Friday, 21 September 2012

The 2007 Credit Crisis: A Summary - Part 2

In the previous part of this topic, we covered the two major factors that set off the crisis. We left off the last part at the burst of the real estate market; In this part, we will explore the causes and costs of this crisis.


After the real estate bubble burst, investors in Mortgage-Backed Securities (MBS) incurred big losses. The value of the MBS tranches created from subprime mortgages was monitored by a series of indices known as ABX. By the end of 2007, these indices showed that the tranches originally rated BBB had lost about 80% of their value. By mid-2009, they lost almost all of their value, at 97%.

The value of the ABS CDO tranches created from BBB tranches was monitored by a series of indices known as TABX. These indiced indicated that tranches rated AAA lost approximately 80% of their value by the end of 2007 and became worthless by mid-2009.

Many financial institutions had big positions in them and had to be rescued  with government funds or taken over by other banks. Among them were UBS, Merrill Lynch, Citigroup, AIG, Bear Sterns, etc. Most notorious was probably Lehman Brothers, which was "allowed" to fail. Many of them are not U.S. banks, and all of them have businesses worldwide. Such huge impact in the largest  economy in the world was felt globally and this is how the credit crisis spread around the world.

What were the other consequences?

The Credit Crisis. The losses on MBS led to a severe credit crisis. Before this crisis, banks were well capitalized, loans were relatively easy to obtain, and credit spreads were low. (Credit Spead is the excess of the loan interest rate over risk-free interest rate.) By 2008, banks' capital had been badly eroded by their losses; therefore, they were more reluctant to lend. Credit spreads increased dramatically. Creditworthy individuals and companies (not just in the U.S.) found themselves in a difficult situation as borrowing became costly and hard to come by. 

So what went wrong?

First and foremost, it was the behavior of investors during the period leading up to the crisis. It could be described as "irrational exuberance,"  a phrase coined by former Chairman of the Federal Reserve Board, Alan Greenspan in the 1990s. This phrase could be interpreted as a warning that the market might be somewhat overvalued (as it was first used during the Dot-com bubble.) Mortgage lenders, investors in ABS & ABS CDO, and companies that sold protection on those instruments thought that the good time would last forever. This was a deadly mistake.

In addition to that, mortgage originators used lax lending standards, trying to earn as much profits as possible. Complex structured products (ABS, ABS CDO) were developed to transfer the risks to investors.

Rating agencies also played a crucial part in the crisis. They moved away from their traditional expertise of rating bonds to rating structured products, which was at the time relatively new and for which there was little historical data. Moreover, the procedures used to rate such products were not properly designed as they were created mirroring they way agencies rate bonds (which were of course different from structured products in many aspects.) The sophisticated products along with the lack of information about the underlying assets made investors rely more on rating agencies rather than their own judgements.

Structured products like those shown in part 1 of this topic are highly dependent on the default correlation between the underlying assets. The Default Correlation measures the tendency for different borrowers to default at about the same time. Even in normal times, mortgages exhibit moderate default correlation, which means that lower tranches of ABSs and ABS CDOs are more prone to losing their principals. although AAA-rated tranches could be fairly safe. However, investors, mortgage originators, along with many other market participants did not properly factor in "panic." In stressed market conditions, default correlations increases dramatically, making high default rates possible.

Agency Cost could be another factor that worsened the crisis. The term is used to describe the cost in a situation where the interests of two parties in a business relationship are not perfectly aligned. 
  • Between the mortgage originators and the investors, their gains and costs were not aligned. For originators, they needed to make more profits by creating and selling as much of their structure products to investors as possible by making bad lending decisions. The crisis might have been less severe if the originators had been required to to keep, for example 20% of each of the tranche created. By having the stake in all the tranches, originators would be more motivated to make the same lending decisions that the investors would.
  • Between financial institutions and their employees, agency costs are created by the employee compensation, which falls into 3 categories: regular salary, end-of-year bonuses, and stock or stock options. Many employees in financial institutions, particularly traders, received their compensations in the form of end-of-year bonuses. This type of compensation focused on short-term performance, which means that the employee would try to make huge profits one year even though his decisions would incur severe loss in the next year. He would be rewarded handsomely in the first year but would not have to pay back his bonus in the second year. This could explain why even when many financial institution employees knew that the real estate bubble would burst soon, they still decided to go on with their ABS CDOs investments. By the end of that year they will get huge bonuses regardless of the losses after that.




