4 - Characteristics of the Opportunity Set Under Risk Flashcards
Frequency Distributions
What are the two measures to capture the relevant information about a frequency function?
Frequency Distribution:
- A frequency distribution is an overview of:
- all distinct values in some variable and
- the number of times they occur.
- Optionally, a frequency distribution may contain relative frequencies:
- frequencies relative to (divided by) the total number of values.
- Relative frequencies are often shown as percentages or proportions.
- Relative frequencies provide easy insight into frequency distributions. Besides, they facilitate comparisons. For example,
- “67.5% of males and 63.2% of females graduated” is much easier to interpret than“79 out of 117 males and 120 out of 190 females graduated”.
Relative Frequencies and Probability
- A special type of relative frequency is a probability.
- A probability is a relative frequency over infinite trials.
- So stating that “a coin flip has a 50% probability of landing heads up” technically means that if we’d flip the coin an infinite number of times, 50% -a relative frequency- of those flips will land heads up.
Two Measures:
- one to measure the average value and
- one to measure dispersion around the average value
What is the central tendency of a frequency distribution?
In MPT = Expected Return
- To calculate the EV for a single discrete random variable, you must multiply the value of the variable by the probability of that value occurring. Take, for example, a normal six-sided die. Once you roll the die, it has an equal one-sixth chance of landing on one, two, three, four, five or six. Given this information, the calculation is straightforward:
(1/6 * 1) + (1/6 * 2) + (1/6 * 3) + (1/6 * 4) + (1/6 * 5) + (1/6 * 6) = 3.5
- If you were to roll a six-sided die an infinite amount of times, you see the average value equals 3.5.
Symmetrical distributions:
- When a distribution is symmetrical, the mode, median and mean are all in the middle of the distribution.
- There are three main measures of central tendency: the mode, the median and the mean.
- Each of these measures describes a different indication of the typical or central value in the distribution.
- Every measure of skewness equals zero for a symmetric distribution.
Skewed Distributions:
- When a distribution is skewed the mode remains the most commonly occurring value, the median remains the middle value in the distribution, but the mean is generally ‘pulled’ in the direction of the tails. In a skewed distribution, the median is often a preferred measure of central tendency, as the mean is not usually in the middle of the distribution.
Positive Skew:
- A distribution is said to be positively or right skewed when the tail on the right side of the distribution is longer than the left side. In a positively skewed distribution it is common for the mean to be ‘pulled’ toward the right tail of the distribution. Although there are exceptions to this rule, generally, most of the values, including the median value, tend to be less than the mean value
- Example:
- Retirement Age Frequency Distribution
- 51-53: 4
- 54-56: 19
- 57-59: 11
- 60-62: 7
- 63-65: 3
- 66-68: 2
- Mode = 54
- Median = 56
- Mean = 57.2
- NOTE – Highest point is on left middle of distribution but more data sets are on the right side making it longer and shorter
- Retirement Age Frequency Distribution
- Example:
Negative Skew
- A distribution is said to be negatively or left skewed when the tail on the left side of the distribution is longer than the right side. In a negatively skewed distribution, it is common for the mean to be ‘pulled’ toward the left tail of the distribution. Although there are exceptions to this rule, generally, most of the values, including the median value, tend to be greater than the mean value.
- Example:
- Retirement Age Frequency Distribution
- 51-53: 1
- 54-56: 3
- 57-59: 6
- 60-62: 10
- 63-65: 18
- 66-68: 4
- Mode = 65
- Median = 63
- Mean = 61.8
- NOTE – Highest point is on right middle of distribution but more data sets are on the left side making it longer and shorter
- Retirement Age Frequency Distribution
Outliers:
- are extreme, or atypical data values that are notable different from the rest of the data
- It is important to detect outliers within a distribution, because they can alter the results of the data analysis. The mean is more sensitive to the existence of outliers than the median or mode.
- Population Mean?
- Sample Mean?
- Arithmetic Mean?
- What is the problem with arithemetic mean
Example 1
- The following are the annual returns on a given asset realized between 2005 and 2015.
