Some New Results on When Extra Risk Strictly Increases an Option's Value

Some New Results on

When Extra Risk Strictly

Increases an Option’s

Value

JAMES HUANG∗

DEYUAN ZHANG

In this paper, we present some new results on when extra risk strictly increases anoption’s value. We give a necessary and sufficient condition for a mean-preservingspread to strictly increase an option’s value. We also give a necessary and sufficientcondition for a risk change to strictly increase the values of options with strikeprices in an open interval while preserving the values of all other options. Thesetwo results significantly improve the results given by Rasmusen (2007) (Whendoes extra risk strictly increase an option’s value? Review of Financial Studies, 20,1647–1667). C© 2012 Wiley Periodicals, Inc. Jrl Fut Mark

1. INTRODUCTION

An interesting topic in finance is how changes in risk affect option values.Merton (1973) discussed it briefly and argued that the value of a call option

We are grateful to the editor and an anonymous referee for very helpful remarks that helped improve thestudy.

∗Correspondence author, Department of Accounting and Finance, Lancaster University, Lancaster,Lancashire LA1 4YX, United Kingdom. Tel: +(44) 1524 593633, Fax: +(44) 1524 847321, e-mail:[email protected]

Received December 2011; Accepted January 2012

� James Huang is with the Department of Accounting and Finance, LancasterUniversity, Lancaster, Lancashire, United Kingdom.

� Deyuan Zhang is with the School of International Business Administration, ShanghaiUniversity of Finance and Economics, Shanghai, China.

The Journal of Futures Markets, Vol. 00, No. 0, 1–11 (2012)C© 2012 Wiley Periodicals, Inc.Published online in Wiley Online Library (wileyonlinelibrary.com).DOI: 10.1002/fut.21556

2 Huang and Zhang

increases with the riskiness of the underlying asset. Jagannathan (1984) made itclear that what really matters should be the riskiness under risk neutral proba-bility measures rather than real world probability measures. This result is widelyaccepted;1 however, as was pointed out by Bliss (2000) and Rasmusen (2007),what Merton’s result really tells us is that additional risk does not reduce anoption’s value. One way to strengthen Merton’s result is to surrender generalityin the class of asset price distributions. Bliss (2001) worked along this line andshowed that a sufficient condition for option values to increase with risk is thatthe underlying asset value has a two-parameter distribution such as the nor-mal or lognormal. Rasmusen (2007) tried to derive some results that can covergeneral distributions. Apart from the two well-known definitions of increasein risk, namely, a mean-preserving spread and Rothschild and Stiglitz (1970)increase in risk, he proposed two additional definitions of changes in risk whichare called pointwise riskier and extremum riskier, respectively. Based on theseconcepts, he presented some necessary conditions and sufficient conditions forextra risk to strictly increase option values.

In this study, we present some new results on the topic. We show that amean-preserving spread strictly increases an option’s value if and only if theoption’s strike price is between the left and right borders of its support. Wealso show that a mean-preserving change in risk strictly increases the values ofoptions with strike prices in an open interval, whereas preserving the values ofother options if and only if the risk change is a sum of mean-preserving spreadssuch that the open set equals the set of the points between the left and rightborders of the supports of these mean-preserving spreads. These two resultssignificantly improve the results given by Rasmusen (2007).

Apart from the aforementioned studies, this study is also related to thework of Franke, Stapleton, and Subrahmanyam (1999) and Grundy (1991).Franke et al. showed that if constant relative risk aversion is changed to de-clining relative risk aversion then it leads to a mean-preserving spread in risksuch that all options increase in value. Grundy examined the relation betweenoption prices and the true distribution of the underlying asset price. Otherrelated work includes some studies which discuss different definitions of mean-preserving increase in risk and their applications such as Eeckhoudt, Dionne,and Gollier (1993), Gollier (1995), and Chateauneuf, Cohen, and Meilijson(2004).

The study’s main contribution is the same as Rasmusen (2007), that is,to tidy up one of the fundamental ideas in finance theory. As to the potentialapplications of this study in finance, we refer to his explanations.

1See, for example, in a standard finance textbook on corporate finance (Copeland & Weston, 3rd ed., p. 243),it is said that the value of an option will increase when the variance of the underlying stock increases.

