Textbook

Statistics are an important factor when measuring the investment worthiness of a product or security. We’ll focus on these descriptive statistics in this chapter:

- Mean
- Median
- Mode
- Range
- Alpha & beta (review)
- Sharpe ratio (review)

The **mean** refers to the average. To calculate the mean, you must add up all the relevant factors and divide by the number of relevant factors. 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?

(spoiler)

Answer = **5.75%**

To find the answer, first add up all the annual returns (10%, 15%, 5%, -7%), which is 23%. Next, divide 23% by the number of returns (4).

23% / 4 = **5.75%**

The **median** refers to the middle number. If there are an odd number of relevant factors, it’s literally the middle number. If there’s an even number of relevant factors, you must average the two middle numbers. 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?

(spoiler)

Answer = **7.5%**

To find the answer, first line up the annual returns from lowest to highest:

-7%, 5%, 10%, 15%

We have an even number of factors, so we’ll need to take the average of the two middle numbers (5% and 10%).

(5% + 10%) / 2 = **7.5%**

Let’s see how it would work with an odd number of factors:

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?

(spoiler)

Answer = **5%**

To find the answer, first line up the annual returns from lowest to highest:

-7%, 3%, 5%, 10%, 15%

**5%** is the middle number, and therefore is the median.

The **mode** refers to the most frequently recurring number. If there is no recurring number, there is no mode. If there are multiple recurring numbers, it’s the one that recurs the most. 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?

(spoiler)

Answer = **There is no mode**

There is no recurring number, 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?

(spoiler)

Answer = **10%**

10% is the only recurring number, making it the mode.

The **range** refers to the difference between the lowest factor and the highest factor. 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?

(spoiler)

Answer = **22%**

To find the answer, first line up the annual returns from lowest to highest:

-7%, 5%, 10%, 15%

Next, identify the lowest and highest factors. -7% is the lowest while 15% is the highest. Last, find the difference between the two factors:

-7% - 15% = **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 determine the effectiveness of a fund manager is through the use of **alpha**. When calculated, alpha determines whether a fund is over or underperforming expectations. If a test question provides the expected return of a fund, the calculation is fairly simple:

$Alpha=actual return - expected return$

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?

$Alpha=17% - 14%$

$Alpha=3$

A positive alpha of 3 means the fund is over-performing expectations by 3%. This is a sign the fund manager is doing a good job of managing the portfolio. If the alpha was negative, the fund would be underperforming expectations by the amount of the alpha. If the alpha was zero, the fund would be meeting expectations.

Math-based alpha questions can be more complicated, and typically involve another figure - **beta**.

A portfolio with a beta of 1.0 has the same volatility as the market, historically speaking. Meaning, this portfolio has generally followed the market in the past. 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 faster than 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 speed of the market. 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 speed of the market, 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 is TUV stock fund’s alpha last year?

Given alpha is a measurement of over or underperformance, we must compare the performance of TUV stock (+19%) to its expected performance. The expected performance is not explicitly provided, but we can make an assumption based on the information provided on BCD stock fund. The only reason it’s included in the question is to tell you the performance of the market in a sneaky way. Remember, a beta of 1 means the investment’s volatility is equal to the market. We can safely assume the market (assumptively the S&P 500) performed equally to BCD stock fund, therefore the market return last year was +14%.

TUV stock fund maintains a beta of 1.5, meaning it historically has moved 1.5 times faster than the market. Because beta is positive, we can assume it’s moving in the same direction as the market. With that information, we can take the beta (1.5) and multiply it times the assumptive market return (14%). This tells us the expected return of TUV stock is 21% (1.5 x 14%).

Now, we can use the original alpha formula:

$Alpha=actual return - expected return$

$Alpha=19% - 21%$

$Alpha=-2$

An alpha of -2 means TUV stock fund underperformed expectations by 2%. The fund manager hopefully will do a better job the following year!

There’s another formula you can utilize to calculate alpha involving a few new components. Here it is:

$Alpha=(PR - RF) - (Beta x (MR - RF))$

$Where:PRRFMR =portfolio return=risk-free return=market return $

The portfolio return and market return should be self-explanatory. The risk-free rate of return measures the return on a relatively risk-free security. The most commonly cited risk-free security is the 3-month Treasury bill. It’s very close to being completely free of risk due to its short-term nature and US government backing, although all securities come with at least some risk potential.

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?

(spoiler)

Answer: **-4**

$Alpha=(PR - RF) - (Beta x (MR - RF))$

$Alpha=(28% - 2%) - (2.5 x (14% - 2%))$

$Alpha=26% - (2.5 x 12%)$

$Alpha=26% - 30%$

$Alpha=-4$

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 utilized. Given the fund is a small-cap stock fund, it’s important to utilize the index that is most correlated with the fund. The S&P 500 contains 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 in determining the effectiveness of an actively managed fund because these fund types aim to outperform their benchmarks (their relevant market index). If a small-cap stock fund manager seeks to pick the top small-cap stocks in the Russell 2000, alpha serves as a good measure of their successes or failures. If the stocks they choose outpace the index on average, they’ll attain a positive alpha. The higher the alpha, the better their investments are performing. And vice versa.

Passively managed funds are built to match the performance of their benchmarks, and therefore should maintain alpha values near zero (meaning they don’t over or underperform the market). The same concept applies to beta as well; passively managed funds should maintain a beta near 1 (meaning they move at the same volatility as the market).

We initially covered the **Sharpe ratio** in a previous chapter. This ratio measures **risk-adjusted returns** for a security or portfolio. In plain terms, this statistic measures “bang for the buck,” or investment efficiency. The best security to seek out is the one with the highest return potential and the lowest risk potential. How much a security or portfolio meets this description can be determined through the Sharpe ratio. Let’s start out with the formula itself:

$Sharpe ratio=Standard deviationActual return - Risk free rate $

The **risk-free rate of return** is equal to the 91-day (3 months) Treasury bill rate. **Standard deviation** measures how far a security deviates from its average price. The higher the standard deviation, the more volatility a portfolio is subject to.

The higher the Sharpe ratio, the more efficient the security or portfolio. Again, the more efficient an investment, the higher its return is in comparison to its risk profile. It’s unlikely you’ll be asked to perform a Sharpe ratio calculation on the exam, but you could encounter questions on the components of the formula or the concept behind it.

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