2.5 Financial Economics Foundations

Binomial tree

A binomial tree projects possible outcomes in a variable such as a security price or interest rate by modeling uncertainty as two movements: an upward movement and a downward movement. A key attribute of most binomial trees is that as the number of steps used increases (i.e., as the total time period is broken into smaller and smaller steps), the terminal distribution of the asset approaches a normal distribution.

key externality of arbitrage activities

A key externality of arbitrage activities is that they tend to drive similar assets toward similar prices which, in turn, improves global economic decisions. Better asset prices serve as signals of more accurate information to producers and consumers, which, in turn, enables supply and demand to be balanced at more efficient levels.

nominal interest rate

A nominal interest rate is the rate of return measured in terms of a given currency without a downward adjustment for the potential effects of positive inflation.

Arbitrage

Arbitrage is the attempt to earn risk-less profits (in excess of the risk-free rate) by identifying and trading relatively mispriced assets.

Arbitrage-Free Pricing in Spot Markets

Arbitrage-free pricing in spot markets involves identifying two sets of transactions with identical outcomes and requiring that their prices be equal. For example, consider an investor wishing to exchange euros for yen. In the spot foreign exchange market, the investor may find that one euro can be exchanged for 140 yen. However, there are numerous sets of transactions for converting euros to yen. For example, the investor may find that one euro can be converted to 1.40 U.S. dollars and that each U.S. dollar can then be converted into 100 yen. Of course, there are many other multiple-transaction paths that would lead to the same result: converting euros to yen.

Six Factors Driving Informational Market Efficiency: 5

Assets will also tend to trade at prices closer to their informationally efficient values when there is easier access to better information, as better information facilitates better financial analysis. In the United States, the Securities and Exchange Commission has as one of its primary goals requiring public companies to disclose meaningful information to the public.

Six Factors Driving Informational Market Efficiency: 6

Assets will also tend to trade at prices closer to their informationally efficient values when there is less uncertainty about their valuation. In other words, better valuation methods lead to better analysis. For example, the development of sound option pricing models in the 1970s led to improved informational market efficiency in options markets.

Six Factors Driving Informational Market Efficiency: 4

Fewer regulatory constraints on trading also tend to lead to improved informational market efficiency by expanding competition and trading. Examples of regulatory constraints that may inhibit competition include restrictions on short selling and leverage.

Strong form informational market efficiency

Finally, the concept of strong form informational market efficiency (or strong level) refers to market prices reflecting all publicly and privately available information. The strong form version of the efficient market hypothesis states that all information—both the information available to the public and any information not publicly known—is completely accounted for in current stock prices, and there is no type of information that can give an investor an advantage on the market. Advocates for this degree of the theory suggest that investors cannot make returns on investments that exceed normal market returns, regardless of information retrieved or research conducted.

Yield to Maturity (YTM)

Fixed-income pricing focuses on yield to maturity. The yield to maturity of a fixed-income instrument is the rate that discounts all of the promised cash flows of the instrument into a summed present value that equals the instrument's market price.

Carry Trades without hedging example

For an unhedged example, consider an investor who observes that a one-year default-free bond in a particular foreign currency offers a 5% yield, whereas a default-free bond in the investor's domestic currency with the same maturity offers a yield of only 4%. The investor shorts the domestic bond (i.e., borrows in the domestic currency) at a cost of 4% and locks in a 5% yield in the foreign currency by purchasing the foreign bond with the borrowed cash. The carry trade offers an interest spread of 1% but is exposed to the risk that the foreign currency will weaken in value relative to the domestic currency. If the foreign currency weakens by more than 1% per year over the lifetime of the trade, the losses will exceed the 1% per year net income. In fact, to the extent that interest rate differentials reflect expectations of different inflation rates, the investor should expect the foreign currency to weaken by an amount that offsets the interest rate spread on a risk-adjusted basis.

An important feature of long-only bond portfolios that are immunized is that they tend to remain immunized as time passes, unless cash is received by the investor (i.e., coupon or principal payments).

