Security equivalent

The individual security equivalent (SÄ or CE, English certainty equivalent ) of an insecure or fortuitous asset , for example a securities account or savings account, denotes that secure , i.e., that is , in financial mathematics and decision theory . H. non-random amount , the benefit of which for the person concerned is equivalent to the expected benefit of the uncertain asset , in other words: the safe payment, for example immediately and in cash, whose “perceived” or subjective benefit for the person concerned is the same expected benefits of uncertain assets : ${\ displaystyle w}$ ${\ displaystyle CE}$ ${\ displaystyle E (u (w))}$ ${\ displaystyle w}$

{\ displaystyle u (CE) = E (u (w))}

.

Accordingly, the value of depends directly on the individual utility function of the person concerned, whereby in principle three cases can be distinguished: ${\ displaystyle CE}$ ${\ displaystyle u (w)}$

CE <E (w) - risk aversion or risk aversion : The individual security equivalent of the uncertain asset w is below its mathematical expected value.

CE = E (w) - Risk neutrality : The individual security equivalent of the uncertain asset w corresponds exactly to its mathematical expected value.

CE> E (w) - risk affinity or risk appetite : The individual security equivalent of the uncertain asset w is above its mathematical expected value.

Formal description

Utility function (left) and inverse utility function (right) of a risk-averse (risk-averse) market participant

CE - safety equivalent; E (U (W)) - expected value of utility (expected utility) of uncertain assets; E (W) - expected value of uncertain assets; U (CE) - utility of the safety equivalent; U (E (W)) - utility of the expected value of the uncertain asset; W ₀ - minimum wealth; U (W ₀ ) - utility of minimum wealth; W ₁ - maximum wealth; U (W ₁ ) - utility of maximum wealth; U ₀ - minimal utility; W ₀ - assets required to achieve the minimum utility; U ₁ - maximum benefit; W ₁ - assets required to achieve maximum benefit; RP - risk premium

Let there be a real, measurable and reversible utility function together with its inverse as well as an uncertain capacity , composed of a certain initial capacity and a random variable with the expected value . The following then applies to the expected value of the uncertain assets : ${\ displaystyle u (w)}$ ${\ displaystyle w (u)}$ ${\ displaystyle x}$ ${\ displaystyle {\ bar {x}} \ in \ mathbb {R}}$ ${\ displaystyle X: \ Omega \ rightarrow \ mathbb {R}}$ ${\ displaystyle E (X) = 0}$ ${\ displaystyle x = {\ bar {x}} + X}$

{\ displaystyle E (x) = E ({\ bar {x}} + X) = {\ bar {x}}.}

Is the equation

{\ displaystyle u (CE) = E (u (x)) = E (u ({\ bar {x}} + X))}

uniquely solvable, the real number thus defined is called the security equivalent of uncertain property . ${\ displaystyle CE ({\ bar {x}}, X)}$ ${\ displaystyle x = {\ bar {x}} + X}$

Is the utility function reversible as required, e.g. B. increasing strictly monotonically, the security equivalent of uncertain assets can be calculated using the inverse utility function as follows: ${\ displaystyle u (w)}$ ${\ displaystyle x = {\ bar {x}} + X}$ ${\ displaystyle w (u)}$

{\ displaystyle CE ({\ bar {x}}, X) = w (E (u (x)))}

Risk premium

The difference between the expected value of uncertain assets and the individual security equivalent of the market participant is called the risk premium : ${\ displaystyle E (x)}$ ${\ displaystyle CE}$ ${\ displaystyle RP}$

{\ displaystyle RP ({\ bar {x}}, X) = E (x) -CE ({\ bar {x}}, X) = E (x) -w (E (u (x)))}

example

The average win of a fair lottery ticket is 50 cents - for someone who knows the value of the ticket "sober", ie. H. Judging by its mathematical expected value alone , this lot will be worth exactly 50 cents. On the other hand, a risk-averse player might prefer in this case, e.g. B. to collect 40 cents immediately and "cash on hand" instead of participating in the lottery yourself. He thus sells the lot for this value. Thus he grants the buyer (together with his risk of loss) at the same time a "risk premium" of an average of 10 cents per lot.

Conversely, in this case a risk- taking player might prefer to talk to someone else, e.g. B. to pay 60 cents immediately and "in cash" just to be able to participate in the lottery (and thus the chances of winning).

In other words, one and the same ticket would be worth a maximum of 40 cents in cash to the risk-averse player (because of the possible loss), but at least 60 cents to the risk-loving player (with a view to possible profit), for the "sober", i.e. H. risk-neutral players end up with exactly 50 cents.

It should be noted that the " risk premium " resulting from the security equivalent can also be negative due to its definition as the "margin between expected value and security equivalent", namely if a risk-loving player himself pays a premium for the possibility of taking on the risk is willing to pay for the expected value instead of charging a premium for it. In the example above, if he buys the ticket for 60 cents, although on average is worth only 50 cents, so he , on average, makes 10 cents per lot loss.

Risk averse and risk-related strategies

Risk-averse strategies are relevant in practice compared to risk-neutral strategies, especially with large potential profits.

The reason for this lies in the decreasing marginal utility , i.e. the right curvature of the risk utility function u (w) of risk-averse market participants. To use a vivid example, it would be rather unwise for a penniless market participant to risk a secure payout of 10 million euros for a profit of 30 million euros that can only be expected statistically, even if the average expected increase in wealth of 20 million euros would be twice as high as the loss of assets of 10 million. Because the disadvantage of possibly not owning anything instead of the secure 10 million euros in the end, will usually outweigh the advantage of a penniless market participant than the advantage of getting another 20 million euros.

On the other hand, strategies with an affinity for risk can also be useful if the framework conditions are appropriate. This is particularly the case when the market participant absolutely needs a certain basic amount, which, however, is above the purely arithmetical expected value of the uncertain payment in question.

Example: A penniless market participant discovers a very valuable gemstone at a flea market, which is offered at a price of only 10 euros due to the seller's ignorance. If the market participant were offered a game with a maximum win of 10 euros, while the expected value was only 5 euros, it could still make sense to speculate on the maximum win and ignore the probability of a total loss. Because in view of the gemstone purchase made possible by means of the maximum profit , the risk of a total loss would be secondary for the market participant - his security equivalent of the average expected game win of 5 euros could if he includes the gemstone purchase made possible with the maximum profit of 10 euros in his calculations, Depending on the actual market value of the stone, they are orders of magnitude higher.

literature

Franz Eisenführ, Martin Weber: Rational decision-making. 4th edition. Springer Verlag, Berlin; Heidelberg; New York 2003, ISBN 3-540-44023-2 .

Individual evidence

↑ Oliver Glück: Glossary: Security equivalent
↑ Helmut Laux: Decision Theory ; Springer-Verlag 2005, ISBN 3-540-23576-0 , p. 215 ff.
↑ Peter Kischka: Lecture Statistics II, chap. IV: Introduction to Decision Theory ; Jena, WS 2005/2006, p. 20.
↑ Hans-Markus Callsen-Bracker, Hans Hirth: Risk Management and Capital Market. 1.2 Risk aversion and risk premiums

[1] Oliver Glück: Glossary: Security equivalent

[2] Helmut Laux: Decision Theory ; Springer-Verlag 2005, ISBN 3-540-23576-0 , p. 215 ff.

[3] Peter Kischka: Lecture Statistics II, chap. IV: Introduction to Decision Theory ; Jena, WS 2005/2006, p. 20.

[4] Hans-Markus Callsen-Bracker, Hans Hirth: Risk Management and Capital Market. 1.2 Risk aversion and risk premiums