Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 79

In the case of S 0 = € 4.38, a k ¼ 0 0395, s 1 ¼ 0, s 2 ¼ 25, and r ¼ 0 03 the

: :

result obtained is Vðs 1 ; s 2 Þ ¼ 123 60 € /(tonnes per year). However if si d ¼ 0 and

:

a k ¼ r ¼ 0 0395 it is Vðs 1 ; s 2 Þ ¼ S 0 ðs 2 s 1 Þ ¼ 109 50/(tonnes per year).

: :

(C) Annuities ( IGBM case )

In this third example, the aim is to deduce the value of an annuity when the price
follows an IGBM type mean reverting process, e.g., in the case of a barrel of oil at
1 year over 20 years.
Based on the future equation:

e ðkþkÞt

kS m
k þ k þ

kS m
k þ k

Fð 0

; tÞ ¼

S 0 ð 24 Þ

An annuity between s 1 and s 2 has a value of:

Z

s 2

e rt d t Vðs 1 ; s 2 Þ ¼ Fð 0

; tÞ

ð 25 Þ

s 1

kS m
kþk

S 0

kS m
rðk þ kÞ ½

e rs 1 e rs 2 e ðkþkþrÞs 1 e ðkþkþrÞs 2 ¼ þ

k þ k þ r ½

Therefore, the current value of a barrel of oil at 1 year over 25 years between
s 1 ¼

kS m

0 and s 2 ¼ 25 when S 0 ¼ 106 42 USD/barrel,

kþk ¼

79 5157

; k þ k ¼

0 5583

: : :

and r ¼ 0 03 is

:

106 42 79 5157 79 5157

: : :

e 14 : 7075 e 0 : 75 Vðs 1 ; s 2 Þ ¼ ½ 1 þ

03 ½

1 ¼ 1444 23

:

0 : 5883 0 :

(D) Annuities ( 3:2:1 Crack Spread, IGBM case )

In this fourth example, the aim is to deduce the value of an annuity with a 3:2:1
crack spread over 25 years with market data. The existence of suf cient liquidity in

the future market means that these re ning margins can be assured.

Leaving aside the effect of seasonality, since periods of higher prices tend to be
offset by periods of lower prices, the current value of a unit with 3:2:1 crack spread
at 1 year over 25 years between s 1 ¼

0 and s 2 ¼ 25 when S 0 ¼ 17 : 3456 USD/

kS m

barrel,

kþk ¼

23 4233

; k þ k ¼

3 0149 and r ¼ 0 03 is

: : :

17 3456 23 4233 23 4233

: : :

e 75 : 3726 e 0 : 75

Vðs 1 ; s 2 Þ ¼ ½ 1 þ

03 ½

1 ¼ 409 97

:

3 0449 0

: :