For example, consider the earlier example of matching a 7.0-year duration target with 60% of a portfolio invested in a five-year bond zero-coupon bond and 40% in a 10-year zero-coupon bond. As one year passes, for example, the 5- and 10-year bonds become 4- and 9-year bonds, leaving a weighted average of six years. It is likely that the investment horizon of the investor has also declined from seven years to six years. The portfolio's duration would tend to decline in line with the investor's target time horizon until the five-year zero-coupon bond matures. The point is that long-only portfolios tend to be somewhat reasonably hedged as time passes, but require rebalancing when cash flows are received.

Six Factors Driving Informational Market Efficiency: 2

Greater trading frequency for the assets increases competition by providing greater incentives for investors, speculators, and arbitrageurs to analyze information and attempt to make favorable trades. Securities that are traded very infrequently typically have large bid-ask spreads due to the reduced profit potential for traders to benefit from mispricing.

Idiosyncratic return

Idiosyncratic return is the portion of an asset's return that is unique to an investment and not driven by a common association.

Idiosyncratic risk

Idiosyncratic risk is the dispersion in economic outcomes caused by investment-specific effects.

The Unbiased Expectations or Pure Expectations Theory

In a risk-neutral world, risk premiums are not required by market participants to bear interest rate risks. In that world, the unbiased expectations theory (i.e., the pure expectations theory) would hold. The unbiased expectations theory hypothesizes that all fixed-income securities offer the same expected return over the same time interval (i.e., there are no risk premiums), therefore serving as a useful tool in risk-neutral modeling in which all interest rates are formed purely on interest rate expectations. Put differently, under the unbiased expectations theory the expected value of every fixed-income security is expected to grow through time at the same rate over the same time interval and the shape of the term structure is driven purely by interest rate expectations (as opposed to being partly driven by risk aversion). Under the unbiased expectations theory the term structure should not be consistently upward sloping—which is inconsistent with historical observations.

pure arbitrage

In its purest sense, often termed pure arbitrage, true arbitrage requires no risk-bearing. Pure arbitrage opportunities exist when identical assets can be traded at different prices, allowing the arbitrageur to buy at the lower price, sell at the higher price, and profit when arbitrage activities force the two identical assets to trade at identical prices.

Six Factors Driving Informational Market Efficiency

The overall driver of informational market efficiency is greater competition among informed buyers and sellers. Thus, markets tend to attain higher degrees of informational market efficiency when there are more traders using all available information, and when those traders can transact with low costs. Let's look at six other major factors. The first four factors serve to facilitate competition and to enhance liquidity; the last two factors facilitate better analysis.

Forward Rate vs. Spot Rate: An Overview

The precise meanings of the terms "forward rate" and "spot rate" are somewhat different in different markets. But what they have in common is that they refer, for example, to the current price or bond yield—the spot rate—versus the price or yield for the same product or instrument at some point in the future—the forward rate. The term structure of interest rates explicitly indicates spot rates—the time value of money from time zero to each prospective point. But it also implicitly indicates forward rates—the time value of money between any two points in time, such as between years 2 and 3. In commodities futures markets, a spot rate is the price for a commodity being traded immediately, or "on the spot". A forward rate is the settlement price of a transaction that will not take place until a predetermined date; it is forward-looking. In bond markets, the forward rate refers to the effective yield on a bond, commonly U.S. Treasury bills, and is calculated based on the relationship between interest rates and maturities.

The Advantage to Binomial Tree Models and Their Extensions

The primary advantage to binomial models is their flexibility to easily incorporate important features such as cash distributions, decisions by corporations to call (i.e., buy back) securities, and decisions by investors to exercise options prior to their expiration or to prepay loans.

real interest rate

The real interest rate is the annualized rate earned on default-free fixed-income investments, after adjusting the nominal rate downward for the effect of inflation. The real interest rate is usually expressed as an annualized short-term rate (e.g., daily or weekly) that is determined in the market by the supply and demand for short-term capital.

Ex Post CAPM equation

The realized return of an asset differs from its expected return due to systematic and idiosyncratic effects, which are illustrated as the right side of the following equation: The left side of the equation is the realized excess return of asset i in time period t. The terms between the equal sign and the plus sign reflect the effect of the market's realized return in time period t, or the effect of systematic risk on the realized return of asset i in time period t. To the extent that the realized return of the market differs from its expected return, an asset with a nonzero beta realizes a return that differs from its expected return proportional to its beta. Finally, εit, the term to the far right, is the portion of the excess return that is due to the effect of idiosyncratic risk. Idiosyncratic returns include any effect on the return of asset i in time period t other than that which is correlated with the return of the market, such as the impact of firm-specific news. Taking the expected value of each side of the ex post CAPM equation and rearranging the terms returns the equation to the ex ante form of the CAPM (Equation 1).