Journal of Futures Markets DOI: 10.1002/fut

Risk Increases and Value of Options 3

The structure of the study is as follows: In Section 1, we present our mainresults. In Section 2, we compare our results with those in Rasmusen (2007).In Section 3, we conclude the study.

2. MAIN RESULTS

In this section, we present our main results on when extra risk strictly increasesan option’s value. We adopt the definition of a mean-preserving spread givenby Machina and Pratt (1997), which is slightly more general than the originalRothschild and Stiglitz’s definition.2

Definition 1 (Machina and Pratt [1997]): A cumulative distribution functionG(x) is said to differ from F (x) by a mean-preserving spread if F (x) and G(x)have the same finite expected value, and there exist x1 ≤ x2 such that G(x) assignsat least as much probability as F (x) to every subinterval of (−∞, x1), G(x) assignsno more probability than F (x) to every subinterval of (x1, x2), and G(x) assigns atleast as much probability as F (x) to every subinterval of (x2, ∞).3

We first present the following lemma.

Lemma 1: Given a cumulative distribution function Q(x) with its support beinga subset of R+, if

∫ ∞0 x d Q(x) < ∞ then the following two equations are true.

∫ ∞

0(1 − Q(x)) dx =

∫ ∞

0x d Q(x). (1)

∫ ∞

K(x − K ) d Q(x) =

∫ ∞

0x d Q(x) − K +

∫ K

0Q(x) dx . (2)

The first equation is well known in statistics and economics, and the secondequation is well known in finance and can be found in Jagannathan (1984); thus,the proof is omitted for brevity. Note that the first equation can also be seen asa special case of the second equation where K = 0.

As an immediate consequence of the above lemma we have the followingresult.

2Unlike Rothschild and Stiglitz’s definition, this definition is valid when probability distributions do not haveprobability density functions.3Following convention, lower case letters are used to denote probability mass functions (in the discrete case)or probability density functions (in the continuous case), and upper case letters are used to denote cumulativeprobability distribution functions.


4 Huang and Zhang

Lemma 2: The value of an option with strike price K is strictly increased (de-creased) by a mean-preserving change in risk S(x) if and only if

∫ K0 S(x) dx >

(<)0.

Proof: Because of the put-call parity, without loss of generality, we assume thatthe option is a call option. Let F (x) and G(x) be the original and changedcumulative distribution functions, respectively. From the second result ofLemma 1,

∫ ∞K (x − K ) dG(x) − ∫ ∞

K (x − K ) d F (x) is equal to(∫ ∞

0x dG(x) −

∫ ∞

0x d F (x)

)+

(∫ K

0G(x) dx −

∫ K

0F (x) dx

).

As the risk change is mean preserving, we have∫ ∞

0 x dG(x) − ∫ ∞0 x d F (x) = 0.

Thus, we obtain∫ ∞

K(x − K ) dG(x) −

∫ ∞

K(x − K ) d F (x) =

∫ K

0S(x) dx,

where S(x) = G(x) − F (x). Thus, the conclusion immediately follows. �

From the above lemma, we have the following corollary.

Corollary 1: Let the left and right borders of the support of a mean-preservingchange in risk be a and b, respectively. Then this risk change preserves the valueof any option with a strike price K /∈ (a , b).

Proof: Let S(x) = G(x) − F (x) be the change in the cumulative probabilitydistribution function. As S(x) is a mean-preserving spread, from the first resultof Lemma 1, we have

∫ +∞0 S(x) dx = 0. Moreover, as a and b are the left and

right borders of the support of the change, we have S(x) = 0, for all x /∈ (a , b).This and the preceding statement imply that

∫ x0 S(y ) dy = 0, for all x /∈ (a , b).

Now applying Lemma 2, we immediately obtain the corollary. �

We now present the following result.

Theorem 1: Let a and b be the left and right borders of the support of a mean-preserving spread (contraction), respectively. The mean-preserving spread (con-traction) strictly increases (decreases) the value of an option if and only if theoption’s strike price K ∈ (a , b).

