predictionmarkets &mdash; Prediction Markets Trading https://prediction-markets.writeas.com/tag:predictionmarkets An Introduction to Trading on Prediction Markets Mon, 10 Aug 2026 19:54:03 +0000 Expected Shortfall in a Prediction Market Setting https://prediction-markets.writeas.com/expected-shortfall-in-a-prediction-market-setting?pk_campaign=rss-feed <![CDATA[How bad can things get? A risk measure provides an answer. Suppose you are presented with a wager costing 0.4 which returns 1 with probability 0.5 and otherwise zero. Such a wager has an expected return of 25%. You can make this wager once a day over the next 10 days. Suppose you do. How bad can things get? We know that there's a small probability ($0.5^{10}$) that you lose 4 (i.e., you lose all 10 wagers). And there's a larger probability, $10 \cdot 0.5^{10}$, that you lose 3 (i.e., win one, lose 9). Although these are useful pieces of information, we might wish for a single-number summary of the riskiness of our planned series of bets. Expected shortfall is one such measure of risk. It describes how much you should expect to lose if things go bad. $\newcommand{\VaR}{\text{VaR}} \newcommand{\E}{\text{E}}$ Expected shortfall, also known as conditional value at risk, is typically defined in terms of value at risk (VaR). Let's begin by defining value at risk. (VaR is also a risk measure, just not a particularly good one.) For a given set of wagers ("portfolio") over a defined period of time and a specified confidence level α, the VaR is the (1-α)-quantile of the portfolio's loss distribution. If the profit-and-loss distribution is X then $\VaRα(X)$ is the (1-α)-quantile of Y := -X. (Losses are negative, profits positive.) Suppose our portfolio X has a standard normal distribution. Then the VaR at level 0.05 of our portfolio is 1.645. 1.645 is the 0.95-quantile of -X. Suppose, alternatively, that our portfolio Z has a probability of 0.05 of losing 2 million or more. The VaR at level 0.05 of the portfolio is 2 million. 2 million is the 0.95-quantile of -Z. asidestrongAside:/strong Have you encountered VaR in an actuarial or engineering setting? Be careful here. Because in such settings we are narrowly concerned with potential losses, conventions are different. In engineering and actuarial settings the distribution of losses is taken as the object of interest, $\VaRα(L)$ where $L = -X$. In this case it is the right tail which is studied. Typical values of α are 0.95 and 0.99./aside Expected shortfall, like value at risk, is defined with respect to a period of time and a confidence level α. It is the expected loss in the worst α of cases. If the profit-and-loss distribution is X and X follows a continuous probability distribution then the expected shortfall of X at level α is the E&nbsp;Ssubα/sub = E[- X | X ≤ - VaRsubα/sub(X)]. This is the left-tail conditional expectation below -$\VaRα(X)$. Suppose again that our portfolio X has a standard normal distribution. Then the expected shortfall at level 0.05 of our portfolio is 2.0627. Expected shortfall, like value at risk, is a single-number measurement of how much we should expect to lose if things go bad. Expected shortfall is a better measure of downside risk than VaR because expected shortfall gives us the average loss when things go bad. VaR, by contrast, tells you a single example of a bad outcome. Consider the following example of a portfolio X which has an expected 2% return. The portfolio costs 1 to acquire and returns 1.6 with probability 0.95, 0 with probability 0.04, and -50 with probability 0.01. The VaR at level 0.05 of this portfolio is 0.!-- technically, given that we have a discrete distribution, I'm not sure this is entirely correct. -- The expected shortfall at level 0.05 is 10. 0 is an example of a bad outcome (VaR). 10 is a measure of the average badness, given a bad outcome. Expected shortfall provides a richer description of downside risk than VaR. For this reason people tend to prefer it whenever it is available. In a prediction market setting, use expected shortfall to characterize the downside risk of a portfolio of wagers. Consider the example of a portfolio which consists of 10 independent wagers. Each wager costs 0.5 and pays out 1 with probability 0.51. (This portfoilio has an expected return of 2%.) To calculate the expected shortfall we calculate the left-tail conditional expectation below $-\VaRα(X) = 2$. Calculating this can be accomplished in at least two ways. Simulating the portfolio is likely the easiest way. We arrive at an expected shortfall of 3.21. When considering different portfolios which we might purchase, it is natural to ask about the typical case: How much do we stand to gain (or lose) on average? It is also natural—or should be natural—to ask how bad things can get. Expected shortfall provides an answer. Expected shortfall describes how much you should expect to lose if things go bad. ----- This post is part of a series. The most recent post in the series is "Virtues of prediction markets: Useful baselines". Learn when new posts appear by subscribing (RSS). You may also follow `@prediction-markets@write.as` in Mastodon or subscribe for email updates. #predictionmarkets #expectedshortfall #conditionaltailrisk #valueatrisk]]> How bad can things get? A risk measure provides an answer.