Duration for Securities with Stochastic Cash Flows

The responsiveness of the price of a floating-rate bond to interest rate shifts approaches zero, as the speed with which the coupon adjusts to the short-term interest rate approaches zero. The idea that a floating-rate bond's duration is based on its coupon-reset period rather than its maturity is best understood by focusing on the sensitivity of the bond's price to interest rate changes. Simply put, fixed-coupon bonds are interest rate sensitive because their cash flows are fixed. Floating-rate bonds are only interest rate sensitive when their coupons adjust slowly or partially to interest rate changes. Duration of Floating-Rate Bond = The Time to the Next Reset Period

Spot Market (Cash Market)

The spot market or cash market is any market in which transactions involve immediate payment and delivery: The buyer immediately pays the price, and the seller immediately delivers the product.

Stochastic

randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely.

Markets tend to be more informationally efficient to the extent that they:

(1) are being traded by large numbers of well-informed and financially sound traders competing for profits, (2) contain securities for which substantial amounts of reliable information are made broadly and quickly available, and (3) are subject to minimal transactions costs, taxes, and other impediments to trade. Large markets in modern economies with institutions that support free trade tend to exhibit high degrees of informational market efficiency.

Skill-based trading strategies are more likely to be successful when:

(1) executed by the most skilled traders in any market and (2) executed in relatively new markets or with relatively new securities that have less competition among skill-based traders.

Two essential attributes of the ex post CAPM are that:

(1) the return from idiosyncratic risk, εit, has an expected value of zero (otherwise, it would appear in the ex ante form of the CAPM), and (2) the return from idiosyncratic risk is not linearly correlated with the return of the market, because any such effects are captured through the beta of the asset.

Duration formula

*where D is the duration, Po is the current bond price, and y is the bond yield.

By way of analogy, consider a plumb line.

A plumb line is a vertical line generally approximated using a suspended string with a weight attached at the bottom. A plumb line can be an important method of ensuring that a building's framework is well constructed. In practice, however, no building has perfect beams or walls. Similarly, the concept of perfect informational market efficiency creates a reference point against which market inefficiencies can be identified and the convergence of prices to the theoretically correct price can be forecast. In other words, perfect market efficiency is how financial analysts predict how prices should behave—allowing traders to identify mispriced assets and estimate their expected return and risk. Skill-based traders base their trades on perceived departures of actual asset prices from their informationally efficient prices.

Hedging or Immunizing a Long-Short Portfolio with Duration through Time

A portfolio with long and short positions (long-short) would be hedged and immunized if the long positions had a duration equal to the duration of the short positions. Further, a long-short portfolio can be immunized if the portfolio has a duration equal to the horizon point at which the investor anticipated using the proceeds from the portfolio. However, long-short portfolios raise a challenge with regard to the passage of time. This is a major reason why duration is said to be a perfect measure of risk when, among other things, there is an instantaneous term structure shift.

Implied Forward Rate

A rate of interest derived from current spot rates that is applicable to a future time period. An implied forward rate, F(t,T), is the annual return between time t and T (with T) inferred from the term structure of interest rates.

relative pricing model

A relative pricing model prescribes the relationship between two prices. A trivial relative pricing model would specify that the price of a troy ounce of gold should sell for about 9.7% more than an avoirdupois ounce because a troy ounce of gold is about 9.7% larger. Note that this relative pricing model implies nothing about the overall price level of gold. Arbitrage-free pricing models are relative pricing models Relative pricing models are typically quite precise. It is the precision of relative pricing models that drives the usefulness of arbitrage-free pricing models. In effect, arbitrage-free pricing models tend to be used wherever relative pricing models are well developed and accurate.

risk-neutral model

A risk-neutral model is a framework for valuing financial derivatives in which risk preferences and probabilities of price changes do not alter the solution and are therefore irrelevant, and in which the analyst selects risk-neutrality as the model's underlying assumption with regard to risk preferences. Given that there exists an infinite number of possible probabilities and risk premiums, an analyst can therefore pick the most convenient scenario—which is the risk-neutral scenario. In the risk-neutral scenario, all discount rates are the riskless rate and all probabilities found are the ones that allow the price of the risky asset to match its model value.

