Research on value and factors of the guarantee payment in the energy performance contracting in China

Abstract

Mistrust of the Energy Using Organisation (EUO) in Energy Services Company (ESCO) is a key factor that hinders the development of Energy Performance Contracting Projects (EPCPs) in China, especially the EPCP is the shared savings type. A payment guarantee that is able to hedge the credit risk of the EUO is the rational choice for the ESCO. Considering an EPCP payment guarantee with a limit on the total amount that is essentially a multi-period European put option portfolio with multiple uncertain exercise prices that the guarantor sells to the ESCO, this paper constructs a valuation formula of an EPCP payment guarantee, in which the exercise price follows the geometric Brownian motion with a Poisson jump, and analyses the effects of the factors on the value of the payment guarantee. The results revealed the following. First, the value of the payment guarantee is affected by the total guaranteed amount, uncertainty in ESCO’s revenue, and uncertainty in EUO’s net cash flow, and consideration of the impact of unexpected shocks on the exercise price can avoid overrating the value. Second, the value of the payment guarantee for each period within the scope of the full guarantee increases with an increase in the ESCO’s initial revenue flow, proportion of the revenue sharing and expected revenue growth rate, and it decreases with an increase in the risk-adjusted discount rate and EUO’s initial net cash flow; the value of the payment guarantee for each period within the scope of the full guarantee first decreases and subsequently increases with an increase in the risk-level of ESCO’s revenue and the EUO’s net cash flow. Third, the results of the Sobol’s sensitivity analysis indicate that, within the scope of the full guarantee, the ESCO’s initial revenue flow, proportion of the revenue sharing, the risk-adjusted discount rate and EUO’s initial net cash flow play relatively important roles in changing the EPCP payment guarantee and its exercise prices. According to the above conclusions, the payment guarantee contract can provide the ESCO with a basis for decision-making in the formulation of a strategy to safeguard energy conservation revenues in accordance with the relationships embodied by the value of the payment guarantee. Finally, the policy recommendations about relieving the mistrust between EUOs and ESCOs are provided.

Appendix

Derivation process of Formula (15):

According to the basic assumptions and Eq. (14), make ${\tilde{G}}_{1i}=\max \left({V}_i-{\tilde{D}}_i,0\right)$ the European call option corresponding to G_1i and ${\tilde{G}}_{0i}$ its current value. As with Merton (1973), a zero investment portfolio Ω is constructed, i.e. the total amount of investment in Ω is zero, which is constituted by the underlying asset V_i (used to hedge the changes in the underlying asset), the hedge security H_i (used to hedge the uncertainty of execution price ${\tilde{D}}_i$) and the call option${\tilde{G}}_{1i}$. The structure of Ω is as follows:

$$ \varOmega ={\tilde{G}}_{1i}+{\varDelta}_1{V}_i+{\varDelta}_2{H}_i $$

(42)

Where Δ₁ and Δ₂ are respectively invested in the share of V_i and H_i. Since Ω is the zero portfolio, according to the no-arbitrage principle, there is dΩ = 0, thus:

$$ d{\tilde{G}}_{1i}+{\varDelta}_1d{V}_i+{\varDelta}_2d{H}_i=0 $$

(43)

In addition, according to the Itô lemma, there is:

$$ {\displaystyle \begin{array}{l}d{\tilde{G}}_{1i}=\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}d{V}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}d{H}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial i} di\\ {}+\frac{1}{2}\left[\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}{\left(d{V}_i\right)}^2+2\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i{H}_i}\left(d{V}_id{H}_i\right)+\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}{\left(d{H}_i\right)}^2\right]\end{array}} $$

(44)