Suppose you are presented with a wager costing 0.4 which returns 1 with probability 0.5 and otherwise zero. Such a wager has an expected return of 25%. You can make this wager once a day over the next 10 days. Suppose you do. How bad can things get? We know that there's a small probability ($0.5^{10}$) that you lose 4 (i.e., you lose all 10 wagers). And there's a larger probability, $10 \cdot 0.5^{10}$, that you lose 3 (i.e., win one, lose 9). Although these are useful pieces of information, we might wish for a single-number summary of the riskiness of our planned series of bets. Expected shortfall is one such measure of risk. It describes how much you should expect to lose if things go bad. $\newcommand{\VaR}{\text{VaR}} \newcommand{\E}{\text{E}}$

Expected shortfall, also known as conditional value at risk, is typically defined in terms of value at risk (VaR). Let's begin by defining value at risk. (VaR is also a risk measure, just not a particularly good one.) For a given set of wagers (“portfolio”) over a defined period of time and a specified confidence level α, the VaR is the (1-α)-quantile of the portfolio's loss distribution. If the profit-and-loss distribution is X then $\VaR_α(X)$ is the (1-α)-quantile of Y := -X. (Losses are negative, profits positive.) Suppose our portfolio X has a standard normal distribution. Then the VaR at level 0.05 of our portfolio is 1.645. 1.645 is the 0.95-quantile of -X. Suppose, alternatively, that our portfolio Z has a probability of 0.05 of losing 2 million or more. The VaR at level 0.05 of the portfolio is 2 million. 2 million is the 0.95-quantile of -Z.

Expected shortfall, like value at risk, is defined with respect to a period of time and a confidence level α. It is the expected loss in the worst α of cases. If the profit-and-loss distribution is X and X follows a continuous probability distribution then the expected shortfall of X at level α is the E Sα = E[– X | X ≤ – VaRα(X)]. This is the left-tail conditional expectation below –$\VaR_α(X)$. Suppose again that our portfolio X has a standard normal distribution. Then the expected shortfall at level 0.05 of our portfolio is 2.0627. Expected shortfall, like value at risk, is a single-number measurement of how much we should expect to lose if things go bad.

Expected shortfall is a better measure of downside risk than VaR because expected shortfall gives us the average loss when things go bad. VaR, by contrast, tells you a single example of a bad outcome. Consider the following example of a portfolio X which has an expected 2% return. The portfolio costs 1 to acquire and returns 1.6 with probability 0.95, 0 with probability 0.04, and -50 with probability 0.01. The VaR at level 0.05 of this portfolio is 0. The expected shortfall at level 0.05 is 10. 0 is an example of a bad outcome (VaR). 10 is a measure of the average badness, given a bad outcome. Expected shortfall provides a richer description of downside risk than VaR. For this reason people tend to prefer it whenever it is available.