short-term interest rates

Although there is some evidence that markets price intraday lending, especially at times of crisis, the short-term interest rate generally refers to holding periods of one day, one week, or perhaps even one month.

absolute pricing model

An absolute pricing model attempts to describe a value or a price level based on its underlying economic factors. For example, the price of a share of common stock typically involves substantial uncertainty with regard to its future growth. Attempts to model the stock's price (such as by using a dividend growth model) are absolute pricing models since they estimate a price based on the stock's underlying fundamental factors. Absolute pricing models tend to be imprecise since the model is based on bold assumptions and estimates about which investors have highly heterogeneous beliefs.

Arbitrage-Free Models

An arbitrage-free model is a financial model with relationships derived by the assumption that arbitrage opportunities do not exist, or at least do not persist.

asset pricing model

An asset pricing model is a framework for specifying the return or value of an asset based on its risk, as well as future cash flows. Although asset pricing models include the term pricing in their name, they are focused on the returns on assets rather than their prices. Also, the term is usually used to describe the returns of equities rather than assets such as bonds.

Bootstrapping the term structure

Bootstrapping the term structure is the process of recursively estimating spot rates using one or more zero-coupon bonds on the short end and coupon bonds on the medium- and long-term regions of the term structure. The idea is to start with a short-term spot rate (i.e., a six-month zero-coupon yield) and use that rate to discount the first coupon of a one-year bond (with semiannual coupon payments). Since the one-year bond has only two cash flows (assuming semiannual coupons), the value of the second cash flow (the principal plus coupon payment in 12 months) can be found by subtracting the discounted value of the six-month cash flow from the total value (price) of the one-year bond.

More issues with CAPM

CAPM is generally faulted for its inability to describe the real world accurately, especially its inability to describe the behavior of alternative investments. Alternative investment analysis often focuses on the potential for multiple sources of systematic risk and on the potential to invest such that the expected idiosyncratic return, E(εit), is positive.

Carry Trades

Carry trades are typically a set of long and short positions intended to generate perceived benefits through time, such as enhanced return, as the positions are "carried." Carry trades can either be hedged or be exposed to the risks of price changes. Carry trades attempt to earn profits from carrying or maintaining long positions in higher-yielding assets and short positions in lower-yielding assets without suffering from adverse price movements.

Carry Trades with hedging example

Continuing from the previous example, the investor faces the risk that the proceeds of the foreign bond may be insufficient to settle the short position in the domestic bond at the end of the trade. The investor may decide to hedge the risk of this carry trade by locking in the exchange rate ahead of time between the foreign and domestic currencies. Specifically, the investor could use derivatives (such as a forward contract, discussed in the session entitled, Derivatives and Risk-Neutral Valuation) to lock in the rate to exchange the principal amount received in the foreign currency when the long position in the foreign bond matures for the amount due in the domestic currency. The key to the hedge is that it must allow the investor the opportunity to exchange the proceeds of the long position to cover the obligation of the short position at a prenegotiated value. However, since the investor is fully hedged against risk, the investor should only be able to receive the riskless return in an informationally efficient market.

Duration vs Time to Maturity

Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates. A bond's duration is easily confused with its term or time to maturity because certain types of duration measurements are also calculated in years. However, a bond's term is a linear measure of the years until repayment of principal is due; it does not change with the interest rate environment. Duration, on the other hand, is non-linear and accelerates as the time to maturity lessens.

Elasticity

Elasticity is a measure of a variable's sensitivity to a change in another variable, most commonly this sensitivity is the change in quantity demanded relative to changes in other factors, such as price. In business and economics, price elasticity refers to the degree to which individuals, consumers, or producers change their demand or the amount supplied in response to price or income changes. It is predominantly used to assess the change in consumer demand as a result of a change in a good or service's price. If demand for a good or service is relatively static even when the price changes, demand is said to be inelastic, and its coefficient of elasticity is less than 1.0 Examples of elastic goods include clothing or electronics, while inelastic goods are items like food and prescription drugs.

ex ante models

Ex ante models such as ex ante asset pricing models, explain expected relationships, such as expected returns. Ex ante means "from before." Ex-ante models provide an understanding of how return expectations or requirements are formed.