Proof: We only need to prove the result for a mean-preserving spread. FromCorollary 1, if a and b are the left and right borders of the support of the mean-preserving spread then, the mean-preserving spread does not change the valueof any option with a strike price K /∈ (a , b). Thus, we need only prove that themean-preserving spread strictly increases the value of an option with any strikeprice K ∈ (a , b). From Lemma 1, this is equivalent to the statement that for all



x ∈ (a , b),∫ x

0 S(y ) dy > 0, where S(x) = G(x) − F (x) denotes the change in thecumulative probability distribution function. But as a is the left border of thesupport of the risk change, we have for all x ∈ (0, a), S(x) = 0. Thus, we needonly prove that for all x ∈ (a , b),

∫ xa S(y ) dy > 0. By contradiction, suppose this

is not true, that is, for some x0 ∈ (a , b),∫ x0

a G(y ) dy − ∫ x0

a F (y ) dy = 0.As S(x) = G(x) − F (x) is a mean-preserving spread, it is well known that

G(x) crosses F (x) once from above.4 As a and b are the left and right bordersof the support of the mean-preserving spread, the crossing must happen atx1 ∈ (a , b). But as a is the left border of the support of the mean-preservingspread, for arbitrarily small ε > 0, we must have G(a + ε) > F (a + ε). This and∫ x0

a G(y ) dy − ∫ x0

a F (y ) dy = 0 imply that x0 > x1. In the meantime, as b is theright border of the support of the mean-preserving spread, for arbitrarily smallε > 0, we must have G(b − ε) < F (b − ε). Thus, we must have

∫ ba G(y ) dy −∫ b

a F (y ) dy < 0. From Lemma 1, this implies that the mean-preserving spreadreduces the value of the option with strike price b, which contradicts the well-known result that a mean-preserving spread never reduces option values. �

We now present a more general result.

Theorem 2: Let (a , b) ⊂ (0, ∞). The following two statements are equivalent.

1. A mean-preserving change in risk S(x) strictly increases the values of optionswith strike prices in (a , b), while it preserves the values of all other options.

2. A change in risk S(x) is the sum of a sequence of mean-preserving spreads withai and bi being the left and right borders of their supports, i = 1, 2, . . ., suchthat (a1, b1) ∪ (a2, b2) ∪ · · · = (a , b).

Proof: The result that Statement 2 ⇒ Statement 1 is an immediate conse-quence of Theorem 1. Thus, we only need to prove the converse. As the mean-preserving change in risk S(x) strictly increases the values of options withstrike prices in (a , b), while it preserves the values of all other options, fromthe second result of Lemma 1, we must have for all x ∈ (a , b),

∫ x0 S(x) dx > 0

and for all x /∈ (a , b),∫ x

0 S(x) dx = 0. Thus, S(x) is a mean-preserving increasein risk with support [a , b]. Applying Machina and Pratt’s (1997) main re-sult, we conclude that there exists a sequence of mean-preserving spreads,Si (x), i = 1, 2, . . ., such that �i=1,2,...Si (x) = S(x).5 Let ai and bi be the left andright borders of Si (x), i = 1, 2, . . .. Obviously we must have infi=1,2,... ai = aand supi=1,2,... bi = b; otherwise we cannot have �i=1,2,...Si (x) = S(x). It followsthat ∪i=1,2,...(ai , bi ) ⊂ (a , b). We now prove (a , b) ⊂ ∪i=1,2,...(ai , bi ). By con-tradiction, suppose this is not true. In this case, there must be a K ∈ (a , b)

4See, for example, Machina and Pratt (1997).5See Theorem 1 in Machina and Pratt’s (1997).


6 Huang and Zhang

such that for all i, K /∈ (ai , bi ). From Theorem 1, this implies that for alli, Si (x) does not change the value of an option with strike price K . FromLemma 2, this further implies that for all i,

∫ K0 Si (x) dx = 0. It follows that∫ K

0 S(x) dx = �i=1,2,...

∫ K0 Si (x) dx = 0. From the second result of Lemma 1,

this implies that S(x) preserves the value of the option with strike price K ,which contradicts Statement 1. This proves Statement 1 ⇒ Statement 2. �

3. COMPARISON WITH RASMUSEN’S(2007) RESULTS

In this section, we compare our results with those given by Rasmusen (2007).We first explain Rasmusen’s results. Rasmusen restricted his discussion to dis-crete probability distributions. Let an asset have terminal value xi with probabil-ity f (xi ), where the values of xi with positive probability are x1 < x2 < ... < xm.In addition to the two well-known definitions of increase in risk, namely, amean-preserving spread and Rothschild and Stiglitz increase in risk, he pro-posed two more definitions of changes in risk which are called pointwise riskierand extremum riskier, respectively. The following are the two definitions.

