Thursday, July 31, 2008

Zero-Sum

In game theory and economic theory, zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Go is an example of a zero-sum game: it is impossible for both players to win. Zero-sum can be thought of more generally as constant sum where the benefits and losses to all players sum to the same value of money and pride and dignity. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others. In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses is either less than or more than zero. Zero sum games are also called Strictly Competitive.

The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game) ^[1]

Situations where participants can all gain or suffer together, are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.

The concept was first developed in game theory and consequently zero-sum situations are often called zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games.

1 Economics and non-zero-sum
2 Psychology and non-zero-sum
3 Complexity and non-zero-sum
4 Solution
5 An example
6 Solving zero-sum games
7 Extensions
8 References
9 More Readings

Economics and non-zero-sum

Many economic situations are not zero-sum, since valuable goods and services can be created, destroyed, or badly allocated, and any of these will create a net gain or loss. Assuming the counterparties are acting rationally, any commercial exchange is a non-zero-sum activity, because each party must consider the goods s/he is receiving as being at least fractionally more valuable to him/her than the goods he/she is delivering. Economic exchanges must benefit both parties enough above the zero-sum such that each party can overcome his or her transaction costs.

See also:

Absolute advantage
Comparative advantage
Free trade

Psychology and non-zero-sum

The most common or simple example from the subfield of Social Psychology is the concept of "Social Traps." In some cases we can enhance our collective well-being by pursuing our personal interests — or parties can pursue mutually destructive behavior as they choose their own ends.

Complexity and non-zero-sum

It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent. As former US President Bill Clinton states:

The more complex societies get and the more complex the networks of interdependence within and beyond community and national borders get, the more people are forced in their own interests to find non-zero-sum solutions. That is, win–win solutions instead of win–lose solutions.... Because we find as our interdependence increases that, on the whole, we do better when other people do better as well — so we have to find ways that we can all win, we have to accommodate each other.... Bill Clinton, Wired interview, December 2000 .[1]

Solution

For 2-player finite zero-sum games, the different game theoretic Solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. In the solution, players play a mixed strategy.

An example

*A zero sum game*
	A	B	C
1	30, -30	-10, 10	20, -20
2	10, -10	20, -20	-20, 20

A game's payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right.

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if Red in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, while Blue should assign the probabilities 0, 4/7 and 3/7 to the three actions A, B and C. Red will then win 20/7 points on average per game.

Solving zero-sum games

The Nash equilibrium for a two-player, zero-sum game can be found by solving a linear programming problem. Suppose a zero-sum game has a payoff matrix $M$ where element $M i, j$ is the payoff obtained when the minimizing player chooses pure strategy $i$ and the maximizing player chooses pure strategy $j$ (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of $M$ is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found by solving the following linear program to find a vector $u$ :

Minimize:

∑	u_i
i

Subject to the constraints:

u

≥ 0

M u

≥ 1

The first constraint says each element of the $u$ vector must be nonnegative, and the second constraint says each element of the $M u$ vector must be at least 1. For the resulting $u$ vector, the sum of its elements is the value of the game. And dividing $u$ by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.

If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.

The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of $M$ (adding a constant so it's positive), then solving the resulting game.

If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.

Extensions

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1) player representing the global profit or loss.^{[citation needed]}

References

^ Samuel Bowles: Microeconomics: Behavior, Institutions, and Evolution, Princeton University Press, pp. 33–36 (2004) ISBN 0691091633

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Variants of Yield to maturity

Given that many bonds have different characteristics, there are some variants of YTM:

Yield to Call: when a bond is callable (can be repurchased by the issuer before the maturity), the market looks also to the Yield to Call, which is the same calculation of the YTM, but assumes that the bond will be called, so the cashflow is shortened.

Yield to Put: same as Yield to Call, but when the bond holder has the option to sell the bond back to the issuer at a fixed price on specified date.

Yield to Worst: when a bond is callable, puttable, exchangeable, or has other features, the yield to worst is the lowest yield of Yield to Maturity, Yield to Call, Yield to Put, and others.

Example

Consider a 30-year zero coupon bond with a face value of $100. If the bond is priced at a yield-to-maturity of 10%, it will cost $5.73 today (the present value of this cash flow, 100*(1/(1.1³⁰) = 5.73)). Over the coming 30 years, the price will advance to $100, and the annualized return will be 10%.

But what happens in the meantime? Suppose that over the first 10 years of the holding period, interest rates decline, and the yield-to-maturity on the bond falls to 7%. With 20 years remaining to maturity, the price of the bond will be $25.84. Even though the yield-to-maturity for the remaining life of the bond is just 7%, and the yield-to-maturity bargained for when the bond was purchased was only 10%, the return earned over the first 10 years is 16.26%. This can be found by evaluating (1+i) = (25.84/5.73)^0.1 = 1.1626.

