This article is an excerpt from the Shortform book guide to "Naked Statistics" by Charles Wheelan. Shortform has the world's best summaries and analyses of books you should be reading.
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How do you calculate investment risk? What do you need to know to calculate the expected payoff of a financial investment?
Investors often use probability to assess risk when making financial decisions. This is typically done with a statistic called an “expected value.” To calculate the expected value of a financial investment, you need to know the probability of each possible outcome and its respective payoff.
Here’s how to use the expected value statistic for calculating investment risk.
Using Probability to Make Financial Decisions
Investors use the “expected value” statistic for calculating investment risk. To determine the risk of a financial investment, we multiply the financial payoff for each possible outcome by its probability and then add them all together.
For example, say you were interested in building a tiny house on your property to rent to tourists. It will cost $50,000 to build the house. Based on location and current market demand, you estimate a 25% chance that the house won’t be rented at all during the first two years, a 50% chance that it will be rented “part-time” and earn $1,500/month, and a 25% chance that it will be in high demand and rented “full time,” earning $3,000/month. To calculate the expected value of the investment during your first two years, you would multiply $0 by 25%, $36,000 (for two years of part-time rentals) by 50%, and $72,000 (for two years of full-time rentals) by 25%, and add them together for an expected value of $36,000.
Wheelan explains that we can represent the calculations for the expected payoff visually in a decision tree. For your tiny house, your decision tree would look like this:
Again, probability dictates that your investment is likely worth $36,000 in the first two years. But looking at your decision tree, you can also see that it’s reasonably likely that you’ll make no money at all and also reasonably likely that you’ll only make $18,000.
The decision tree highlights an important lesson in probability: Probability is a tool, not a guarantee. Unless the probability of an event is zero or 100%, we should always be prepared for an unlikely outcome. Wheelan reminds us that statistically improbable things happen all the time (someone’s “illogical” ticket purchase will win them the lottery!).
Since no outcome is guaranteed, Wheelan explains that we all use probability differently depending on our goals, motivation, risk tolerance, and so on. For example, Wheelan points out that the expected value becomes a more and more useful statistic as the number of financial risks one takes increases. Real-estate developers, for instance, can use this tool to make sure that their multiple investments are likely to make money as a whole. Even if one property loses money or underperforms in a given year, as long as the expected value of their portfolio is profitable overall, they are likely to make money.
Decision Trees for Purchasing Stock
As Wheelan explains, probability is an effective tool for managing risk. Unfortunately, many of us underutilize probability when investing in the stock market. Research shows that we tend to overestimate the probability of rare events and our ability to foresee them. For example, people will often invest in a single stock that they think will be the next Apple instead of spreading their investment across a diverse portfolio. This desire to “hit the jackpot” instead of earn a more modest yet statistically likely return on their investment leads people to under-diversify their stock to the detriment of their wallets. Data suggests that these errors in judgment cost the average investor $2,500 per year.
Using a tool to assess the probable outcomes of our financial investments, such as the decision tree described in this section, is likely a good idea before investing in the stock market. With a decision tree, our “gut feeling” about a stock can be tempered by math, and we may make smarter investments.
Calculating Investment Risk Using Probability
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An explanation and breakdown of statistics into digestible terms
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Why learning statistics is an exercise in self-empowerment
Risk Assessment Matrix Basic risk equation: Risk = Probability x Consequence (1) Basic risk equation to person(s): Risk = Probability x (Consequence level x Human Exposure) (2) Equation of risk to person(s) from explosives events -Annual risk: Pf = Pe x Pf/e x Ep (3) Where: Pf -Probability of fatality; Pe-Probability ...
The expected return is calculated by multiplying the probability of each possible return scenario by its corresponding value and then adding up the products. The expected return metric – often denoted as “E(R)” – considers the potential return on an individual security or portfolio and the likelihood of each outcome.
The five measures include alpha, beta, R-squared, standard deviation, and the Sharpe ratio. Risk measures can be used individually or together to perform a risk assessment. When comparing two potential investments, it is wise to compare similar ones to determine which investment holds the most risk.
lated through appropriate accounting procedures to obtain simulated values for the net present value of the proposed investment. By repeating the process many times, a probability distribution for net present value is generated, thus allowing an investor to evaluate risk associated with a particular investment.
Probability is the likelihood that a risk will occur. The impact is the consequence or effect of the risk, normally associated with impact to schedule, cost, scope and quality. Rate probability and impact using a scale such as 1 to 10 or 1 to 5, where the risk score equals the probability multiplied by the impact.
Investors often use probability to assess risk when making financial decisions. This is typically done with a statistic called an “expected value.” To calculate the expected value of a financial investment, you need to know the probability of each possible outcome and its respective payoff.
To calculate the risk in the portfolio, you can use the formula: σ P = w A 2 ⋅ σ A 2 + w B 2 ⋅ σ B 2 + 2 ⋅ w A ⋅ w B ⋅ σ A ⋅ σ B ⋅ ρ A B where: - stands for the portfolio risk, - and are the weights of investment in asset A and asset B, - and are the standard deviations of returns of asset A and asset B respectively, - ...
To calculate probability, you must divide the number of favorable events by the total number of possible events. This generates a sample, and the calculation can be performed from the data obtained.
We can also say that it measures the financial risk of the business firm. The formula can be calculated in the following ways: DFL = % Change in Net Income / % Change in Earnings Before Interest and Taxes (EBIT) DFL = % Change in Earnings per Share (EPS) / % Change in EBIT.
One common method is to use a risk rating system. This system assigns a numerical value to each degree of risk associated with an investment. For example, a security that is considered to have low risk would be given a rating of 1, while a security that is considered to have high risk would be given a rating of 5.
To work out an expected value for a significant risk, multiply the probability of the risk happening by the size of the consequence. The result provides the risk premium – the estimated cost of accepting the risk. The disadvantage to this is that it does not account for variability in outcomes.
Risk—or the probability of a loss—can be measured using statistical methods that are historical predictors of investment risk and volatility. Commonly used risk management techniques include standard deviation, Sharpe ratio, and beta.
For businesses, technology risk is governed by one equation: Risk = Likelihood x Impact. This means that the total amount of risk exposure is the probability of an unfortunate event occurring, multiplied by the potential impact or damage incurred by the event.
Here are three commonly used formulas for VaR calculation: Historical VaR: VaR = -1 x (percentile loss) x (portfolio value) Parametric VaR: VaR = -1 x (Z-score) x (standard deviation of returns) x (portfolio value) Monte Carlo VaR: VaR = -1 x (percentile loss) x (portfolio value)
The formula for RR equals (risk of an event in one group)/(risk of an event in a second group). It is an indication of the strength of the association between the 2 risks. Its interpretation is intuitive.
Let Θ denote nature's parameter space, A the statistician's action space, and L(θ, a) a specific loss function. Then, the risk function of a decision procedure δ(X1,···,Xn) is the average loss incurred by δ: R(θ, δ) = Eθ[L(θ, δ(X1,···,Xn))].
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