and electricity delivered at a distant future date, the forward curve is more strongly determined by expectations of future supply and demand, expected fuel and carbon costs, generation availability, weather, seasonality and risk premia.

A simplified representation of the expected marginal cost of thermal generation, for example, is:

$$MC_T=HR\cdot P_{G,T}+EF\cdot P_{C,T}+VOM$$

which links the future economics of electricity generation to forward gas and carbon prices.

This does not imply that electricity forwards must equal expected marginal generation costs. The equation instead illustrates how the forward relationships introduced in the previous section can contribute to the formation of power prices.

Forward curves as relative prices through time

A forward curve can ultimately be interpreted as a set of relative prices for delivery of the same commodity at different points in time.

For (n) maturities, the curve observed at time (t) can be represented as:

$$\mathbf{F}_t=[F(t,T_1),F(t,T_2),\ldots,F(t,T_n)]$$

Changes in the market can affect the entire curve or only particular sections of it. A parallel movement can be approximated as:

$$\Delta F(t,T_i)\approx c$$

for all (i), while a change in relative prices produces movements in calendar spreads:

$$\Delta S=\Delta F(t,T_1)-\Delta F(t,T_2)$$

This distinction matters because many physical and financial energy positions are exposed to the shape of the forward curve rather than only to its overall level.

Storage, seasonal supply contracts and generation portfolios are examples of positions whose economic value depends on relative prices across delivery periods. The forward curve therefore provides the natural extension from analysing individual energy prices to analysing the value of energy through time.

5. Profiles, weighted prices and capture rates

Energy prices vary across delivery intervals, while physical production and consumption are rarely constant over time. The economic value of an energy position therefore depends on the interaction between its volume profile and the corresponding market-price profile.

This is particularly relevant for renewable generation, where production is determined by the availability of the underlying resource and may be systematically concentrated in hours characterised by specific price conditions.

Average market price

For (n) delivery intervals, the arithmetic average market price is:

$$\bar P=\frac{1}{n}\sum_{t=1}^{n}P_t$$

If hourly electricity prices over four hours are 50, 60, 80 and 90 EUR/MWh, the average price is:

$$\bar P=\frac{50+60+80+90}{4}=70\text{ EUR/MWh}$$

This measure assigns the same weight to every delivery interval, independently of the amount of energy produced or consumed during each period.

For a physical asset with a variable production profile, this may not represent the price actually realised by the asset.

Volume-weighted average price

Let $(Q_t)$ denote the energy produced during interval (t). Total revenue is:

$$R=\sum_{t=1}^{n}P_tQ_t$$

while total production is:

$$Q=\sum_{t=1}^{n}Q_t$$

The average price actually obtained for the generated energy is therefore:

$$P_{VWAP}=\frac{\sum_{t=1}^{n}P_tQ_t}{\sum_{t=1}^{n}Q_t}$$

In power-generation analysis, this quantity is commonly referred to as the capture price:

$$P_{capture}=\frac{\sum_{t=1}^{n}P_tQ_t}{\sum_{t=1}^{n}Q_t}$$

Consider the same four electricity prices:

$$P=[50,60,80,90]$$

and suppose a solar plant produces:

$$Q=[0,4,2,0]\text{ MWh}$$

Its total revenue is:

$$R=50\cdot0+60\cdot4+80\cdot2+90\cdot0=400\text{ EUR}$$

while total production is:

$$Q=0+4+2+0=6\text{ MWh}$$

The capture price is consequently:

$$P_{capture}=\frac{400}{6}\approx66.67\text{ EUR/MWh}$$

Although the average market price is 70 EUR/MWh, the plant captures only 66.67 EUR/MWh because most of its production occurs during the lower-priced producing hour.

Capture rate

The relationship between the capture price and the average market price is commonly expressed through the capture rate:

$$CR=\frac{P_{capture}}{\bar P}$$

For the previous example:

$$CR=\frac{66.67}{70}\approx0.9524$$

or approximately 95.24%.

A capture rate below one indicates that the production profile is, on average, concentrated in intervals with prices below the arithmetic market average:

$$CR<1$$

Conversely:

$$CR>1$$

indicates that production tends to occur during relatively high-priced intervals.

The capture rate therefore provides a compact measure of the economic relationship between a generation profile and the temporal structure of market prices.