Over the remaining 20 years of the bond, the annual rate earned is not 16.26%, but 7%. This can be found by evaluating (1+i) = (100/25.84)^.05 = 1.07. Over the entire 30 year holding period, the original $5.73 invested matured to $100, so 10% annually was made, irrespective of interest rate changes in between.

Yield Spread Premium, YSP

The yield spread premium (YSP) is the cash rebate paid to a mortgage broker based on selling an interest rate above the wholesale par rate that the borrower qualifies for.

For example, If a mortgage broker offers a borrower a loan of $100,000 at an interest rate of 6.25%, and the par rate is 6%, the broker may earn a YSP equal to 1.0% of the loan amount. This $1,000 fee is paid by the lender directly to the broker as a "rebate." The mortgage broker earns "one point" directly from the lender "POC" (Paid outside Closing). Although the borrower is not charged the fee directly, the borrower does pay the fee indirectly by accepting a higher interest rate in exchange for lower fees.

In the U.S., mortgage brokers are required to disclose YSP as a fee "POC" (Paid Outside Closing) on page 2 of the HUD1 Settlement statement, inside the margin, away from the column marked "Paid from Borrower's funds at Settlement."

YSPs as a financial instrument are not controversial. What is controversial is how they are applied, and how and when brokers and lenders have to disclose their existence and their amount to the borrower.

Consumer groups such as the Center for Responsible Lending contend that disclosing the YSP to borrowers informs borrowers that the broker might be charging them a higher interest rate than they might otherwise qualify for.^[1] They point out that the YSP amount to a fee paid to the broker, and therefore its exact amount should be made known when the borrower commits to a broker ("locks in the rate"), rather than later in the loan process.

Some mortgage brokers contend that this disclosure requirement puts them at a disadvantage when compared to Institutional ("Retail") Lenders, who do not have to disclose their YSP. In addition, they point out that there are truly legitimate reasons for a YSP, such as help in offsetting closing costs for borrowers who are short of cash. For those borrowers, brokers use the YSP to help pay closing costs, as outlined below. Conventional mortgage brokers also point out that if only they were required to disclose their YSP, borrowers might not save money, but simply be steered to retail lenders who would charge the same amount for a loan, but would appear better at first glance because they did not have to list those fees explicitly.

These arguments are further outlined below.

1 Banks versus Mortgage Brokers and YSP
2 "Giveaway to Big Business?"
3 No Closing Cost Loans Explained
4 Upfront mortgage brokers
5 References
6 More Readings

Banks versus Mortgage Brokers and YSP

Lenders that fund loans and then sell them after closing do not need to disclose the amount of the yield spread premium they make. This is a HUD rule (RESPA 1974). On the other hand, brokers are forced to disclose the amount of yield spread they get from the bank. HUD's stated rationale is that Institutional lenders sell their loans in a true "Secondary Market Transaction" sometime after the loan is closed. This means that the loan is sold at a later time and their true "Yield Spread" or additional revenue is not yet known. Although the exact amount they earn may not be known, the fact is that they are earning revenue that they do not have to disclose to the borrower.

The Federal Reserve Bank has proposed significant changes to the Home Ownerships and Equity Protection Act (HOEPA) and Regulation Z, regarding yield spread premium disclosure, on December 12, 2007. The related part of the proposal will require a mortgage brokerage to negotiate a dollar-specific fee, in writing, for their services or waive the right to receive yield spread premium from a lender. This will allow a consumer to "shop" mortgage brokers on "price" and specifically guarantee that compensation harvesting is eliminated by mortgage brokers.[1]

This difference gives institutional lenders an unfair advantage over a mortgage broker because when a borrower closes a loan with a mortgage broker, the borrower is fully aware of the revenue that the broker business earns on their loan. On the other hand, consumers are never made aware of the amount of profit that large institutional lenders generate when they in turn sell servicing rights to another lender. Further, many mortgage brokers argue that the use of the term "par rate" in describing YSP is misleading; if a true "par rate" is provided to the customer, the only way for a mortgage broker to earn compensation would then be to charge additional origination and/or discount points to the consumer, whereas the institutional lenders commonly set their zero-point loans at a significantly higher rate than undisclosed "par rates" of the secondary market.

Many brokers maintain that the best way to serve the interest of the consumer is with fair, full, and uniform disclosure of loan amount, interest rate, and settlement costs. Toward that end, they argue that the single best way to compare different rates and fees is through examination of the Good Faith Estimate and Truth-In-Lending Disclosure, and further, through comparison of the Annual Percentage Rate that is disclosed in the Truth-In-Lending. YSP has no impact on APR; the variables impacting APR include loan amount, rate and fees. Therefore, with respect to obtaining the most cost-effective financing in both the short and long term, APR is the best index available to the consumer.

