CFA Study Guide

Quantitative Methods: Statistical Concepts

Regular variance telsl you the probability of both greater and lesser outcomes compared to the mean. Since most investors are more heavily influenced by downside risk of their investments than the possible upside gains, often times it will be imperative to show the variance only below the mean instead of around both sides of it. Obviously the mean is still an important factor. Although, since you will be using only some the variance data (the lower bounded), you'll have to work only with the subset of values that fall below the mean, and this is where it gets the designation "Semi", or part of.

Semivariance of a normal curve is essentially figured as equal to its regular variance. You can see in the illustration that the lower bounded subset (negative variance) is a mirror image of the upper bound. Semivariance becomes more complicated to compute when the distribution is skewed and those pieces are not mirror images. Even when the distribution is still normal, if the target at which the lower bound is set below the mean, you cannot rely on semivariance equaling the variance. For these two instances you'd have to measure the dispersion below the mean or target by squaring the differences of those observations below that point.

I've not seen any problems in the curriculuum asking for a computation of a skewed or target semivaiance. You should probably get by just knowing the effect of skewness and targets on the variance in relation to a normal instance.

Quantitative Methods: Statistical Concepts

There are two categories of risk measurements: Absolute and Relative. I've spoken earlier about CV and Sharpe Ratios which measure risk in relation to expected returns (risk÷return; return÷risk). As you might guess, those methods would fall under the Relative category. Absolute measures of risk don't account for any returns. The two Absolute measures are Variance and Mean Absolute Deviation and they only measure a deviation compared to the number of observations (Dev÷N **). The difference between the two is how they handle negative errors.

Variance:
Notice that the deviations from the mean (errors) are squared before summing so that no negative numbers are incorporated. This amplifies the effect of any outliers.


MAD:
Accounts only for the absolute value of the errors and keeps the dispersion results closer to the mean than variance does.

Key Point:
Variance is more sensitive to outliers in the observations.


** Please be aware of the actual equations for these two formulas. Dev÷N is an oversimplification for illustration purposes.

Economics: Elasticity Curves



Elastic Curve
Like a rubber band, it is very sensitive to price - a small change in price equals a much larger change in qty. numerically this produces a Numerator that is larger than the denominator, so the answer will be greater 1 and Visually this produces and horizontal line.

Key Point:
Quantity is larger, Numerator Larger, Elasticity of Demand, Larger than 1. Also Infinity is larger than 1.


Inelastic Curve
Not as sensitive to price - even a large change in Price has little effect on qty. Price changed is larger than Qty changed, the Numerator is smaller than denominator, and the answer will be smaller than one. Visually this produces a vertical line.

Key Point:
Quantity is smaller, Numerator smaller, Elasticity of Demand Smaller than 1, and 0 is always smaller than 1.

Economics: Price Elasticity of Demand

When you start putting the curves into numbers and a formula you can get the specific Price Elasticity of Demand (or Supply depending on what you are charting). The answer that is figured from this equation is used to measure just how much the price of an item can be manipulated before demand will change.

The perfect instances are obviously ∞ or 0 as I covered before. For the imperfect instances, the answers obviously aren't as restrictive, but they can be seen as answers that are either greater than one or less than one. The actual answer is found by dividing the percent change in price into the percent change of quantity. Looking at it theoretically, if the Numerator of a fraction is smaller than the denominator the answer is less than 1 and if reversed, the answer is greater than 1 for example 1 divided by 2 is one half and two divided one is two. So if you look a bit further into to this concept you may be thinking to yourself, "if the change in quantity is negative then that numerator will be negative, so the answer will always be less than one", that is a completely true observation, but in this case we're just going to be looking at the absolute value of the changes and answers. because either way it's the same effect.

Economics: Elastic & Inelastic Curves

Price vs. Quantity is a common chart for most economics curves. There are two basic curve shapes you will need to know that can be made with a chart like this. What they both illustrate is the general concept of demand/supply. The two types of elasticity curves are called either an Elastic curve or an Inelastic curve which by nature of their names you can tell are the exact opposites of each other in terms of how quantity is affected. The basic underlying fact is that, in terms of demand, as a price increases, quantity demanded decreases. Furthermore the shape of the curve (elastic or inelastic) tells you to what extent price has an effect on demand.

In an INelastic curve:
a large change in price really only affects the quantity demanded very little. Examples of this kind of curve are most often Gasoline, Cigarettes and Home heating oils - things that people need or will pay for no matter how the price fluctuates, and remember this also holds true to price decreases because a person only has so many cars to fill tanks on and so many cigarettes he can smoke in one day, so demand really stays more or less constant with these kinds of items and this is what makes the Inelastic curve more vertical. A large change in price really doesn’t make much movement in quantity demanded on the X axis.

Key Point:
A vertical curve = INelastic curve

You can see that it almost even looks like the chart could be made into the letter "N" if you try real hard.

Quantitative Methods: Statistical Concepts

Measuring the risk of error in any set of values incorporates basic statistical equations to find the variance. The variance, in terms of financial analysis, measures how wide or narrow your margin of risk will be from the normally expected values. The larger that variance, the larger the risk, which means investors will expect a significant opportunity for gains with such a risky venture.

The two equations for measuring risk in portfolios are the Coefficient of Variation, and the Sharpe Ratio. Both of these formulas include the component of risk , standard deviation (√variance), compared to the expected return. It should be noted that investors will always want risk to be as low as possible and return to be as high as possible. The important fact to realize is where (numerator or denominator) those factors are placed.

Coefficient of Variation (CV): Risk is found in the numerator; when you want the numerator smaller than the denominator, this means you want the answer of your equation to be as low as possible


Sharpe Ratio: Risk is found in the denominator; when you want the denominator as small as possible, this means you want the answer of your equation to be as large as possible.



Key Point:
CV: Smaller numerator: Smaller Answer is better
Sharpe: Smaller denominator: Larger answer is better

Ethics & Professional Standards: Code of Standards