Definition 2 (Rasmusen [2007]): A distribution g (x) is said to be pointwise riskierthan distribution f (x) if s (x) ≡ g (x) − f (x) is a mean-preserving spread whosesupport contains the support of the original distribution f (x).

Definition 3 (Rasmusen [2007]): A distribution g (x) is said to be extremum riskierthan distribution f (x) if the cumulative distribution G(x) is strictly larger thanF (x) at the immediate right of the left border of the support of f (x) and is strictlysmaller than F (x) at the immediate left of the right border of the support of f (x).

Rasmusen’s Proposition 1 states that a mean-preserving increase in riskdoes not reduce any option values, which is Merton’s (1973) Theorem 8.6 HisProposition 1a states that a mean-preserving increase in risk strictly increasesall option values, which he proved to be false. His Proposition 1b states thata mean-preserving increase in risk strictly increases some option values andreduces no option values. These two results are relatively straightforward. Thus,we focus on his main results, namely, Propositions 2, 3, 3a, and 4.

3.1. Rasmusen’s Proposition 2

In his Proposition 2, Rasmusen stated that a mean-preserving change in riskstrictly increases all option values only if the probability distribution becomes

6It is also part of Jagannathan’s (1984) main theorem.



extremum riskier after the risk change. This proposition is, however, not true.We give the following counterexample.

Example 1: Let the original probability distribution f (x) be a two-point distribu-tion as follows: f (4) = 0.5 and f (8) = 0.5. Let the change in risk be as follows:s (x) = s1(x) + s2(x), where s1(x) is defined as s1(1) = 0.1, s1(4) = −0.2, s1(7) =0.1 and s2(x) is defined as s2(6) = 0.1, s2(8) = −0.2, s2(10) = 0.1.

We have s1(1) + s1(4) + s1(7) = 0, 1 × s1(1) + 4 × s1(4) + 7 × s1(7) = 0,and s1(1) > 0, s1(4) < 0, s1(7) > 0. We also have s2(6) + s2(8) + s2(10) =0, 6 × s2(6) + 8 × s2(8) + 10 × s2(10) = 0, and s2(6) > 0, s2(8) < 0, s2(10) >

0. Hence both s1(x) and s2(x) are mean-preserving spreads. Thus, it is straight-forward that g (x) = f (x) + s (x) is riskier than f (x) in Rothschild and Stiglitz’ssense. Moreover, we have (1, 7) ∪ (6, 10) = (1, 10), where 1 and 7 are the leftand right borders of the support of s1(x), respectively, whereas 6 and 10 are theleft and right borders of the support of s2(x), respectively. Applying Theorem 2,we conclude that s (x) strictly increases the values of all options with strike pricesin (1, 10). But as s1(1) + s1(4) = 0.1 − 0.2 = −0.1, for any ε ∈ (0, 1), we haveG(5 + ε) − F (5 + ε) = −0.1 < 0, which contradicts the necessary condition inRasmusen’s Proposition 2 that for arbitrary small ε > 0, G(5 + ε) > F (5 + ε).

However, the proposition will be right for discrete distributions with afinite number of atoms if we slightly change the definition of extremum riskieras follows.

Definition 3’: A distribution g (x) is said to be extremum riskier than distributionf (x) if the cumulative distribution G(x) is strictly larger than F (x) at the im-mediate right of the left border of the combined support of f (x) and g (x) and isstrictly smaller than F (x) at the immediate left of the right border of the combinedsupport.

We now verify that with the above new definition, Rasmusen’s Proposition2 is true for discrete distributions with a finite number of atoms. As the riskchange s (x) strictly increases all option values, from Theorem 2, the supportof the risk change must have the same left and right borders as the combinedsupport of f (x) and g (x). As both distributions have a finite number of atoms,this implies that s (x1) �= 0 and s (xm) �= 0, where x1 and xm are the left andright borders of the combined support of f (x) and g (x), respectively. If s (x1) <

0 then there exists arbitrarily small ε > 0 such that G(x1 + ε) < F (x1 + ε),which implies that for all K ∈ (x1, x1 + ε),

∫ Kx1

G(S) d S <∫ K

x1G(S) d S. This and

Lemma 1 imply that s (x) strictly decrease the values of options with strike pricesin (x1, x1 + ε), which causes a contradiction. Thus, we must have s (x1) > 0.Similarly, we can show that s (xm) > 0. It immediately follows that for arbitrarilysmall ε > 0, G(x1 + ε) > F (x1 + ε) and G(xm − ε) < F (xm − ε).