This argument ignores, however, that consumers do not receive a Good Faith Estimate of Truth-In-Lending Disclosure when they shop for a mortgage. The GFE and TIL Disclosure are only issued after a borrower has applied, which involves significant effort on both the borrower's and the broker's part. Most borrowers instead shop for mortgages by asking for quotes for both the APR and the closing costs. This fact makes it difficult for borrowers to obtain exact quotes that are enforceable. The following pitfalls exist:

Market volatility and Obsolete Prices. The price for a loan may change throughout the day, and a quote given in the morning may no longer be honored in the afternoon. This may happen without any malice on the part of the broker: many lenders send two price sheets per day to the brokers doing business with them. A borrower must trust the broker that indeed the price has gone up when the broker says so. Since most brokers do not share their price sheets with their borrowers, borrowers cannot independently verify if their broker is honest with them or pretends that rates have gone up in order to pocket a larger YSP.

Market Niche Misclassification. When quoting a price, a broker may not know all circumstances about a borrower. It may turn out the borrower does not qualify for the quoted price before applying. Since the mortgage market is highly departmentalized, this can in fact often occur, even without malice on the broker's part - but the borrower again has no way of verifying whether a broker's not honoring an originally quoted rate is in fact due to such misclassification or whether the broker intentionally misclassified to be able to quote more favorable terms.

Mortgage Price Low-balling. Rogue brokers (so-called sunshine blowers) may deliberately quote lower prices than they are willing to honor. Borrowers cannot verify whether a given quote is low-balled or not since they do not see the price sheet.

Fake rate-locks. Rogue brokers may say that a rate was locked in with the lender when in fact it was not, or was locked in for a shorter period of time. Again, the borrower has no way of telling whether a broker is rogue or not when shopping for a mortgage. (They can ask, however, to see the rate commitment letter from the lender, which honest brokers will provide.)

In summary, whilst in theory it is sound advice to compare total settlements and APR between brokers, in practice this would require borrowers to complete a loan application with several brokers, in essence keeping all their options open for a significant portion of the closing process, be prepared to back out of any of the deals, and finally pick a "winner." This would be a disaster to honest brokers, who would then have to invest a significant amount of work and time into deals they would only have a small chance of closing.

Another competitive disadvantage, seen from the point of view of mortgage brokers competing with institutional lenders, is that brokers, unlike institutional lenders, have to disclose that they earn YSP on the Good Faith Estimate. There are two separate Good Faith Estimates, one for the broker, one for the institutional lender. The difference is one section that exists only on the Good Faith Estimate used by the mortgage broker. It states:

"Compensation To Broker, Not Paid Out Of Loan Proceeds"

The exact disclosure requirements vary between states. In some states brokers must disclose a range, typically 0%-2% of the loan amount, but not the exact amount^{[citation needed]}. Some states simply require that a warning is appended^{[citation needed]}. Some states require the broker to disclose an estimated dollar amount, such as $2,500^{[citation needed]}.

As of October 2007, Florida requires that the maximum compensation a broker might make on a loan be disclosed as a dollar amount (not a range). Furthermore, once the loan is rate locked and the amount of compensation is known, this amount must be re-disclosed to the borrower within 3 days of knowing the exact amount. The disclosure must be earlier than 3 days prior to closing. Florida Statute 494.0038. Retrieved on 2007-11-02.

Institutional lenders are exempt from this requirement.

"Giveaway to Big Business?"

Sometime on or around 2002, The Secretary of HUD, now Senator Mel Martinez, R-Florida, tried and failed to pass sweeping RESPA (Real Estate Settlement Procedures Act) reform legislation that would have further put mortgage brokers at a disadvantage by requiring mortgage brokers to credit borrowers the amount of yield spread premium that they earned, and then charge the borrowers a fee to recover the yield spread premium. Lenders would have still been exempt from that requirement.

For example, if a borrower went to his regular bank or to a large lender, they may receive a quote from them of 6.25%, Zero Points. The lender presumably would earn additional compensation when they sold the loan at some later date. For the sake of this example, we'll call it 1%.

If a mortgage broker offered the borrower the same rate, 6.25%, the borrower would receive the one point YSP as a "Lender Credit to Borrower" Then, in order to earn the same fee as the institutional lender does above, the mortgage broker would have to charge the borrower a 1% "mortgage broker fee".