- 12%
- 13%
- 11.5%
- 14%
- 9.5%
- 17%
- 16.1%
- 13%
- 11%
- 14%
- The population mean is the summation of all the observed values in the population divided by the total number of observations, N.
- The population mean defers from the sample mean, which is based on a few observed values ‘n’ that are chosen from the population
- The population mean and the sample mean are both arithmetic means. The arithmetic mean for any data set is unique and is computed using all the data values. Among all the measures of central tendency, it is the only measure for which the sum of the deviations from the mean is zero
- A commonly-sighted demerit of the arithmetic mean is that it’s not resistant to the effects of extreme observations, or what we call ‘outsider values.’ For instance, consider the following data set: {1 3 4 5 34}
- The arithmetic mean is 9.4, which is greater than most of the values. This is due to the last extreme observation i.e. 34.
Example 1
- Population Mean
- (0.12 + 0.13 + 0.115 + 0.14 + 0.098 + 0.17 + 0.161 + 0.13 + 0.11 + 0.14)/10
- = 0.1314 or 13.14%
- Sample Mean
- (0.13 + 0.11 + 0.14)/3
- = 0.1267 or 12.67%
Weighted Mean = ΣXiWi
Example 1
A portfolio consists of 30% ordinary shares, 25% T-bills and 45% preference shares with returns of 7%, 4%, and 6% respectively. Compute the portfolio return
The weighted mean takes into account the weight of every observation. It recognizes that different observations may have disproportionate effects on the arithmetic mean.
Example 1
The return of any portfolio is always the weighted average of the returns of individual assets. Thus:
- Portfolio return = (0.07 * 0.3) + (0.04 * 0.25) + (0.06 * 0.45) = 5.8%
Geometric Mean
How do you handle non-negative values?
Example
An ordinary share from a certain company registered the following rates of return over a 6-year period:
- -4% 2% 8% 12% 14% 15%
Compute the compound annual rate of return for the period
It’s a measure of central tendency, mainly used to measure growth rates. We define it as the nth root of the product of n observations i.e.
GM = (((1+RerturnX1) * (1+RerturnX2) * (1+RerturnX3) * (1+RerturnX4) 1/4 -1) x 100
The formula above only works when we have non-negative values. To solve this problem especially when dealing with percentage returns, we add 1 to every value and then subtract 1 from the final result.
Example - Geometric Mean
- GM = (1 + .04 * 1 + .02 * 1 + .12 * 1 + 0.14 * 1 + .15)<span>1/6</span> - 1
- GM = ((0.96 * 1.02* 1.08 * 1.12 * 1.14 * 1.15)1/6 - 1) * 100=
- (1.0761 – 1) * 100 =
- 7.61%
Example Arithmetic Mean
- (-4 + 2 + 8 + 12 + 14 + 15) / 6 =
- 7.833%
TAKE AWAYS
- The geometric mean is most appropriate for series that exhibit serial correlation. This is especially true for investment portfolios.
- Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums. The longer the time horizon, the more critical compounding becomes, and the more appropriate the use of geometric mean.
- For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding.
Geometric Mean is always ______ than the Arithmetic Mean
the geometric mean is always less than the arithmetic mean and the gap between the two widens as the variability of values increases
Arithmetic Mean
- In the world of finance, the arithmetic mean is not usually an appropriate method for calculating an average. Consider investment returns, for example. Suppose you’ve invested your savings in the financial markets for five years. If your portfolio returns each year were 90%, 10%, 20%, 30% and -90%, what would your average return be during this period?
- With the arithmetic average, the average return would be 12%, which appears at first glance to be impressive—but it’s not entirely accurate.
Geometric Mean
- The formula appears to be quite intense, but on paper, it’s not that complex. Returning to our example, let’s calculate the geometric average: Our returns were 90%, 10%, 20%, 30%, and -90%, so we plug them into the formula as:
- (1.9 * 1.1 * 1.2 * 1.3 * 0.1)^1/5 - 1
- =.7992 - 1
- =-0.2008
- =20.08%
- The result gives a geometric average annual return of -20.08%. The result using the geometric average is a lot worse than the 12% arithmetic average we calculated earlier, and unfortunately, it’s also the number that represents reality in this case.
