Search
Create
Log in
Sign up
Log in
Sign up
Get ahead with a $300 test prep scholarship
| Enter to win by Tuesday 9/24
Learn more
investments!!! L1-8
STUDY
Flashcards
Learn
Write
Spell
Test
PLAY
Match
Gravity
Terms in this set (279)
3 different measures of return:
holding period return (total return on an asset or portfolio over a period during which it is held. HPR for a stock= dividend yield plus capital gains yield
expected return ER / average return: value of a random variable one could expect if the process of finding the random variable could be repeated an infinite number of times. subjective probability assessment on the return of the stock.
geometric return: calculates average rate per period on investments that are compounded over multiple periods
holding period return
rate of return over a given investment period.
the HPR of a share of stock depends on the increase/decerase in the price of the share over the investment period as well as any dividend income the share has provided.
HPR= (ending price - beginning price + cash dividend) / beginning price.
(P1-P0+D1)/P0
arithmetic average return
sum of returns in each period divided by the number of periods.
geometric average return
single per-period return that gives the same cumulative performance as the sequence of actual returns
terminal value (TV) of 1/n - 1
variance
the expected value of the squared deviation from the mean.
=stdev^2 = (sum(r- r mean)/n
value at risk
=
equation: Mean of 10%, investment 100$, 1 year VaR=
measure of downside risk. worst loss that will be suffered with a given probability often of 5%
the 5% VaR is the 5th percentile rate of return. for a sample of 100 returns with rates ordered from high to low, found the 5th observation from the bottom.
VaR= mean - Fa * stdev
Mean of 10%, investment 100$, 1 year VaR=
initial investment of 100, make negative bc it is a loss. .10 is the mean, - the inverse CDF of 1.645 x volatility of 0.15= $14.675.
this is for one daay, but you are looking for 7 days. multiple by square root of T (T is the number of days you are trying to find). so here T= square root of 7
• How to scale to monthly loss? Daily Var to monthly Var: apply the square root rule. 22 trading days in one month. So multiply this number by square root of 22
VaR and CTE are complements or substitutes?
complements.
VaR is a threshold measure- very popular with new Basel rules. Content is different for CTE.
you have option to take a bet on the toss of a coin. 50% H or T.
head you win 1 euro, tail 0 euro.
what are you willing to pay if you are risk averse? and what is this called?
gambler?
fair game?
up to 50 cents, bc there is an expected payoff of 50 cents. risk averse student won't spend over 50.
we call this a speculation0 you want to receive a risk premium for taking on this risk.
gambler: willing to spend over 50 cents. they expect a negative risk premium.
fair game: risk premium is 0. most people tend to be risk averse0 reject fair game or worse opportunities.
does low medium or high risk portfolio generate highest utility?
medium risk.
how are systematic risk and firm specific risk correlated?
uncorrelated to each other
If two investments have the same expected return, but one has a lower variance, which once is the better choice?
the one with the lower variance is the better choice.
Different levels of diversification can be achieved in a portfolio by combining stocks with different variances and expected returns.
ranking portfolios by their sharpe ratios is called
mean-variance analysis
is it enough to just know averages?
no- you need more info than just averages.
this is why we use VaR value at risk.
3 measures of dispersion
variance: how much a set of observations differ from each other. represents how spread out the data set numbers are.
bias-corrected variance:
volatility: degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns
two measures to describe the shape of return distributions
skewness: measure of the asymmetry of a probability distribution.
averaged cubed deviation from the mean divided by the stdev cubed. negatively skewed distributions have a long left tail, meaning greater chance of extremely negative outcomes
kurtosis: aka fat tail risk. measures thickness of the tilt. measure of fatness of the tails of a probability distribution relative to that of a normal distribution. indicates likelihood of extreme outcomes.
degree of peak in a distribution. more peak than normals means that a distribution also has fatter tails and that there are more chances of extreme outcomes compared to a normal distribution
as investor going long for an asset, would you prefer an investment skewed to left or right?
skewed right bc higher chance for positive return
fat tailed distribution
exhibits large skewness or kurtosis.
important bc fatness of tails shows a riskier investment- higher probability for a negative return. this is bad to investors.
kurtosis is actually a measure of excess kurtosis.