Source: Chapter 8: Securitization and the Credit Crisis of 2007, Fundamentals of Futures and Options Markets. John C. Hull.


Wednesday, 19 September 2012

The 2007 Credit Crisis: A summary - Part 1

Whether you're having a class project related to this topic or new to economics/finance and trying to understand how this crisis came about, this post will provide a solid and easy to understand explanation for it. This post is organized based on the content of Chapter 8: "Securitization and the Credit Crisis of 2007" in the book Fundamentals of Futures and Options Markets by John C. Hull. In part 1 of this topic, we'll cover the major factors and the onset of the crisis (real estate buble). In part 2, we'll explore the causes and costs (credit crisis) of it.

To better understand how the crisis started, we can start by asking and answering questions concerning the two factors that set the stage for it: Securitization and the US Housing Market.

a. What is securitization?

Traditionally, banks fund their loans primarily with deposits. However, in 1960s US banks found that they could not keep pace with the rising demand of mortgage with this type of funding any more. This led to the development of mortgage-backed securities (MBS) market, where cash flows generated from the mortgage portfolios were packaged and sold to investors as securities. MBS is a type of Asset-backed Securities (ABS) and the process of creating them are called securitization. The Government National Mortgage Association (Ginnie Mae), created in 1968, guaranteed interest and principal payments on qualifying mortgages and created the securities that were sold to investors. Theoretically, Ginnie Mae's guarantee protected MBS investors against defaults by borrowers.

b. How is an Asset-Backed Security created?

The Asset-backed securities, used during the period of 2000-2007, were typically created as shown in the picture below.
How an ABS is created
A portfolio of income-producing assets (such as loans) are bundled together and sold to a special purpose vehicle (SPV) by the originating banks. Cash flows from these assets - payments by mortgage borrowers - are then allocated into different tranches: Senior tranche, Mezzanine tranche and Equity tranche. 

The portfolio has a principal of $100 million, which is divided into the tranches as in the picture above. For example: the Senior tranche has a share of $80 million from the portfolio's principal and promises a return of LIBOR + 60 basis points or bp (1bp = 0.01%). 

c. How does it work? 

Think about the way cash flows are allocated to tranches as a waterfall (see picture below). Investors in the senior tranche would first receive their promised return. After the Senior investors have received their return in full, cash flows will then be allocated to the other two tranches, starting with Mezzanine. In the same manner, Equity investors will receive the residual amount (if any) after Mezzanine investors have received their return. Since the payments of principal and interest are not guaranteed, to compensate for the risk, the equity tranche has the highest expected return while the senior tranche has lowest expected return. As a result, Senior tranche and Mezzanine tranche are rated AAA and BBB respectively by rating agencies (S&P, Moody's, and Fitch) while Equity tranche is typically unrated.


Allocation of cash flows in an ABS

d. What happens when the mortgage borrowers fail to pay their loans?


If there are losses on the underlying assets, the first 5% of the principal ($5 mil) is borne by the equity tranche. Losses over 5%, Equity tranches loses all its principal and the amount exceeding 5% will be borne by the Mezzanine tranche. Accordingly, when the losses go over 20%, Equity and Mezzanine tranches lose all their principals, the loss amount exceeding 20% is borne by the Senior tranche.

e. How do they make riskier tranches attractive to investors?

Most investors are willing to buy the Senior tranches (AAA-rated) while the Equity tranches are retained by the originators or sold to hedge funds. To make the less desirable Mezzanine tranches appealing to investors, Asset-Backed Security Collateralized Debt Obligations (ABS CDO or Mezz ABS CDO) are created, very much similarly to the way an ABS is created. In addition to tranching out the whole portfolio, banks split the Mezzanine tranche of an ABS into smaller Mezzanine tranches as shown in the picture below.
A simplified ABS CDO
As you can see, the Senior tranche of an ABS CDO makes up 65% of the principal of the ABS Mezzanine tranche - which accounts for 15% of the whole portfolio). Mezzanine and Equity tranches of the ABS CDO accounts for 25% and 10% of the ABS Mezzanine respectively.By design, the Senior tranche of ABS CDO is rated AAA. This makes the total percentage of AAA-rated instruments created from the portfolio 89.75% (i.e. 80% + 65% x 15%)

f. What are the risk of the new tranches created in the ABS CDO?

