multiperiod risk premiums - univerzita karlovakopa/mrp.pdf · 2015. 9. 17. · multiperiod risk...

Multiperiod risk premiums

Milos Kopa

Dept. of Probability and Mathematical Statistics, Faculty of Mathematics and Physics,Charles University in Prague, Czech Republic

[email protected]

Milos Kopa Multiperiod risk premiums

Univariate risk premium

The traditional Arrow-Pratt measure of (absolute) risk aversion:Ru(x) = −u′′(x)/u′(x)Risk aversion:

Eu(w + x) < u(w + E (x)) (1)

An investor is risk averse ⇔ Ru(x) > 0The risk averse decision maker is willing to pay a risk premium π toeliminate the risk x .

u(w + E (x)− π(w ,Px)) = Eu(w + x) (2)

π(w ,Px) > 0 ⇔ Ru(x) > 0 ⇔ u is concave


Univariate risk premium with random initial wealth

Euw (w + E (x)− π(w ,Px)) = Eux,w (w + x) (3)

π(w ,Px) > 0 6⇔ Ru(x) > 0


Multivariate risk premium I

Suppose a decision maker with utility function u(w) and initial wealthw = (w1,w2, ...,wn)′.Assume that u(w) is continuous and increasing in all variables.The multivariate risk premium π:

u(w + Ex− π) = Exu(w + x) (4)

The risk aversion at level w:

u(w + Ex) > Exu(w + x) (5)

for any given gamble x.


Multivariate risk premium II

If u is concave than there exists a nonnegative risk premium for anygamble x.If there exists a nonnegative risk premium for any gamble x then aninvestor is risk averse.The absolute risk aversion matrix measure:

R(x) =−uij(x)

ui (x)


Multivariate risk premium with random initial wealth

The decision maker is indifferent between two random vectors: (w − π)and (w + x):

Ewu(w + Ex− π) = Ew,xu(w + x). (6)

The decision maker is risk averse at level w with respect to gamble x if:

Ewu(w + Ex) > Ew,xu(w + x). (7)


Example 1

Let (w1, x1,w2, x2) = ( 12 , 1, 0,−

12 ) or (1,−1, 1

2 ,12 ) with the same

probabilities.Consider u(w) = log(w1 + w2).

Ew,xu(w + x) > Ewu (w + Ex) .

Moreover if (w1, x1,w2, x2) = ( 12 , 1,

12 ,−

32 ) or (1,−1, 0, 3

2 ) with the sameprobabilities then

Ew,xu(w + x) < Ewu (w + Ex) .

The risk premium in the i-th direction is a solution of (6) with theproperty that j-th component of π satisfies

πj = 0 j 6= i

= πi j = i .


Direction risk premiums

Direction risk premiums π1, π2 in Example 1 are:

Ew,xu(w + x) = Ewu(w − π); π = (π1, 0)′

0 =1

2log

(1

2− π1

)+

1

2log

(3

2

)π1 = −1

6

Ew,xu(w + x) = Ewu(w − π); π = (0, π2)′

0 =1

2log

(1

2

)+

1

2log

(3

2− π2

)π2 = −1

2

Concavity of u does not guarantee nonnegativity of directional riskpremiums.


Multiperiod risk premium

Let u(w) be an increasing utility function. We interpret the arguments ofw as the random amounts of cash measured at times 1, . . . , n. It is thevector of initial wealth in each period.We would like to define i-th element of multiperiod risk premium Π suchthat the decision maker is indifferent between accepting the gamble xi

and paying Πi − Ex in i-th time period.The i-th element of multiperiod risk premium depends on the initialwealth at time i and on the probabilistic distribution of x. The initialwealth wi depends on wi−1 and on the decision of investor at time i − 1,whether he accepted gamble xi−1 or paid Πi−1 − Exi−1.We assume that Ex = 0.


Multiperiod problem

Finally, we assume that a history of decisions does not depend on x andall possible histories of decisions are described by the following scenarioswhere an investor has only two possibilities in each time period: toaccept the gamble or to pay risk premium.

〈

accept x1 〈accept x2 〈 . . .

accept xn−1

pay Πn−1

pay Π2 〈 . . .accept xn−1

pay Πn−1

pay Π1 〈accept x2 〈 . . . ...

pay Π2 〈 . . .accept xn−1

pay Πn−1

If the decision maker accepts a gamble in i-th time period then letksi = 1 else ks

i = 0. The scenario s is represented by vector

K s = (ks1 , k

s2 , . . . , k

sn−1)


Multiperiod problem

The initial wealth in j-th time period along scenario s:

w sj = w1 +

j−1∑i=1

[ksi xi − (1− ks

i )Πi ]. (8)

In a formal way, we would like to define multiperiod risk premium by thesystem of equations:

Exu(ws + x) = Exu (ws −Π) ∀s ∈ S . (9)

However, this system of 2n−1 equations and n variables does not usuallyhave a solution unless n ≤ 2.