Capture price and covariance

The relationship between production and prices can be expressed more formally using covariance.

The capture price is:

$$P_{capture}=\frac{\sum_{t=1}^{n}P_tQ_t}{\sum_{t=1}^{n}Q_t}$$

Define the arithmetic means:

$$\bar P=\frac{1}{n}\sum_{t=1}^{n}P_t$$

and:

$$\bar Q=\frac{1}{n}\sum_{t=1}^{n}Q_t$$

Using the population covariance over the observed intervals:

$$Cov(P,Q)=\frac{1}{n}\sum_{t=1}^{n}(P_t-\bar P)(Q_t-\bar Q)$$

we can use the identity:

$$Cov(P,Q)=\frac{1}{n}\sum_{t=1}^{n}P_tQ_t-\bar P\bar Q$$

and therefore:

$$\frac{1}{n}\sum_{t=1}^{n}P_tQ_t=\bar P\bar Q+Cov(P,Q)$$

Since:

$$\sum_{t=1}^{n}Q_t=n\bar Q$$

the capture price can be written as:

$$P_{capture}=\bar P+\frac{Cov(P,Q)}{\bar Q}$$

This expression provides a direct mathematical interpretation of profile value.

If production and prices are uncorrelated:

$$Cov(P,Q)=0$$

then:

$$P_{capture}=\bar P$$

If production is concentrated in high-price periods:

$$Cov(P,Q)>0$$

then:

$$P_{capture}>\bar P$$

while if production is concentrated in low-price periods:

$$Cov(P,Q)<0$$

then:

$$P_{capture}<\bar P$$

The difference between capture price and average market price can consequently be expressed as:

$$P_{capture}-\bar P=\frac{Cov(P,Q)}{\bar Q}$$

This relationship shows that the value of a generation profile depends not only on total production, but also on how production co-moves with market prices.

Profile value

Consider two generators (A) and (B\ producing the same total amount of energy:

$$\sum_tQ_t^A=\sum_tQ_t^B$$

Their revenues need not be equal:

$$\sum_tP_tQ_t^A\neq\sum_tP_tQ_t^B$$

and consequently:

$$P_{capture}^A\neq P_{capture}^B$$

The difference results entirely from the temporal distribution of production relative to prices.

This is particularly important when comparing dispatchable and non-dispatchable generation. A dispatchable generator may have some ability to concentrate production in economically favourable periods, subject to technical constraints and input costs. Wind and solar generation are primarily determined by weather conditions and therefore have substantially less control over their production profiles.

Cannibalization

The same framework helps describe the price cannibalization effect associated with increasing penetration of renewable generation.

Consider aggregate solar generation $(Q_t^{solar})$. If high solar output contributes to lower electricity prices during the same delivery intervals, increasing solar penetration can produce a negative relationship between production and price:

$$Cov(P,Q^{solar})<0$$

From the previous identity:

$$P_{capture}^{solar}=\bar P+\frac{Cov(P,Q^{solar})}{\bar Q^{solar}}$$

a more negative covariance reduces the solar capture price relative to the average market price.

The corresponding capture rate is:

$$CR^{solar}=\frac{P_{capture}^{solar}}{\bar P}$$

and may decline as additional generation with a similar production profile enters the system.

This is a simplified representation of cannibalization. Actual electricity prices are determined by the interaction of demand, generation technologies, transmission constraints, storage, interconnection, fuel and carbon prices, and other system conditions. The covariance relationship does not establish causality by itself.

Revenue and profile effects

Revenue can also be decomposed using the capture price:

$$R=P_{capture}Q$$

where:

$$Q=\sum_tQ_t$$

Substituting the covariance representation gives:

$$R=Q\left(\bar P+\frac{Cov(P,Q_t)}{\bar Q}\right)$$

Since:

$$Q=n\bar Q$$

this can equivalently be written as:

$$R=n\bar P\bar Q+nCov(P,Q)$$

The first component,

$$n\bar P\bar Q$$

represents the revenue that would result from combining average production with the average market price, while:

$$nCov(P,Q)$$

captures the contribution associated with the interaction between the production and price profiles.

A negative covariance therefore creates a negative profile effect, while a positive covariance creates a positive profile effect.

This decomposition illustrates why total MWh alone are insufficient to determine the economic value of an energy asset. The timing of those MWh, and their statistical relationship with market prices, is part of the value itself.

