Exam 2: 12, 13, 14, 9, 10
Capital market assumption 3
: Capital market theory assumes that a risk-free asset exists in which investors can invest. Moreover, it assumes that investors can borrow funds at the same interest rate offered on that risk-free asset. That is, it assumes that investors can lend and borrow at some risk-free rate.
credit default swaps
A credit default swap (CDS) is a derivative that can be used to buy or sell protection against particular types of events that can adversely affect the credit quality of a debt obligation, such as the default or bankruptcy of the borrower. The two parties to a CDS: the protection buyer and protection seller. Over the life of the CDS, the protection buyer agrees to pay the protection seller a payment at specified dates to insure against the impairment of the debt of a reference entity as the result of a credit events. If a credit event does occur, the protection buyer makes a payment only up to the credit event date and makes no further payment. At this time, the protection seller is obligated to perform; the contract will call for the protection seller to compensate for the loss in the value of the debt obligation.
underlying
A derivative is an instrument that derives its value from the value of an
Futures Contract
A futures contract is an agreement that requires a party to the agreement either to buy or to sell something at a designated future date at a predetermined price. Commodity Futures: traditional agriculture commodities, imported foodstuffs, and industrial commodities. Financial Futures: stock index futures, interest rate futures, currency futures. The price at which the parties agree to transact in the future is called the futures price. The designated date on which the parties must transact is called the settlement or delivery date. The contract with the closest settlement date is called the nearby futures contract. A party to a futures contract has two choices to liquidate the position: The position can be liquidated prior to the settlement date; the party must take an offsetting position in the same contract. The party can wait until the settlement date, and either accept delivery or deliver the underlying at the agreed-upon price. Futures contracts settled in cash only are referred to as cash settlement contracts. Associated with every futures exchange is a clearinghouse, which performs several functions. One of these functions is guaranteeing that the two parties to the transaction will perform. The risk that the parties to a futures contract—or any derivative contract, for that matter—face is that the counterparty will default, also referred to ask the counterparty risk. The clearinghouse interposes itself as the buyer for every sale and the seller for every purchase. initial margin and is required as deposit for the contract. Maintenance margin is the minimum level (specified by the exchange) by which an investor's equity position may fall as a result of an unfavorable price movement before the investor is required to deposit additional margin.
leptokurtic distribution
A probability distribution with this characteristic is said to be a
short and long hedge
A short hedge is used to protect against a decline in the future cash price of a financial instrument or portfolio. To execute a short hedge, the hedger sells a futures contract (agrees to make delivery). Consequently, a short hedge is also known as a sell hedge. A long hedge is undertaken to protect against an increase in the price of a financial instrument or portfolio to be purchased in the cash market at some future time. In a long hedge, the hedger buys a futures contract (agrees to accept delivery). A long hedge is also known as a buy hedge. The cash price is also called the spot price.
caps and floors
Agreements are available in the financial market in which one party, for a fee, agrees to compensate the other if a designated reference is different from a predetermined level, also called the strike. When the seller agrees to pay the buyer if the designated reference exceeds a predetermined level, the agreement is referred to as a cap. The agreement is referred to as a floor when the seller agrees to pay the buyer if a designated reference falls below a predetermined level. The designated reference could be a specific interest rate, the rate of return on some domestic or foreign stock market index, or an exchange rate. The payment made by the seller of the cap to the buyer on a specific date is determined by the relationship between the designated reference and the strike. In a cap or floor, the buyer pays a fee that represents the maximum amount the buyer can lose and the maximum amount the seller of the agreement can gain. The buyer of a cap benefits if the designated reference rises above the strike price because the seller must compensate the buyer. The buyer of a floor benefits if the designated reference falls below the strike price because the seller must compensate the buyer. In essence, the payoff of these contracts is the same as in an option.
Capital Market theory 1
An investor should create portfolio with the highest expected return for a given level of risk
exotic options
An option can be created with an exercise style in which the option can be exercised at several specified dates, as well as at the expiration date of the option. Such options are referred to as limited exercise options, Bermuda options, or Atlantic options. An alternative option, also called an either-or option, provides a payoff that is the best independent payoff of two distinct assets. An outperformance option is an option whose payoff is based on the relative payoff of two assets at the expiration date.
option
An option is a contract in which the writer of the option grants the buyer of the option the right, but not the obligation, to purchase from or sell to the writer something at a specified price within a specified period of time (or at a specified date). The writer, also referred to as the seller, grants this right to the buyer in exchange for a certain sum of money, which is called the option price or option premium. The price at which the asset may be bought or sold is called the exercise or strike price. The date after which an option is void is called the expiration date.
feasible portfolio
Any portfolio that an investor can construct given the universe of candidate assets is referred to as a
Capital Market theory 3
As an example, we'll compare portfolio 𝑷_𝑨, which is on the Markowitz efficient frontier, with portfolio 𝑷_𝑩 which is on the line and therefore has some combination of the risk-free rate and the Markowitz efficient portfolio M. Notice that for the same risk, the expected return is greater for 𝑷_𝑩 than for 𝑷_𝑨. In fact, this dominance is true fro all but one portfolio on the line, portfolio M, which is on the Markowitz efficient frontier. Recognizing this preference, we must modify the conclusion from portfolio theory that an investor will select a portfolio on the Markowitz efficient frontier, depending on the investor's level of risk aversion.