Expected Value - Probability Weighted Value
In a probability distribution, the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities, is known as the expected value
MPT = Uses Probability Weighted Average for Expected Value
= R1P1 + R2P2 + … + RnPn
NOTE - No dividing by the total numbers; summation only; the “N” is taken care of in each probability applied to each return!
Variance - With Same Probabilities
Variance - With Unequal Probabilities
The arithmetic average of the squared deviations away from the mean.
For well-diversified equity portfolios, symmetrical distribution is a reasonable assumption, so the variance is an appropriate measure of downside risk.
- Variance = [(Each Annual Return - Average Return)2] / Total Number of Returns
Variance with Unequal Probabilities: The summation of:
- Variance = [Probability of Annual Return (Each Annual Return - Average Return
Steps:
- Work out the mean of the set
- For each number, subtract the Mean and Square the Result
- Then work out the mean of those squared differences
- For Standard Dev, take the square root of that and we are done!
Semi-Variance
Measures downside risk relative to a benchmark given by expected return. It is equal to the average of the squared deviations below the mean.
Semivariance =
(1/n) x Summation of (Average - rt)2
Where:
- n = total number of observations below the mean
- rt = the observed value
- Average = mean or target value of the dataset
Semivariance is a useful tool in portfolio or asset analysis because it provides a measure for downside risk.
While standard deviation and variance provide measures of volatility, semivariance only looks at the negative fluctuations of an asset. Semivariance can be used to calculate the average loss that a portfolio could incur because it neutralizes all values above the mean, or above an investor’s target return.
For risk-averse investors, determining optimal portfolio allocations by minimizing semivariance could reduce the likelihood of a large decline in the portfolio’s value
If returns on an asset are symmetrical, semivariance is proportional to the variance.
Value at Risk Measure
Value at Risk Measure:
- widely used by banks to measure their exposure to adverse events and to measure the least expected loss that will be expected with a certain probability.
Three Components:
- Time Period = a day, month, or year
- Confidence Level = typically either 95% or 99%
- Loss Amount = typically expressed in dollar or percentage terms
Example
- If 5% of the outcomes are below -30%, and if the decision maker is concerned about how poor the outcomes are 5% of the time, then -30%
This metric is most commonly used by investment and commercial banks to determine the extent and occurrence ratio of potential losses in their institutional portfolios.
Using a firm-wide VaR assessment allows for the determination of the cumulative risks from aggregated positions held by different trading desks and departments within the institution. Using the data provided by VaR modeling, financial institutions can determine whether they have sufficient capital reserves in place to cover losses or whether higher-than-acceptable risks require them to reduce concentrated holdings.
PROBLEMS:
- There is no standard protocol for the statistics used to determine asset, portfolio or firm-wide risk
- For example, statistics pulled arbitrarily from a period of low volatility may understate the potential for risk events to occur and the magnitude of those events
- Risk may be further understated using normal distribution probabilities, which rarely account for extreme or black-swan events
2.
What are the three methods of measuring VaR?
- Historical Method
- If we calculate each daily return, we produce a rich data set of more than 1,400 points
- Let’s put them in a histogram that compares the frequency of return “buckets.”
- For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%
- Notice the red bars that compose the “left tail” of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the “left tail”). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%.
- That is VAR in a nutshell. Let’s re-phrase the statistic into both percentage and dollar terms:
- With 95% confidence, we expect that our worst daily loss will not exceed 4%.
- If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).
- You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to “move to the left” on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:
- With 99% confidence, we expect that the worst daily loss will not exceed 7%.
- Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.
- The Variance-Covariance Method
- This method assumes that stock returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve.
- The idea behind the variance-covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve.