So, the mortgage broker deal would be 6.25%, 1 Pt. Fee paid by borrower to broker, plus 1 points paid by lender to borrower, whereas the institutional lender could offer the equivalent at 6.25%, zero points. It is likely that borrowers would prefer the simpler deal from the lender, even though they would pay the same number of points (zero), and get the same interest rate. These concerns are corroborated by an FTC study, which found customers more likely to take a more expensive non-broker product under this disclosure method.^[2]

HUD proposed broker compensation disclosures as part of its July 2002 RESPA reform proposal (HUD 2002a, 49134). Mortgage brokers would be required to disclose, in the Good Faith Estimate (GFE) provided to borrowers, any compensation received from the lender in connection with the origination of the loan. A major part of the compensation is any Yield spread premium (YSP) paid by the lender for a loan originated at an above-par interest rate. The YSP reflects the additional value to the lender of a loan originated at the higher interest rate. The proposed disclosure was motivated by a concern that brokers were placing borrowers in above par loans without their knowledge, and keeping the YSPs rather than passing them through to consumers in the form of reduced settlement costs. Direct lenders would not be required to make the same disclosure, even though they may be charging the same interest rate and settlement costs and earning the same compensation as a broker.

The compensation disclosures had a significant adverse impact on the respondents perception of loan costs and on respondents’ choice of loans. The disclosures caused a significant proportion of respondents to choose more expensive loans by mistake and caused a substantial bias against broker loans even when the broker loans cost the same or less than direct lender loans.

The findings of this study indicate that broker compensation disclosures are likely to harm rather than help consumers and competition in the mortgage market. --The disclosures are likely to lead a significant proportion of borrowers to choose more expensive loans by mistake. --The disclosures are likely to cause a substantial bias against broker loans that may reduce competition and increase the cost of all mortgages. --All three versions of the compensation disclosure tested in the study resulted in significant consumer confusion about loan costs and a substantial bias against broker loans. This included versions that moved the disclosure to a second page of the cost information.^[2]

No Closing Cost Loans Explained

YSP can also be used by a mortgage broker to offer "No Closing Cost" loans. For example, if a borrower takes a $800,000 loan and the total closing costs amount to $5,000, the broker could increase the interest rate that pays the broker a YSP of say 1% and the broker could then credit the borrower $5,000 of the $8000 made in YSP towards his closing costs. The broker would still earn a $3,000 -paid by the YSP. This is almost always the case for loans advertised as "no closing cost" or "no fee." The only way for a broker to provide a loan without fees (and stay in business) is to charge the borrower a rate which pays a sufficient YSP to cover the closing costs, as well as earn some money for themselves.

Note that in that example, the key expression is "could" - the broker is under no obligation to share their YSP with the borrower, though it is disclosed on the est. HUD-1 by law. So, if the broker steers the borrower towards a mortgage with a higher interest rate, the broker might receive a YSP of say 1.5%, resulting in $12,000 paid by the lender to the broker. Subtracting the $5,000 spent towards closing costs, the broker would have made $7,000 rather than $3,000 as in the original example. The borrower should always check his/her estimated HUD-1 at the time he/she signs loan documents in order to ensure the loan fees and charges are in line with what they were told/expected. Unlike a Good Faith Estimate (which is provided by the broker), a HUD-1 is a document provided by the escrow company handling escrow for the loan. Though technically an estimate until the loan is funded, an est HUD cannot be changed after loan docs have been signed unless the borrower resigns certain documents disclosing finance charges associated with the loan ("re-disclosure").

During the refinancing boom in 1998-2005, rogue mortgage brokers used the YSP to defraud wholesale lenders by colluding with borrowers as follows. A borrower would accept a much higher interest rate that would result in a huge YSP paid to the broker. For instance, for the $800,000 loan discussed above, a YSP of 3 points would result in a $24,000 cash payment to the broker. The broker would share the YSP with the borrower. Shortly after closing the loan, the borrower would refinance the loan at a lower interest rate, essentially avoiding paying back the YSP received in a higher interest rate over time. This practice, also known as "Churning" if used intentionally, defrauds the investors who paid the YSP in expectation of receiving higher interest payments while the loan is being paid back.

Wholesale lenders have introduced practices to combat this type of fraud. Most wholesale mortgage broker agreements specifically require that borrowers make at least 4 payments on their new loan, or the broker receives an Early-Pay-Off notice (EPO). An EPO requires the broker (subject to their wholesale brokerage agreement with the lender) to pay back the entire rebate they earned. It is the case that many brokers will refinance the same clients over and over again, typically in 6 month increments, and they will share the YSP with the borrower, while the borrower agrees to a higher rate on what amounts to a permanent basis, since they keep refinancing the same type of loan the same way. However, with the current softening of housing values (as of 4-2007) much of that type of refinancing has come to an end. Another measure is that most lenders (and many states) have limits on the amount of YSP they will pay out (currently most lenders limit this is no more than 3 points). Lastly, brokers who repeatedly refinance borrowers in this way will almost certainly be cut off from doing business with lenders who they had been approved with.