Security return = Systematic return + Unsystematic return.
As the systematic return is proportional to the market return, it can be expressed as the symbol beta (β) times the market return, RM. The proportionality factor beta is a market sensitivity index, indicating how sensitive the security return is to changes in the market level.
nonlinear payoff derivative
Asymmetric Payoff: Derivatives with this type of payoff are called options. Nonlinear payoff derivatives are equivalent in terms of their economic function insurance arrangements.
Hersh Shefrin has noted the following three themes in the behavioral finance literature:
Behavioral Finance Theme 1: Investors err in making investment decisions because they rely on rules of thumb. Behavioral Finance Theme 2: Investors are influenced by form as well as substance in making investment decisions. Behavioral Finance Theme 3: Prices in the financial market are affected by errors and decision frames. The first behavioral finance theme involves heuristics. This term means a rule-of-thumb strategy or a good guide to follow to shorten the time it takes to make a decision. For example, here are three rules of thumb provided on the MSN Money website for increasing the likelihood of success when investing in common stock: (1) ignore guru predictions, (2) avoid cheap stocks, and (3) follow the big players.
Suppose the futures contract costs $92. Consider this strategy:
Buy the futures contract at $92. Short asset XYZ in the cash market for $100. Invest $100 for three months at 8% per year.
Tests of the CAPM 2
CAPM equation asserts that over the period of time analyzed, (1) a linear relationship exists between the average risk premium return on the market and the average risk premium return on a stock or portfolio, and its slope is βi; and (2) the linear relationship should pass through the origin. Moreover, according to the CAPM, beta is a complete measure of a stock's risk. Consequently, alternative risk measures that might be proposed, the most common being the standard deviation of return, should not be significant contributors to the explanation of a stock's return. We recall here that the standard deviation measures a stock's total risk, which includes both systematic and unsystematic components. The CAPM holds for both individual securities and portfolios. Therefore, the empirical tests can be based on either. Tests based on individual securities, however, are not the most efficient method of obtaining estimates of the magnitude of the risk/return trade-off, for two reasons.
Behavioral assumption 3
Capital market theory assumes all investors make investment decisions over some single-period investment horizon. The length of that period (six months, one year, two years, etc.) is not specified. In reality, the investment decision process is more complex, with many investors looking at more than one investment horizon. Nonetheless, the assumption of a one-period investment horizon is necessary to simplify the mathematics of the theory.
Capital Market Assumption 2
Capital market theory assumes no transaction costs or impediments interfere with the supply and demand for an asset. Economists refer to these various costs and impediments as "frictions". The costs associated with frictions generally result in buyers paying more than in the absence of frictions and sellers receiving less. In the case of financial markets, frictions include commissions charged by brokers and the bid-ask spreads charged by dealers. They also include taxes and government-imposed transfer fees.
Behavioral assumption 4
Capital market theory assumes that all investors have the same expectations with respect to the inputs that are used to derive the Markowitz efficient portfolios, namely, asset returns, variances, and correlations.
Behavioral assumptions
Capital market theory assumes that investors make investment decisions based on two parameters: the expected return and the variance of returns. This why the portfolio theory introduced previously is referred to as a mean-variance model. This assumption tells us what investors use as inputs in making their investment decisions. Their specific behavior follows the assumptions that to accept greater risk, they must be compensated by the opportunity of realizing a higher return. We refer to such investors as risk averse. This definition is oversimplified; a more rigorous definition of risk aversion is described by a mathematical specification of an investor's utility function. This complexity need not concern us here.
Capital Market Assumption 1
Capital market theory assumes that the capital market is perfectly competitive.
Behavior assumption 2
Capital market theory assumes that the risk-averse investor ascribes to the method of reducing portfolio risk by combining assets with non-unity correlations.
Suppose the futures contract costs $107.
Consider this strategy: Sell the futures contract at $107. Purchase asset XYZ in the cash market for $100. Borrow $100 for three months at 8% per year. From settlement of the futures contract: Proceeds from sale of asset XYZ: $107 Payment received from investing in asset XYZ for 3 mos.: $3 Total proceeds: $110 From the loan: Repayment of principal of loan: $100 Interest on loan 2% for 3 months: $2 Total outlay: $102 This cash and carry trade guarantees a profit of $8
How does an investor construct portfolio M?
