Textbook

Securities can be analyzed in a number of different ways. We’ll explore the following analytical methods in this chapter:

- Price to earnings (PE) ratio
- Price to book ratio
- Dividend payout ratio
- Time value of money concepts, including:
- Present value
- Net present value
- Internal rate of return

- Descriptive statistics, including:
- Mean
- Median
- Mode
- Range

- Alpha
- Beta
- Sharpe ratio

**Price to earnings (PE) ratios** are used by investors to determine if a company is overvalued or undervalued. The price references the market price per share of the company, while the earnings represent the profits made by the company on a per-share basis. Earnings are a line item on a corporate **income statement**.

$PE ratio=Earnings per shareCommon stock market price $

The higher the PE ratio, the more likely that an investment is overvalued. For example, if the PE ratio was 100, the company’s market price is 100 times what it makes in annual earnings. Unless the company’s profits are expected to grow considerably, this investment may be overpriced. On average, PE ratios range between 15-25, depending on the company and industry.

**Growth companies** typically maintain higher PE ratios. These businesses usually have an evolving and expanding business, and are expected to make larger profits in the future. Therefore, the investment may seem “overpriced” today, but it may be a good deal in the long-term.

For example, Roku stock (symbol: ROKU) reflects a PE ratio of 180 as of October 2021. The company’s stock price is 180 times the amount it makes in annual earnings. If you were to purchase the entire company and they maintained the same earnings level, it would take 180 years to recoup your original investment. While it sounds overpriced, what if Roku could substantially increase its profits in the next few years? Their investors are betting on this occurring, which is the only justification for investing in a company with a PE ratio this high.

**Value companies** typically have lower PE ratios. These businesses are usually large, well-established, and with a long track record of profits. Given their large size, investors don’t expect value companies to increase in size significantly. Without a reason to bet on large future growth, investors are generally unwilling to “overpay” for a value stock, which leads to their lower PE ratios.

For example, Allstate stock (symbol: ALL) reflects a PE ratio of 6 as of October 2021. The company’s stock price is only 6 times the amount it makes in annual earnings. If you were to purchase the entire company and they maintained the same profit level, it would take only 6 years to recoup your original investment. This is a stark difference as compared to Roku stock, making Allstate stock seem cheap and undervalued.

Value companies don’t grow as fast as growth companies, and therefore value stocks don’t typically experience high levels of capital gains. So, how do investors make a return? Value companies commonly pay cash dividends. Allstate is an example, which consistently pays quarterly dividends to shareholders.

There are a number of methods to calculate a company’s value. **Book value** utilizes the company’s accounting measures (in particular, assets and liabilities) to determine its worth. There are numerous ways to calculate book value, but the specifics aren’t important for the exam. Generally speaking, assume book value represents a company’s overall value from the perspective of an accountant.

The **price to book ratio** compares a company’s stock price to its book value. Although it’s unlikely you’ll be asked to perform a calculation, here’s the formula:

$PB ratio=Book value per shareCommon stock market price $

The higher a company’s stock price in comparison to its book value, the higher its price-to-book ratio. High price-to-book ratios may indicate a company is overvalued. Think about it - what if an accountant thought a company was worth $2 million (book value), when it was being sold for $100 million (market price)? This example would result in a price-to-book ratio of 50. For reference, most stocks don’t exceed a ratio of 5:1, and the average ratio in the S&P 500 is roughly 3.

The lower a company’s stock price in comparison to its book value, the lower its price-to-book ratio. Low price-to-book ratios may indicate a company is undervalued. Think about it - what if an accountant thought a company was worth $2 million (book value), when it was being sold for $1 million (market price)? The example would result in a price-to-book ratio of 0.5, well below the average ratio of 3 in the S&P 500.

After corporations pay for the cost of goods sold, operational expenses, interest & principal on outstanding debts, and taxes, it has **net earnings** (profits). To better understand this, let’s look at the sample income statement from the fundamental analysis chapter:

Line item | Amount |
---|---|

Sales revenue | +$200,000 |

Cost of goods sold (COGS) | -$80,000 |

Gross profit |
$120,000 |

Operating expenses | -$30,000 |

*Income from operations (EBIT) |
$90,000 |

Interest (bonds & loans) | -$25,000 |

*Income before taxes (EBT) |
$65,000 |

Taxes | -$10,000 |

Net income |
$55,000 |

**EBIT = earnings before interest & taxes*

**EBT = earnings before taxes*

The corporation in this example has $55,000 of net income, better known as net earnings (profit). When profits exist, corporations can utilize them in one of three ways:

- Retain the profits for future business expenses
- Distribute the profits to shareholders (cash dividend)
- Retain part, distribute part

Growth companies tend to retain profits so they have capital (money) to expand the business further. Value companies, which tend to be large and profitable, typically distribute part of their profits to shareholders by cash dividends and keep part of their profits for future business expenses. This is common for dividend-paying companies. In fact, it’s very rare to find a company that distributes 100% of its earnings to shareholders.

