WorldQuant-Financial Markets

01-CreditRisk-and-Financing.md

Credit Risk and Financing

Bond Pricing

Consider a two-year bond. You receive a coupon, say, in six months. You receive another six months later, another six months later, and the last one in two years, along with the principal amount. So, if we think about the total amount, what we must consider is that the separate times will result in those amounts having different value today. In other words, we simply cannot add cash flows at all different time points in the future and expect to have a straightforward way to sum them today.

Suppose we have $1,000 invested for one year at 4%. What we're doing is something called future valuing. This helps us to calculate what that $1,000 will be 1 year from now. The answer is $1,040. What if we did that problem in reverse? What if we discounted $1,040 that we receive one year from now to today’s dollars? This is known as present-valuing. What is that worth today?

Discounting

When you take an amount in the present and imagine how much it's worth in the future, we call this future valuing.

When you take an amount in the future—an amount that has not occurred yet—and bring it back to today, then this is called present valuing. It is also known as discounting.

Higher Interest Rates Mean Lower Price

Yield to Maturity (YTM)

It is the total return anticipated on a bond if it is held until it matures. YTM is expressed as an annual percentage rate (i.e., a rate per annum).

In the example you provided, you have two cash flows. The first one, $40, is expected one year from now, and the second one, $1,040, is expected two years from now. The interest rate, which you've left blank in your description, is what we would use to discount these cash flows to the present. The process of discounting is necessary to account for the time value of money - the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity.

The Yield to Maturity assumes that all coupon payments (in this case, the $40 after one year) are reinvested at the same rate as the bond's current yield and takes into account both the capital gain or loss at the end of the bond's life and the potential income collected along the way.

To calculate YTM, you solve for the interest rate in the bond's present value formula which equates the present value of its future cash flows (coupons and principal repayment) to its current market price.

If bond yields (interest rates) rise after you've bought a bond, the price of the bond generally drops. This happens because when rates increase, the fixed interest payments of the bond become less attractive compared to what could be earned in other investments. Hence, to attract buyers, the price of the bond has to decrease to offer the same return as the prevailing interest rates. This inverse relationship between bond prices and yields is fundamental in bond investing.

Buy-Side vs. Sell-Side

The buy-side, is professionally managing client money. At times, those investment managers may decide to buy investments. At times, they want to realize the capital gains on those investments and sell the investments. As such, the buy side is both buying and selling securities, but they are doing so on behalf of investors.

The sell-side, on the other hand, is NOT acting on behalf of investors but rather is acting on behalf of the investment bank. The sell-side’s primary responsibility is to make markets. Market-making means that they are willing to engage with the buy-side (or even other sell-side firms) by both offering securities to sell and bidding for securities to buy. You see, the buy-side is trading on behalf of customers; they are looking for opportunities that appeal to their customer base. The sell-side, on the other hand, is simply trading according to the needs and desires of the buy-side and other sell-side firms who want to trade. Sell-side trades make what is known as a two-sided market.

How does each side effectively make money? In broad terms, the buy-side makes profit by earning a fee for managing the funds. In addition to a management fee, they may collect fees related to the trading in the account or consulting with clients, and they might collect a performance incentive fee. If the fund earned above a certain amount (which may be 0%), then the fund may take a percentage of the amount earned. Effectively, the buy-side acts as a fiduciary for its clients by offering professional management and is paid both a fixed fee and potentially an incentive fee for good performance. The sell-side, however, makes money in a more complicated way. To understand this, let us look at an example by considering a buy-side firm called PDQ. Let us say that they wish to set up a brokerage account with a sell-side firm called ABC. ABC is willing to do two things at any given time:

• ABC is willing to sell shares of our risky bond XYZ at $101.00
• ABC is willing to buy shares of our risky bond XYZ at $100.50.

The price at which ABC is offering to sell is known as the offer price. Like everything there is also another way to refer to this—the ask price. We will use offer price and ask price interchangeably since they mean the same thing.

The price at which ABC is willing to buy is known as the bid price.

Bid Ask Spread

The bid price is the price at which someone is willing to buy. The ask price is the price at which someone is willing to sell. The difference between these two prices is known as the bid-ask spread. The sell-side makes a profit by this bid-ask spread.

If you have ever been to an airport kiosk that provides a foreign exchange service, then you will understand this. Suppose you arrive at the airport wanting to convert euros to Japanese yen. Your 100 euros bought you 12,800 yen. Then, you find out that your flight got cancelled. So you return to the airport kiosk and convert the 12,800 yen back to euros—but you only get 95 euros. The kiosk makes money by selling you yen at a high price and buying back the yen at a low price.

The same ideas apply to sell-side market makers. The bid-ask spread, and any transaction costs, are part of the revenue of the sell-side. You can see that the sell-side has a tough job because to buy and sell, they must have inventory amounts that may drive them to hold either too much of a security or run out of it and borrow the security. Indeed, it is possible to sell things that you don't even have.

Shorting

The idea of buying low and selling high is very straightforward, but less so is that the order can be reversed. In other words, what if you were to “sell high, then buy low?” This is indeed a trading strategy. But how can you sell something you don’t own? This is known as shorting. Shorting refers to borrowing a security that you do not own, selling it in the marketplace and receiving cash for it. The short seller then hopes and waits for the price to drop, and if and when it does, the short seller buys the security at this lower price.

Financing Short Positions

Whoever lends out the security will want to receive an amount of income for it. This is known as the financing cost. You can agree that your broker on the sell-side can lend out your bonds to another customer. When you do so, you will receive whatever the financing cost is from that short seller. Financing provides extra income to the owner of a security.

Shorting is a remarkably interesting part of the market because it serves the purpose of preventing assets from getting overvalued. If there's consensus that a financial asset is overvalued, then short sellers come into the market and put downward pressure on that asset, helping it to achieve its natural equilibrium state. If there is no mechanism to short, then a market can become frozen. That means that sellers have prices that are too high, and there are no buyers willing to pay the prices.