Eugene Fama answered this question by demonstrating that M must consist of all assets available to investors, and each asset must be held in proportion to its market value relative to the total market value of all assets. So, for example, if the total market value of some asset is $500 million and the total market value of all assets is $X. Because portfolio M consists of all assets, it is referred to as the market portfolio.
theoretical futures price
F= P+P(r-y)
market price of risk.
For this reason, the slope of the CML is referred to as the
reservse cash and carry trade
From settlement of the futures contract: Price paid for purchase of asset XYZ: $92 Proceeds to lender of asset XYZ in order to borrow the asset: $3 Total outlay: $95 From the loan: Proceeds received from maturing of investment: $100 Interest earned from the 3-month loan investment: $2 Total proceeds: $102 In this strategy a security is sold short and the proceeds received from the short sale are invested.
two fund separation theorem,
The theoretical result that all investors will hold a combination of the risk-free asset and the market portfolio is known as the ____________________
capital asset pricing model (CAPM):
If we substitute 𝜷_𝒊 into the CAPM equation, we have the beta version of the SML, popularly referred to as the _____________ 𝑬[𝑹_𝒊 ]=𝑹_𝑭+𝜷〗_𝒊 (𝑬[𝑹_𝑴 ]−𝑹_𝑭 )
efficient portfolios
In constructing a portfolio of assets, investors seek to maximize the expected return from their investment, given some level of risk they are willing to accept. (Alternatively stated, investors seek to minimize the risk they are exposed to given some target expected return.) Portfolios that satisfy this requirement are called______________________________
Multifactor CAPM 3
In the multifactor CAPM, in addition to investing in the market portfolio, investors will also allocate funds to something equivalent to a mutual fund that hedges a particular extra-market risk. Although not all investors are concerned with the same sources of extra-market risk, those that are concerned with a specific extra-market risk will basically hedge them in the same way. We have just described the multifactor model for a portfolio. How can this model be used to obtain the expected return for an individual security? Since individual securities are nothing more than portfolios consisting of only one security, our multifactor CAPM must hold for each security, i. That is, 𝑬[𝑹_𝒊 ]=𝑹_𝑭+\ 𝜷_(𝒊.𝑴) (𝑬[𝑹_𝑴 ]−𝑹_𝑭 )+𝜷_(𝒊,𝑭𝟏) (𝑬[𝑹_𝑭𝟏 ]−𝑹_𝑭 )+𝜷_(𝒊,𝑭𝟐) (𝑬[𝑹_𝑭𝟐 ]−𝑹_𝑭 )+⋯ +𝜷_(𝒊,𝑭𝑲) (𝑬[𝑹_𝑭𝑲 ]−𝑹_𝑭 )
systematic risk(nondiversifianle risk =
Market risk
Asset return
R=(p1-p0+C)/Po
Tests of the CAPM VII
Since 1977 a number of studies have purported either to support or to reject the CAPM. These tests attempt to examine implications of the CAPM other than the linearity of the risk/return relation as the basis of their methodology. Unfortunately, none provides a definitive test, and most are subject to substantial criticism, suffering from the same problem of identifying the "true" market portfolio.
To construct efficient portfolios, the following assumptions about how investors select assets are made:
Mean-Variance Assumption: Only the expected value and the variance are used by investors in making asset selection decisions. Risk-Aversion Assumption: Investors are risk averse, which means that when faced with a decision about which of two assets in which to invest when both have the same expected return but different risks, investors will prefer the asset with the lower risk. Homogeneous Expectations: Assumption All investors have the same expectations regarding expected return, variance, and covariance for all risky assets. One-Period Horizon: Assumption All investors have a common one-period investment horizon. Optimization Assumption: In constructing portfolios, investors seek to achieve the highest expected return for a given level of risk.
Capital Market Theory 5
Now we can restate how a risk-averse investor who makes investment decisions as suggested by portfolio theory and who can borrow and lend at the risk-free rate should construct efficient portfolios. This process combines an investment in the risk-free asset with the market portfolio. The theoretical result that all investors will hold a combination of the risk-free asset and the market portfolio is known as the two fund separation theorem, with one fund consisting of the risk-free asset and the other consisting of the market portfolio. Although all investors will select a portfolio on the CML, the optimal portfolio for a specific investor is the one that will maximize that investor's risk preference.
exchange traded vs. OTC options
Options, like other financial instruments, may be traded either on an organized exchange or in the over-the-counter (OTC) market. Exchange-traded options offer three advantages. The exercise price and the expiration date of the contract are standardized. The direct link between buyer and seller is severed after order execution. Hence, counterparty risk is minimal. The transaction costs are lower for exchange-traded options than for OTC options. OTC options are sometimes referred to as dealer options. Investors use futures to protect against symmetric risk and options to protect against asymmetric risk.
theoretical futures price
Suppose the futures contract costs $99. Consider either one of the previous strategies by either buying or selling the futures contract at $99. The other activities remain the same (i.e. long or short asset XYZ at $100 and borrow or invest in $100). Repeat proceeds and outlay calculations. The profit in either strategy will be $0. Neither strategy results in an arbitrage profit. Hence, a futures price of $99 is the equilibrium price because any higher or lower futures price would permit riskless arbitrage profits. This equilibrium price is also called the theoretical futures price.