Upfront mortgage brokers

Upfront mortgage brokers (UMBs) are a group of mortgage brokers who have agreed to disclose the total fee they earn upfront. At closing time, they will refund any YSP they receive to the borrower, except that which is required to cover their fee, which is negotiated upfront. In essence, UMBs practice voluntarily what the 2002 RESPA reform would have required all mortgage brokers to do. Contrary to the belief that this would result in customers choosing too expensive loans, UMBs provide loans at wholesale ("at par") rates to consumers with a known markup negotiated between the borrower and the broker. A UMB typically provides the borrower with direct access to the lender's wholesale price sheet, allowing the borrower to decide what interest rate to buy and whether to pay points to receive a lower rate, or whether to accept a higher rate in exchange for a YSP from the lender. In both cases, the broker's fee is the same. Therefore, the broker has no interest in steering the borrower towards higher interest loans that might result in higher YSPs, and therefore a higher fee for the broker. The borrower can pick exactly what rate to buy and how many - if any - points to pay for a lower interest rate or receive in exchange for accepting a higher rate. Non-UMBs (aka traditional mortgage brokers) only provide the interest rate and points and hide how much YSP they will receive until closing, in effect hiding their true fees until it is too late (or very difficult for the borrower to back out.)

The argument for UMBs was made by Jack M. Guttentag, Professor of Finance Emeritus at the Wharton School of the University of Pennsylvania. Prof. Guttentag's argument is that preparing a mortgage is providing a service, and as such, should be subject to a negotiated fee. The opposing argument is that brokering a mortgage is like selling a product, in which case the broker doesn't have to make known their markup upfront. In addition, non-UMBs point out, having to declare their fee might put them at a competitive disadvantage when compared to institutional lenders who are allowed to roll the costs associated with the loan preparation into the interest rate without having to disclose that fact, as outlined above.

References

^ Yield Spread Premiums: A Powerful Incentive for Equity Theft. Center for Responsible Lending.
^ ^a ^b The Effect of Mortgage Broker Compensation Disclosures on Consumers and Competition: A Controlled Experiment. Federal Trade Commission.

The typical shape of the yield curve

Yield curves are usually upward sloping asymptotically; the longer the maturity, the higher the yield, with diminishing marginal growth. There are two common explanations for this phenomenon. First, it may be that the market is anticipating a rise in the risk-free rate. If investors hold off investing now, they may receive a better rate in the future. Therefore, under the arbitrage pricing theory, investors who are willing to lock their money in now need to be compensated for the anticipated rise in rates — thus the higher interest rate on long-term investments.

However, interest rates can fall just as they can rise. Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). Risk premium should be paid, since with longer maturities, more catastrophic events might occur that impact the investment. This explanation depends on the notion that the economy faces more uncertainties in the distant future than in the near term, and the risk of future adverse events (such as default and higher short-term interest rates) is higher than the chance of future positive events (such as lower short-term interest rates). This effect is referred to as the liquidity spread. If the market expects more volatility in the future, even if interest rates are anticipated to decline, the increase in the risk premium can influence the spread and cause an increasing yield.

The opposite situation — short-term interest rates higher than long-term — also can occur. For instance, in November 2004, the yield curve for UK Government bonds (i.e. the bonds which the UK Government issues to borrow money - see gilts) was partially inverted. The yield for the 10 year bond stood at 4.68% but only 4.45% on the thirty year bond. The market's anticipation of falling interest rates causes such incidents. Negative liquidity premiums can exist if long-term investors dominate the market, but the prevailing view is that a positive liquidity premium dominates, so only the anticipation of falling interest rates will cause an inverted yield curve. Strongly inverted yield curves have historically preceded economic depressions.

The yield curve may also be flat or hump-shaped, due to anticipated interest rates being steady, or short-term volatility outweighing long-term volatility.

Yield curves move on a daily basis, reflecting the market's reaction to news. A further "stylized fact" is that yield curves tend to move in parallel (i.e., the yield curve shifts up and down as interest rate levels rise and fall).

Types of yield curve

There is no single yield curve describing the cost of money for everybody. The most important factor in determining a yield curve is the currency in which it is denominated. The economic situation of the countries and companies using each currency is a primary factor in determining the yield curve. For example the sluggish economic growth of Japan throughout the late 1990s and early 2000s has meant the yen yield curve is very low (rising from virtually zero at the three month point to only 2% at the 30 year point). By contrast the British pound curve ranges from 4-5% along its curve.