6. Conclusions

The examples developed in this article cover different areas of energy-market analysis, but rely on a relatively small set of mathematical relationships.

At the physical level, energy is obtained by integrating power over time:

$$E=P\Delta t$$

Market value then results from combining prices with the corresponding physical quantities:

$$V=\sum_tP_tQ_t$$

Differences between prices describe economic relationships across time, locations or commodities, while transformation margins extend the same principle to assets that convert one energy product into another:

$$M=P_{output}-\sum_ia_iP_i$$

Finally, when quantities vary over time, their interaction with prices determines the effective economic value of the physical profile:

$$P_{capture}=\frac{\sum_tP_tQ_t}{\sum_tQ_t}$$

These relationships illustrate a common feature of quantitative energy-market analysis: market prices acquire economic meaning only after the relevant physical dimensions have been specified. Unit, currency, delivery period, location, conversion efficiency and production or consumption profile can all affect the interpretation of a quoted price.

As a consequence, significant quantitative errors do not necessarily originate from sophisticated models. Comparing inconsistent units, confusing MW with MWh, applying percentage or logarithmic returns to unsuitable price series, ignoring conversion efficiency in generation margins, or using an arithmetic average where a volume-weighted price is required can materially alter the result of an otherwise elementary calculation.

The same consideration applies to forward markets. A forward curve contains information about the relative value of delivery across time, but its interpretation depends on storage possibilities, seasonality, physical constraints and risk premia. Standard financial relationships such as cost of carry remain useful, provided that their underlying assumptions are consistent with the commodity being analysed.

Coming next

The framework considered here has deliberately focused on prices and physical economics. Introducing uncertainty extends the analysis to another set of quantitative problems. Price changes become risk factors, relationships between commodities require covariance and correlation, physical exposures create hedging problems, and flexible assets generate nonlinear payoffs.

At a general level, the resulting change in portfolio value can be represented through its sensitivities to the relevant market variables:

$$\Delta V\approx\sum_i\frac{\partial V}{\partial x_i}\Delta x_i$$

where the $(x_i)$ may represent power, gas, carbon, oil, foreign-exchange rates, volumes or other relevant factors.

These extensions provide the natural subject for a second part, focused on the mathematics of energy-market risk, hedging and optionality.

References

Geman, H. (2005). Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals and Energy. Wiley.
Eydeland, A., & Wolyniec, K. (2003). Energy and Power Risk Management: New Developments in Modeling, Pricing, and Hedging. Wiley.
Hull, J. C. (2022). Options, Futures, and Other Derivatives, 11th ed. Pearson.
Weron, R. (2006). Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach. Wiley.
Bunn, D. W. (Ed.). (2004). Modelling Prices in Competitive Electricity Markets. Wiley.
Hirth, L. (2013). “The market value of variable renewables: The effect of solar wind power variability on their relative price.” Energy Economics, 38, 218–236.
Hirth, L. (2015). “Market value of solar power: Is photovoltaics cost-competitive?” IET Renewable Power Generation, 9(1), 37–45.

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Essential Math of Energy Markets

Introduction

Essential relations

1. Units and dimensional consistency

Power and energy

Energy-unit conversions

Currency conversion

From power and price to monetary value

Dimensional checks

2. Prices, returns and normalization

Absolute price changes

Simple returns

Logarithmic returns

Zero and negative prices

Normalized price indices

Volatility from price changes or returns

Price levels and statistical transformations

3. Spreads and generation margins

Calendar spreads

Location and basis spreads

From commodity spreads to transformation margins

Efficiency and heat rate

Clean spark spread

Clean dark spread

Break-even electricity price

Spreads as tradable economic relationships

4. Forward curves and term structure

Forward-curve shape

Spot and forward prices

Storage and convenience yield

Forward prices are not forecasts

Seasonality

Electricity and the limits of cost of carry

Forward curves as relative prices through time

5. Profiles, weighted prices and capture rates

Average market price

Volume-weighted average price

Capture rate

Capture price and covariance

Profile value

Cannibalization

Revenue and profile effects

6. Conclusions

Coming next

References

Further reading on Aziona blog

Scarica l’ebook “La guida definitiva alla comprensione del Debito Tecnico”

Essential Math of Energy Markets

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Energy Trading Essentials