Linear payoff derivative
Symmetric Payoff: every $1 change in the price of the underlying results in an identical gain.
Kurtosis
The "fatness" of the tails of a probability distribution is related to the peakedness of the distribution about its mean. The joint measure of peakedness and tail fatness is called kurtosis. Kurtosis, denoted by α, is the fourth moment of a probability distribution and determines the tail weight. As with skewness, there is no standard measure for kurtosis. Fisher's kurtosis and Pearson's kurtosis are two common measures. Fisher's kurtosis, also referred to as excess kurtosis, is found by subtracting three from Pearson's kurtosis
arbitrage pricing theory (APT) model.
The APT model postulates that a security's expected return is influenced by a variety of factors, as opposed to just the single market index of the CAPM. Specifically, look at the CAPM equation, which states the return on a security is dependent on its market sensitivity index and an unsystematic return. The APT, in contrast, states that the return on a security is linearly related to H "factors". The APT does not specify what these factors are, but it is assumed that the relationship between security returns and the factors is linear.
three ways to classify derivatives
The Underlying of Derivatives Commodity Derivatives: Hard (energy, precious & industrial metals) vs. Soft (wheat, livestock, grains, coffee, sugar, oranges) Financial Derivatives: securities, reference financial index, currency pair, reference interest rate The Nature of the Underlying Risks (eg. Equity Price Risk, Interest Rate Risk, Credit Risk, Foreign Exchange Risk) The Type of Payoff (Linear, Nonlinear)
buying put option
The buying of a put option is referred to as a long put position. The loss is limited to the option price and the theoretical maximum profit is generated if asset XYZ's price falls to zero.
hedging with futures
The major function of futures markets is to transfer price risk from hedgers to speculators. Hedging in this case is the employment of a futures transaction as a temporary substitute for a transaction to be made in the cash market. As long as cash and futures prices move together, any loss realized on one position (whether cash or futures) will be offset by a profit on the other position. When the profit and loss are equal, the hedge is called a perfect hedge. In a market where the futures contract is correctly priced, a perfect hedge should provide a return equal to the risk-free rate.
factors that influence the option price
The current price of the underlying asset. The strike price. (call decrease, put increase) The time to expiration of option. (call increase, put increase) Expected price volatility of the underlying asset over the life of the option. (call increase, put increase) The short-term risk-free interest rate over the life of the option. (call-increase, put decrease) Anticipated cash payments on the underlying asset over the life of the option. (call decrease, put increase)
risks of hedging
The difference between the cash price and the futures price is called the basis. If a futures contract is priced according to its theoretical value, the difference between the cash price and the futures price should be equal to the cost of carry. The risk that the hedger takes is that the basis will change, called basis risk. Hedging involves the substitution of basis risk for price risk, that is, the substitution of the risk that the basis will change for the risk that the cash price will change. When a futures contract is used to hedge a position where either the portfolio or the individual financial instrument is not identical to the instrument underlying the futures, it is called cross-hedging. Cross-hedging is common in asset/liability and portfolio management because no futures contracts are available on specific common stock shares and bonds. Cross-hedging introduces another risk—the risk that the price movement of the underlying instrument of the futures contract may not accurately track the price movement of the portfolio or financial instrument to be hedged. This is called cross-hedging risk.
CAPM 3
The first problem is called the "errors in variables bias" By carefully grouping the securities into portfolios of securities with similar betas, much of this measurement error problem can be eliminated. The errors in individual stocks' betas cancel out so that the portfolio beta can be measured with much greater precision. In turn, tests based on portfolio returns will be more efficient than tests based on security returns.
increased volatitity bad?