Inflation

Inflation is the rate of change in the value of a currency relative to a basket of real assets with a positive inflation rate indicating that the value of the currency is declining.

Informational market efficiency

Informational market efficiency refers to the extent to which asset prices reflect available information. An informationally efficient market is a market in which assets are traded at prices that equal their values based on all available information. The concept of informational market efficiency is sometimes referred to as efficient market theory or the efficient market hypothesis.

Interest rate immunization

Interest rate immunization is the process of protecting the value of a position against shifts in interest rates. A portfolio is said to be immunized with respect to specified shifts (or in some cases all interest rate shifts), if the value of the portfolio is independent of the shifts. The traditional approach to interest rate risk management involves managing a portfolio's duration.

Factors Influencing Informational Efficiency in Alternative Asset Markets

It is primarily with regard to the fifth and sixth factors that many alternative markets possess features that lend themselves to less efficient pricing: substantial nonpublic information and substantial uncertainty with regard to valuation methods. The practices and tools for investing in traditional assets tend to be better developed and more widely accepted. Market participants tend to better understand the relationship between traditional asset values (such as bond prices) and information (such as expected inflation rates) than the relationship between alternative asset values (such as intellectual property values) and information (such as technological innovations). The complex trading strategies inherent in some alternative investments rely on the discovery and exploitation of market inefficiencies in order to be successful.

The Liquidity Preference or Liquidity Premium Theory

Longer-term bonds tend to experience greater price volatility than shorter-term bonds, leading to the hypothesis that longer-term bonds are riskier and therefore require higher expected rates of return (i.e., higher risk premiums). Given risk aversion in a well-functioning market, the liquidity preference theory (i.e., the liquidity premium theory) would hold. The liquidity preference theory hypothesizes that longer-term fixed-income securities offer higher expected returns over the same time interval as shorter-term bonds, that risk premiums are positive and increasing in the bond's longevity, that all interest rates are formed based on both interest rate expectations and risk premiums, and that fixed-income management reflects a trade-off between risk and return. The liquidity preference theory hypothesizes that the expected return on zero-coupon fixed-income securities is an increasing function of the security's maturity, and that the shape of the term structure is formed as the sum of the term structure that would exist in a risk-neutral world (i.e., under the unbiased expectations theory) and the risk premiums associated with each maturity. The consistent upward slope to the term structure is consistent with the liquidity preference theory.

Six Factors Driving Informational Market Efficiency: 3

Low levels of trading frictions facilitate higher competition by encouraging arbitrage and speculation with the lowering of total trading costs. Reduced trading frictions include lower transaction costs, such as brokerage fees, exchange fees, regulatory fees, and taxes.

Macaulay Duration

Macaulay Duration is an estimate of a bond's interest rate sensitivity based on the time, in years, until promised cash flows will arrive. Not useful measure for bonds with embedded options. Macaulay duration estimates how many years it will take for an investor to be repaid the bond's price by its total cash flows.

Modified Duration

Modified duration is a risk measure used with discrete compounding applications in which the traditional duration formula is adjusted through division by (1+y/m). Modified duration measures the price change in a bond given a 1% change in interest rates.

The Market Segmentation or Preferred Habitat Theory

Recall that the liquidity preference theory hypothesizes that fixed-income securities offer monotonically increasing expected returns (higher risk premiums) to securities with longer maturities. The market segmentation theory (i.e., preferred habitat theory) is based on an assumption that there may be localized imbalances in the supply and demand for bonds with different longevities. Specifically, some investors, such as pensions and insurance companies, may prefer or better tolerate the risks of longer-term bonds while others may prefer holding short-term bonds. Similarly, borrowers have their preferred longevities for obtaining funding. The market segmentation theory hypothesizes that the preferred habitats of borrowers and lenders influence the expected returns of each maturity range, resulting in varying risk premiums and varying expected returns across maturity ranges that form humps and other non-monotonic shapes that are not eliminated by arbitrageurs (because the market is segmented). Economic theorists argue that the activities of speculators willing to form hedges that are long bonds within relatively underpriced maturity ranges and short bonds within relatively overpriced maturity ranges should minimize or eliminate expected return differentials based on habitat preferences.

market weight

The market weight of an asset is the proportion of the total value of that asset to the total value of all assets in the market portfolio. Thus, if the combined market value of all shares of XYZ Corporation is $250 billion, and if the combined market value of all investable assets in the world is $250 trillion, then the market weight of XYZ's equity would be 0.10%.