The Senior tranche of the ABS Mezzanine is riskier than the Senior tranche of the ABS although they are both rated AAA. An investor in the Senior tranche of an ABS may still get his principal and interest when the loss on the underlying portfolio is 20% or less; However, for an investor in the Senior tranche of an ABS CDO, he needs the loss to be at most 10.25% (i.e. 100% - 89.75%). A loss of 20% would wipe out the whole Mezzanine tranche and of course all of his investment in it. In the same way, we can figure out the risk of the rest of the tranches of an ABS CDO. The following table will summarize the losses to an AAA-rated tranches of ABS CDO in different scenarios:

Losses on Underlying Assets
Losses to Mezzanine tranche of ABS
Losses to Equity tranche of ABS CDO
Losses to Mezzanine tranche of ABS CDO
Losses to Senior tranche of ABS CDO
10%
33.3%
100.0%
93.3%
0.0%
13%
53.3%
100.0%
100.0%
28.2%
17%
80.0%
100.0%
100.0%
69.2%
20%
100.0%
100.0%
100.0%
100.0%

Now that we've had a clear picture of the development of sophisticated financial instruments and the increasing risk associated with them, let's have a look at the second factor - the U.S. Housing Market.

g. What's wrong with the US Housing Market?

The US House prices from the year 2000 onwards had been rising much faster than they had in the previous decades as you can see from the P&P/Case-Shiller composite-10 index of U.S real estate. The composite-10 index measures the change in value of residential real estate in 10 metropolitan areas of the U.S.

S&P/Case-Shiller 10-City Composite Home Price Index

What happened to the US housing market in the period of 2000 to 2006 was the increase in subprime mortgage lending, which was significantly riskier than average. As a consequence, houses got a lot more expensive.

h. Why were subprime mortgages highly risky? And why were there so many of them? 

To mortgage brokers and mortgage lenders, more lending meant more profits. And to be able to lend more, more people needed to be qualified for mortgages. As a results, families that had previously been considered not creditworthy enough for mortgages then had access to mortgage loans. Those were called subprime mortgages. This relaxation of lending standards created increase in demand, which in turn, led to the increasing prices of real estates. This again resulted in more aggressive lending activities. And this cycle went on and on.
The problem was that when prices rose, it was more difficult for first-time buyers to acquire houses. In order to attract new entrants into the market, brokers had to find ways to relax the lending standards even more. Their solutions: the amount lent as a percentage of house prices increased. Adjustable-rate mortgages (ARMS) were developed where mortgage borrowers would first enjoy very low interest rate - a "teaser" rate (about 6%) that lasted for two or three years and for the subsequent years paid at a rate that was much higher (LIBOR + 6%). The risk of subprime mortgages was very high because the possibility that subprime mortgage borrowers default on their loans was much higher than that of prime mortgage borrowers. However, this risk was not high enough for brokers and lenders to ignore the huge profits coming from subprime mortgages.

i. How do subprime mortgages relate to securitization?

Subprime mortgages were securitized (into MBS) in the ways shown in questions (b) and (e). Investors in MBS usually did not know or care about vital information such as creditworthiness of the borrowers. The only pieces of information that they received were the loan-to-value ratio and the borrowers' FICO score.
  • Loan-to-value ratio is the ratio of the size of the loan to the assessed value of the house.
  • FICO is a credit score developed by the Fair Isaac Corporation and is widely used in mortgage lending.
Surprisingly, both the ratio and the score were of doubtful quality as they could easily be modified to give a better financial standing of the borrower. 

j. Why was the government not regulating the behavior of mortgage lenders? 

The government actually played quite a big part in the crisis. The U.S. government had since the 1990s been trying to expand home ownership and had been applying pressure to mortgage lenders to increase loans to low and moderate-income people.  

k. How was the onset of the crisis? 