Shorting is a risky business because of the potential of an unlimited loss. When you think about the fact that prices can jump up, there are potentially catastrophic losses to the short seller. We have been discussing bonds, but imagine you apply this to other asset classes for which there are no natural ceilings to the prices—like stocks or Bitcoin. During the global recession, there was a lack of confidence in the financial institutions themselves. It got to a point where so many investors were shorting the financial institutions that the government intervened and prohibited short selling of financials.

Central Banks

While we have discussed distinct kinds of banks, there is one type of bank we have yet to discuss—the central bank. Central banks are financial institutions whose purpose is to maintain stability and transparency for countries' economies. Central banks help to keep inflation in check, provide maximum employment opportunities, and enforce healthy monetary policy for a stable economy.

02-Return-and-volatility.md

Return and Volatility

Stocks vs. Bonds

When a corporation issues a bond, the bond buyer is effectively giving a loan to the company with the repayment date of the maturity date. During the life of the bond, the bondholder receives periodic interest payments as a way of receiving interest. When a corporation issues stock, the stock buyer is effectively giving a loan to a company, but with no repayment date. That is, the money collected when stocks are issued is never paid back by the company to the stockholder.

That is the main conceptual difference between the bond buyer and the stock buyer. . Their funds will almost assuredly be paid back, so long as the company has sufficient assets. . In fact, many companies will declare bankruptcy, pay their bondholders first, and then have little if any funds to pay their stockholders. Typically, this stock will not go to zero, but it can be much, much less than what the price was when the stock traded. For this reason, bonds are safer than stocks. This explains why bondholders are paid first and paid fixed amounts at specified times—all before any money can be distributed to stockholders.

For the bond buyer to receive interest and principal, the company merely needs to remain solvent. For a stockholder to receive dividends, the company must remain solvent, be profitable, and decide to pay dividends (as opposed to retaining the earnings). Since that is more difficult and uncertain, stocks tend to be more volatile.

Finally, for a stockholder to receive capital gains, the company has to create the perception (even if it is not real) that there is greater value, which justifies a higher share price. Of course, companies might fail to do this, which creates more risk. The lack of guarantees on dividends or price gains makes stocks riskier.

Market capitalization

Market capitalization = Number of Shares outstanding * Current Share Price

The right-hand side consists of two terms. The first is the number of shares the company has outstanding. Typically, these shares are created at the time of the IPO. However, a company may issue additional shares. When shares are repurchased by the company itself, these are known as treasury stock. In total, all of the shares combined should represent 100% of the ownership of the firm.

Stock ownership represents an exclusive and complete ownership of the firm. Typically, the number of shares does not change over time. Doing so would dilute existing investors. For example, if a firm had issued a million shares and decides to issue a million more, then they would be diluting the original investors, watering down the amount of ownership those owners have. When a company needs to raise additional funds, it is possible to have new shares issued, but typically the existing shareholders have the right to maintain their proportion of ownership (for a fee).

MODELING THE PERFORMANCE OF STOCKS

Normally, if you were to pull up a financial stock series, you would see its price over time. That is the y-axis would indicate the price range, the x-axis a period of time, say, the last two years. For stocks, prices never go below zero. For bonds, yield would be on the y-axis, and it is possible that yields can go negative.

Volatility in financial markets is a broader concept than just standard deviation and encompasses the fluctuation and variation in market prices over time. Volatility is influenced by various factors such as market participants, news events, and economic conditions.

When visualizing financial time series, it's important to consider the timescales and match them appropriately. Shorter timescales, such as intraday or daily returns, emphasize financial properties and shorter-term trading opportunities, while longer timescales, such as monthly or quarterly returns, highlight larger economic cycles. The choice of timescale depends on the specific analysis and objectives.

Different intervals of returns have different characteristics and can emphasize different aspects of the data. Daily returns may contain noise and short-term fluctuations, while monthly returns might miss some trading opportunities. It is generally preferred to have higher frequency data and then decide on the appropriate interval of returns for analysis.

Analyzing the distribution of returns is important in assessing whether they follow a normal distribution. The normal distribution is convenient because it simplifies data representation by summarizing it with a mean and standard deviation. However, it cannot be assumed that financial data follows a normal distribution, as market behavior is dynamic and can deviate from the assumptions of independence and identical distribution required by the Central Limit Theorem. Therefore, it is necessary to assess the basic features of the distribution to determine if it is Gaussian or normal.

When examining the distribution of returns, it is important to assess whether the distribution is symmetric or not. Symmetry is a characteristic of the Gaussian or normal distribution, where the distribution is the same on both sides of the mean. However, in finance, it is common to observe distributions that are not symmetric, with more extreme values on one side compared to the other.

The presence of more extreme values on the left side of the distribution, indicating larger losses, is a common trend in equity markets. This reflects the occurrence of panic days or market downturns that result in significant negative drops. This asymmetry is influenced by behavioral biases and the tendency of individuals to remember and respond more strongly to negative events than positive ones.

Finance is not solely governed by physical laws like physics but is influenced by human behavior, cognitive biases, and social interactions. The complex interactions between humans and computers in financial markets make it challenging to model returns purely as if they were from physical systems.

To visually assess symmetry, two approaches can be used:

• Draw a histogram of the returns and check if the bars are equally populated on both sides.
• Create a density plot. If the resulting shape resembles a normal distribution, the overall distribution can be considered approximately normal. Departures from normality will be visually apparent.
• These visual techniques help in identifying departures from symmetry and deviations from a normal distribution in financial returns.

Another way to assess the assumption of normality is to examine the constancy of volatility. While calculating the standard deviation of a distribution yields a single value, it does not provide information about the uncertainty associated with that standard deviation.

The second moment of a distribution, which measures variance, gives the squared variation around the mean. To assess the accuracy of this squared variation, we can examine the excess kurtosis. Excess kurtosis serves as a quality check on the variance. A normal distribution has no excess kurtosis because the variance is known with certainty. However, if an empirical distribution exhibits excess kurtosis, it likely indicates that the distribution did not come from a normal distribution.