Systematic Risk vs. Unsystematic Risk

Systematic Risk - These are market risks—that is, general perils of investing—that cannot be diversified away. Interest rates, recessions, and wars are examples of systematic risks. Unsystematic Risk - Also known as "specific risk," this risk relates to individual stocks. In more technical terms, it represents the component of a stock's return that is not correlated with general market moves. Modern portfolio theory shows that specific risk can be removed or at least mitigated through diversification of a portfolio. The trouble is that diversification still does not solve the problem of systematic risk; even a portfolio holding all the shares in the stock market can't eliminate that risk. Therefore, when calculating a deserved return, systematic risk is what most plagues investors.

Systematic return

Systematic return is the portion of an asset's return driven by a common association.

Systematic risk

Systematic risk is the dispersion in economic outcomes caused by variation in systematic return.

The previous section noted that long-only bond portfolios matched to a horizon point duration tend to experience a decline in their duration that roughly matches the rate at which the time-to-the-horizon point is declining (until a cash flow occurs). The same cannot be said of a long-short portfolio.

Take, for example, the portfolio in Application C in which a five-year duration $1,000,000 portfolio is hedged with a $500,000 short position in a 10-year duration portfolio. Note that after one year the long side of the portfolio would tend (in the absence of intervening cash flows) to decline to having a duration of 4.0, while the short side would tend toward a duration of 9.0. The 2-1 hedge would no longer provide immunization.

single-factor asset pricing model

The CAPM is an example of a single-factor asset pricing model. A single-factor asset pricing model explains returns and systematic risk using a single risk factor. Whereas the CAPM describes the entire economy, other single-factor models may simply describe relative prices and returns among a subset of the economy.

The Fisher effect (or Fisher equation)

The Fisher effect (or Fisher equation) states that the nominal interest rate (r) is equal to the sum of the real interest rate (i) and the expected inflation rate (π), when interest rates are expressed as continuously compounded rates.

anticipated inflation rate (π)

The anticipated inflation rate (π) is generally defined as a measure of the expected rate of change in the value of a currency measured through changes in overall price levels. Expectations of inflation rates vary across market participants and are generally unobservable. Accordingly, indications of anticipated inflation are often based on surveys of consensus estimates, derived from past inflation, or inferred from other market information such as interest rates.

risk premium

The asset's risk premium, βi[E(Rm)−Rf], is the product of the asset's risk, or beta, and the market risk premium, meaning the amount investors demand for bearing each unit of risk.

Beta (CAPM)

The beta of a potential investment is a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have a beta greater than one. If a stock has a beta of less than one, the formula assumes it will reduce the risk of a portfolio. A stock's beta is then multiplied by the market risk premium, which is the return expected from the market above the risk-free rate. The risk-free rate is then added to the product of the stock's beta and the market risk premium. The result should give an investor the required return or discount rate they can use to find the value of an asset.

Capital Asset Pricing Model (CAPM)

The capital asset pricing model (CAPM) provides one of the easiest and most widely understood examples of single-factor asset pricing by demonstrating that the risk of the overall market index is the only risk that offers a risk premium. The CAPM is a general equilibrium model, meaning that it prices all assets rather than simply describing one or more relative pricing relationships. Investors expect to be compensated for risk and the time value of money. The risk-free rate in the CAPM formula accounts for the time value of money. The other components of the CAPM formula account for the investor taking on additional risk. The goal of the CAPM formula is to evaluate whether a stock is fairly valued when its risk and the time value of money are compared to its expected return.

Semistrong form informational market efficiency

The concept of semistrong form informational market efficiency (or semistrong level) refers to market prices reflecting all publicly available information (including not only past prices and volumes but also any publicly available information such as financial statements and other underlying economic data). The semi-strong form efficiency theory follows the belief that because all information that is public is used in the calculation of a stock's current price, investors cannot utilize either technical or fundamental analysis to gain higher returns in the market. Those who subscribe to this version of the theory believe that only information that is not readily available to the public can help investors boost their returns to a performance level above that of the general market.