The relaxation of lending standard created a bubble in house prices.  As prices kept increasing from 2000 to 2006, by 2007 many mortgage holders found that they could no longer afford their mortgage when the "teaser" rates period ended. This led to foreclosures of a large number of households in the market. Additionally, the nonrecourse feature of the US household market (i.e. in case of default, lender are able to take possession of the house but other assets of the borrowers are off-limits) effectively created  free American-style put options for the borrowers. This means that the borrowers can at any time sell the house to lender for the principal outstanding on the mortgage. Consequently, many houses were then put back on the market, making house prices plunge.

l. what of the mortgage defaulters?

Not all mortgage defaulters were in the same positions. Some had to give up their only homes when they defaulted. Others were speculators, which means that they purchased multiple homes as rental properties. When the real estate market crashed, they would reasonably exercised their put options. However, some got creative. After selling their homes to the lenders, they right away bought other houses that were in for foreclosure (at a real bargain). Here's an example to illustrate their strategy. Two people living in identical houses in the same neighborhood have mortgages of $250,000 each. Both houses are worth $200,000 and in foreclosure can be expected to sell for $170,000. The optimal strategy is, and this was what they did, to sell their houses back to the lenders and buy each other's houses after foreclosure.

m. What happened when the bubble burst?

As foreclosures increased, the loss on mortgages also increased. Houses in foreclosure were often in poor condition and sold for a small fraction of their value prior to the crisis. in 2008 and 2009, average losses were as high as 75% of the original values. 


Lenders, brokers, and investors also suffered greatly from this crisis. In part 2, we'll look into the causes and costs of it.

Thursday, 13 September 2012

CFA Level I - Financial Statement Analysis: An Introduction

For other CFA Level I topics, click here.

This topic will cover the following items:
  • basic accounting methods
  • 3 important financial statements
  • footnotes and supplementary information
  • audit and internal controls
  • financial statement analysis framwork

1. Basic Accounting Methods:
  • Cash-basis accounting - this method recognizes revenue and expenses when payments are made or cash is received.
  • Accrual accounting - This method recognizes  revenue in the accounting period in which it is earned (revenue is recognized when the company provides a product or service to a customer, regardless of when the company gets paid). Expenses are recorded when they are incurred instead of when they are paid.

2. The Financial Statements (3)

The Income statement shows the performance of a company over a specific period of time. The main elements of income statement include:
  • Revenues measures how much company earns by delivering goods or services in a specific period.
  • Expenses measures how much cost incurs when delivering goods and services in a specific period
  • Other Income includes gains that may or may not arise in the ordinary course of business
 
The Balance sheet shows the firm’s financial position at a point of time.
  • Assets are economic resources controlled by the firm.
  • Liabilities represents economic obligations (debts) payable to a person or organization outside the business.
  • Owner’s equity is the residual interest in the net assets of a firm after deducting liabilities.

Fundamental accounting equation: Asset = Liabilities + Owners’ equity


The Cash flow statement is the report of company’s cash receipts or payments during a period of time. Cash flows are classified as follows:
  • Operating cash flow involves transactions from the day-to-day business activities of the company.
  • Investing cash flow is associated with the acquisition and disposal of fixed or long-term assets of the companies.
  • Financing cash flow is related to the receipt or payment of capital to be used in business such as from retirement of long-term debt, issuance of new shares or dividend payment. 


3. Financial statement notes and supplementary information

Footnotes contain information about the methods of calculation or assumptions used in preparing financial statements. Information such as acquisition and disposals, legal actions, or sales to related parties or segments of the firm is also included. Footnotes are audited.

Supplementary schedules are additional disclosures such as operating sales by regions or business segments, reserves for an oil and gas company, or information about hedging activities and financial instruments. Supplementary schedules are not audited.

Management discussion and analysis (MD&A): Under U.S GAAP, publicly held companies are required to include in their financial reports a section for MD&A. It is also possible to include this section in the financial statements of companies reporting under IFRS. Management must highlight any favorable or unfavorable trends and identify significant events that affect the company’ solvency and performance and operation. MD&A must include:
  • Effects of inflation and changing prices of material.
  • Impact of off-balance-sheet obligations and contractual obligations such as purchase commitments.
  • Accounting policies that require significant judgement by management.
  • Forward-looking expenditures and divestitures.