Assessing constancy of volatility and examining measures such as excess kurtosis helps determine if the assumption of normality holds and whether the distribution of returns deviates from a purely Gaussian distribution.

There are distributions available that introduce skewness and kurtosis, such as the F distribution, skew normal distribution, log normal distribution, and others. Skewness and kurtosis can be formally tested for and there are robust measures that can handle them, such as focusing on ranks instead of values or using one side of returns (e.g., negative returns) while ignoring positive returns.

Assessing whether returns follow a normal distribution or not is crucial, and it requires examining the data to determine if normality still holds. If returns are normally distributed, estimation of percentiles and probabilities becomes easier using the mean and standard deviation. However, if the normal distribution is inappropriate, alternative distributions that capture skewness, kurtosis, or both may be used, which may involve additional modeling skills or nonparametric methods.

03-Correlation.md

Correlation

Risk Premium

This is essentially the extra return that an investor requires to hold a risky asset instead of a risk-free asset. For instance, if the risk-free rate of return is 2% and an investor requires a 5% return to hold a specific risky asset, the risk premium would be 3% (5% - 2%). The size of the risk premium varies depending on an individual investor's risk tolerance. Risk-averse investors will require a larger risk premium to hold a risky asset, while risk-neutral or risk-loving investors would require a smaller risk premium or none at all.

Certainty Equivalent Income

This is the guaranteed amount of money that an investor would accept rather than taking a chance on a higher, but uncertain, amount. For instance, an investor might be indifferent between receiving a guaranteed $50 and a 50/50 chance of receiving either $100 or nothing. In this case, the certainty equivalent income is $50. It's called a "certainty equivalent" because it makes the investor just as happy as the uncertain prospect does.

The relationship between risk premium, certainty equivalent income, and an individual's risk preference can be expressed as:

y_CE = E(y) - (λ/2) * σ_y^2

Here, y_CE is the certainty equivalent income, E(y) is the expected return, λ is the risk preference measure of the individual (also known as the degree of risk aversion), and σ_y^2 is the variance of returns (a measure of risk).

Expected Value-Variance Criterion

The Expected Value-Variance Criterion is a method of ranking investment options for risk-averse investors. It involves ranking investments based on their expected returns (E(y)) and variances (σ^2).

The key assumptions in this criterion are:

• Investors prefer more to less – meaning they prefer higher expected returns.
• Investors are risk averse – meaning they prefer lower variance or risk.

However, when one investment has a higher expected return and higher variance than another investment, we can't definitively rank the two investments using this criterion. This is because some investors may be willing to accept the higher risk (variance) for a higher expected return, while others may not.

The set of investment options that are ranked as preferred for risk-averse decision makers is known as the Efficient Frontier or the Expected Value-Variance (EV) efficient set. In an EV Frontier graph, any point that lies on the line of the Efficient Frontier represents an investment that offers the maximum possible expected return for a given level of risk.

Direct and Indirect Outcome Variables

Typically, we connect a risky event ϵ to a random variable y(ϵ). For example, ϵ may represent uncertain prices and y(ϵ) may represent income which depends on uncertain prices ϵ. We will refer to ϵ as the direct outcome variable and y(ϵ) as the indirect outcome variable over which the firm’s utility is defined.

In most risk models, the relationship between ϵ and y(ϵ) is monotonic, if the direct outcome variable goes up (down) so does the indirect outcome variable. If prices increase, so does income; if the variance of ϵ increases, so does the variance of y(ϵ). However, one can easily construct examples in which the linkage is not so direct. For example, suppose the firm faces financial stress and that only very favorable outcomes will permit the firm to meet its cash flow obligations and survive. Under such circumstances, the firm may increase its expected income and its probability of surviving by choosing a strategy that increases its variance of direct outcomes. Consider such a problem by defining an indirect outcome variable w = 0 if the firm fails to survive and w = 1 if the firm survives. In this model the direct outcome variable is y. Then we connect the indirect outcome variable w to the direct outcome variable y by defining a survival income yd and defining w in terms of y as follows:

w =

• 0 for y < y_d
• 1 for y ≥ y_d

The ad hoc decision rule can be expresses as:

V(y) =

• U(0) for y < y_d
• U(1) for y ≥ y_d

Since U(0) and U(1) are arbitrarily assigned values, let U(0) = 0 and U(1) = 1. Then, the ad hoc decision rule to be maximized is:

Max 1 - Pr(y < y_d)

which calls for minimizing Pr(y < yd). This rule is known as Roy’s safety-first rule. When direct outcomes are defined in terms of winning or losing, Roy’s safety-first rule is consistent with an expected utility model.

Of course, one can think of other direct and indirect outcome variable relationships. For example, y could represent uninsured income and w could represent insured income, or y could represent unhedged income and w could represent hedged income, or y could represent income produced without risk reducing inputs and w could represent income produced with risk reducing inputs.

So, what have we learned? We learned that risk responses are defined over direct outcome variables. Failure to distinguish between indirect and direct outcome variables may lead us to view responses to indirect outcome variables as risk preferring when they are indeed risk averting.

Firm response to Risk

The extent of risk aversion determines the choice of investments. Those with high risk aversion choose investments with lower expected values and variances, whereas less risk-averse individuals opt for investments with higher expected values and variances.

Firms strive to move towards a risk position on the expected value-variance (EV) frontier, representing optimal trade-offs between risk and return. This movement may involve purchasing insurance to shift risk, or adjusting the balance of safe and risky investments in the portfolio.

Sharing risky outcomes

The average of multiple random variables can also create a new random variable. For instance, when two business owners combine their operations and agree to share average earnings, the expected value each owner receives remains the same, but the variance reduces by half, reducing their individual risk. As more partners join, the variance of each partner's income reduces further.

This concept is illustrated by a hypothetical scenario where a business' earnings are determined by coin toss outcomes. When the outcomes of two independent tosses are averaged, the standard deviation (which measures risk) reduces significantly, even though the expected value remains the same. This reduction in risk becomes even more significant as the number of people sharing outcomes increases.