8 Huang and Zhang

Even with the new definition of extremum riskier, Rasmusen’s Proposition2 is not true for discrete distributions with an infinite number of atoms orcontinuous distributions. A counterexample can be constructed in the samespirit as the Cantor distribution; but it is omitted for brevity.


In his Proposition 3, Rasmusen stated that a mean-preserving change in riskincreases the values of all options if (1) g (x) is extremum riskier than f (x) and(2) g (x) is riskier than f (x). Here we give the following counterexample to showthat this proposition is not true.

Example 2: Let the original probability distribution f (x) be a four-point distri-bution as follows: f (2) = f (3) = f (7) = f (8) = 0.25. Let the change in risk beas follows: s (x) = s1(x) + s2(x), where s1(x) is defined as s1(2) = 0.05, s1(3) =−0.1, s1(4) = 0.05 and s2(x) is defined as s2(6) = 0.05, s2(7) = −0.1, s2(8) =0.05.

We have s1(2) + s1(3) + s1(4) = 0, 2 × s1(2) + 3 × s1(3) + 4 × s1(4) = 0,and s1(2) > 0, s1(3) < 0, s1(4) > 0. We also have s2(6) + s2(7) + s2(8) = 0, 6 ×s2(6) + 7 × s2(7) + 8 × s2(8) = 0, and s2(6) > 0, s2(7) < 0, s2(8) > 0. Henceboth s1(x) and s2(x) are mean-preserving spreads. Thus, it is straightforwardthat g (x) = f (x) + s (x) is riskier than f (x) in Rothschild and Stiglitz’s sense.Moreover, we have g (2) = 0.3, g (3) = 0.15, g (4) = 0.05, g (6) = 0.05, g (7) =0.15, g (8) = 0.3. It is straightforward that for sufficiently small ε > 0, G(2 +ε) = 0.3 > 0.25 = F (2 + ε) and G(8 − ε) = 0.7 < 0.75 = F (8 − ε). Hence ac-cording to Rasmusen’s definition, g (x) is extremum riskier than f (x). Thus,both conditions in Rasmusen’s Proposition 3 are satisfied, and according to thisproposition, the change of risk s (x) should increase the values of all optionswith strike prices in (2, 8).

Now consider a call option with strike price K ∈ [4, 6]. We have K /∈ (2, 4),where 2 and 4 are the left and right borders of the mean-preserving spread, s1(x);thus according to Corollary 1, s1(x) does not change the value of this option.Moreover, we have K /∈ (6, 8), where 6 and 8 are the left and right borders ofthe mean-preserving spread, s2(x); thus, according to Corollary 1, s2(x) doesnot change the value of this option either. Hence s (x) = s1(x) + s2(x) preservesthe values of options with strike prices in [4, 6]. This shows that Rasmusen’sProposition 3 is not true.

We can apply Theorem 2 to explain the error in Rasmusen’s Proposition 3.One of the two conditions in Rasmusen’s Proposition 3 requires that the riskchange S(x) = G(x) − F (x) is a mean-preserving increase in risk. Thus, thereexists a sequence of mean-preserving spreads Si (x) with ai and bi being the left



and right borders of their supports, i = 1, 2, . . ., such that S1(x) + S1(x) + · · · =S(x). But the other condition in Rasmusen’s Proposition 3, which requires thatg (x) is extreme riskier than f (x), cannot guarantee that (a1, b1) ∪ (a2, b2) ∪ · ·· = (a , b). This is exactly the case in the above counterexample. As was shownabove, in this example the risk change s (x) is the sum of two mean-preservingspreads, namely, s1(x) and s2(x) with left borders a1 = 2 and a2 = 6 and rightborders b1 = 4 and b2 = 8, respectively. It is clear that (a1, b1) ∪ (a2, b2) =(2, 4) ∪ (6, 8) �= (a , b) = (2, 8), where a and b are the left and right borders ofthe combined support of f (x) and g (x), respectively. And the mean-preservingincrease in risk s (x) fails to increase the values of the options with strike pricesin [(a , b) − (a1, b1) ∪ (a2, b2)] = [4, 6].