In finance, fat tails are considered undesirable because of the additional risk they imply. For example, an investment strategy may have an expected return, after one year, that is five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions that emerge in finance generally do so because the factors influencing an asset's value or price are mathematically "well-behaved", and the central limit theorem provides for such a distribution. However, traumatic "real-world" events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically well-behaved.
two measures of comovement
covariance: measure of the degree to which returns on two risky assets move in tandem. positive moves together, negative move inversely. (covariance=correlation coefficient x stdev stock A x stdev stock B) =
=*
betaA
betaB*stdevOfMarket^2=
=sum probaiility x (R1-Rmean1)x(Rt-RmeanT)
the probability-weighted average of the products is called covariance and measures the average tendency of the asset returns to vary in tandem (to co-vary).Cov(Rs,RT)= sum of all: Prob
(Rs-Ers)
(Rt-Ert)
correlation: how investments move in relation to one another and when
serial correlation
correlation between return of yesterday and today.
high correlation means there was high distribution today and yesterday.
why does correlation fall close to zero? efficiency-markets tend to be efficient. so if there are high returns, everyone will find out about it and want to invest.
if a market is versatile today, high chance it'll be versatile tomorrow and next months
risk premium
expected excess return over the risk free rate (risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss, over a given period of time)
minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset in order to induce an individual to hold the risky asset rather than the risk-free asset
sharpe ratio
excess return per unit of risk*
useful in understanding how returns increase relative to risk increases
examines performance of an investment by adjusting for its risk.
calculates risk-adjusted return
a dimensionless score to compare similar investments that may vary both in riskiness and returns without needing to know the investor's risk tolerance. It does this by separating the task of valuing an investment (which can be made independent of the investor's risk tolerance) from the task of allocating/valuing a portfolio (which must depend on the investor's preferences).
aka reward-to-volatility (for portfolios) =
risk premium / ST of excess return
sharpe ratio = (avg return of the portfolio - risk free rate) / stdev
the 'excess' return of an asset over the return of a risk free asset (risk premium) divided by the variability or standard deviation of returns (STD of excess return)
normal distribution and why is it useful?
68, 95, 99
its symmetric. mix of normally distributed variables is also normal.
its entire shape can be defined with only two parameters (mean and st dev)
dependence of normally distributed variables can fully be characterized by correlation
serial correlation of daily returns is close to what? returns are therefore easy/hard to predict from the past.
on the other hand, variance has what correlation for small horizons?
close to zero. hard to predict
positive autocorrelation. variance aka squared returns
variance vs volatility
Variance is a measure of distribution of returns and is not neccesarily bound by any time period. Volatility (R^2) is a measure of the standard deviation (square root of the variance) over a certain time interval. In finance, variance and volatility both gives you a sense of an asset's risk
is VaR ir CTE a more conservative measure of downside risk?
CTE because it takes an average return of worst cases, while VaR takes the highest return from the worst cases
"R squared"
percent of variance explained by the model.
squared returns
represents the percentage of a fund or securitys movements that can be explained by movements in a benchmark index.
r-squared values range from 0 to 1 and are commonly stated as percentages from 0 to 100%. r squared of 100% means all movements of a security are completely explained by movements in the index.
a high r squared indicated the funds performance patterns have been in line with the index. a fund with a low r squared indicates the security does not act much like the index
the distribution of daily returns has smaller/fatter tails than the normal
fatter. higher probability of large losses. increasing the holding horizon (month, year) brings the distribution closer to normal
the distribution of daily returns is what?
asymmetric, typically negatively skewed. there are more large drops in returns that upward moves
value-at-risk
aka VaR: a measure of downside risk.