Given x, letf s(Π) = |Exu(ws + x)− Exu (ws −Π) |

We are interested to find Π which minimizes f s(Π) jointly for all s ∈ S .We are looking for a vector which minimizes the maximal value of f s(Π):

minΠ

maxs∈S

f s(Π)



An equivalent form:

minΠ

d (10)

s.t. f s(Π) ≤ d ∀s ∈ S .

The multiperiod risk premium is a solution of the problem:

minΠ

d (11)

s.t. − d ≤ Exu(ws + x) − Exu (ws −Π) ≤ d ∀s ∈ S .


Multiperiod risk aversion

The decision maker is multiperiod risk averse at wealth level w withrespect to gamble x if

Exu(ws + x) < Exu (ws) ∀s ∈ S . (12)

We define i -th directional multiperiod risk premium Πi as a solution ofthe following problem:

minΠ

d (13)

s.t. − d ≤ Exu(ws + x) − Exu (ws −Π) ≤ d ∀s ∈ S

Πj = 0 j 6= i .


Theorem 1:

If the decision maker is risk averse at wealth level w with respect togamble x then all directional risk premiums are positive.

Sketch of the proof:Choose i ∈ {1, 2, . . . , n}. Let Πs

i be a solution of equation:Exu(ws + x) = Exu (ws −Π) under conditions: Πj = 0 for all j 6= i .Assumption of risk aversion at wealth level w with respect to gamble x isequivalent to positivity of Πs

i for all s ∈ S . Let

Πi = mins∈S

Πsi .


Proof con’t

It is easy to show that

f s(Π) = |Exu(ws + x)− Exu (ws −Π) |

is a decreasing function in variable Πi on (−∞,Πi ) for all s ∈ S underconditions: Πj = 0 for all j 6= i .

Therefore Πi ≥ Πi > 0.


Additional notation

Let A be the set of considered time periods in multiperiod risk premiumconstruction. If i ∈ A then let yi = −Πi else yi = xi . We will denote bySA the subset of S which consist of scenarios with the property:if i ∈ {1, 2, . . . , n} \ A then ks

i = 1.


Partial multiperiod risk premium

The partial multiperiod risk premium ΠA is a solution of the problem:

minΠA

d (14)

s.t. − d ≤ Exu(ws + x) − Exu (ws + y) ≤ d ∀s ∈ SA

yi = −ΠAi i ∈ A

yi = xi i /∈ A.


Example 2

Consider u(w1,w2,w3) = log(w1 + w2 + w3). Let x1, x2, x3 be anindependent random variables: xi = ± 1

2 with the same probabilities,i = 1, 2, 3. Finally, set w1 = 2.It is clear that S consists of four scenarios: s1 ∼ (1, 1), s2 ∼ (1, 0),s3 ∼ (0, 1) and s4 ∼ (0, 0).

The multiperiod risk premium is:

Π = (1.252, 1.27,−2.319) and d∗ = 6.10−4.


Example 2

The direction risk premiums:

Π1 = 0.1367 and d∗ = 0.0124

Π2 = 0.1368 and d∗ = 0.0124

Π3 = 0.1368 and d∗ = 0.0124

The partial multiperiod risk premium:we assume that the insurance possibility does not exist in the secondperiod, i.e. A = {1, 3}. Thus y = (−Π1, x2,−Π3).

Π1 = 1.638 and Π3 = −1.5


Modification for random initial wealth

We define the multiperiod risk premium for random w1 as a solution ofthe problem:

minΠ

d

s.t. − d ≤ Ex,w1u(ws + x) − Ex,w1u (ws −Π) ≤ d ∀s ∈ S .


References

M. Kopa: Risk premiums for multiperiod risks, Bulletin of the CzechEconometric Society, 21 (2004), 67 - 79.

R. Ambarish, J. G. Kallberg: Multivariate risk premiums, Theory andDecision 22 (1987), 77-96.

J. A. Chalfant, I. Finkelshtain: Multivariate risk premiums, PortfolioChoices in the Presence of Other Risks, Management Science 39, 8,(1993), 925-936.

G. T. Duncan: A matrix measure of multivariate local risk aversion,Econometrica, 45 (1977), 895-903.

R. E. Kihlstrom, L. J. Mirman: Risk aversion with many commodities,Journal of Economic Theory 8 (1974), 361-388.


References con’t

R. E. Kihlstrom, D. Romer, S. Williams: Risk aversion with randominitial wealth, Econometrica, 49, 4, (1981), 911-920.

J. W. Pratt: Risk aversion in the small and in the large, Econometrica 32(1964), 122-136.

S. F. Richard: Multivariate risk aversion, utility dependence andseparable utility functions, Management Science 21 (1975), 12-21.

S. A. Ross: Some stronger measures of risk aversion in the small and thelarge with applications, Econometrica, 49, 3, (1981), 621-638.


multiperiod risk premiums - univerzita karlovakopa/mrp.pdf · 2015. 9. 17. · multiperiod risk...

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