Portfolio return

Portfolio Standard Deviation

rho: is the correlation coefficient between the returns of stocks A and B.

Eg:

• Stock A: 5% return, 3% standard deviation.
• Stock B: 11% return, 8% standard deviation.
• The correlation between A and B is 70%.

Portfolio Return and Variance

Recall from the last module that we took a stock’s volatility and divided it by the stock’s expected return. This is the coefficient of variation. We can do the same for a portfolio.

We could use this number to compare to other portfolios, and determine the number of ‘vol’ points we pay per unit of return. As always, the smaller this number, the greater the "bang for the buck." Generally, this number is used in relative comparisons rather than absolute comparisons. Could we compare it to Security B itself? For Security B,

Interestingly, the portfolio with A and B has a lower "vol per unit of return cost" than stock B alone. By including stock A in the portfolio, the portfolio somehow has a better risk-return relationship than B alone.

Portfolio Sharpe Ratio

Likewise, we took the stock’s excess return (the return minus the risk-free rate) divided by the stock’s standard deviation. This is known as the Sharpe ratio. We can do the same for a portfolio:

To calculate the portfolio’s Sharpe ratio, we need the risk-free rate. Let’s suppose this is 1%.

Let’s compare it to the Sharpe ratio of stock A this time.

Interestingly, the portfolio with A and B has a higher Sharpe ratio, that is, a higher return per unit of volatility than stock A alone. By including stock B in the portfolio, the portfolio seems to have a better risk-return relationship than A alone.

mark-to-market/model

When we use two stocks, it is likely very easy to ensure that they have observable prices. But what if we combined one asset that is exchange traded and another that rarely trades, such as real estate. Since exchanges have official closes, it is easy to get the stock price correct. Each day we can mark our position using the market’s close. This process is known as mark-to-market. We mark our security prices to levels indicated by the market. But how would we know the price of an asset without exchange closes or OTC closing prices? We may need a mark-to-model price. We would mark our security at levels indicated by a model, provided by our company or perhaps even regulatory requirements.

Diversification in investment

Adding a risky investment may reduce overall portfolio risk. It uses the example of a company investing in both sunglasses and umbrellas. Both investments provide a 10% expected return, but their returns fluctuate with weather conditions - when one investment does poorly (due to unfavorable weather conditions), the other does well. This results in a consistent portfolio return and zero standard deviation, indicating no risk, thanks to their perfect negative correlation.

A correlation of -1 means that you have perfect negative correlation. In financial terms this is known as a perfect hedge! Let us remember that -1 is the maximum amount of diversification one could ever hope to achieve. For the above example, portfolio can have a smaller standard deviation (or variance) than either of the individual standard deviations (or variances).

• Having a negative correlation lowers the variance. With 0 correlation, we are simply not adding to the variance.
• When there is perfect negative correlation, then the variances offset each other. 8% - 3% = 5%. Since we have 50% of each, we effectively have half that number: 2.5%.
• When there is perfect positive correlation, then the variances combine with each other. 8% + 3% = 11%. The average of these is effectively half that number: 5.5%.

Purchase Risk Reducing Investments

This text discusses the concept of risk-reducing investments, which are undertaken primarily to decrease the variability of returns, even though they may also affect expected incomes. This is illustrated with a case study of an investment in an irrigation system for a farm.

The firm in the case study is assumed to face five possible moisture states (normal, low stress, moderate stress, high stress, and drought), each with an equal probability of occurring.

With the irrigation system, the expected return per acre is lower ($87.60 vs. $89.60 without), but so is the standard deviation of returns, indicating reduced risk. A risk-neutral decision maker would opt not to invest in the irrigation system due to the higher expected returns without it. However, a risk-averse decision maker might still choose to invest in the system, as it reduces the variability of returns.

This is demonstrated using the concept of certainty equivalent income, which takes into account both expected returns and the variance of returns. The certainty equivalent income without the irrigation system is higher due to the higher expected returns. However, with the irrigation system, the certainty equivalent income is lower but has less variability.

The decision to invest in the irrigation system or not depends on the risk aversion of the decision maker, denoted by λ. A calculation is provided to find the break-even λ that would make a decision maker indifferent between the two options. In this case, the break-even λ is .009. If the decision maker's actual λ is higher than .009, indicating higher risk aversion, they would be better off with the irrigation system despite its lower expected returns.

Correlation as Linear Association

Correlation measures linear dependency. This is known as Pearson correlation.

This is known as the Pearson correlation coefficient. It emphasizes the linear relationship between two series. Pearson’s coefficient measures the degree of linear association. One way to think of it is to assume you have two variables X & Y. You graph one versus the other. Then, you fit the best regression line to them. Correlation asks the following questions: How tightly wrapped around that line are the data points? Suppose all the points were along the line. Then, you have perfect correlation. If that line is sloping upwards, then you'd have a perfect positive correlation. If the line were downward sloping, then you'd have perfect negative correlation.

Spearman Correlation

Let's consider a functional relation between X and Y such that y = x cubed. This is clearly a cubic rather than a linear association.

The correlation in this example is 91.7%. It is high to be sure, but it is not 100%. However, we can say that every time x is larger, y is larger. In finance, it is helpful to know that two things move in the same direction, even if they move by different amounts. The Pearson correlation fails to capture that by focusing on linear association. The Pearson correlation is penalizing our cubic association because it is nonlinear.

Now, let’s consider the Spearman correlation. Imagine replacing the data points with the ranks of the data points. In other words, imagine if you had 10 data points in each set. You're going to replace the value of x with its rank: 1 refers to the largest data point, 2 to the second largest data point, and so on. Similarly, you do the same for y: replace the largest value of y with 1, replace the second largest value of y with 2, and so on. In the case that there are ties, then all those ties are assigned the same number. In our case, suppose there are two numbers tied for the largest. Ordinarily, these would have ranks 1 and 2. We average the ranks (1.5) and then assign each of these numbers to 1.5. Now, we have a set of ranks for the x values and a set of ranks for the y values. Now we apply the Pearson correlation to the ranks of the data. This method is exactly what the Spearman correlation does.