The greater volatility resulting from an innovation may simply more faithfully reflect the actual variability of fundamental values. In this case, "more" asset volatility need not be bad but rather may be a manifestation of a well-functioning market. Of course, to say that more volatility need not be bad does not mean that it is good. Clearly, price volatility greater than what can be justified by relevant new information or fundamentals (or by standard asset pricing models) is undesirable. By definition, it makes prices inefficient and is referred to as "excess volatility." No one has been able to test whether recent financial innovations actually increased or decreased excess volatility. Moreover, as Franklin Edwards has pointed out, "Too little volatility is equally bad, although this concept does not seem to have generated enough interest to have been given the label of 'deficient volatility.'"
pricing of option
The intrinsic value of an option is the economic value of the option if it is exercised immediately. The option price reflects the option's intrinsic value and any additional amount over its intrinsic value, often referred to as the time value or time premium. The intrinsic value of a call option is the difference between the current price of the underlying asset and the strike price if positive; it is otherwise zero. For a put option, the intrinsic value equals the amount by which the current asset price is below the strike price.
economic meaning of the slope
The numerator is the expected return of the market beyond the risk-free return. It provides a measure of the risk premium, or the reward for holding the risky market portfolio rather than the risk-free asset. The denominator is the risk of the market portfolio. Thus, the slope measures the reward per unit of market risk.
Capital market theory 4
The particular efficient portfolio that the investor will select on the line will depend on the investor's risk preference. the opportunity to borrow or lend at the risk-free rate implies a capital market in which risk-averse investors will prefer to hold portfolios consisting of combinations of the risk-free asset and some portfolio M on the Markowitz efficient frontier.
buying call option
The purchase of a call option is referred to as a long call position. Even though the breakeven point and the loss depend on the option price and the strike price, the profile shown in figure 10.1 holds for all buyers of call options. The shape indicates that the maximum loss is the option price and is subject to substantial upside potential.
Tests of CAPM V
The relationship between risk and return appears to be linear. The studies give no evidence of significant curvature in the risk/return relationship. Tests that attempt to discriminate between the effects of systematic and unsystematic risk do not yield definitive results. Both kinds of risk appear to be positively related to security returns, but substantial evidence supports the proposition that the relationship between return and unsystematic risk is at least partly spurious—that is, partly a reflection of statistical problems rather than the true nature of capital markets. Obviously, we cannot claim that the CAPM is absolutely right. On the other hand, the early empirical tests do support the view that beta is a useful risk measure and that high-beta stocks tend to be priced so as to yield correspondingly high rates of return.
put call parity relationship
The relationship between the price of a call option and the price of a put option on the same underlying instrument, with the same strike price and the same expiration date is commonly referred to as the put-call parity relationship. Put option price − Call option price = Present value of strike price + Present value of cash distribution − Price of underlying asset. This relationship is the put-call parity relationship for European options, though it is approximately true for American options. If this relationship does not hold, arbitrage opportunities exist(i.e. portfolios that consist of long and short positions in the asset and related options will provide an extra return with certainty).
CAPM 4
The second problem relates to the obscuring effect of residual variation. Realized security returns include a large random component, which typically accounts for about 70% of the variation of return (the diversifiable or unsystematic risk of the stock). By grouping securities into portfolios, we can eliminate much of this "noise" and thereby get a much clearer view of the relationship between return and systematic risk. The evidence shows a significant positive relationship between realized returns and systematic risk as measured by beta. The average market risk premium estimated is usually less than predicted by the CAPM, however.
The Multifactor CAPM II
The total extra-market sources of risk is equal to: 𝜷_(𝒑,𝑭𝟏) (𝑬[𝑹_𝑭𝟏 ]−𝑹_𝑭 )+𝜷_(𝒑,𝑭𝟐) (𝑬[𝑹_𝑭𝟐 ]−𝑹_𝑭 )+⋯+𝜷_(𝒑,𝑭𝑲) (𝑬[𝑹_𝑭𝑲 ]−𝑹_𝑭 ) This expressions says that investors want to be compensated for the risk associated with each source of extra-market risk, in addition to market risk.
writing (selling) call option
The writer of a call option is said to be in a short call position. The maximum profit is the option price and the maximum loss is not limited.
For example, suppose that the market value of the investor's portfolio is initially $100,000, consisting of $20,000 in security 1, $30,000 in security 2, and $50,000 in security 3. Suppose an investor changes the initial portfolio to $35,000 in security 1, $25,000 in security 2, and $40,000 in security 3.
Then the percentage changes would be as follows: 𝑽_𝟏=($𝟑𝟓,𝟎𝟎𝟎−$𝟐𝟎,𝟎𝟎𝟎)/($𝟏𝟎𝟎, 𝟎𝟎𝟎)=𝟎.𝟏𝟓. 𝑽_𝟐=($𝟐𝟓,𝟎𝟎𝟎−$𝟑𝟎,𝟎𝟎𝟎)/($𝟏𝟎𝟎, 𝟎𝟎𝟎)=−𝟎.𝟎𝟓. 𝑽_𝟑=($𝟒𝟎,𝟎𝟎𝟎−$𝟓𝟎,𝟎𝟎𝟎)/($𝟏𝟎𝟎, 𝟎𝟎𝟎)=−𝟎.𝟏𝟎.