It measures the potential loss over a specified horizon such that
there is a (low) probability α that the actual loss will be larger
• VaR as a quantile of the projected distribution of returns
• The 5% VaR, commonly estimated in practice, is the return at the
5th percentile when returns are sorted from high to low.
it estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day.
typically used by firms and regulators to gauge amount of assets needed to cover possible losses.
You can see how the "VAR question" has three elements: a relatively high level of confidence (typically either 95% or 99%), a time period (a day, a month or a year) and an estimate of investment loss (expressed either in dollar or percentage terms)
Value at Risk (VAR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence. We looked at three methods commonly used to calculate VAR. But keep in mind that two of our methods calculated a daily VAR and the third method calculated monthly VAR
the most popular and traditional measure of risk is ____. the main problem with _______ however is that :
The most popular and traditional measure of risk is volatility. The main problem with volatility, however, is that it does not care about the direction of an investment's movement: a stock can be volatile because it suddenly jumps higher. Of course, investors are not distressed by gains!
VAR using normal distribution for mean (u) and st dev (o)
VaR of alpha (a) = u - Fa^-1o
= mean - inverse N(0,1) CDF for probability a
compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VAR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:
With 95% confidence, we expect that our worst daily loss will not exceed 4%.
If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).
You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns. this moves up to 99% confidence.
for an investment of $100 in a stock with annual mean of 10% and stdev of 15%, the 5% 1-year VaR is:
(stdevs from the mean for 95% confidence: 1.645)
VaR of alpha (a) = u - Fa^-1o
= mean - inverse N(0,1) CDF for probability a
5% 1 year var means 95% confidence level.
VaR(0.95)= -100 x (0.10-1.645x0.15) = $14.675
= -$ x (mean-(stdevZscore x stdev))
if we increase the time horizon, does VaR increase or decrease?
increase
the variance of a T-day return :
T times the variance of a 1-day return
if returns are independent and identically distributed:
VaR(95%,T periods)=
VaR(95%, 10 days)=
=VaR(95%, 1 period) x square root of T
=VaR (95%, 1 day) x square root of 10
assumption of var using historical returns?
distribution of tomorrows returns( R(t+1) ) is well approximated by the empirical distribution of past observations (m)
expected shortfall aka...
aka conditional tail expectation CTE, conditional VAR (CVAR), mean excess loss, mean shortfall, average value at risk
risk measure- a concept used to evaluate the market risk or credit risk of a portfolio.
the "expected shortfall at q% level" is the expected return on the portfolio in the worst % of cases.
it quantifies the expected value of the loss given that an event outside a given probability level has occurred
a risk assessment technique often used to reduce the probability that a portfolio will incur large losses. This is performed by assessing the likelihood (at a specific confidence level) that a specific loss will exceed the value at risk. Mathematically speaking, CVaR is derived by taking a weighted average between the value at risk and losses exceeding the value at risk.
VaR measures ....
CTE measures....
number of losses. VaR takes the highest return for the worst cases.
magnitude of losses. CTE takes an average return of the worst cases. more conservative measure of downside risk than VaR
The VaR model allows managers to limit the likelihood of incurring losses caused by certain types of risk, but not all risks. The problem with relying solely on the VaR model is that the scope of risk assessed is limited, since the tail end of the distribution of loss is not typically assessed. Therefore, if losses are incurred, the amount of the losses will be substantial in value.
CVaR was created to calculate the average of the losses that occur beyond the VaR cutoff point in the distribution. The smaller the value of the CVaR, the better. (CVAR=CTE)
is the absolute value of CTE or VaR greater?
CTE always greater, because it is always located more negatively on the timeline
risk averse investor
averse=dislike/opposition to risk.
accepts risk free or speculative prospects with positive risk-premiums
rejects portfolios that are fair games (or worse)
how to rank portfolios based on investors preferences over their risk-return trade-off?
