Abstract
Time unit invariance is introduced as an additional requirement for multiperiod risk measures: for a constant portfolio under an i.i.d. risk factor process, the multiperiod risk should equal the one period risk of the aggregated loss, for an appropriate choice of parameters and independent of the portfolio and its distribution. Multiperiod Maximum Loss over a sequence of Kullback–Leibler balls is time unit invariant. This is also the case for the entropic risk measure. On the other hand, multiperiod Value at Risk and multiperiod Expected Shortfall are not time unit invariant.
Highlights
Risk measures for stochastic processes have been a popular research topic over the last decade
If the correct parameter value does not depend on the underlying loss distribution, we say that the risk measure is time unit invariant
The same holds for T periods and if the portfolio loss at time t is given by a linear function of n risk factors, Lt = lt ·, which are independent and normally distributed with mean μt and covariance ma√trix period Maximum Loss for radius k equals 2k t, rt ∼ Pt = N
Summary
Risk measures for stochastic processes have been a popular research topic over the last decade. Time unit invariance is introduced as an additional requirement for multiperiod risk measures: for a constant portfolio under an i.i.d. risk factor process, the multiperiod risk should equal the one period risk of the aggregated loss, for an appropriate choice of parameters and independent of the portfolio and its distribution. If the correct parameter value does not depend on the underlying loss distribution, we say that the risk measure is time unit invariant.
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