Discounted maximum loss is the present value

Present value

Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

 of the worst case scenario for a financial portfolio

Portfolio (finance)

Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...

.

An investor must consider all possible alternatives for the value of his investment. How he weights the different alternatives is a matter of preference. One might require a pension fund

Pension fund

A pension fund is any plan, fund, or scheme which provides retirement income.Pension funds are important shareholders of listed and private companies. They are especially important to the stock market where large institutional investors dominate. The largest 300 pension funds collectively hold...

 never to go bankrupt. If this is the case, the manager of its portfolio must consider the worst alternative as the benchmark. Finally, as the investment takes place today he must evaluate the alternatives in their present value

Present value

Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

, hence the discounting.

Definition

Given a finite state space , let be a portfolio with payoff for . If is the order statistic

Order statistic

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference....

 the maximum loss is simply , where is the discount factor.

Properties

The Discounted maximum loss is the 1-expected shortfall

Expected shortfall

Expected shortfall is a risk measure, a concept used in finance to evaluate the market risk or credit risk of a portfolio. It is an alternative to value at risk that is more sensitive to the shape of the loss distribution in the tail of the distribution...

. It is therefore a coherent risk measure

Coherent risk measure

In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have...

.

Example

As an example, assume that a portfolio is currently worth 100, and the discount factor is 0.8 (corresponding to an interest rate

Interest rate

An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

of 25%):

probability	value
of event	of the portfolio
40%	110
30%	70
20%	150
10%	20

In this case the maximum loss is from 100 to 20 = 80, so the discounted maximum loss is simply