Utility is enhanced by higher expected returns and diminished by higher risk
assign a utility (welfare) score.**
utility function is an important concept that measures preferences over a set of goods and services. Utility is measured in units called utils, which represent the welfare or satisfaction of a consumer from consuming a certain number of goods. Because satisfaction or welfare is a highly abstract concept, economists measure utility in terms of revealed preferences by observing consumer choices and creating an ordering of consumption baskets from least desired to the most preferred. Economists create a parametric functional form for the utility function based on the assumption of observed consumer behavior, with the amount of goods as variables and certain fixed parameters. After that, utility is calculated by substituting certain numerical values for the consumption of goods in the utility function.
measures welfare or satisfaction of a consumer as a function of consumption of real goods, such as food, clothing and composite goods rather than nominal goods measured in nominal terms
mean variance utility=
U= Er - .5Ao^2
U=utility.
A is risk-aversion coefficient.
investors are ... risk averse if A>0.
risk-neutral if A=0 (decisions are only based on Er).
risk loving if A<0.
process of weighing risk (variance) against expected return. By looking at the expected return and variance of an asset, investors attempt to make more efficient investment choices - seeking the lowest variance for a given expected return or seeking the highest expected return for a given variance level.
variance and expected return. Variance represents how spread-out the data set numbers are, such as the variability in daily or weekly returns of an individual security. The expected return is a subjective probability assessment on the return of the stock. If two investments have the same expected return, but one has a lower variance, the one with the lower variance is the better choice.
Different levels of diversification can be achieved in a portfolio by combining stocks with different variances and expected returns
choice of your portfolio depends on...
risk aversion
Consider the case when the investor has to allocate her
portfolio between risk-free money-market securities (with
return rf) and a risky one (with return rE).
• The return of the portfolio is:
Rp= wxRe + (1-w)Rf
w=weight on the risky asset
portfolio of one risky and a risk-free asset.
the expected rate of return of the portfolio:
standard dev of the portfolio:
E(Rp)= wE(Re)+(1-w)Rf = Rf+w(E(Re)-Rf)
op=w(oe)
o=sigma=stdev
oe=stdev of the risky asset
optimal portfolio allocation with one risky asset
In order to find the optimal allocation to the risky asset,
the investor solves the following utility maximization
problem:
optimal portfolio of risky assets is found in the weights that result in the steepest CAL
The optimal CAL is the one that is tangent to the efficient frontier. This CAL offers the highest reward-to- variability ratio (sharpe), which is the slope of the CAL. It will also allow the investor to reach his highest feasible level of utility.
maxU= E(Rp) - .5Ao^2 =
= Rf + w(E(Re)-Rf)-.5Aw^2o^2
Solving for "w" gives the following expression for the
optimal portfolio weights invested in the risky asset:
w*= E(Re)-Rf / Ao^2
Find the optimal portfolio weights of a mean-variance investor with risk aversion coefficient of 6 who chooses
between a risk-free and a risky asset. Assume a risk free rate of 0.001
• Using monthly Heineken data between 2006 and 2015:
• Applying the formula for the optimal risky weight
Mean St Dev
0.014 0.062
w*= 0.014-0.001 / (6 x (0.062)^2 ) = 0.56
w*= (mean - risk free rate) / (risk aversion coefficient x (stdev)^2))
optimal portfolio allocation with one risky asset.
how will the graph look?
utility as a function of the allocation to the risky assets.
so x axis: weight in risky asset.
y axis: utility
line is like an umbrella. optimal portfolio at highest relative max on umbrella line.
The optimal CAL is the one that is tangent to the efficient frontier. This CAL offers the highest reward-to- variability ratio, which is the slope of the CAL. It will also allow the investor to reach his highest feasible level of utility. it will be the CAL with the steepest slope.
what are the required portfolio returns for varying op levels, such that utility remains constant (the investor is indifferent between the risk-return combos)
recall that: E(Rp)=U+.5Ao^2
thus, we can find the required expected portfolio return for different levels of risk (through op) and for a given utility score
indifference curves
the higher the indifference curve, the higher the utility level.**
the steeper the indifference curve, the higher the risk aversion **(ex: higher compensation required for the same level of risk).
represents a series of combinations between two different economic goods, between which an individual would be theoretically indifferent regardless of which combination he received
Indifference curves represent the trade-off between two variables. In portfolio building, the choice is between risk and return. The investor is indifferent between all possible portfolios lying on the indifference curve. However, indifference curves are contour maps, with all curves parallel to each other. The curve plotting in the most northwest position is the curve offering the greatest utility to the investor. However, this most desirable curve may not be attainable at the market place. The point of tangency between an indifference curve (representing what is desirable) and the capital allocation line **
(representing what is possible), is the optimum portfolio for that investor.