4. Audit activities on financial statements and internal controls

Audit refers to the activities of reviewing company financial statements by independent parties such as public accounting practices or audit firms. The audit process provides auditors with opinion on fairness and accuracy the company’s accounting and internal control systems. Standard auditor’s opinion contains 3 parts:  
  • Auditors perform independent review whereas the financial statements are prepared by management.
  • General auditing standards are followed, thus providing reasonable assurance that the  financial statements contains no material errors.
  • The auditors are satisfied that the statements were prepared according with accepted accounting principles and that the principles chosen and estimates should be reasonable. Auditor’s report must contain additional explanation when accounting method is not used consistently between periods.

Under Us General Accepted Accounting Principles (GAAP), auditor’s opinion must be indicated in companies’ internal control. There are:
  • Unqualified opinion indicates that the auditor believe that the statements are sound, i.e. free from material errors or omission 
  • Qualified opinion is issued when there are exceptions to the accounting principles and these must be explained in audit reports.
  • Adverse opinion shows that the financial statements are not presented fairly and contains materials not in compliance with accounting standards.

Auditor’s opinion should also contain explanatory paragraph when a material loss is probable but the amount cannot be reasonably estimated. These uncertainties may relate to the going concern assumption, the valuation or realization of asset values.


Internal controls are the process by which a company ensures accuracy in financial reports. Under US GAAP, auditors must state opinion on the company’s internal control.


5. Other sources of information useful for financial statement analysis

Quarterly or Semi-annual Report: these are interim financial reports that include major updates in financial statements and footnotes. These statements are not audited.

Securities and Exchange Commission (SEC) filings are documents that need to be filed to the Securities and Exchange Commission annually or quarterly regarding the consolidated financial statements of a company. Common forms are:
  • Form 10-K are required annual fillings. It includes information about the business and its managements, audited financial statements and disclosures about legal issues.
  • Form 10-Q contains quarterly financial statements and disclosure of important events
  • Form 8-K includes disclosure of company’s events such as acquisition, disposals of major assets or changes in governance

Proxy statements are issued to shareholders when there are matters that require a shareholder vote. They are also filed with SEC and are a major source of information about board members, compensations, management qualifications and stock options.

Corporate report and press releases includes information used for public relation or sales materials and are not audited.


6. Financial statement analysis framework

Step 1: State the purpose and context of the analysis. Identify questions that the analysis needs to answer, information that needs to be presented, the time and resources to perform the analysis.

Step 2: Collect data from financial statements, industry and economic research, customers, suppliers and site visits.

Step 3: Process Data. Make appropriate adjustments to the financial statements, calculate ratios, and prepare necessary exhibits (graphs, common-size balance sheets, etc.)

Step 4: Analyze and Interpret Data. Use the processed data to answer the questions that are specified in the first step and decide the appropriate conclusion and recommendation

Step 5: Report Conclusions or Recommendations to the intended audience.

Step 6: Update the Analysis. Update data in the analysis and change the conclusions or recommendations if necessary.

Sunday, 9 September 2012

CFA Level I: Quantitative Methods - Statistical Concepts and Market Returns (Part 2)

For list of other CFA Level I topics, click here.
For the previous part of this topic, click here.

In part 2 of this topic, we are going to cover the following items:
- Calculations with quantiles
- measures of central tendency (mean, mode, median, range,  standard deviation, variance,  etc.)
- skewness and kurtosis of distributions

1. Calculations with quatiles

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies.
  • Quartile - the distribution is divided into quarters
  • Quintile - the distribution is divided into fifths
  • Decile - the distribution is divided into tenths
  • Percentile - the distribution is divided into hundredths (percents) 


The equation for the position of the observation at a given percentile y , with n data points sorted in ascending order is:
Ly = (n + 1)y/100

The following example is taken from the CFA Level I curriculum (2011) as an illustration of the concepts above.