Different institutions borrow money at different rates, depending on their creditworthiness. The yield curves corresponding to the bonds issued by governments in their own currency are called the government bond yield curve (government curve). Banks with high credit ratings (Aa/AA or above) borrow money from each other at the LIBOR rates. These yield curves are typically a little higher than government curves. They are the most important and widely used in the financial markets, and are known variously as the LIBOR curve or the swap curve. The construction of the swap curve is described below.

Besides the government curve and the LIBOR curve, there are corporate (company) curves. These are constructed from the yields of bonds issued by corporations. Since corporations have less creditworthiness than governments and most large banks, these yields are typically higher. Corporate yield curves are often quoted in terms of a "credit spread" over the relevant swap curve. For instance the five-year yield curve point for Vodafone might be quoted as LIBOR +0.25%, where 0.25% (often written as 25 basis points or 25bps) is the credit spread.

Normal yield curve

From the post-Great Depression era to the present, the yield curve has usually been "normal" meaning that yields rise as maturity lengthens (i.e., the slope of the yield curve is positive). This positive slope reflects investor expectations for the economy to grow in the future and, importantly, for this growth to be associated with a greater expectation that inflation will rise in the future rather than fall. This expectation of higher inflation leads to expectations that the central bank will tighten monetary policy by raising short term interest rates in the future to slow economic growth and dampen inflationary pressure. It also creates a need for a risk premium associated with the uncertainty about the future rate of inflation and the risk this poses to the future value of cash flows. Investors price these risks into the yield curve by demanding higher yields for maturities further into the future.

However, a positively sloped yield curve has not always been the norm. Through much of the 19th century and early 20th century the US economy experienced trend growth with persistent deflation, not inflation. During this period the yield curve was typically inverted, reflecting the fact that deflation made current cash flows less valuable than future cash flows. During this period of persistent deflation, a 'normal' yield curve was negatively sloped.

Steep yield curve

Historically, the 20-year Treasury bond yield has averaged approximately two percentage points above that of three-month Treasury bills. In situations when this gap increases (e.g. 20-year Treasury yield rises relatively higher than the three-month Treasury yield), the economy is expected to improve quickly in the future. This type of curve can be seen at the beginning of an economic expansion (or after the end of a recession). Here, economic stagnation will have depressed short-term interest rates; however, rates begin to rise once the demand for capital is re-established by growing economic activity.

Flat or humped yield curve

A flat yield curve is observed when all maturities have similar yields, whereas a humped curve results when short-term and long-term yields are equal and medium-term yields are higher than those of the short-term and long-term. A flat curve sends signals of uncertainty in the economy. This mixed signal can revert back to a normal curve or could later result into an inverted curve. It cannot be explained by the Segmented Market theory discussed below.

Inverted yield curve

An inverted yield curve occurs when long-term yields fall below short-term yields. Under unusual situations, long-term investors will settle for lower yields now if they think the economy will slow or even decline in the future. An inverted curve has indicated a worsening economic situation in the future 5 out of 6 times since 1970. The New York Federal Reserve regards it as a valuable forecasting tool in predicting recessions two to six quarters ahead. In addition to potentially signaling an economic decline, inverted yield curves also imply that the market believes inflation will remain low. This is because, even if there is a recession, a low bond yield will still be offset by low inflation. However, technical factors, such as a flight to quality or global economic or currency situations, may cause an increase in demand for bonds on the long end of the yield curve, causing long-term rates to fall. This was seen in 1998 during the Long Term Capital Management failure when there was a slight inversion on part of the curve.

Theory

There are four main economic theories attempting to explain how yields vary with maturity. Two of the theories are extreme positions, while the third attempts to find a middle ground between the former two.

Market expectations (pure expectations) hypothesis

(1 + i_lt)ⁿ = (1 + i_st^{year 1})(1 + i_st^{year 2})...(1 + i_st^{year n})

This hypothesis assumes that the various maturities are perfect substitutesand suggests that the shape of the yield curve depends on market participants' expectations of future interest rates. These expected rates, along with an assumption that arbitrage opportunities will be minimal, is enough information to construct a complete yield curve. For example, if investors have an expectation of what 1-year interest rates will be next year, the 2-year interest rate can be calculated as the compounding of this year's interest rate by next year's interest rate. More generally, rates on a long-term instrument are equal to the geometric mean of the yield on a series of short-term instruments. This theory perfectly explains the stylized fact that yields tend to move together. However, it fails to explain the persistence in the shape of the yield curve.

