Risk premium

The risk premium (RP, English riskpremium ), depending on the sign also called risk premium or risk premium referred to in financial mathematics and decision theory, the difference between the mathematical expectation of an uncertain assets E ( w), such as security ( Lottery Ticket, stock, bond, savings account ), and the individual certainty equivalent ( CE, English certainty equivalent) of such property, that is, those secure payment CE, for example, immediately and in cash, the subjectively promises to the person concerned the same benefits (and thus has equal value ) as the uncertain wealth w:

Decisive for the magnitude and sign of the risk premium RP is therefore primarily the relationship between the a and w are the same assets always equal the mathematical expectation E ( W) and the individual certainty equivalent CE of the operator shall:

  • If E ( w)> CE, the risk premium RP is positive, ie the person is ready, the person who relieves him of the risk of uncertain assets ( and thus the risk of a potentially real property loss ), but to pay a premium. The best known example of such transactions are insurance contracts in which one denotes the risk premium RP as an insurance premium. Market participants whose certainty equivalent CE usually smaller than the expected value E (w) is its uncertain assets are risk-averse or risk-averse called. Decisive for risk-averse decisions is the higher weighting of possible loss of assets from possible wealth gains.
  • If E (w) = CE, the risk premium RP is equal to zero, ie the person is not willing to pay someone else a premium for the acquisition of own assets risk nor vice versa abzukaufen someone else whose assets risk. Market participants whose certainty equivalent CE is usually the expected value E ( w) of their insecure assets covers, are called risk neutral. Decisive for risk-neutral decisions is the equal weighting of possible asset losses and gains.
  • If E ( w) < CE, the risk premium RP is negative, ie the person is now reversed willing that which he risk his uncertain assets (and thus the prospect of a potentially real asset profit) assigns, to pay for a premium. The best known example of such transactions are virtually all real, ie mathematically considered always "unfair " lotteries the ticket price this is regularly above its expectation E (L). Market participants whose certainty equivalent CE is greater usually than the expected value E ( w ) of its assets is uncertain, risikoaffin be called risk- loving or. Decisive for risk- affine decisions is the higher weighting of possible capital gains against potential losses assets.
  • 2.1 Example 1
  • 2.2 Example 2
  • 3.1 Example 1
  • 3.2 Example 2
  • 4.1 Example 1
  • 4.2 Example 2
  • 5.1 Example 1
  • 5.2 Example 2

Formal Description

Given a real, measurable and reversible utility function u ( w ), together with its inverse w ( u ) and an insecure assets x, composed of a safe output power and a random variable with the expected value E ( X ) = 0 for the expected value of the uncertain assets then:

If the equation

Uniquely solvable, we call the real number defined by the risk premium (or the certainty equivalent of the random variable X) for a given output power.

If the utility function u ( w) as required reversible, eg, strictly increasing, the risk premium by means of the inverse utility function u ( w) can be calculated as follows:

Interpretation

  • The positive risk premium is the discount that a risk-averse decision-maker ( with True for him concave utility function ) willing to accept in order to avoid the risk of a random variable X with a fixed -average earnings.
  • The negative risk premium is the awarding to a risikoaffiner decision maker is willing to pay ( with True for him convex utility function ) in order to take on the additional risk of the random variable X with a fixed average yield may.

Risk premium and Arrow -Pratt measure of absolute risk aversion

As John W. Pratt showed in 1964, the risk premium can ( the minimum required risk premium ) for small values ​​of the variance and the expected value for any continuously differentiable utility functions as follows be approximated:

Examples

It should be thrown a coin, and you get depending on the outcome of the coin toss either a payout of € 1.00 or nothing. The expected value E ( w) would therefore be € 0.50, the price of a lot in fairness the lottery also € 0.50.