**
what is the highest attainable utility level given the investment opportunity set?
fix the level of risk aversion of the investor.
the tangent indifference curve to the CAL gives the highest possible utility.
the tangent point gives the optimal portfolio.
markowitz portfolio selection model
we can generalize the two asset case to many risky assets. solving for optimal portfolio involves following steps:
identify the risk-return opportunities available to the investor.
find the optimal risky portfolio on the efficient frontier which provides the highest reward-variability (sharpe) ratio.
choose the approrpatite mix btwn the risky and risk0free asset depending on the risk aversion of the investor.
markowitz portfolio selection model:
mean-variance frontier.
optimal risky portfolio.
risky-riskless asset mix.
minimize variance for each target level of expected return. draw efficient frontier.
search for the CAL with the highest reward-to-variability ratio. locate point P.
solve for the optimal mix btwn the risk-free and the risky assets, which depends on indidivudal investor preferences. locate point C.
challenges in applying markowitz portfolio selection model
Challenges in applying the model
A large number of parameters. For n stocks
N estimates of expected return
N estimates of variance
(n^2 - n)/2 estimates of covariances
Errors in estimating correlation coefficients
the single index model
accounts in a tractable way for the sources of risk- due to common factors (business cycle, IRs).
decomposes uncertainty to: systematic risk and firm-specific risk.
Ri=excess return = Ri-Rf= BiRm + Ei
BiRm=return due to movement in overall market
Bi=response of stocks excess return to changes in market index's excess return. securitys responsiveness to market.
ei= firm specific / residual risk (expected value is 0)
ai: not a risk measure. expected return on stock beyond any return induced by market index movements. security alpha. positive alpha= attractive to investors (suggests underpriced sucruity). stocks expected excess return if market factor is neutral(if market index excess return is zero)
Rf rate = Alpha of stock + (b x Rf rate)
single factor model
decomposes the risky assets return into the sum of an expected and unexpected components
Ri=E(Ri)+BiM+Ei
m=common factor. captures the uncertainty about the economy.
Ei=unexpected return. captures the uncertainty about the particular firm.
Bi: exposure to the common factor.
decomposition of risk: the variance of the risky asset has how many components?
2: systematic and firm specific
the Information Ratio
*measures the extra return we obtain from security analysis per unit of firm specific risk we are exposed to if we under or over weight securities relative to the market index.
abnormal return (alpha) per unit of non-systematic risk (tracking error).
alpha of portfolio is the (Avg Return - CAPM) / residualstdev
IR= alpha of portfolio / stdev of Ep
IR= Avg Return - CAPM = x , x/Residual stdev = IR
assumptions behind the CAPM
investors are price takers: individual trades don't affect prices.
single period investment horizon.
investments are limited to traded assets.
no taxes or transaction costs.
all investors are rational mean0variance optimizers.
info is costless and available to all.
investors have homogenous expectations.
All investors choose to hold the same portfolio: the market portfolio
• The proportion of each stock in the market portfolio is the market value of the stock expressed as a percentage of total market value
• The market portfolio is the tangency portfolio: the Capital Market Line (CML) is the best attainable CAL
The market is composed of many small investors, who are price-takers; i. e., perfect
competition. In reality this assumption was fairly realistic until recent years when institutional investors increasingly began to influence the market with their large transactions.
(b) All investors have the same holding period. Obviously, different investors have different goals, and thus have different holding periods.