Liquidity preference theory

The Liquidity Preference Theory, an offshoot of the Pure Expectations Theory, asserts that long-term interest rates not only reflect investors’ assumptions about future interest rates but also include a premium for holding long-term bonds, called the term premium or the liquidity premium. This premium compensates investors for the added risk of having their money tied up for a longer period, including the greater price uncertainty. Because of the term premium, long-term bond yields tend to be higher than short-term yields, and the yield curve slopes upward. Long term yields are also higher not just because of the liquidity premium, but also because of the risk premium added by the risk of default from holding a security over the long term.

Market segmentation theory

This theory is also called the segmented market hypothesis. In this theory, financial instruments of different terms are not substitutable. As a result, the supply and demand in the markets for short-term and long-term instruments is determined independently. Prospective investors would have to decide in advance whether they need short-term or long-term instruments. Due to the fact that investors prefer their portfolio to be liquid, they will prefer short-term instruments to long-term instruments. Therefore, the market for short-term instruments will receive a higher demand. Higher demand for the instrument implies higher prices and lower yield. This explains the stylized fact that short-term yields are usually lower than long-term yields. This theory explains the predominance of the normal yield curve shape. However, because the supply and demand of the two markets are independent, this theory fails to explain the observed fact that yields tend to move together (i.e., upward and downward shifts in the curve).

In an empirical study, 2000 Alexandra E. MacKay, Eliezer Z. Prisman, and Yisong S. Tian found segmentation in the market for Canadian government bonds, and attributed it to differential taxation.

For a brief period in the last week of 2005, and again in early 2006, the US Dollar yield curve inverted, with short-term yields actually exceeding long-term yields. Market segmentation theory would attribute this to an investor preference for longer term securities, particularly from pension funds and foreign investors who prefer guaranteed longer term yields.

Preferred habitat theory

The Preferred Habitat Theory states that in addition to interest rate expectations, investors have distinct investment horizons and require a meaningful premium to buy bonds with maturities outside their "preferred" maturity, or habitat. Proponents of this theory believe that short-term investors are more prevalent in the fixed-income market and therefore, longer-term rates tend to be higher than short-term rates, for the most part, but short-term rates can be higher than long-term rates occasionally. This theory represents a middle ground between the Market Segmentation Theory and the Market Expectations Theory. Moreover, it seems to explain both the persistence of the normal yield curve shape and the tendency of the yield curve to shift up and down while retaining its shape.

Malinvestment Theory

Austrian School economist Dr. Paul Cwik claims that the yield curve's shape depends mostly upon actions by a monetary authority that promote an atmosphere of malinvestment resulting from periods of loose monetary policy. He claims that the process of liquidating those malassets inverts the curve.

Historical development of yield curve theory

On 15 August 1971, U.S. President Richard Nixon announced that the U.S. dollar would no longer be based on the gold standard, thereby ending the Bretton Woods system and initiating the era of floating exchange rates.

Floating exchange rates made life more complicated for bond traders, including importantly those at Salomon Brothers in New York. By the middle of the 1970s, due to the prodding of the head of bond research at Salomon, Marty Liebowitz, traders began thinking about bond yields in new ways. Rather than think of each maturity (a ten year bond, a five year, etc.) as a separate marketplace, they began drawing a curve through all their yields. The bit nearest the present time became known as the short end -- yields of bonds further out became, naturally, the long end.

Academics had to play catch up with practitioners in this matter. One important theoretic development came from a Czech mathematician, Oldrich Vasicek, who argued in a 1977 paper that bond prices all along the curve are driven by the short end (under risk neutral equivalent martingale measure), and accordingly by short-term interest rates. The mathematical model for Vasicek's work was given by an Ornstein-Uhlenbeck process, and has since been discredited because the model predicts a positive probability that the short rate becomes negative and is inflexibile in creating yield curves of different shapes. Vasicek's model has been superseded by many different models including the Hull-White model (which allows for time varying parameters in the Ornstein-Uhlenbeck process), the Cox-Ingersoll-Ross model, which is a modified Bessel process, and the Heath-Jarrow-Morton framework. There are also many improvements on each of these models, but see the article on short rate model. Another modern approach is the LIBOR Market Model, introduced by Brace, Gatarek and Musiela in 1997 and advanced by others later.