In these circumstances, investors are often interested in knowing their company’s **dividend payout ratio**. Here’s the formula:

$Dividend payout ratio=Net income (earnings)Total dividends paid $

With this formula in mind, let’s re-share the example income statement, but with two added lines at the bottom:

Line item | Amount |
---|---|

Sales revenue | +$200,000 |

Cost of goods sold (COGS) | -$80,000 |

Gross profit |
$120,000 |

Operating expenses | -$30,000 |

*Income from operations (EBIT) |
$90,000 |

Interest (bonds & loans) | -$25,000 |

*Income before taxes (EBT) |
$65,000 |

Taxes | -$10,000 |

Net income |
$55,000 |

Dividends paid | -$20,000 |

Retained earnings |
$35,000 |

**EBIT = earnings before interest & taxes*

**EBT = earnings before taxes*

Can you calculate the dividend payout ratio?

(spoiler)

Answer = **36.4%**

To calculate the dividend payout ratio, we need two items - net income and dividends paid. The corporation reported $55,000 in net income, but only paid out $20,000 in dividends. Now we can perform the calculation:

$Dividend payout ratio=$55,000$20,000 $

$Dividend payout ratio=36.4%$

It’s possible to be asked to perform this calculation with components displayed on a per-share basis. Let’s look at an example question:

An investor is researching a stock and performing several calculations to determine its quality. They find the following pieces of data:

- Quarterly dividend = $1.00
- EPS = $10.00

When information is presented in this manner, the dividend payout ratio formula slightly changes:

$Dividend payout ratio=EPSAnnual dividends $

**Earnings per share (EPS)** measures the profitability of a corporation on a per-share basis. For example, a company reporting $10,000,000 of net earnings with 1,000,000 shares outstanding would report an EPS of $10. Here’s the formula for EPS:

$EPS=Shares outstandingNet earnings $

$EPS=1,000,000$10,000,000 $

$EPS=$10$

Now that we’ve covered EPS, let’s go back to the original question on the dividend payout ratio:

An investor is researching a stock and performing several calculations to determine its quality. They find the following pieces of data:

- Quarterly dividend = $1.00
- EPS = $10.00
What is the stock’s dividend payout ratio?

Can you figure it out?

(spoiler)

Answer = **40%**

Be careful here! Dividends are reported on a quarterly basis, but the dividend payout ratio is measured on an annual basis. If the stock pays a $1.00 quarterly dividend, it must pay a $4.00 annual dividend. The EPS is already stated on an annual basis, so no adjustment is needed to our EPS of $10.00.

Now, do the formula:

$Dividend payout ratio=EPSAnnual dividends $

$Dividend payout ratio=$10.00$4.00 $

$Dividend payout ratio=40%$

In summary, the dividend payout ratio measures the level of profits shared with investors. Growth companies tend to have low dividend payout ratios (or no DPR if they don’t pay dividends), while value companies typically have higher payout ratios.

In the dividend models and discounted cash flow chapters, we discussed the concept of the **time value of money**. A dollar received today is worth more than a dollar received in the future due to **opportunity cost**. Money received in the future misses out on potential returns if it were invested today.

For the rest of this chapter, we’ll discuss this concept further with a specific emphasis on the following:

- Present value (review)
- Net present value (NPV)
- Internal rate of return (IRR)

*This section (Present value) is a repeat from the previous discounted cash flow chapter, but the NPV and IRR sections in this chapter are new. Regardless, you should re-read this section as the following sections build upon the example discussed below.*

We already discussed **present value** in the discounted cash flow chapter, but let’s go ahead and refresh ourselves on the topic. An investor can determine the current value of future money received through this present value calculation:

$PV=(1+DR)_{n}FV where:PVFVDRn =present value=future value=discount rate=# of years $

Let’s break down the components of this formula, then work through some numbers to better understand this topic. **Future value** is the amount of return (money) to be received in the future. The **discount rate** represents the average rate of return in the market. This is an important factor as it demonstrates the missed opportunity a person experiences if they must wait to receive money in the future. For example, it can be assumed the investor is missing out on a 5% return if the average rate of return in the market is 5%. The ‘**n**’ in the formula refers to the number of years the investor must wait to receive the future return.