For each data point, Spearman computes a value, d, between the 2 ranks.

Think of Spearman as the correlation of ranked data. Ranking doesn’t care about linear or non-linear; it simply cares about maintaining the order of the data points. In our example, the ranks of x all line up with the ranks of y. There are no discrepancies in ranks. When we have the nth largest x value, we have the nth largest y value. Therefore, our d matrix is 0. The entire second term on the right-hand side drops out, and we have a correlation of 1.

The advantage of Spearman’s correlation over Pearson’s correlation is that we can simply see if things move together, instead of adding linearity to the requirement. Imagine if you are a trader and you wanted to know if one security goes up whether the other is also likely to go up. Which correlation would be more useful? Well, Pearson’s correlation would help. Spearman's correlation focuses entirely on ranks, so it would be a more useful way to measure co-movements. This method of replacing data with ranks is a technique from non-parametric statistics and can be helpful if data contains outliers or skewness, problems that we discussed in the previous module.

While Pearson’s correlation is meant to measure linear relationships between two random variables, the advantage of Spearman’s correlation over Pearson’s correlation is that it can detect any monotonic relationship and not just linear ones.

Correlation changes

The relationship between volatility and correlation depends on the level of volatility: when volatility is low, correlations tend to be lower between two series than when those series have high volatility.

The sobering fact from this finding affects what we know about diversifying a portfolio. Diversification—the very heart of what low correlation promises—can be yanked away when it is most needed: in highly volatile markets.

04-levarage-nonLinearity.md

Leverage and Nonlinearity

Call Option

A call option gives the holder the right, but not the obligation, to buy an underlying asset at a predetermined price within a specific timeframe. Call options are typically purchased when the investor expects the price of the underlying asset to increase.

Put Option

A put option gives the holder the right, but not the obligation, to sell an underlying asset at a predetermined price within a specific timeframe. Put options are typically purchased when the investor expects the price of the underlying asset to decrease.

Derivatives as a Whole

A derivative is a security whose value depends on the value of another security. It’s a derivative because its value is derived from the prices of another security. If we know the value of the underlying security, then through the mathematics of stochastic processes and the art of securities hedging and replication, we are likely going to know the price of the derivative. More precisely, a derivative is a security whose value depends on another security—or even another measurable quantity.

Option Dependancy

1. Underlying Security

The value of an option is dependent on its underlying security (such as a specific stock). In the case of a call option, as the price of the underlying security increases, the value of the call option also increases. Conversely, for a put option, as the price of the underlying security increases, the value of the put decreases.

2. Strike Price

The strike price, which is set at the time of purchasing the option, is a threshold that the price of the underlying security must reach for the option to be profitable (in-the-money). Higher strikes make call options worth less and put options worth more. Strike selection is crucial as it impacts the likelihood of the option being in-the-money at expiration.

3. Expiration Time

The value of an option is dependent on its expiration time. As options get closer to their expiration date, they generally lose time value (known as time decay), all else being equal. An option with a longer expiration date will be worth more due to the increased optionality it offers. However, if an option expires out-of-the-money, it becomes worthless.

4. Risk-Free Rate

The value of options is also affected by the prevailing interest rate. For call options, an increase in interest rates tends to increase their value, while for put options, an increase in interest rates tends to decrease their value.

• When interest rates increase, the opportunity cost of tying up funds in an option also increases. Investors have alternative investment opportunities with higher returns due to higher interest rates. This makes holding call options less attractive, leading to a decrease in their demand and subsequently their value.
• An increase in interest rates raises the opportunity cost of holding a put option because investors can potentially earn higher returns elsewhere. As a result, the demand for put options may decrease, leading to a reduction in their value.

5. Dividend Yield

The income-generating power of the underlying stock (dividend yield) also influences option prices. Specifically, call options decrease in value if dividend yields increase, while put options increase in value if dividend yields increase.

6. Volatility

The volatility of the underlying security is one of the most crucial factors affecting an option's price. Higher volatility increases the option's value as it increases the potential for more extreme price movements of the underlying, hence affecting the likelihood of the option ending up in-the-money at expiration.

In essence, the value of an option is a complex interplay of several variables, each of which must be considered when trading or investing in options.

Levarage

What is leverage? One definition of leverage is that it borrows capital to invest more deeply in an investment than merely by investing with the cash on hand. The return on capital is expected to exceed the rate of borrowing, so leverage is a double-bet: that the investment itself will have a positive return, and that the rate of return exceeds the cost of borrowing so that it will have a net positive return on borrowed funds as well.

The beauty of options is indeed that they are a leveraged investment because they leverage the returns. How? Not by borrowing funds. Indeed, an option can only be paid for in full. Instead, options use leverage through a conversion factor greater than 1. The conversion factor is the number of shares of stock that can be traded for one option. For stock options, the conversion factor is 100. One long equity option entitles the holder to trade 100 shares of the underlying stock. Buying one call option would grant the buyer the right (and not the obligation) to buy 100 shares of stock at the strike level. Note that this option costs less than buying 100 shares. So for a lower initial investment than buying stocks, or even buying stock on margin, the equity option offers a leveraged approach to investing.

Buying Stock Using Cash

The first thing we could do is buy the stock using cash. Suppose that the stock price is $500 per share. We want to trade 100 shares. We need $500/share * 100 shares = $50,000. Recall our expected return is 10%. So our $50,000 would increase 10% or $5,000. Our return is $5,000 on a $50,000 investment or 10%. Buying stocks for cash is unleveraged. The returns are in no way magnified because no debt was incurred to borrow funds and no derivatives were used to increase the exposure size. This is the most expensive way to make $5,000 because we need $50,000 of capital upfront to do so.