- They are a function of our desired confidence and the standard deviation
- The blue curve above is based on the actual daily standard deviation of the QQQ, which is 2.64%. The average daily return happened to be fairly close to zero, so we will assume an average return of zero for illustrative purposes.
- Monte Carlo Simulation
- The third method involves developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulationrefers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology.
- For most users, a Monte Carlo simulation amounts to a “black box” generator of random, probablistic outcomes. Without going into further details, we ran a Monte Carlo simulation on the QQQ based on its historical trading pattern. In our simulation, 100 trials were conducted. If we ran it again, we would get a different result–although it is highly likely that the differences would be narrow. Here is the result arranged into a histogram (please note that while the previous graphs have shown daily returns, this graph displays monthly returns):
- To summarize, we ran 100 hypothetical trials of monthly returns for the QQQ. Among them, two outcomes were between -15% and -20%; and three were between -20% and 25%. That means the worst five outcomes (that is, the worst 5%) were less than -15%. The Monte Carlo simulation therefore leads to the following VAR-type conclusion: with 95% confidence, we do not expect to lose more than 15% during any given month.
If returns are symmetrical, what metrics can be used?
- In cases where distribution of returns is symmetrical, the ordering of portfolios in mean-variance space will be the same as the ordering of porfolios in mean semivariance space or mean and any of the other measures of downside risk discussed earlier.
- If returns on an asset are symmetrical, the semivariance is proportional to the variance. Thus, in most of the portfolio literature, the variance, or equivalently the standard deviation, is used as a measure of dispersion.
The very essence of Modern Portfolio Theory:
How does variance change for a combination of assets vs. single asset?
The risk of a combination of assets is very different from a simple average of the risk of the individual assets. Most dramatically, the variance of a combination of two assets may be less than the variance of either of the assets themselves.
Remember, the risk of a combination of assets is very different from a simple average of the risk of individual assets. Most dramatically, the variance of a combination of two assets may be less than the variance of either asset by itself.
Condition of Market / Return of A2 / Return of A3 / Combination of A2 and A3
Good / $1.16 / $1.01 / $1.10
Average / $1.10 / $1.10 / $1.10
Poor / $1.04 / $1.19 / $1.10
Suppose the investor invests $0.60 in A2 and $0.40 in A3, what is the return in each condition of the market?
Good=
- $0.696 + $0.404 = $1.10
Average
- $0.66 + $0.44 = $1.10
Poor =
- $0.624 + $0.476 = $1.10
When assets have their good and bad outcomes at different times (assets 2 and 3), investment in these assets can radically reduce the dispersion obtained by investing in one of the assets by itself. In our example, the good and bad outcomes are always negatively correlated at -1, then the dispersion is reduced to 0.
NOTE - If the good outcomes of an asset are not always associated with the bad outcomes of a second asset, but the general tendency is in this direction, then the reduction in dispersion still occurs, but the dispersion will not drop all the way to zero, as it did in our example.
Asset 4 has three Possible Returns which depends on rainfall. Therefore, if the rainfall is plentiful, we can have good-, average-, or poor-security markets. Plentiful rainfall does not change the likelihood of any particular market condition occurring.
Asset 4 returns (rainful):
- Plentiful = 16%
- Average = 10%
- Poor = 4%
Asset 2 returns (market conditions):
- Good = 16%
- Average = 10%
- Poor = 4%
Consider an investor with $1,00 who invests $0.50 in each asset. Because we have assumed that each possible level of rainfall is equally likely as each possible condition of the market, there are nine equally likely outcomes.
Ordering them from highest to lowest, we have:
- $1.16
- $1.13
- $1.13
- $1.10
- $1.10
- $1.10
- $1.07
- $1.07
- $1.04
NOTE = The mean is the same as an investment in Asset 2 by itself, however, the dispersion around the mean is less! Now, instead of a 1/3 chance of $1.16 and $1.04, there is a 1/9 probability of $1.16 and $1.04. Less dispersion.
NOTE = With independent returns, extreme observations can still occur, they just occur less frequently.