Let’s work through this example:

An investor is considering the purchase of a $1,000 par, 2-year, 5% corporate debenture currently trading at 97. The rate of return in the market is 6%. What is the present value of the debenture?

With the information provided in the question, we can calculate the bond’s present value. This is a two-year bond, so we’ll need to do two present value calculations - one for the return received after one year, and another for the return received after two years.

**Present value - year 1**

This bond pays a 5% coupon, which is always based on the bond’s par value ($1,000). Therefore, this bond will pay $50 of annual interest to the investor. In the first year of ownership, this is the only return the investor will gain. Let’s do the first year’s present value calculation:

$PV=(1+DR)_{n}FV $

$PV=(1+0.06)_{1}$50 $

$PV=1.06$50 $

$PV=$47.17$

If the investor must wait a full year to receive $50 when the average rate of return in the market is 6%, then the future return is only worth $47.17 in “today dollars.” Here’s another way to think about it - if the investor had $47.17 today and obtained an average 6% return, they would earn a return of roughly $2.83 ($47.17 x 6%). Earning $2.83 on an original investment of $47.17 results in a total of $50 after one year ($47.17 + $2.83). That’s why $50 received after one full year is considered equivalent to $47.17 today. It’s all about the missed opportunity!

**Present value - year 2**

The bond will pay another $50 in the second year, plus the investor will also receive the $1,000 par value at maturity. With this bond trading at a discount, the investor officially “earns” the discount at maturity. Bottom line - the investor receives $1,050 at the end of year two due to the combination of interest and par value. Let’s do the second year’s present value calculation:

$PV=(1+DR)_{n}FV $

$PV=(1+0.06)_{2}$1,050 $

$PV=1.06_{2}$1,050 $

$PV=1.1236$1,050 $

$PV=$934.50$

If the investor must wait two full years to receive $1,050 when the average rate of return in the market is 6%, then the future return is only worth $934.50 in “today dollars.” Here’s another way to think about it - if the investor had $934.50 today and obtained an average 6% return (compounded over two years), they would earn a return of roughly $115.50 (represents a compounded 6% return on $934.50 over two years). Earning $115.50 on an original investment of $934.50 results in a total of $1,050 after one year ($934.50 + $115.50). That’s why $1,050 received after two full years is considered equivalent to $934.50 today. Again, it’s all about the missed opportunity!

**Putting it all together**

To determine the total present value of the bond, we will now add the two years of present value we just calculated:

$Total PV=Year 1 PV + Year 2 PV$

$Total PV=$47.17 + $934.50$

$Total PV=$981.67$

From a pure “time value of money” perspective, the present value of the bond is $981.67. We discounted the future cash flow from this bond back to its value in today’s dollars. By doing so, we have an indicator of the bond’s value. We can now utilize this information to determine whether the bond is a good or bad deal at its current market price. Here’s a quick clue - it might be underpriced!

Once a bond’s present value is determined, it should be compared to its market value (a.k.a. cost) to determine if an investment should be made. This can be accomplished through a **net present value (NPV)** calculation:

$NPV=Present value - investment cost$

In the present value section above, we determined the following:

- Bond’s market price =
**$970.00** - Bond’s present value =
**$981.67**

With this information, we can perform an NPV calculation:

$NPV=Present value - investment cost$

$NPV=$981.67 - $970.00$

$NPV=$11.67$

Our present value calculation estimated the value of the bond based on its future cash flows. The present value - $981.67 - is higher than the market price of the bond ($970.00). At the current market price, this bond is underpriced according to our time value of money analysis. Specifically, it’s underpriced by $11.67. A positive NPV indicates an investment is a “good deal” and should be acquired. If an investor is doing a discounted cash flow analysis on multiple investments, the one with the highest positive NPV should be purchased.

A negative NPV indicates the opposite. Let’s reset our numbers and assume the following:

- Bond’s market price =
**$990.00** - Bond’s present value =
**$981.67**

Now, let’s perform the NPV calculation:

$NPV=Present value - investment cost$

$NPV=$981.67 - $990.00$

$NPV=-$8.33$

Again, the present value estimates the value of the bond based on its future cash flows. The present value - $981.67 - is lower than the market price of the bond ($990.00). At the current market price, this bond is overpriced according to our time value of money analysis. We think it’s only worth $981.67, but it’s trading for $990.00. Simply put, we believe it’s overpriced by $8.83. A negative NPV indicates an investment is a “bad deal” and should be avoided.