Leverage multiplies a raw return by the ratio of the total investment to your original investment. Leverage is indifferent to whether the return is positive or negative; it is a mere multiplier. Consequently, leverage can make a lot of money quickly, or it can also lose a lot of money quickly. Indeed, there are levered investments that cause losses people cannot sustain and can wipe out an entire investment portfolio. Being overleveraged is a risk.

Buying Options

Enter options. Options can only be bought fully with cash. That is, the option itself cannot be purchased using borrowed funds. This is because options already have leverage in them through a conversion factor rather than through borrowed funds. Recall that the conversion factor is 100: One option controls 100 shares of stock. If the stock price went up a dollar, in theory, the option should be worth about $100 more because each of those shares that it controls increases by one. It's not as simple as that because the pricing is a complex combination of stock price and the other five factors.

Stocks or Options?

So clearly if you were bullish on a stock, you could decide to use all the funds to buy an option rather than buying the underlying or buy extra underlying with borrowed funds. What are the differences among these strategies? Let's consider timing. Suppose we continued to believe the stock was going to increase 10%, but now we put a time horizon on that: within 1 month. So our three strategies would be:

• Buy the stock with cash
• Buy the stock with 50% cash and 50% borrowed funds
• Buy a call option that expires within the month at a suitable strike Now, suppose we get the time horizon wrong. It took 1 month and 2 days. The stocks make the same returns as in the previous example, but the option expires worthless. The stocks have the same returns because the underlying has no time constraint on earning a return—there is no expiration date.

Direction of Stock or Volatility

There is one way to avoid picking a direction. Suppose you are better at determining volatility than you are at estimating the direction the stock is headed. Instead of getting direction by choosing a call or a put, you can simply try to estimate the direction of volatility. In doing so, you can become a volatility trader. Suppose you were to buy a call and a put at the same strike. Here is what the payoff diagram looks like:

Now, what you are betting is the cost of two premiums for the stock's price to move a sizeable amount in either direction. Your total cost is the premium of a call and the premium of a put, both at the same strike. Since the cost is two premiums, the underlying stock has to move a considerable distance before there is a net profit. If the stock price moves off the strike level, then one of the options is in the money. However, this doesn’t mean that a payout amount is going to cover the cost of the two premiums. One of the two options will be exercised, but it may be insufficient to cover the payoff. A volatility trader is picking a direction of volatility, rather than direction of the stock. The volatility trader who opts for this strategy, known as a straddle, believes volatility will increase. They are long volatility. If volatility does increase, they will make some money; it if increases enough, then they should make a profit.

Option Payoff: Non-linear

The option payoff—whether for calls or puts—looks like hockey sticks. This is what is known as a nonlinear payoff. On the left side we have a flat line. This flat line goes from 0 to K. (Since stock prices are non-negative, we set the lower limit of the x-axis to 0). This means when the stock price is less than the strike level, the payoff for that call option is zero. The next part of the payoff is where it slopes upwards. This goes from K to infinity. Anytime the stock price exceeds the strike level, the option is in the money. When the stock price equals the strike level, the option is at the money (ATM). Even there, the payoff is zero, so the only positive payoff is when the stock price exceeds the strike level.

Home Equity as an Option

The Merton model states that you have some equity ownership in the house provided the home’s price exceeds the mortgage. In this case, your home equity is like a call option that is in the money. Note that the horizontal axis is the market value of the house. It’s not what you paid but the current market value. Since housing prices change with market conditions, this can be more or less than your purchase price.

Let’s use the same examples:

• Suppose the home’s market price is 120. You could sell the house at 120, repay the mortgage of 100, and pocket the 20 difference as P&L. Thus, as an investor, you could have enjoyed a capital gain of 20 on a house.
• Suppose instead the home’s market price is 100. When you sell the price at 100, you use the entire proceeds to pay off the mortgage of 100. There is nothing leftover; there is no home equity. You have no capital gain.
• Suppose instead the home’s market price is 90. When you sell at 90, you use the proceeds to pay the mortgage, but you fall short. When a house is worth less than the mortgage on it, we say the house is "underwater." This is the same meaning as out of the money. Instead of paying 100, you only have 90. Will the bank want to get the other 10 from you?

Well, according to the option payoff, the answer is no. The option framework simply flattens out. The non-linearity protects you on the downside.

Problems

Suppose you combined non-recourse loans with the ability to enter the mortgage with virtually no money down. In other words, suppose you are able to buy a home with a non-recourse mortgage with 100% financing. Now you have a bet.

• If the house price increases, flip the house: Sell the house, pay off the mortgage, and collect the difference.
• If the house price decreases, walk away.

In the first case, you make money. In the second case, you do not lose any money. See the problem? This is an amazing deal. It’s like a free lottery ticket. There’s a positive chance to make profits and virtually no chance to lose money. (Well, the reality is there are high carry costs for real estate, so there are chances to lose money). This type of market can and will invite speculators of all kinds. Speculators will simply buy homes with little to no money down, wait and hope for the home prices to increase, and then close the trade and exit with a profit. They try to repeat this as many times as possible until the trend reverses and housing prices decline. Then. they simply walk away, leaving banks with the keys to the house.

OPTION STRATEGIES AND SCENARIOS

Choosing the option’s strike is likely the most difficult choice to make. Choosing the strike is effectively choosing the size move within the option’s lifetime. The higher the strike, the less expensive the call, but choosing too high a strike may result in the option never getting in the money.

Hedging with Options

Let’s consider the trader who wishes to minimize risk with options. We’ll consider risk as volatility. We would like to minimize the standard deviation (or variance) of outcomes. Suppose you're a farmer and you need to purchase seed for this year’s harvest. You're willing to pay $50 a bushel for seed. If you were to pay more, you're unsure that you could sell it at a price that would keep the farm profitable. Therefore, you want to lock in the price of $50. You might first consider going to the futures market. The futures contract would cost nothing to enter. Perhaps you could find the price at $50 bushel in three months' time. But what if the price of the seed dropped over the next three months to $30? Your competition could buy it at that lower price, and sell their crops at lower rates than you because you were locked in at $50. Your higher costs would inevitably result in higher prices, weakening your competitiveness on price. Instead of a future, what if you had bought a call option? Compared to futures, options look expensive. The futures require no upfront capital but could lead to very unfavorable conditions. Although options have a premium, they never lead to trading at unfavorable prices. Suppose instead you were to purchase a call option struck at 50 to buy seed. One of two scenarios could occur.