Duration for a Bond Portfolio

The duration for a portfolio is simply a weighted average of the durations of the portfolio's constituent assets, much like the duration for a coupon bond is a weighted average of the durations of its prospective cash flows.

duration of a fixed-coupon bond

The duration of a fixed-coupon bond is the weighted average of the longevities of the cash flows to a coupon bond where the weight of each of the bond's cash flows is the proportion of the bond's total value attributable to that cash flow.

Duration as the Longevity of a Zero-Coupon Bond of Equivalent Risk

The duration of a zero-coupon bond is equal to its time to maturity, as is easily verified by examining the formula for duration in Equation 2 when the coupon rate is set to zero. There is only one cash flow (the principal at maturity) and so the weighted-average life of the zero-coupon bond is T—its time to maturity.

Efficiently inefficient markets

The enigma as to how markets can become efficient when efficiency destroys the incentives to process information has led to the proposition that markets tend toward being efficiently inefficient. The idea is that each market tends toward its own equilibrium degree of informational inefficiency, where that amount of inefficiency balances the marginal costs of additional skill-based trading with the marginal revenues from the skill-based trades.

excess return

The excess return of an asset refers to the excess or deficiency of the asset's return relative to the periodic risk-free rate.

Duration

The general definition of duration is the elasticity of a bond price with respect to a shift in its yield (or a uniform shift in the spot rates, corresponding to each prospective cash flow) The most popular single-factor approach to modeling default-free bond returns is traditional duration. Like any other single-factor measure of interest rate risk, duration is limited in its effectiveness to the type or types of interest rate risks for which it is designed.

Six Factors Driving Informational Market Efficiency: 1

The greater the value of the assets being traded, the greater the competition for potential profits and losses from mispricing, within limits. Higher profit potential motivates market participants to use more information and better analysis. Everything else equal, a $100 trade mispriced by 1% transfers only $1 of wealth between traders, whereas a $1,000,000 trade mispriced by only 0.1% transfers $1,000 of wealth. However, very large asset values, such as huge equity deals, may reduce competition if there are relatively few traders who have the resources to acquire the assets.

market portfolio

The market portfolio is a hypothetical portfolio containing all tradable assets in the world (except riskless financial assets). Each asset in the market portfolio is held in a quantity based on its market weight. The market portfolio that is used to find the market risk premium is only a theoretical value and is not an asset that can be purchased or invested in as an alternative to the stock. Most of the time, investors will use a major stock index, like the S&P 500, to substitute for the market, which is an imperfect comparison.

Managing the Duration of a Long-Only Bond Portfolio to a Target

The target duration of the investor may emanate from several goals: (1) the investor has a horizon point in time at which she is concerned with the uncertainty of the portfolio's value, such as the end of a reporting period, (2) the investor has a set of projected cash needs and wants to be assured those needs can be funded by the portfolio, or (3) the investor views duration as a measure of short-term price risk and wishes to control the short-term price risk by controlling the duration—perhaps in tandem with efforts to predict interest rate shifts and use market timing in an attempt to add return.

The term structure of implied forward rates

The term structure of implied forward rates is the relationship between implied forward rates and the starting point of each rate and is often superimposed on the term structure of spot rates. Given the relation for spot and forward rates derived in the previous section, F(t,T)=(rTT−rtt)/(T−t), it is clear that the forward rate will lie above (below) the spot rate in the case of an upward (downward) sloping spot rate. The gap between the two tends to diminish from left to right. Also, the forward rate tends to "exaggerate" slope changes in the spot rate because the spot is an averaged rate while the forward rate is a marginal rate.

term structure of interest rates

The term structure of interest rates is the relationship between spot interest rates and their associated longevities (i.e., times-to-maturity). Given a large set of default-free zero-coupon bonds with a spectrum of maturities, the term structure can be estimated at each longevity corresponding to the maturities of the zero-coupon bonds.

Managerial Implications of the Three Term-Structure Theories

The three theories of the term structure prescribe different fixed-income strategies. The unbiased expectations hypothesis implies that borrowing and lending decisions should focus on issues of convenience such as cash-flow matching because all longevity-related choices offer equal expected returns to lenders and costs to borrowers. The liquidity premium theory asserts that lenders should seek longer longevities until the marginal aversion to risk offsets the higher expected returns, while borrowers should seek shorter maturities until risks associated with cash flow mismatches (funding risks) offset the lower expected costs. Finally, the market segmentation hypothesis introduces the complexities that different longevities offer expected returns to lenders and costs to borrowers that are driven by supply and demand factors that differ across maturity ranges and that may vary substantially through time.