In a prediction market setting, use expected shortfall to characterize the downside risk of a portfolio of wagers. Consider the example of a portfolio which consists of 10 independent wagers. Each wager costs 0.5 and pays out 1 with probability 0.51. (This portfoilio has an expected return of 2%.) To calculate the expected shortfall we calculate the left-tail conditional expectation below $-\VaR_α(X) = 2$. Calculating this can be accomplished in at least two ways. Simulating the portfolio is likely the easiest way. We arrive at an expected shortfall of 3.21.

When considering different portfolios which we might purchase, it is natural to ask about the typical case: How much do we stand to gain (or lose) on average? It is also natural—or should be natural—to ask how bad things can get. Expected shortfall provides an answer. Expected shortfall describes how much you should expect to lose if things go bad.

This post is part of a series. The most recent post in the series is “Virtues of prediction markets: Useful baselines”. Learn when new posts appear by subscribing (RSS). You may also follow @prediction-markets@write.as in Mastodon or subscribe for email updates.

#predictionmarkets #expectedshortfall #conditionaltailrisk #valueatrisk

]]> https://prediction-markets.writeas.com/expected-shortfall-in-a-prediction-market-setting Sun, 08 Sep 2019 13:27:00 +0000 Virtues of prediction markets: Useful baselines https://prediction-markets.writeas.com/virtues-of-prediction-markets-useful-baselines?pk_campaign=rss-feed <![CDATA[Participation in prediction markets has a social benefit: a market's prediction provides a useful baseline against which other predictions (outside the market) may be judged. A market's prediction can also be useful to individuals making decisions that depend on the outcome of the predicted event. In both cases, the social utility provided by the market remains even if the market's prediction is not perfect. The market is useful even if it only provides a general assessment of the relative probabilities of different outcomes. Prediction markets provide baseline predictions about uncertain events. Consider the task of predicting which candidate will receive the US Democratic party's presidential nomination in 2020. In May 2019 there were 23 candidates for the nomination. The thought that each candidate is equally likely to win the nomination does not merit being described as a baseline prediction. Some candidates have a far better chance than others. Here the prediction market gives the public a general sense of which candidates are better positioned than others (Figure 1). And the public gets this for no cost. The probability of one of these five candidates winning is 87%, in the judgement of PredictIt participants in late May 2019. There are many other candidates in this race. The prediction market here tells us that some candidates stand a far better chance than other candidates. Figure 1: The probability of one of these five candidates winning is 87%, in the judgement of PredictIt participants in late May 2019. There are many other candidates in this race. The prediction market here tells us that some candidates stand a far better chance than other candidates. The utility of having a baseline prediction is easy to appreciate in the case of political events. With political events, we have political commentators and polling firms which deliver forecasts of what will happen. It is natural to ask whether or not these commentators or firms are skilled at forecasting. Given that polling is expensive and salaries for political commentators are non-zero, we anticipate that their predictions will approach or exceed a certain standard of utility. One standard we can use is the following: ask that their predictions tend to be more accurate than those made by prediction markets. Having prediction markets provide a standard here is particularly valuable because there are typically no other standards we can use. We cannot, for example, ask the commentator or firm to do better than the "prediction" which assumes each candidate has an equal chance of winning because virtually anyone can make better predictions than that. Prediction markets prove their worth by providing a baseline in settings where non-trivial baselines are scarce. Consider participating in a prediction market. Even if your wagers fail to pay out, the public stands to benefit. ----- This post is part of a series. The most recent post in the series is "Are there sure bets on PredictIt?". Learn when new posts appear by subscribing (RSS). You may also follow `@prediction-markets@write.as` in Mastodon or subscribe for email updates. #predictionmarkets #predictit ]]> Participation in prediction markets has a social benefit: a market's prediction provides a useful baseline against which other predictions (outside the market) may be judged. A market's prediction can also be useful to individuals making decisions that depend on the outcome of the predicted event. In both cases, the social utility provided by the market remains even if the market's prediction is not perfect. The market is useful even if it only provides a general assessment of the relative probabilities of different outcomes.

Prediction markets provide baseline predictions about uncertain events. Consider the task of predicting which candidate will receive the US Democratic party's presidential nomination in 2020. In May 2019 there were