EXHIBIT 39.9  Valuation Result for Mixed Price and Quantity Risk
Price risk event
Quantity risk event
Example 1: High technology risk
Example 2: Low technology risk
Price
Size
Cash flow
Investment
Net cash flow
ROV
DTA
Price
Size
Cash flow
Investment
Net cash flow
ROV
DTA
 p* = 0.45
p = 0.50
1 – p = 0.50
 1 – p* = 0.55
q = 0.33
1 – q = 0.67
q = 0.33
1 – q = 0.67
150
2.50
375
(105)
270
38.6
39.8
150
1.50
225
(105)
120
17.1
17.3
150
0.26
39
(105)
–
–
–
150
0.75
113
(105)
8
2.3
0.9
50
2.50
125
(105)
20
3.4
2.3
50
1.50
75
(105)
–
–
–
50
0.26
13
(105)
–
–
–
50
0.75
38
(105)
–
–
–
Value
42.0
42.0
Value
19.5
18.2
Note: p = probability of mineral price being high
p* = binomial (risk-neutral) probability
q = probability of mine size being large
776

Methods for Valuing Flexibility  777
Summing these values over all possible scenarios leads to an ROV of $42.14
Taking a DTA approach instead, we would multiply the conditional payoffs 
by the normal probabilities for price and quantity scenarios and then separately 
discount the cash inflows at the mine’s cost of capital and the investment cash 
flow at the risk-free rate. For the large-mine, high-price scenario, for example:
0 50
0 33
375
1
10
105
1
5
39 8
.
.
$
%
$
%
$
.
×
×
+
−
+



=
Similarly, summing the results over all scenarios, we obtain a DTA value of 
$42, which is exactly equal to the ROV result. The reason is that the decision 
is driven entirely by the diversifiable, technological risk related to the size of 
mine. The nondiversifiable price risk leads to different cash flows but does not 
matter for the investment decision.15
The DTA and ROV approaches both provide the theoretically correct an-
swer when the contingent decisions are (predominantly) driven by diversifi-
able underlying risk. Examples are geological risks, such as the size of an 
undeveloped oil field, and even some forms of marketing risk, such as con-
sumer acceptance of a new product. As in the numerical illustration, these 
risks often have more impact on value than nondiversifiable risks. For ex-
ample, the driver of the decision to invest in drug development is whether the 
drug passes the trials, not whether the drug—once successfully developed—is 
worth more or less, depending on general economic conditions.
In contrast, only the ROV approach is theoretically correct if the contingent 
decision is affected by nondiversifiable risk; the DTA result would lead to an 
approximate result. For the same numerical illustration as before, the nondi-
versifiable price risk does affect the investment decision if the quantity risk is 
smaller (see Exhibit 39.9, Example 2). As shown in the exhibit, the variation in 
price becomes the prevailing risk that drives the decision to start production in 
the case of lower variation in potential mine size outcomes. For example, if the 
price turns out to be high, production is started whether the mine size ends up 
at the higher or lower end of its range. As a result, the DTA approach now only 
provides an approximation ($18.20) of the correct ROV value ($19.50).16 Simi-
larly, for some real-world investments, the nondiversifiable risks outweigh any 
technological, regulatory, or other diversifiable risks. For example, decisions to 
invest in the expansion of a power plant are typically driven by the difference 
between fuel and power prices and by overall demand for power.
15 If the probability distribution of the commodity price was continuous rather than discrete as in this 
example, there would always be some price outcome overturning the production decision. But the 
point remains that if the probability of reaching such price levels is small, the difference between the 
ROV and DTA outcome would be small, too.
16 For the ROV result, the present value of the large-mine and high-price scenario would now be (0.45 
× 0.33 × $120)/(1 + 5%) = $17.1. Adding this to the present value for the small-mine and high-price 
scenario, or (0.45 × 0.67 × $7.5)/(1 + 5%) = $2.3, leads to total ROV of $19.5 when rounded.
14 Note that this value is higher than in the original example, because we now develop the mine only if 
the reserves are large; we have introduced additional flexibility.