Statistics are an important tool for evaluating the investment worthiness of a product or security. This chapter focuses on four common descriptive statistics:
The mean is the average value in a set.
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%) to get 23%. Then divide by the number of returns (4).
23% / 4 = 5.75%
The median is the middle value in a set after the values are ordered from lowest to highest.
To calculate the median:
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, order the returns from lowest to highest:
-7%, 5%, 10%, 15%
There are four values (an even number), so average the two middle values (5% and 10%).
(5% + 10%) / 2 = 7.5%
Here’s 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, order the returns from lowest to highest:
-7%, 3%, 5%, 10%, 15%
5% is the middle value, so it’s the median.
The mode is the value that occurs most often.
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.
Here’s 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’s 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, order the returns from lowest to highest:
-7%, 5%, 10%, 15%
The lowest return is -7% and the highest return is 15%. The range is the difference between them:
-7% - 15% = 22%
This section is a direct copy of what you already learned in the alpha and beta 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 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 fund’s alpha?
A positive alpha of 3 means the fund outperformed expectations by 3%. If alpha is negative, the fund underperformed by that amount. If alpha is zero, the fund met expectations.
More math-based alpha questions typically introduce another figure: beta.
A portfolio with a beta of 1.0 has historically had the same volatility as the market. In other words, it has generally moved with the market. If the S&P 500 was up 10% last year, this portfolio was up 10% (10% x 1.0) as well.
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 was up 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 at half the market’s pace. If the S&P 500 was up 10% last year, this portfolio was up 5% (10% x 0.5).
Last, a portfolio with a negative beta moves opposite to the market. A portfolio with a beta of -2.0 moves at twice the market’s pace, but in the opposite direction. If the S&P 500 was up 10% last year, this portfolio was down 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 was TUV stock fund’s alpha last year?
Because alpha measures overperformance or underperformance, we need TUV’s actual return and its expected return.
The question includes BCD stock fund to help you infer the market return. Since BCD has a beta of 1.0, it has market-level volatility, so we can assume its return matches the market return. That implies the market return last year was +14%.
TUV has a beta of 1.5, meaning it has historically moved 1.5 times as much as the market (in the same direction, since beta is positive). So its expected return is:
Now apply the alpha formula:
An alpha of -2 means TUV underperformed expectations by 2%.
There’s another formula you can use to calculate alpha that includes a few additional components:
The portfolio return and market return are usually given in the question. 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’s considered close to risk-free because of its short maturity and U.S. government backing, although all securities carry at least some risk.
Here’s an example that uses 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. Since the fund is a small-cap stock fund, you want the index that best matches (is most correlated with) small-cap performance. The S&P 500 is primarily large- and mid-cap stocks, while the Russell 2000 is a small-cap stock index. Therefore, the S&P 500 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). For example, if a small-cap stock fund manager tries to beat the Russell 2000 by selecting small-cap stocks, alpha helps measure whether those choices added value.
Passively managed funds are designed to match their benchmarks, so they should generally have alpha values near zero (meaning they don’t meaningfully overperform or underperform the benchmark). A similar idea applies to beta: passively managed funds that track the market should typically have a beta near 1 (meaning they tend to move with market volatility).
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 “bang for the buck,” or investment efficiency: how much return you’re getting for the amount of risk you’re taking.
Here’s the formula:
The higher the Sharpe ratio, the more efficient the 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 ratio measures or what each part of the formula represents.
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