multifactor CAPM
These extra-market sources of risk are also referred to as "factors." Hence the model derived by Merton is called a _________________ 𝑬[𝑹_𝒑 ]=𝑹_𝑭+\ 𝜷_(𝒑.𝑴) (𝑬[𝑹_𝑴 ]−𝑹_𝑭 )+𝜷_(𝒑,𝑭𝟏) (𝑬[𝑹_𝑭𝟏 ]−𝑹_𝑭 )+𝜷_(𝒑,𝑭𝟐) (𝑬[𝑹_𝑭𝟐 ]−𝑹_𝑭 )+⋯ +𝜷_(𝒑,𝑭𝑲) (𝑬[𝑹_𝑭𝑲 ]−𝑹_𝑭 )
The Security Market Line III
To see how this relationship is developed, let's look at the decomposition of our security return. In a well-diversified portfolio (i.e., the Markowitz diversified), the unique or unsystematic risk is eliminated. Consequently, it can be demonstrated that 𝝈^𝟐 (𝑹_𝒊 )=𝜷_𝒊^𝟐 𝝈^𝟐 ( 𝑹_𝑴 ) And the standard deviation is: 𝝈(𝑹_𝒊 )=𝜷_𝒊 𝝈( 𝑹_𝑴 )⇒𝜷_𝒊=𝝈( 𝑹_𝒊 )/𝝈( 𝑹_𝑴 ) If we substitute 𝜷_𝒊 into the CAPM equation, we have the beta version of the SML, popularly referred to as the capital asset pricing model (CAPM): 〖𝑬[𝑹_𝒊 ]=𝑹_𝑭+𝜷〗_𝒊 (𝑬[𝑹_𝑴 ]−𝑹_𝑭 )
lending vs. borrowing
Typically the borrowing rate is greater than the lending rate. Therefore, the actual theoretical futures price is bounded by the two futures prices as calculated from the cash-and-carry (using the borrowing rate) and reverse cash-and-carry (using the lending rate) trades. Transaction costs widen the boundaries for the theoretical futures price. include entering into and closing the cash position transaction costs for the futures contract
option pricing model
Using arbitrage arguments, it can be shown that the minimum price for an American call option is its intrinsic value. Call option price ≥ Max (0, Price of asset − Strike price). Of the several models available to determine the theoretical value of an option, perhaps the most popular is that developed by Fischer Black and Myron Scholes in 1973 for valuing European call options. The idea behind the arbitrage argument is that if the payoff from owning a call option can be replicated by purchasing the asset underlying the call option and borrowing funds, the price of the option is then (at most) the cost of creating the replicating strategy. The theoretical option price can be calculated using the binomial option pricing model based on arbitrage arguments.
utility function
assigns a (numerical) value to all possible choices faced by a decision maker, and the larger the assigned value of a particular choice, the greater the utility derived from that choice.
safety first risk measure
VAR- value at risk VaR is defined as the minimum level of loss at a given, sufficiently high, confidence level for a predefined time horizon. A VaR's predefined time horizon could be a time period of any length. The confidence level that is often used in practice is either 95% or 99%. Banks, for example, calculate a daily VaR. Suppose a portfolio has a one-week 95% VaR equal to $10 million. This means that over the horizon of one week, the portfolio may lose more than $10 million with probability equal to 1 minus the confidence interval, 5% in our example. The probability of 1 minus the confidence level is called the tail probability. Although this example is in terms of a portfolio's dollar loss, the VaR for a portfolio can be calculated in terms of percentage returns.
binomial option pricing model
We begin by constructing a hedged portfolio consisting of (1) a long position in a certain amount of the asset and (2) a short call position in the underlying asset. A hedged portfolio is one in which the amount of the underlying asset purchased is such that the position will be hedged against any change in the price of the asset at the expiration date of the option. That is, given the two possible asset prices (binomial), the hedged portfolio is riskless. To derive the price of a call option, we can rely on the basic principle that the hedged portfolio, being riskless, must have a return equal to the riskless rate. Our strategy should correctly price the call regardless of its price, which affects only the magnitude of the outcome.
The CML says that the expected return on a portfolio is equal to the risk-free rate plus a risk premium.
We seek a measure of the risk premium. According to the capital market theory, the risk premium is equal to the market price of risk times the quantity of risk for the portfolio (as measured by the standard deviation of the portfolio). That is, 𝑬[𝑹_𝒑 ]=𝑹_𝑭+𝐌𝐚𝐫𝐤𝐞𝐭 𝐩𝐫𝐢𝐜𝐞 𝐨𝐟 𝐫𝐢𝐬𝐤 𝐱 𝐀𝐦𝐨𝐮𝐧𝐭 𝐨𝐟 𝐫𝐢𝐬𝐤
call or call option
When an option grants the buyer the right to purchase the designated instrument from the writer (seller), it is referred to as a call option, or call.
The portfolio return of interest is known as a joint probability distribution.
