Statistics are an important tool for evaluating the investment worthiness of a product or security. This chapter focuses on these descriptive statistics:
The mean is the average value in a set of numbers.
To calculate the mean:
For example:
A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the mean of the annual returns?
Can you figure it out?
Answer = 5.75%
Add the annual returns:
10% + 15% + 5% + (-7%) = 23%
Divide by the number of returns (4):
23% / 4 = 5.75%
The median is the middle value in a set of numbers after you sort them from lowest to highest.
How you find it depends on how many values you have:
For example:
A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the median of the annual returns?
Can you figure it out?
Answer = 7.5%
First, sort the returns from lowest to highest:
-7%, 5%, 10%, 15%
There are 4 values (an even number), so average the two middle values (5% and 10%):
(5% + 10%) / 2 = 7.5%
Let’s see how it works with an odd number of values:
A security obtains annual returns of 10%, 15%, 5%, -7%, and 3% over the past five years. What is the median of the annual returns?
Can you figure it out?
Answer = 5%
First, sort the returns from lowest to highest:
-7%, 3%, 5%, 10%, 15%
With 5 values (an odd number), the median is the single middle value: 5%.
The mode is the value that occurs most often.
Key points:
For example:
A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the mode of the annual returns?
Can you figure it out?
Answer = There is no mode
None of the returns repeat, so there is no mode.
Let’s look at another example:
A security obtains annual returns of 10%, 15%, 5%, and -7%, 10%, 8%, and 12% over the past seven years. What is the mode of the annual returns?
Can you figure it out?
Answer = 10%
10% is the only value that repeats, so it is the mode.
The range is the difference between the highest value and the lowest value.
For example:
A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the range of the annual returns?
Can you figure it out?
Answer = 22%
First, sort the returns from lowest to highest:
-7%, 5%, 10%, 15%
Identify the lowest and highest values:
Compute the difference (highest minus lowest):
15% - (-7%) = 22%
This section is a direct copy of what you already learned in the pooled investments suitability chapter. This should serve as a review.
A common way to evaluate the effectiveness of a fund manager is by using alpha. Alpha measures whether a fund overperformed or underperformed its expected return.
If a question gives you the expected return, the calculation is straightforward:
A question could sound something like this:
An investor determines the expected return of a large-cap stock mutual fund over a year to be +14%. At the end of the year, the actual return was +17%. What is the alpha of the fund?
A positive alpha of 3 means the fund outperformed expectations by 3%. If the alpha were negative, the fund would have underperformed expectations by that amount. If the alpha were zero, the fund would have met expectations.
Some alpha questions are more involved and typically bring in beta.
Beta measures how volatile a portfolio has been relative to the market (historically speaking):
A portfolio with a beta of 1.0 has the same volatility as the market. That means it has generally moved with the market. If the S&P 500 was up 10% last year, this portfolio would be up about 10% (10% x 1.0).
A portfolio with a beta above 1.0 is more volatile than the market. A portfolio with a beta of 1.5 moves 1.5 times as much as the market. If the S&P 500 was up 10% last year, this portfolio would be up about 15% (10% x 1.5).
A portfolio with a beta between zero and 1.0 is less volatile than the market. A portfolio with a beta of 0.5 moves about half as much as the market. If the S&P 500 was up 10% last year, this portfolio would be up about 5% (10% x 0.5).
Last, a portfolio with a negative beta tends to move opposite the market. A portfolio with a beta of -2.0 moves about twice as much as the market, but in the opposite direction. If the S&P 500 was up 10% last year, this portfolio would be down about 20% (10% x -2.0).
Here’s a table summarizing what we just discussed:
| S&P 500 return | Portfolio beta | Portfolio return |
|---|---|---|
| Up 10% | 1.0 | Up 10% |
| Up 10% | 1.5 | Up 15% |
| Up 10% | 0.5 | Up 5% |
| Up 10% | -2.0 | Down 20% |
There are two types of math-based questions involving both alpha and beta to be aware of. First, let’s explore this question:
An investor is comparing two different funds in an investment analysis. BCD stock fund maintains a beta of 1.0, while TUV stock fund maintains a beta of 1.5. Last year, BCD stock fund’s performance was +14%, while TUV stock fund’s performance was +19%. What is TUV stock fund’s alpha last year?
Because alpha measures over- or underperformance, you need TUV’s actual return and its expected return.
BCD has a beta of 1.0, which means it moves with the market. The question includes BCD so you can treat its +14% return as the market return for the year.
Now use TUV’s beta (1.5) to estimate its expected return:
Then apply the alpha formula:
An alpha of -2 means TUV underperformed expectations by 2%.
There’s another formula you can utilize to calculate alpha involving a few new components. Here it is:
The portfolio return and market return are the returns for the fund and its benchmark. The risk-free rate of return is the return on a relatively risk-free security. The most commonly cited risk-free security is the 3-month Treasury bill. It is considered close to risk-free due to its short maturity and U.S. government backing, although all securities carry some level of risk.
Here’s an example of a question involving this formula:
An investor is analyzing the market and the returns of a small-cap stock fund held in their portfolio. The fund was up 28% while maintaining a beta of 2.5 last year. During the same year, the S&P 500 was up 10%, the Russell 2000 was up 14%, and the 3-month Treasury bill gained 2%. What is the small-cap stock fund’s alpha?
This is a tough question, but can you figure it out using the formula above?
Answer: -4
This fund manager underperformed expectations by 4%, leading to an alpha of -4.
One note to point out in the question: both the S&P 500 and the Russell 2000 returns were provided, but only the Russell 2000 was used. Because the fund is a small-cap stock fund, you should use the index most closely correlated with small-cap performance. The S&P 500 is primarily large-cap (and some mid-cap), while the Russell 2000 is a small-cap index. Therefore, the S&P 500 return should be disregarded.
Alpha is most relevant when evaluating an actively managed fund, because active managers aim to outperform a benchmark (a relevant market index). If a small-cap fund manager is trying to beat the Russell 2000 by selecting small-cap stocks, alpha is a direct measure of whether that goal was achieved.
Passively managed funds are designed to match their benchmarks, so they should have alpha values near zero (meaning they don’t consistently over- or underperform the market). A similar idea applies to beta: passive funds that track the market should generally have a beta near 1.
We initially covered the Sharpe ratio in a previous chapter. This ratio measures risk-adjusted returns for a security or portfolio. In plain terms, it measures investment efficiency: how much return you receive for the amount of risk you take.
Here is the formula:
In this formula:
A higher Sharpe ratio indicates a more efficient security or portfolio (more return per unit of risk). It’s unlikely you’ll be asked to calculate the Sharpe ratio on the exam, but you may see questions about what the components mean or what a higher Sharpe ratio implies.
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