• In scenario 1, the price of seed in three months goes well above 50. Having the option struck at 50, you have locked in that price and will buy the seed at a level much lower than the market. Later, you will be able to sell the crops at a very competitive price since your costs were relatively low.

• In scenario 2, the price of seed in three months goes below 50. Having an option to buy at 50 is unhelpful, so you simply don’t exercise it. You buy your seed at the lower price along with everyone else. The option served as an insurance against seed inflation.

When options are used to minimize risk, you can think of them as insurance. When futures are used, they can help to minimize risk, but they can create unfavorable terms due to price changes between the locked-in future price and the actual future price. Options minimize risk more because the premium is the maximum amount you could lose.

Similarly, we could construct an example of someone buying a put to minimize risk by guaranteeing a minimum price for their outputs. The same farmer might purchase a put to sell corn at $90 a bushel. For the cost of a premium, the farmer guarantees a sale at 90 even if it drops in the future, while still having the flexibility to get an even higher price than 90 if future prices increase.

Speculating with Options

• In strategy 1, you would simply buy calls. You have to pay the cost of the premium (with no financing). If the stock price goes up, within the option’s time frame, and past the strike you chose, then the option is ITM. Of course, you have to pick the right strike and the right expiration date.
• In strategy 2, you simply sell puts. Selling anything generates cash flow. Selling a put means you are exposed if the stock price drops. But if you strongly believe the stock will go up, you make money, within the option’s time frame, so long as it ends up higher than the put’s strike. This strategy does not require any upfront capital (except that if the price sussequently drops, then some cash will be needed to satisfy margin requirements).
• In strategy 3, you buy a call at a strike, say 100, and then sell a call at a higher strike, say 110. This is known as a bull spread strategy. This means that you're paying money for the first call but you're actually collecting money for the second call because it’s getting sold. Compared to strategy 1, this strategy is cheaper because you are "refunded" the cost of the second option. However, this limits the upside of your investment to the difference between the two strikes: in our example, $10. If you wanted to have a higher potential profit, then you could sell the second option at a higher strike. Strategy 3 is known as a bull spread. At the second strike, the two calls offset each other, so the P&L comes from the first call being (K2 - K1) dollars in the money. The bull spread is appropriate if you have “bounded” optimism rather than “unbounded” optimism about the stock’s upside.

05-liquidity-regulation.md

Liquidity and Regulation

5 Cs: capacity, capital, character, collateral, and conditions

Mortgage Origination

Jacob receives a mortgage loan of $400,000, which is a $400,000 debt or liability, and in combination with the down payment of $100,000, he can now afford to purchase the home. The bank has originated a mortgage, which consists of a corresponding $400,000 asset.

The bank can now hold on to this mortgage loan asset and wait for Jacob to make monthly payments of principal and interest until he has fully paid back the loan. However, there are at least two reasons that this may be undesirable from the bank's perspective:

• The mortgage term is likely 30 years, which is a long time for the bank to have its capital tied up in a single transaction. If the bank needs money during that time (say if depositors in the bank want to withdraw money), it could be hard for this bank to sell a single mortgage. A single mortgage is considered illiquid because most investors and other financial players do not want a single mortgage; there isn't a market for single mortgages. If the bank desperately needed to sell the mortgage loan because it needed money to pay its liabilities (e.g., the depositors want to withdraw their money), another bank might buy the mortgage but for less than it is worth. The risk of not being able to sell an asset at all or having to sell an asset at a steep discount is called liquidity risk.
• This retail bank may not have much capital, and $400,000 may be a sizable portion of their assets. Let's consider an extreme example and say that the bank only has $4 million in capital. That means the bank loaned Jacob 10% of their capital. If Jacob is the only borrower who defaults (doesn't make the monthly mortgage payments) on the loan, the bank's capital decreases to $3.6 million, which may be unacceptable to investors in the bank as well as the bank's senior management. Too much of their capital, and too much risk, is concentrated in a single borrower. This is known as concentration risk.

If the bank does not want to keep the mortgage because of these risks, it can sell Jacob's loan to an investment bank.

The Investment Bank

Investment banks have a major role in the securitization of mortgages and other assets.

The following motto summarizes the strategy of investment banks active in mortgage securitization: "We are in the moving business, not the storage business."

Since the investment bank moves these mortgages into an SPE, the investment bank doesn't keep the liquidity or concentration risks.

The Securitization Transformation

As you can see, a single mortgage-backed security is backed or collateralized by many thousands of individual mortgages. If an investor buys $1 million worth of MBSs, a single mortgage will only comprise a fraction of a percent of that investment. The concentration risk we discussed above has been diversified away. The MBS has diversified underlying assets (mortgages) in the sense that there is no concentration in a single mortgage borrower. However, the MBS is still completely concentrated in the larger mortgage market

Because this concentration risk has been at least partly eliminated, more investors want to buy MBSs—that is, there is an active market for MBSs. As long as there is an active market for buying and selling a security, it is considered liquid. The illiquid, concentrated single mortgage has been transformed into a relatively liquid security with diverse underlying mortgages.

Tranches and the Payment Waterfall

The differences between senior, mezzanine, and equity tranches of the collateralized debt obligation (CDO), especially in terms of risk and priority of cash flows.

continue reading -- https://learn.wqu.edu/my-path/courses/financial-markets/modules/m-5-liquidity-and-regulation/tasks/lesson-2-valuation-challenges-market-frictions-and-model-risk-lesson-notes

11-Misc.md

Long short market neutral strategy

A long-short market-neutral strategy is an investment strategy that involves taking long positions (buying securities that are expected to increase in value) and short positions (selling securities that are expected to decrease in value) in equal measure with the goal of creating a portfolio that is neutral to the overall market movements.