No.
Company
Div Yield (%)
No.
Company
Div Yield (%)
1
AstraZeneca
0.00
26
UBS
2.65
2
BP
0.00
27
Tesco
2.95
3
Deutsche Telekom
0.00
28
Total
3.11
4
HSBC Holdings
0.00
29
GlaxoSmithKline
3.31
5
Credit Suisse Group
0.26
30
BT Group
3.34
6
L’Oreal
1.09
31
Unilever
3.53
7
SwissRe
1.27
32
BASF
3.59
8
Roche Holding
1.33
33
Santander Central Hispano
3.66
9
Munich Re Group
1.36
34
Banco Bilbao Vizcaya Argentina
3.67
10
General Assicurazioni
1.39
35
Diageo
3.68
11
Vodafone Group
1.41
36
HBOS
3.78
12
Carrefour
1.51
37
E.ON
3.87
13
Nokia
1.75
38
Shell Transport and Co.
3.88
14
Novartis
1.81
39
Barclays
4.06
15
Allianz
1.92
40
Royal Dutch Petroleum Co.
4.27
16
Koninklije Philips Electronics
2.01
41
Fortis
4.28
17
Siemens
2.16
42
Bayer
4.45
18
Deutsche Bank
2.27
43
DaimlerChrysler
4.68
19
Telecom Italia
2.27
44
Suez
5.13
20
AXA
2.39
45
Aviva
5.15
21
Telefonica
2.49
46
Eni
5.66
22
Nestle
2.55
47
ING Group
6.16
23
Royal Bank of Scotland Group
2.60
48
Prudential
6.43
24
ABN-AMRO Holding
2.65
49
Lloyds TSB
7.68
25
BNP Paribas
2.65
50
AEGON
8.14

a. Caluclate the 10th and 90th percentile
b. Calculate first, second, and third quartile
c. Find Median

Answers

a. In this example: n = 50, using the equation Ly = (n + 1)y/100 for the position of the yth percentile (Py)

For the 10th percentile: L10 = (50 + 1)(10/100) = 5.1
L10 is between the 5th and 6th observations with values X5 = 0.26 (Credit Suisse Group) and X6 = 1.09 (L’Oreal). The estimate of the 10th percentile (first decile) for the dividend yield is
P10 ≈ X5 + (L10 – 5)(X6 – X5) = 0.26 + (5.1 – 5)(1.09 – 0.26) = 0.34% 

For the 90th percentile:  L90 = (50 + 1)(90/100) =45.9
L90 is between the 45th and 46th observations with X45 = 5.15 and X46 = 5.66. The estimate of the 90th percentile is
   P90 ≈ X45 + (L90 – 45)(X46 – X45) = 5.15 + (45.9 – 45)(5.66 – 5.15) = 5.61%

Note: In the calculations above, P10 shows that 10th percentile lies (5.1 – 5) = 10% of the distance between the 5th and 6th observations. The distance between the 5th and 6th observations is 1.09 – 0.26 = 0.83, 10% of that distance is 0.083. We obtain P10 by adding this value (0.083) to the closest observation before L10 (i.e. X5).  The calculation for P90 is exactly the same.

b. The first, second, and third quartile correspond to P25, P50, and P75 respectively. 
L25 = (50 + 1)(25/100) = 12.75
L50 = (50 + 1)(50/100) = 25.50
L75 = (50 + 1)(75/100) = 38.25
Using the same way we calculate the positions of the 10th and 90th percentile in the previous question, we obtain the following results
P25 = Q1 = 1.69%         P50 = Q2 = 2.65%         P75 = Q3 = 3.93%

c. The median is the 50th percentile, 2.65%.

2. Range, Mean Absolute Deviation, Variance, Standard Deviation, and Chebyshev's Inequality

Range is the distance between the largest and the smallest value in a data set
range = max value – min value

The Mean Absolute Deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean
Population Variance (σ2) is the average of squared deviations from the mean.  
Population Standard Deviation (σ) is a measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance. 