Construction of the full yield curve from market data

**Typical inputs to the money market curve**
Type	Settlement date	Rate (%)
Cash	Overnight rate	5.58675
Cash	Tomorrow next rate	5.59375
Cash	1m	5.625
Cash	3m	5.71875
Future	Dec-97	94.24
Future	Mar-98	94.23
Future	Jun-98	94.18
Future	Sep-98	94.12
Future	Dec-98	94.00
Swap	2y	6.01253
Swap	3y	6.10823
Swap	4y	6.16
Swap	5y	6.22
Swap	7y	6.32
Swap	10y	6.42
Swap	15y	6.56
Swap	20y	6.56
Swap	30y	6.56
A list of standard instruments used to build a money market yield curve.
The data is for lending in US dollar, taken from 6 October 1997

The usual representation of the yield curve is a function P, defined on all future times t, such that P(t) represents the value today of receiving one unit of currency t years in the future. If P is defined for all future t then we can easily recover the yield (i.e. the annualized interest rate) for borrowing money for that period of time via the formula

Y(t) = (1 / P(t))^1/t - 1

The significant difficulty in defining a yield curve therefore is to determine the function P(t). P is called the discount factor function.

Yield curves are built from either prices available in the bond market or the money market. Whilst the yield curves built from the bond market use prices only from a specific class of bonds (for instance bonds issued by the UK government) yield curves built from the money market uses prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for t ≤ 3m, futures which determine the mid-section of the curve (3m ≤ t ≤ 15m) and interest rate swaps which determine the "long end" (1y ≤ t ≤ 60y).

In either case the available market data provides with a matrix A of cash flows, each row representing a particular financial instrument and each column representing a point in time. The (i,j)-th element of the matrix represents the amount that instrument i will pay out on day j. Let the vector F represent today's prices of the instrument (so that the i-th instrument has value F(i)), then by definition of our discount factor function P we should have that F = A*P (this is a matrix multiplication). In actual fact noise in the financial markets means it is not possible to find a P that solves this equation exactly, and our goal becomes to find a vector P such that

A * P = F + ε

where $ε$ is as small a vector as possible (where the size of a vector might be measured by taking its norm, for example).

Note that even if we can solve this equation, we will only have determined P(t) for those t which have a cash flow from one or more of the original instruments we are creating the curve from. Values for other t are typically determined using some sort of interpolation scheme.

Practitioners and researchers have suggested many ways of solving the A*P = F equation. It transpires that the most natural method - that of minimizing $ε$ by least squares regression - leads to unsatisfactory results. The large number of zeroes in the matrix A mean that function P turns out to be "bumpy".

In their comprehensive book on interest rate modelling James and Webber note that the following techniques have been suggested to solve the problem of finding P:

Approximation using Lagrange polynomials
Fitting using parameterised curves (such as splines, the Nelson-Siegel family or the Svensson family of curves). Van Deventer, Imai and Mesler summarize three different techniques for curve fitting that satisfy the maximum smoothness of either forward interest rates, zero coupon bond prices, or zero coupon bond yields
Local regression using kernels
Linear programming

In the money market practitioners might use different techniques to solve for different areas of the curve. For example at the short end of the curve, where there are few cashflows, the first few elements of P may be found by bootstrapping from one to the next. At the long end, a regression technique with a cost function that values smoothness might be used.

References

Jessica James & Nick Webber (2001). Interest Rate Modelling. John Wiley & Sons. ISBN 0-471-97523-0.
Riccardo Rebonato (1998). Interest-Rate Option Models. John Wiley & Sons. ISBN 0-471-97958-9.
Nicholas Dunbar (2000). Inventing Money. John Wiley & Sons. ISBN 0-471-89999-2.
N. Anderson, F. Breedon, M. Deacon, A. Derry and M. Murphy (1996). Estimating and interpreting the yield curve. John Wiley & Sons. ISBN 0-471-96207-4.
John C. Hull (1989). Options, futures and other derivatives. Prentice Hall. ISBN 0-13-015822-4. See in particular the section Theories of the term structure (section 4.7 in the fourth edition).
Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models - Theory and Practice. Springer. ISBN 3-540-41772-9.
Donald R. van Deventer, Kenji Imai, Mark Mesler (2004). Advanced Financial Risk Management, An Integrated Approach to Credit Risk and Interest Rate Risk Management. John Wiley & Sons. ISBN-13: 978-0470821268.
Ruben D Cohen (2006) "A VaR-Based Model for the Yield Curve [download]" Wilmott Magazine, May Issue.
Paul F. Cwik (2005) "The Inverted Yield Curve and the Economic Downturn [download]" New Perspectives on Political Economy, Volume 1, Number 1, 2005, pp. 1-37.

Benefits of membership

Benefits that come with being a member of the workforce are said to be financial independence, a feeling of usefulness, self-confidence, and respect from fellow citizens. However, they do not necessarily follow directly from being a member of the workforce. One's finances are commonly dependent upon one's employer, customers, wages and inflation; furthermore, one can feel useful without being a member of the workforce.|^{[citation needed]}|

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Thursday, July 31, 2008

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