The strategy involves buying stocks that are expected to outperform and short selling stocks that are expected to underperform. The idea is to make money from the difference in returns between the long and short positions, while simultaneously reducing the overall risk of the portfolio by being market-neutral.

For example, suppose a portfolio manager believes that the technology sector will outperform the broader market while the financial sector will underperform. The manager might buy technology stocks and sell short financial stocks to create a market-neutral portfolio. If the technology sector outperforms the market and the financial sector underperforms, the manager would make a profit on the long positions and the short positions, resulting in a positive return for the portfolio overall.

For example, in 2019, the Emkay Alpha Fund identified a mispricing in the stock of Sun Pharmaceutical Industries, an Indian pharmaceutical company. The fund took a long position in Sun Pharma's stock while simultaneously taking a short position in the stock of Dr. Reddy's Laboratories, a competitor of Sun Pharma. At the time, Dr. Reddy's was viewed as the stronger company and had a higher market valuation than Sun Pharma. However, the Emkay Alpha Fund's analysis suggested that Sun Pharma had a stronger competitive position and was undervalued by the market. The fund's long position in Sun Pharma's stock generated significant returns, while the short position in Dr. Reddy's stock helped to reduce the overall risk of the portfolio.

The Efficient Market Hypothesis (EMH)

It suggests financial markets are efficient and that asset prices fully reflect all available information. According to the EMH, it is not possible to consistently outperform the market by using any publicly available information because prices already reflect all relevant information.

It has implications for investment strategies. If markets are efficient, it suggests that it is difficult to consistently beat the market through stock picking, market timing, or other investment techniques based on publicly available information. Therefore, the most effective approach may be to invest in a diversified portfolio of assets, such as index funds, and to focus on factors such as asset allocation and risk management rather than trying to predict or beat market movements.

The Capital Asset Pricing Model (CAPM)

It is a financial model that attempts to explain the relationship between the expected return of an asset and its systematic risk. It provides a framework for estimating the expected return on an investment based on its risk characteristics, specifically its sensitivity to market movements.

CAPM Formula: The CAPM formula calculates the expected return of an asset. It is expressed as: Expected Return = Risk-Free Rate + Beta × (Market-Return - Risk-Free Rate).

The CAPM provides a theoretical framework for estimating the expected return of an asset based on its risk characteristics. It assumes that investors are rational and risk-averse, and that they seek to maximize their expected return for a given level of risk.

Arbitrage Pricing Theory (APT)

It is a financial theory that attempts to explain the pricing of assets by considering multiple factors or risk sources that affect their expected returns. APT is based on the concept of arbitrage, which involves taking advantage of price discrepancies to make riskless profits.

Here are the key features and concepts of Arbitrage Pricing Theory:

• Multi-Factor Model: APT assumes that the expected return of an asset is influenced by multiple factors rather than just the market risk factor used in the CAPM. These factors can include macroeconomic variables, industry-specific factors, interest rates, inflation, and other market influences.
• Factor Sensitivities: APT analyzes the sensitivity of asset returns to various factors by estimating the factor loadings or sensitivities of the asset to each factor. These loadings represent the asset's exposure to systematic risks associated with each factor.
• Risk Premiums: APT asserts that the expected return on an asset is determined by the risk premiums associated with the different factors. The risk premium is the additional return investors require for holding an asset with exposure to a specific factor.
• Arbitrage Opportunities: APT assumes that if an asset is mispriced relative to its expected return based on the factor sensitivities, arbitrageurs will step in to exploit the mispricing and drive the asset's price back to its fair value. Arbitrageurs will construct portfolios that offset the factor exposures of the mispriced asset to generate riskless profits.
• No Specific Market Portfolio: Unlike the CAPM, APT does not assume the existence of a specific market portfolio. Instead, it focuses on the relationships between assets and their sensitivities to various factors.

The APT framework allows for a more flexible and nuanced analysis of asset pricing compared to the CAPM. By considering multiple factors, APT attempts to capture the sources of risk that drive asset returns more comprehensively.

The general formula for APT can be expressed as follows:

Expected Return = Risk-Free Rate + (Factor 1 Coefficient × Factor 1) + (Factor 2 Coefficient × Factor 2) + ... + (Factor n Coefficient × Factor n) + Error Term

The Greater Fool Theory

It is an investment concept that suggests that the price of an asset can be justified by the belief that there will always be a "greater fool" willing to buy it at a higher price, regardless of its intrinsic value. In other words, investors who follow the Greater Fool Theory buy an asset not based on its fundamental value, but rather with the expectation that they can sell it to someone else at a higher price.

Active Funds vs Passive Funds

Index funds are “passively managed”. They simply buy and hold stocks in the same proportion as an index such as Sensex 30. If a stock drops out of the index, they sell too. They copy, they mimic, and they mirror the index. That’s it.

So if the index was dumping JFS, they’d have to exit too. They didn’t care if the stock had a sound business or not.

Now some folks might point to this ‘attack’ on JFS and remark, “Look at the harm caused by the mindless mimicry by index funds. It’s distorting the market.”

But what if we told you that these events might actually be good for active fund managers? You know, the ones who spend hours poring through annual reports and speaking to management before making a buy or sell decision.

You see, the breed of active management has been going through some tough times. Their performance has been quite underwhelming and they just can’t seem to outperform the index.

But think of these situations— 👉🏼When the stock is being deleted from an index. Passive funds have no option but to sell en masse. It could knock down the share price. And if the fundamentals are strong, it gives a perfect buying opportunity for an active fund. 👉🏼When a stock is added to an index. Active fund managers can start buying the stock in the lead-up to that event. Because they know that passive funds have to wait on the sidelines. They’ll take action only once the stock makes its appearance in the index. And once it does, the massive influx of passive fund money can drive up the price. It’s called the Index Inclusion Effect. And the active fund manager who entered earlier benefits.

So, the active fund manager can basically ride the passive wave for their own benefit.