Time at risk

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Time at Risk (TaR) is a time-based risk measure designed for corporate finance practice.

In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.


TaR represents certain quantile for a given probability distribution, so is similar to Value at Risk (VaR). [1] However, TaR measures risk amount as time(time until an adverse event) rather than value (loss amount).

Quantile cutpoint dividing a set of observations into equal sized groups

In statistics and probability quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one less quantile than the number of groups created. Thus quartiles are the three cut points that will divide a dataset into four equal-sized groups. Common quantiles have special names: for instance quartile, decile. The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.

Definition and examples

Mathematical definition of TaR is same as that of VaR. [2]

However, value-based random variable is replaced with time-based one, and given time-horizon is replaced with given finance structure.

Examples comparing VaR and TaR are as below.

This means it is 90% probability that insurance claim payout would be below 10 million dollars; so if the insurer has accumulated 10 million dollars in cash, it would be 90% safe.
This means it is 90% probability that net liquid assets(= liquid assets - volatile liabilities) would not be run out within 3 years; so for 3 years, the insurer under current finance structure would be 90% safe.

For confidence level α,

Thus for same α, lower VaR means lower risk and higher TaR means lower risk.


TaR is a simple measure for whom are familiar with VaR, so is easy to communicate by. TaR also can be used for supplementary purpose to VaR analysis.

Applying TaR in financial models, practitioners can analyze sources of risks and take remedial actions in corporate finance planning; not only for liquidity risk mentioned above, but also for any risks that demands time-based analysis.

When TaR is applied to a household's financial planning it can measure longevity risk, and TaR in this case is referred to as Age at Risk (AaR).

A longevity risk is any potential risk attached to the increasing life expectancy of pensioners and policy holders, which can eventually result in higher pay-out ratios than expected for many pension funds and insurance companies.

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The Liquidity-at-Risk is a quantity to measure financial risks and is the maximum net liquidity drain relative to the expected liquidity position which should not be exceeded at a given confidence level. The LaR is analog to the Value-at-Risk (VaR) where a quantile of the EBIT-distribution is considered, however it does take stochastic cash flows into account.


  1. Jorion, Philippe (2007). Value at risk : the new benchmark for managing financial risk (3. ed.). New York [u.a.]: McGraw-Hill. ISBN   978-0-07-146495-6.
  2. Embrechts, Alexander J. McNeil, Rüdiger Frey, Paul (2005). Quantitative risk management : concepts, techniques and tools. Princeton, N.J.: Princeton University Press. ISBN   978-0-691-12255-7.

See also