Time Value of Money (TVM)

The time value of money (TVM) is the concept that a sum of money is worth more now than the same sum will be at a future date due to its earnings potential in the interim. The formula for computing the time value of money considers the amount of money, its future value, the amount it can earn, and the time frame. For savings accounts, the number of compounding periods is an important determinant as well.

modified Fisher equation

The traditional Fisher equation is sometimes modified for anticipated income taxes by assuming a uniform tax rate, T, on nominal interest income. The modified Fisher equation expresses the nominal interest rate as the combination of the after-tax real interest rate, r, and the anticipated rate of inflation (π), with an adjustment for the income tax rate, T, as shown in Equation 6 Equation 6 assumes that r, i, and π are expressed as continuously compounded rates. If annualized rates are used a cross-term is introduced. Note that the Fisher equation is equal to the modified Fisher equation for the case of T = 0.

Problems With the CAPM

There are several assumptions behind the CAPM formula that have been shown not to hold in reality. Modern financial theory rests on two assumptions: (1) securities markets are very competitive and efficient (that is, relevant information about the companies is quickly and universally distributed and absorbed); (2) these markets are dominated by rational, risk-averse investors, who seek to maximize satisfaction from returns on their investments. Despite these issues, the CAPM formula is still widely used because it is simple and allows for easy comparisons of investment alternatives. The CAPM is frequently and correctly criticized for failing to explain and predict financial returns accurately The most serious critique of the CAPM is the assumption that future cash flows can be estimated for the discounting process. If an investor could estimate the future return of a stock with a high level of accuracy, the CAPM would not be necessary.

The Three Primary Theories of the Term Structure of Interest Rates

There are three primary theories (i.e., hypotheses) for the shape of the term structure of spot interest rates (default-free). The theories propose explanations of the relationship between interest rates corresponding to different longevities. These theories are vital tools that cut to the heart of fixed-income investment management. 1. The Unbiased Expectations or Pure Expectations Theory 2. The Liquidity Preference or Liquidity Premium Theory 3. The Market Segmentation or Preferred Habitat Theory

ex post model

This section discusses an ex post (meaning "from afterward" or realized) form of the CAPM. An ex post model describes realized returns and provides an understanding of risk and how it relates to the deviations of realized returns from expected returns.

duration of a fixed-coupon bond formula

This standard formula for duration, D, is shown in Equation 2 for a fixed-coupon bond (with annual coupon payment and annual compounding for exposition simplicity), where t is time, C(t) is the bond's promised cash flow for time t, y is the bond's yield to maturity, and Po is the bond's current value. Note that in the case of a fixed-coupon bond, the cash flows prior to T are coupons and the cash flow at period T is the principal payment.

Certain factors can affect a bond's duration, including:

Time to maturity: The longer the maturity, the higher the duration, and the greater the interest rate risk. Consider two bonds that each yield 5% and cost $1,000, but have different maturities. A bond that matures faster—say, in one year—would repay its true cost faster than a bond that matures in 10 years. Consequently, the shorter-maturity bond would have a lower duration and less risk. Coupon rate: A bond's coupon rate is a key factor in calculation duration. If we have two bonds that are identical with the exception of their coupon rates, the bond with the higher coupon rate will pay back its original costs faster than the bond with a lower yield. The higher the coupon rate, the lower the duration, and the lower the interest rate risk.

Weak form informational market efficiency

Weak form informational market efficiency (or weak level) refers to market prices reflecting available data on past prices and volumes (i.e., historical trading data). The weak form suggests that today's stock prices reflect all the data of past prices and that no form of technical analysis can be effectively utilized to aid investors in making trading decisions. Advocates for the weak form efficiency theory believe that if the fundamental analysis is used, undervalued and overvalued stocks can be determined, and investors can research companies' financial statements to increase their chances of making higher-than-market-average profits. If weak form efficiency is violated, then all three forms will be violated, because the semistrong and strong forms include the information set used in the weak form.