Where ${\left(d{V}_i\right)}^2={\sigma}_V^2{V_i}^2 di$,${\left(d{H}_i\right)}^2={\tilde{\sigma}}_D^2{H_i}^2 di$,$\left(d{V}_id{H}_i\right)={\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D{V}_i{H}_i di$. Substitute Eqs. (3) and (8) into Eq. (44) to obtain:

$$ {\displaystyle \begin{array}{l}d{\tilde{G}}_{1i}=\left(\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}{\alpha}_v{V}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}{r}_H{H}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial i}\right) di\\ {}+\frac{1}{2}\left(\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}{\sigma}_V^2{V_i}^2+2\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i{H}_i}{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D{V}_i{H}_i+\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}{\tilde{\sigma}}_D^2{H_i}^2\right) di\\ {}+\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}{\sigma}_V{V}_id{z}_t+\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}{\tilde{\sigma}}_D{H}_id{z}_D\end{array}} $$

(45)

Substitute Eq. (45) into Eq. (43) to obtain:

$$ {\displaystyle \begin{array}{l}\left[\left(\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}+{\varDelta}_1\right){\alpha}_v{V}_i+\left(\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}+{\varDelta}_2\right){r}_H{H}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial i}\right] di\\ {}+\frac{1}{2}\left(\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}{\sigma}_V^2{V_i}^2+2\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i{H}_i}{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D{V}_i{H}_i+\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}{\tilde{\sigma}}_D^2{H_i}^2\right) di\\ {}+\left(\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}+{\varDelta}_1\right){\sigma}_V{V}_id{z}_t+\left(\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}+{\varDelta}_2\right){\tilde{\sigma}}_D{H}_id{z}_D\\ {}=0\end{array}} $$

(46)

Since the right side of the above equation is the income of the zero risk portfolio, the coefficient of dz_t and dz_D on the left side of the above equation must be 0, there is:

$$ {\varDelta}_1=-\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i},{\varDelta}_2=-\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i} $$

(47)

Substituting the above results into Eq. (46) and eliminating di, yields that ${\tilde{G}}_{1i}$ meets the partial differential equation:

$$ \frac{\partial {\tilde{G}}_{1i}}{\partial i}+\frac{1}{2}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}{\sigma}_V^2{V_i}^2+\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i{H}_i}{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D{V}_i{H}_i+\frac{1}{2}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}{\tilde{\sigma}}_D^2{H_i}^2=0 $$

(48)

It can be seen that Eq. (48) is similar to the multi-asset option Black-Scholes formula, and there is:

$$ {\tilde{G}}_{1i}=\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}{V}_i+\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}{H}_i $$

(49)

Then Eq. (48) can be rewritten as:

$$ \frac{\partial {\tilde{G}}_{1i}}{\partial i}+\frac{1}{2}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}{\sigma}_V^2{V_i}^2+\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i\partial {H}_i}{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D{V}_i{H}_i+\frac{1}{2}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}{\tilde{\sigma}}_D^2{H_i}^2=0 $$

(50)

According to the boundary condition ${\tilde{G}}_{1i}\left({V}_i,{H}_i,i;{\tilde{D}}_i\right)=\max \left({V}_i-{\tilde{D}}_i,0\right)$, perform the following equation transformation:

$$ {\zeta}_i=\frac{V_i}{H_i{\tilde{D}}_i} $$

(51)

$$ {\tilde{G}}_{1i}\left({V}_i,{H}_i,i;{\tilde{D}}_i\right)={H}_i{\tilde{D}}_iu\left({\zeta}_i,i\right)={H}_i{\tilde{D}}_i\max \left({\zeta}_i-\frac{1}{H_i},0\right) $$

(52)

Then, there are:

$$ {\displaystyle \begin{array}{c}\frac{\partial {\tilde{G}}_{1i}}{\partial {V}_i}={H}_i{\tilde{D}}_i\frac{\partial u}{\partial {\zeta}_i}\frac{1}{H_i{\tilde{D}}_i}=\frac{\partial u}{\partial {\zeta}_i}\\ {}\frac{\partial {\tilde{G}}_{1i}}{\partial {H}_i}={\tilde{D}}_iu-{H}_i{\tilde{D}}_i\frac{\partial u}{\partial {\zeta}_i}\frac{V_i}{H_i^2{\tilde{D}}_i}={\tilde{D}}_i\left(u-{\zeta}_i\frac{\partial u}{\partial {\zeta}_i}\right)\\ {}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V_i}^2}=\frac{\partial^2u}{\partial {\zeta_i}^2}\frac{1}{H_i{\tilde{D}}_i}\\ {}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {V}_i\partial {H}_i}=-\frac{\partial^2u}{\partial {\zeta_i}^2}\frac{V_i}{{H_i}^2{\tilde{D}}_i}=-\frac{\zeta_i}{H_i}\frac{\partial^2u}{\partial {\zeta_i}^2}\\ {}\frac{\partial^2{\tilde{G}}_{1i}}{\partial {H_i}^2}={\tilde{D}}_i\left(-\frac{\partial u}{\partial {\zeta}_i}\frac{V_i}{H_i^2{\tilde{D}}_i}+\frac{V_i}{H_i^2{\tilde{D}}_i}\frac{\partial u}{\partial {\zeta}_i}+{\zeta}_i\frac{V_i}{H_i^2{\tilde{D}}_i}\frac{\partial^2u}{\partial {\zeta_i}^2}\right)=\frac{{\tilde{D}}_i}{H_i}{\zeta_i}^2\frac{\partial^2u}{\partial {\zeta_i}^2}\end{array}} $$

Substitute all of the above into the Eq. (50) and eliminate the expression ${H}_i{\tilde{D}}_i$ to obtain:

$$ \frac{\partial u}{\partial i}+\frac{1}{2}\left({\sigma}_V^2-2{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D+{\tilde{\sigma}}_D^2\right){\zeta_i}^2\frac{\partial^2u}{\partial {\zeta_i}^2}=0 $$

(53)

Equations (53) and (52) make up a problem with definitive solution, make ${\widehat{\sigma}}^2={\sigma}_V^2-2{\rho}_{DV}{\sigma}_V{\tilde{\sigma}}_D+{\tilde{\sigma}}_D^2$, then yields:

$$ u\left({\zeta}_i,i\right)={\zeta}_iN\left\{\frac{\ln {\zeta}_i+\frac{1}{2}{\widehat{\sigma}}^2i}{\widehat{\sigma}\sqrt{i}}\right\}-\frac{1}{H_i}N\left\{\frac{\ln {\zeta}_i-\frac{1}{2}{\widehat{\sigma}}^2i}{\widehat{\sigma}\sqrt{i}}\right\} $$

(54)

Back to the original function, there is:

$$ {\tilde{G}}_{1i}={H}_i{\tilde{D}}_iu\left({\zeta}_i,i\right)={V}_iN\left\{\frac{\ln {\zeta}_i+\frac{1}{2}{\widehat{\sigma}}^2i}{\widehat{\sigma}\sqrt{i}}\right\}-{\tilde{D}}_iN\left\{\frac{\ln {\zeta}_i-\frac{1}{2}{\widehat{\sigma}}^2i}{\widehat{\sigma}\sqrt{i}}\right\} $$

(55)

Combining $E\left({\tilde{D}}_t\right)={D}_0{e}^{\tilde{\alpha}t}$ with H₀ = 1 yields ${\tilde{G}}_{0i}$ (the present value of ${\tilde{G}}_{1i}$):

$$ {\displaystyle \begin{array}{l}{\tilde{G}}_{0i}={V}_0\cdot N\left\{\frac{\ln \left(\frac{V_0}{D_0}\right)+\left({r}_H-\tilde{\alpha}+\frac{{\widehat{\sigma}}^2}{2}\right)i}{\widehat{\sigma}\sqrt{i}}\right\}\\ {}-{D}_0{e}^{-\left({r}_H-\tilde{\alpha}\right)i}\cdot N\left\{\frac{\ln \left(\frac{V_0}{D_0}\right)+\left({r}_H-\tilde{\alpha}-\frac{{\widehat{\sigma}}^2}{2}\right)i}{\widehat{\sigma}\sqrt{i}}\right\}\end{array}} $$

(56)

where N(⋅) is the cumulative distribution function of the standard normal distribution, ${\widehat{\sigma}}^2$ represents the variance of the rate of change for $\frac{V_i}{{\tilde{D}}_i}$.