  • If the player draws it before now, to leave in place the uncertain dividend pay out an amount < € 0.50 in cash, eg his own Los someone else to sell for such lesser amount, he is risk-averse or risk-averse called of that and the risk premium, which buys him the lot is positive (it is statistically make a profit ).
  • Sale of the players on the other hand someone else his lot for just € 0.50, it is thus itself a draw ( indifferent ) if he is to participate in the lottery or not, it is called risk- neutral, and the risk premium of the person who buys him the lot, remains zero (it is statistically neither a profit nor loss making ).
  • If the player is finally only willing to sell his lot to someone else, if this for an amount > € 0.50 paid him on the spot, such a player is risk- loving or risikoaffin called, and that of the risk premium, which buys him the lot is negative (it is statistically make a loss ).

Depending on the risk premium on the risk type

Example 1

A risk-averse player with the risk utility function and its inverse function will take part in a raffle in which the chances for a top prize of 2500 € at 1%, representing a consolation price of only 25 € on the other hand, in the remaining 99%.

The expected value of the uncertain wealth w and the expected benefit when participating in the raffle are thus:

Certainty equivalent of the uncertain assets w and risk premium are calculated so that the raffle for the player as follows:

The risk-averse players would be willing to pay a maximum of € 29.70 for one lot or, conversely, for € 29.70 (or more) to resell, and the buyer would make a profit of 20.05 € on average, since the average yield lot yes, as shown, is € 49.75.

Example 2

A risk-averse player with the risk utility function and its inverse function will participate in the same raffle in which the chances for a top prize of 2500 € again at 1%, representing a consolation price of only 25 € on the other hand, in the remaining 99%.

The expected value of the uncertain wealth w and the expected benefit when participating in the raffle are thus:

Certainty equivalent of the uncertain assets w and risk premium calculated for the player so that now as follows:

The risk-loving players would be willing to pay a maximum of € 251.23 for one lot or, conversely, for € 251.23 (or more) to resell, and the buyer would make a loss of € 201.48 on average, since the average yield lot yes, as shown, is only at € 49.75.

Depending on the risk premium from the output capacity

The location of the influent into the formula for the risk premium expected value of the uncertain assets w is among other things determined by the initial wealth w0.

Example 1

A risk-averse player with the risk utility function and its inverse function possesses only a lottery ticket on the with a probability p = 0.5 is a gain of 7 € paid, its output capacity w0, however, was equal to zero.

The expected value of the uncertain assets w = w0 L and the expected benefit when participating in the lottery are thus:

Certainty equivalent of the uncertain assets w = w0 L = L and risk premium are calculated so the player as follows:

As can be seen, the lottery ticket the penniless players would be € 1.75 worth less than that corresponding to the purely computational value: Although the Los promises a profit of € 3.50 on average, the destitute players would be prepared already for € 1.75, the Los someone else or sell it yourself to buy for no more than this € 1.75, as the risk of total loss of the stake in this case outweighs the prospect of gain.

Example 2

Another risk-averse players with the same risk utility function and its inverse function also possess the same again lottery ticket on that with a probability p = 0.5 is a gain of 7 € paid, but now a safe output capacity w0 of 9 €.

The expected value of the uncertain assets w = w0 L and the expected benefit when participating in the lottery are thus:

Certainty equivalent of the uncertain assets w = w0 L and risk premium are calculated so the player as follows:

As can be seen, the same lottery ticket would be the " wealthy " players only € 0.25 worth less than that corresponding to the purely computational value: Although the Los promises a profit of € 3.50 on average, which would be " wealthy " players due to its risk aversion but also only willing to spend even € 3.25 for it and it already for € 3.25 (or more) resell.

Depending on the risk premium of the profit margin

Another factor that influences the position of the influent in the risk premium formula expected value of the uncertain assets w, is the span of the measure in view of profit.