Example: Find MAD, variance, and standard deviation of the following set of investment returns [5%, 15%, 22%, 12%, 7%]

Mean = (5 + 15 + 22 + 12 + 7)/5 = 12.2%
MAD = (|5 – 12.2| + |15 – 12.2| + |22 – 12.2| + |12 – 12.2| + |7 – 12.2|)/5 = 5.04%
This result can be interpreted to mean that, on average, an individual return deviate +/- 5.04% from the mean return of 12.2% 
Variance = σ2 = [(5 – 12.2)2 + (7 – 12.2)2 + (12 – 12.2)2 + (15 – 12.2)2 + (22 – 12.2)2]/5 = 36.56 (%2) 
Standard Deviation = σ = 6.05%

Sample variance (s2) is the measure of dispersion that applies when we evaluate a sample of n observations from a population.
Sample Standard Deviation (s) is the square root of sample variance


Chebyshev's Inequality states that for any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of observations that lie within k standard deviations of the mean is at least 1 – 1/k2 for all k > 1

According to Chebyshev's Inequality, the following relationships hold for any distribution. At least:
  • 36% of observations lie within ± 1.25 standard deviations of the mean
  • 56% of observations lie within ± 1.50 standard deviations of the mean 
  • 75% of observations lie within ± 2 standard deviations of the mean
  • 89% of observations lie within ± 3 standard deviations of the mean 
  • 94% of observations lie within ± 4 standard deviations of the mean
Example: find out the minimum percentage of any distribution that will lie within ± 2.5 standard deviations of the mean.


3. Coefficient of Variance, Sharpe Ratio, 

Coefficient of Variantion (CV) is a statistical measure of the dispersion of data points in a data series around the mean. In the investing world, the coefficient of variation allows you to determine how much volatility (risk) you are assuming in comparison to the amount of return you can expect from your investment.
Example: Given monthly the mean return on T-bills is 0.25% (usually represents risk-free rate) with a standard deviation of 0.36% and the mean monthly return for S&P500 is 1.09% with a standard deviation of 7.3%. Calculate and interprete the CVs of these 2 investments.

CVT-bills = 0.36/0.25 = 1.44
CVS&P500 = 7.3/1.09 = 6.70
The reults indicate that there is less dispersion (risk) per unit of monthly return for T-bills than for S&P500

The Sharpe Ratio (Reward-to-variability ratio) measures excess return per unit of risk. Investments with large positive Sharpe ratios are preferred to portfolios with smaller ratios.

Note: Limitations of the Sharpe Ratio
  • If 2 porfolios have negative Sharpe ratios, it is not necessarily true that the higher Sharpe ratio means better risk-adjusted performance.
  • Sharpe ratio is useful when standard deviation is an appropriate measure of risk. However, investment strategies with option characteristics have asymmetric return distributions (i.e. large probability of small gains and small probability of large losses). In such cases, standard deviation may underestimate risk and produce high Sharpe ratios. 
4. Skewness and Kurtosis

A distribution is symmetrical if it is shaped identically on both sides of its mean. In finance, it means that intervals of losses and gains will exhibit the same frequency. 
Skewness refers to the extent to which a distribution is not symmetrical. This depends on the occurrence of outliers in the data set. Outliers are the observations with extraordinary large values, either positive ornegative
  • A positively skewed distribution is chracterized by many outliers in the upper region (right tail).
  • A negatively skewed distribution has many outliers in the lower region (left tail)
The skewness affects the location of the mean, median, and mode of a nonsymmetrical, unimodal distribution.

Kurtosis is a measure of the degree to which a distribution is more or less "peaked" than a normal distribution.
  • Leptokurtic - more peaked than a normal distribution
  • Platykurtic - flatter than a normal distribution
  • Mesokurtic - same kurtosis as a normal distribution

The kurtosis for normal distribution is 3. If a distribution has more or less kurtosis than the normal distribution, it is said to exhibit excess kurtosis.
  • Normal distribution has excess kurtosis = 0
  • Leptokurtic distribution has excess kurtosis > 0
  • Platykurtic distribution has excess kurtosis < 0
To find out the skewness of a sample, apply the following formula
 Note: if |SK| > 0.5, the distribution has a significant level of skewness
Sample Kurtosis is measured using the following formula
The sample kurtosis is measured relative to the kurtosis of a normal distribution, which is 3
 
Excess Kurtosis = Sample Kurtosis 3

Excess kurtosis > 0, the distribution is leptokurtic (more peaked, fat tail)
Excess kurtosis < 0, the distribution is platokurtic (less peaked, thin tail)
Excess kurtosis > 1 in absolute value is considered large.