When dealing with joint probability distributions, an investor is faced with the interdependence between the two return distributions. For example, in the case of the returns for the common stock of companies A and B, do large returns for company A imply large or small returns for the stock of company B? This property is referred to as the dependence of random variables. When there is no dependence between two random variables, the two random variables are said to be independently distributed. The two random variables are said to be independently distributed if the value of one random variable does not provide any information about the value of the other random variable.
put or put option
When the option buyer has the right to sell the designated instrument to the writer, the option is called a put option, or put. An option is also categorized according to when the option buyer may exercise the option, also referred to as the option's exercise style. American option. May be exercised at any time up to and including the expiration date. European option. May be exercised only at the expiration date. The maximum amount that an option buyer can lose is the option price and the maximum profit that the option writer can realize is the option price.
writing (selling) put option
Writing a put option is referred to as a short put position. The maximum profit is the option price and the theoretical maximum loss can be be as large as the strike price less the option price.
Capital Market Assumption 3:
a risk-free asset exists in which investors can invest. investors can borrow funds at the same interest rate offered on that risk-free asset. That is, investors can lend and borrow at some risk-free rate.
Behavioral assumtpion 4
all investors have the same expectations with respect to the inputs: asset returns, variances, and correlations. This is the homogeneous expectations assumption.
asset pricing models
allows one to determine the theoretical value of an asset knowing the expected cash flow and the expected return. some simplifying assumptions. These assumptions simplify matters a great deal, some of them may even seem unrealistic. they make economic theories more tractable from a mathematical standpoint. criticized by proponents of behavioral finance.
Capital Market Theory 2
assuming that investors can borrow and lend at the risk-free rate, the conclusion of portfolio theory can be illustrated by the figure on the previous slide. Every combination of the risk-free asset and the Markowitz portfolio M is shown on the tangent line in the figure. This line is drawn from the vertical as the risk-free rate tangent to the Markowitz efficient set, also referred to as the efficient frontier. The point of tangency is denoted by M. All of the portfolios on the line are feasible for the investor to construct. Portfolios to the left of M represent combinations of risky assets and the risk-free asset. Portfolios to the right of M include purchases of risky assets made with funds borrowed at the risk-free rate.
continuous random variable
can take on any possible value within the range of outcomes.
At a portfolio size of about 20 randomly selected
common stocks, the level of total portfolio risk is reduced such that what is left is systematic risk. For individual stocks, the average ratio of systematic risk to total risk is about 30%. On average, approximately 40% of the single-security risk is eliminated by forming randomly selected portfolios of 20 stocks. Additional diversification yields a rapidly diminishing reduction in risk. The improvement is slight when the number of securities held is increased beyond, say, 10. The return on a diversified portfolio follows the market closely, with the ratio of systematic risk to total risk exceeding 90%.
Capital Market theory
deals with the effects of investor decisions on security prices. More specifically, it shows the relationship that should exist between security returns and risk, if investors construct portfolios as indicated by portfolio theory. Together, portfolio theory and capital market theory yield a framework to specify and measure investment risk and develop an equilibrium relationship between risk and expected return (and hence between risk and the required return of an investment).
Portfolio theory
deals with the selection of portfolios to maximize expected returns consistent with individually acceptable levels of risk. Portfolio theory covers how investors under assumed conditions select the assets to be included in a portfolio. The theory presented is referred to by various names: modern portfolio theory, Markowitz portfolio theory, and mean-variance theory.
Systematic risk can be quantified by
dividing security return into two parts, one part perfectly correlated with and proportionate to the market return and a second part independent from (uncorrelated with) the market.
Behavioral assumption 1:
investors make investment decisions based on two parameters: the expected return and the variance of returns. to accept greater risk, they must be compensated by the opportunity of realizing a higher return. We refer to such investors as risk averse
sharpe ratio
most popular reward-risk ratio that measures reward on absolute basis.
Capital Market Assumption 2:
no transaction costs or impediments ("frictions"). The costs associated with frictions generally result in buyers paying more and sellers receiving less. In the case of financial markets, frictions include commissions charged by brokers and the bid-ask spreads charged by dealers. They also include taxes and government-imposed transfer fees.
swaps
is an agreement whereby two parties (called counterparties) agree to exchange periodic payments. The dollar amount of the payments exchanged is based on some predetermined dollar principal, called the notional amount or notional principal. Types of Swaps. Interest Rate Swaps. The counterparties swap payments in the same currency based on a fixed or floating interest rate, also referred to as the reference rate. Interest-Equity Swaps. One party exchanges a payment based on an interest rate and the other party exchanges a payment based on the return of some equity index. Equity Swaps. Both parties exchange payments in the same currency based on some equity index. Currency Swaps. Both parties agree to swap payments based on different currencies. Swaps can be decomposed into a package of forward contracts. In many markets with forward and futures contracts, the longest maturity does not extend out as far as that of a typical swap. A swap is a more transactionally efficient than forward contracts, which means that in one transaction an entity can effectively establish a payoff equivalent to a package of forward contracts. The liquidity of the swap market continues to grow since its beginning in 1981; it is now more liquid than many forward contracts, particularly long-dated (i.e., long-term) forward contracts.