Example 1

A risk-averse players will participate in the final round of a TV show in which the players must eventually decide between two doors behind which nothing once, the other times are hidden € 1,600. Alternatively, each player but also has the ability, instead of having to choose between the doors instantly receive € 800 in cash as a consolation prize. Both these cash as well as playing with the doors so have the same mathematical expectation value of 800 €. An absorbed. a risk-averse players, however, is always the secure € 800 - risk-neutral players, the risk to choose the wrong door, it would not matter, would now draw ( indifferent ), whether to opt for the game with the doors or secure cash prefer.

Supposing are the risk utility function of the risk-averse goal and its inverse function and can be calculated expected value of the gain at the doors rates w = T and the expected benefits thereof, as follows:

Certainty equivalent and risk premium of doors - counseling are then obtained as follows:

As can be seen, for risk-averse players with a risk utility function as the above no reason to opt for the doors - rates: The "perceived" benefit of the expected average game profit of 800 € is just the same as a secure instant payment from € 400, far lower than that offered by the emcee alternative of 800 €.

Example 2

Had the show master now dealing only with players of such ( and most people are risk averse ), the show would soon end. One of the ways the players yet to move to accommodate the risk could be the doubling of profits from 1600 to 3200 € in view of this, and thus his expected value from 800 to 1600 €:

Certainty equivalent and risk premium of doors - double guessing also:

In the new situation, it would be still not clear whether the players actually end up deciding for the average profit of the doors - guessing by now € 1,600 or rather for the safe withdrawal of 800 €, as their " perceived " benefit to now just keep the balance with the security equivalent of the expected profit rate e ( T). Definitely in favor of the doors - guessing the tide would not therefore apply only for gains > € 3200.

Depending on the risk premium on the course of individual risk utility function

Other than the location of the expected value and variance of the uncertain assets w plays the course of the risk utility function u ( w ) itself, including its slope and / or curvature of their behavior, a crucial role in determining the risk premium.

Example 1

A risk-averse market participants with an accumulated capacity of 100,000 € learns from his doctor that he carried a disease whose treatment costs are not covered by his health insurance, if the disease were to break with him, and at worst can lose 90 % of its assets, albeit with a probability of 1:10. The choice, he is faced with this is:

The expected value E ( w) of the uncertain assets w of the operator is thus calculated, taking into account the above starting values ​​and probabilities, as follows:

Everything now depends more on the individual utility function of the operator - is it a risk-averse market participants with one of the two adjacent utility functions or, the following scenarios would be possible, for example:

As can be seen, the certainty equivalent of the uncertain assets is for the market participants in the first case again € 4,212 below the expected value of its assets in the amount of € 91,000 - he would be, where applicable, a total of up to € 13,212 for the prevention of disease cost risk ( in spend of € 9,000 ). In the second case, the certainty equivalent of the operator is even deeper - the price of insurance here could increase to up to € 28,460 due to the risk aversion of the insured, of which € 19,460, the average net premium of the insurer for this would be that he the insured whose illness cost risk ( in the amount 9,000 € ) decreases.

Example 2

A risk-seeking market participants with an accumulated assets of € 10,000 will be offered to participate in a risk bet where he could increase tenfold its assets, even if only with a probability of 1:10. The choice, he is faced with this is:

The expected value E ( w) of the uncertain assets w of the operator is thus calculated, taking into account the above starting values ​​and probabilities, as follows:

Everything now depends more on the individual utility function of the operator - is it a risk-seeking market participants with one of the two adjacent utility functions or, the following scenarios would be possible, for example:

As can be seen, the certainty equivalent of the uncertain asset is w of the operator in the first case again about € 14,015 over the expected average assets of 19,000 € itself - the market participants would be necessary, ready to spend up to € 33,015 for the chance to be tenfold assets. In the second case, however, the certainty equivalent is only about € 8,085 higher than the expected value - in this case, therefore, the price of the betting slip could only be a maximum of € 27,085, of which € 8,085, the average net premium of the Sweepstakes organizer of this would he the player the chance to win ( in the amount 90,000 € ) grants.

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