3.3. Rasmusen’s Proposition 3a

Rasmusen’s Proposition 3a states that if a mean-preserving change in riskdoes not reduce any option values then it must be a mean-preserving in-crease in risk. This proposition is actually part of Jagannathan’s (1984) mainresult.7 Rasmusen presented a counterexample to show that the Proposition3a is false. In the following, we explain the error in his proof of the falsity ofProposition 3a.

In his counterexample, f (x) is a four-point distribution: f (2) = f (4) =f (6) = f (8) = 0.25, and g (x) is a three-point distribution: g (1) = 0.3, g (5) =0.4, g (9) = 0.3. In the following, we show that contrary to Rasmusen’s claim,the risk change s (x) = g (x) − f (x) is a mean-preserving increase in risk.

Let s1(x), s2(x), and s3(x) be as follow: s1(1) = 0.25, s1(2) = −0.25, s1(4) =−0.25, s1(5) = 0.25; s2(5) = 0.25, s2(6) = −0.25, s2(8) = −0.25, s2(9) =0.25; s3(1) = 0.05, s3(5) = −0.1, s3(9) = 0.05. We have s1(1) + s1(2) + s1(4) +s1(5) = 0, 1 × s1(1) + 2 × s1(2) + 4 × s1(4) + 5 × s1(5) = 0, and s1(1) > 0,s1(2) < 0, s1(4) < 0, s1(5) > 0; s2(5) + s2(6) + s2(8) + s2(9) = 0, 5 × s2(5) +6 × s2(6) + 8 × s2(8) + 9 × s2(9) = 0, and s2(5) > 0, s2(6) < 0, s2(8) < 0,s2(9) > 0; s3(1) + s3(5) + s3(9) = 0, 1 × s3(1) + 5 × s3(5) + 9 × s3(9) = 0, ands3(1) > 0, s3(5) < 0, s3(9) > 0. Hence s1(x), s2(x), and s3(x) are all mean-preserving spreads. It is not difficult to see that g (x) = f (x) + s1(x) + s2(x) +s3(x). Thus, it is straightforward that g (x) is riskier than f (x) in Rothschild andStiglitz’s sense, and this example is not a counterexample to Proposition 3a.


We now discuss Rasmusen’s Proposition 4. In his Proposition 4, he stated thata change in risk strictly increases the values of all options if the probability

7See the main theorem in Jagannathan (1984).


10 Huang and Zhang

distribution becomes pointwise riskier after the change, that is, the risk changeis a mean-preserving spread whose support contains the support of the originaldistribution. If the support of a mean-preserving spread contains the supportof the original distribution, and if we only consider options with strike pricesin the combined support of the original and the changed distributions, thenfrom Theorem 1, the values of all these options are strictly increased. Thus, thisproposition can be obtained as a corollary of the above theorem.

4. CONCLUSIONS

It is well known that a mean-preserving increase in risk does not reduce optionvalues; however, as was pointed out by Rasmusen (2007), such an increasein risk cannot guarantee an increase in option values. In this study, we havepresented a necessary and sufficient condition for a mean-preserving spreadto strictly increase an option’s value. We have also presented a necessary andsufficient condition for a risk change to strictly increase the values of optionswith strike prices in an open interval, while preserving the values of all otheroptions. These two results significantly improve Rasmusen’s (2007) answer tothe question when extra risk strictly increases an option’s value.

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Franke, G., Stapleton, R. C., & Subrahmanyam, M. G. (1999). When are options over-priced: The Black-Scholes model and alternative characterizations of the pricingkernel. European Finance Review, 3, 79–102.

Gollier, C. (1995). The comparative statics of changes in risk revisited. The Journal ofEconomic Theory, 66, 522–535.

Grundy, B. D. (1991). Option prices and the underlying asset’s return distribution.Journal of Finance, 46, 1045–1070.

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Some New Results on When Extra Risk Strictly Increases an Option's Value

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