cash and carry trade
is so named because the strategy involves borrowing cash to purchase a security and "carrying" that security to the futures settlement date. Assumptions: In the cash market, asset XYZ is selling for $100. Asset XYZ pays the holder (with certainty) $12 per year in four quarterly payments of $3, and the next quarterly payment is exactly three months from now. The futures contract requires delivery three months from now. The current three-month interest rate at which funds can be loaned or borrowed is 8% per year. Question: What should the futures price be?
systematic risk
is the portion of an asset's return variability that can be attributed to a common factor. It is also called undiversifiable risk. Systematic risk is the minimum level of risk that can be obtained for a portfolio by means of diversification across a large number of randomly chosen assets. systematic risk results from general market and economic conditions that cannot be diversified away. The portion of an asset's return variability that can be diversified away is referred to as unsystematic risk. It is also sometimes called diversifiable risk, residual risk, idiosyncratic risk, or company-specific risk.
forward contract
just like a futures contract, is an agreement for the future delivery of something at a specified price at the end of a designated period of time. A forward contract differs nonstandardized (i.e., the terms of each contract are negotiated individually between buyer and seller), no clearinghouse coordinates forward contract trading, secondary markets are often nonexistent or extremely thin over-the-counter instrument, as opposed to exchange traded intended for delivery. The parties in a forward contract are exposed to counterparty risk because either party may default on the obligation.
discrete random variable
limits the outcomes such that the random variable can only take on discrete values.
Four measures are commonly used to describe a probability distribution:
location, dispersion, asymmetry, and concentration in tails.
Behavioral finance
looks at how psychology affects investor decisions and the implications not only for the portfolio theory we describe in this chapter but also for asset pricing theory, option pricing theory, and market efficiency.
asymmetry of risk
means that it is reasonable to expect that risk is an asymmetric concept related to downside outcomes. Consequently, any realistic candidate for an investment risk measure should value upside and downside differently. The standard deviation considers positive and negative deviations from the mean as a potential risk, so it is unsatisfactory on this front.
Volatility clustering
means that large price changes tend to be followed by large price changes and small price changes tend to be followed by small price changes.
autoregressive behavior
means that price changes depend on price changes in the past (e.g., positive price changes tend to be followed by positive price changes).
temporal behavior of tail thickness
means that the probability of extreme price changes through time is smaller in normal markets and much larger in turbulent markets.
Relativity of risk
means that the risk should be related to performing worse than some alternative investment or benchmark.
Dispersion
measure of how the spread out the potential outcomes are that can be realized.
Behavioral assumption 3
single-period investment horizon. The length of that period (six months, one year, two years, etc.) is not specified. In reality, the investment decision process is more complex, with many investors looking at more than one investment horizon.
Stability property
states that the same of a number of N random variables that follow a normal distribution will again be a normal distribution, provided that the random variables behave independently of one another.
Capital Market Assumption 1:
the capital market is perfectly competitive. In general, the number of buyers and sellers is sufficiently large, and all investors are small enough relative to the market, that no individual investor can influence an asset's price. all investors are price takers, the market price is determined where supply equals demand.
Behavorial assumption 2
the risk-averse investor reduces portfolio risk by combining assets with non-unity correlations.
CML 4
𝑬[𝑹_𝒑 ]=𝑹_𝑭+(𝑬[𝑹_𝑴 ]−𝑹_𝑭)/𝝈(𝑹_𝑴 ) 𝝈(𝑹_𝒑 )
multifactor CAPM:
𝑬[𝑹_𝒑 ]=𝑹_𝑭+\ 𝜷_(𝒑.𝑴) (𝑬[𝑹_𝑴 ]−𝑹_𝑭 )+𝜷_(𝒑,𝑭𝟏) (𝑬[𝑹_𝑭𝟏 ]−𝑹_𝑭 )+𝜷_(𝒑,𝑭𝟐) (𝑬[𝑹_𝑭𝟐 ]−𝑹_𝑭 )+⋯ +𝜷_(𝒑,𝑭𝑲) (𝑬[𝑹_𝑭𝑲 ]−𝑹_𝑭 )
CML 1 formula
𝒘_𝑭+𝒘_𝑴=𝟏 or 𝒘_𝑭=𝟏−𝒘_𝑴
CML 3
𝝈^𝟐 (𝑹_𝒑 )=𝒘_𝑴^𝟐 𝝈^𝟐 (𝑹_𝑴 )