Present value calculations are based on the missed opportunity, measured by the average rate of return in the market (the discount rate). While NPV calculations tend to focus on dollar amounts (present value vs. market value), it’s actually a reflection of an investment’s rate of return as compared to average market returns. Even a bond with a negative NPV will likely result in a profit. Assuming the $990 market price above, the investor would still receive $50 in annual interest, plus keep the $10 discount. NPV isn’t as much of. a reflection of profitability as it is a reflection of returns above market averages.

With that being said, a positive NPV indicates an investment’s returns are better than the market average. This relates to the investment being considered underpriced; the lower the price, the higher the return. On the other hand, a negative NPV indicates an investment’s returns are worse than the market average. This relates to the investment being considered overpriced; the higher the price, the lower the return.

You might be asking this - what if the NPV is zero? Good question! Quick answer - it means the investment is appropriately* priced. With a present value equivalent to its market value, the investment isn’t a good or bad deal. We can also relate this to market returns; a zero NPV indicates the investment’s returns are equivalent to the average market return. This will be important to note for our next section.

**When an investment is appropriately priced, the market it trades in is efficient. The more efficient a market, the more its prices reflect true value. On the other hand, an inefficient market has over and/or underpriced investments, which would reflect positive and/or negative NPVs.*

An investment’s **internal rate of return (IRR)** measures its overall rate of return. The term ‘internal’ means this measurement only focuses on the specifics of the investment, not external forces (e.g. inflation, other market risks). There’s a commonly accepted “textbook” definition of IRR:

The IRR is the discount rate that results in the NPV of all future cash flows being equal to zero

What’s this statement saying? When an investment’s NPV is equal to zero, its overall rate of return is equal to the average market return. To better understand this, let’s refresh ourselves with the numbers referenced above:

An investor is considering the purchase of a $1,000 par, 2-year, 5% corporate debenture currently trading at 97. The rate of return in the market is 6%.

After performing a few calculations, we found the present value of the bond to be $981.67. If the bond were to be trading exactly at $981.67, it would have an NPV of zero. Assuming this were true, we could safely assume the IRR (overall rate of return) of the bond to be * equal* to the average market return (6%).

Now, let’s go back to the original market price - $970.00. At this price, we calculate a positive NPV of $11.67 (we calculated this above in the NPV section). When an investment demonstrates a positive NPV, we can assume its IRR to be * higher* than the average market return (6%). Remember, the lower the price (or the more underpriced), the higher the return.

What if the market price was above the present value? For example, let’s assume the market price was $990.00. This would demonstrate an NPV of -$8.33 ($981.67 - $990.00). When an investment demonstrates a negative NPV, we can assume its IRR to be * lower* than the average market return (6%). Remember, the higher the price (or the more overpriced), the lower the return.

Let’s go ahead and summarize what we’ve learned:

NPV | IRR |
---|---|

Positive | Greater than average market return |

Zero | Equal to average market return |

Negative | Lower than average market return |

A bond’s IRR is equal to its yield to maturity (YTM). As we discussed in the investment vehicles unit, YTM represents a bond’s overall rate of return if held to maturity. This is being mentioned because test questions may use the terms IRR and YTM interchangeably.

As we’ve demonstrated, we can discount the value of future cash flows back to today’s present value. With investments like bonds, it’s very easy to determine future cash flows. Bonds pay fixed semi-annual interest, plus the par value at maturity. There’s no guessing as to what the future cash flow will be. However, these tools become less useful when future cash flow is unpredictable. This is why present value, NPV, and IRR calculations are not typically associated with securities like common stock*. Some common stocks don’t pay cash dividends at all, and therefore calculating future cash flows is impossible. Even companies that pay dividends on their common stock tend to increase their dividend payments at various rates, and some even suspend or cancel their dividend payments.

**While present value, NPV, and IRR calculations are not typically utilized for common stock due to its unpredictable future cash flow, it can be used for preferred stock. As a reminder, preferred stock pays a fixed, predictable dividend rate.*

Bottom line - time value of money calculations are most appropriate when future cash flow is predictable. The less predictable it is, the less relevant and accurate present value, NPV, and IRR calculations are.

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

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-month) 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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