(c) Investments are limited to those that are publicly traded. In addition, it is assumed that investors may borrow or lend any amount at a fixed, risk-free rate. Obviously, investors may purchase assets that are not publicly traded; however, the dollar volume of publicly traded assets is considerable. The assumption that investors can borrow or lend any
amount at a fixed, risk-free rate obviously is false. However, the model can be modified
to incorporate different borrowing and lending rates.
(d) Investors pay no taxes on returns and incur no transaction costs. Obviously, investors do
pay taxes and do incur transaction costs.
(e) All investors are mean-variance efficient. This assumption implies that all investors make
decisions based on maximizing returns available at an acceptable risk level; most investors probably make decisions in this manner. However, some investors are pure wealth maximizers (regardless of the risk level); and other investors are so risk averse that avoiding risk is their only goal.
(f) All investors have homogeneous expectations, meaning that given the same data all investors would process the data in the same manner, resulting in the same risk/return assessments for all investment alternatives.
capital market line
its the CAL using the market index portfolio as the risky asset.
CML. it is the best attainable CAL capital allocation line
the CAL:investment opportunity set formed with a risky asset and a risk-free asset. The CAL has an intercept equal to the risk-free rate. It is a straight line through the point representing the risk-free asset and the risky portfolio, in expected-return/standard deviation space.
CML:the market portfolio will be on the efficient frontier and it is also the tangency portfolio to the optimal CAL
line from Rf rate through market portfolio M is also best attainable CAL
why do all investors hold the market portfolio?
passive strategy: investing in the market index is efficient.
mutual fund theorem (separation property): all investors choose to hold a market index mutual fund. the allocation btwn the mutual fund and the risk free asset depends on individual investors risk aversion.
The mutual fund theorem (the separation property): All investors choose to hold a market index mutual fund
The allocation between the mutual fund and the risk-free asset depends on individual investor's risk aversion
CAPM
Tests of the CAPM that use regression techniques are subject to inaccuracies because the slope coefficient of the regression equation is biased downward. This would be a problem even if it were possible to use the returns on the true market portfolio in these regressions. It is due to the fact that the independent variable (the beta that is found in the first-pass regression and used as the independent variable in the second-pass regression) is measured with error.
applies to all portfolio and individual securities only. they don't have to be efficient. capital asset pricing model. determines theoretically appropriate required return of an asset, to make decisions about adding assets to a well diversified portfolio.
only uses market risk value.
Er=Rf + B (Rm-Rf)
expected return of capital asset= risk free rate of interest + beta x (expected return of market - risk free rate of interest.
OR Er=Rf + B x market premium.
market premium is the difference between the expected market rate of return and the risk free rate of return.
zero-beta model of Black
extension of CAPM.
Absence of a risk-free asset (i.e. restrictions on borrowing or investing in the risk-free asset)
Combinations of portfolios on the efficient frontier are also efficient
Any portfolio on the efficient frontier has a companion (zero-beta) portfolio with which it is uncorrelated
Returns on the efficient frontier can be expressed as linear combinations of any two frontier portfolios:
ICAPM
intertemporal capital asset pricing model.
linear factor model with wealth and state variable that forecast changes in the distribution of future returns or income.
value stocks typically ha
ve higher returns than growth stocks.
relaxes assumptions that we only have a single period in our economy.
A financial model that takes into account major sources of risk when optimizing consumption over a period of time. The intertemporal capital asset pricing model (ICAPM) assumes that security returns are normally distributed over multiple time periods, and that all future consumption will be funded by security returns.
ICAPM is a consumption-based asset-pricing model, and it goes a step further than CAPM in taking into account how investors participate in the market. Most investors do not participate in financial markets for one year, but instead for multiple years. Over longer time periods, investment opportunities might shift as expectations of risk change, resulting in situations in which investors may wish to hedge. For example, an investment may perform better in bear markets, and an investor may consider holding that asset if a downturn in the business cycle is expected.
ICAPM uses mean-variance analysis to create normal distribution of consumption risk over time. Because ICAPM covers multiple time periods, multiple beta coefficients are used to determine how many security concerns covary with a basket of risky securities.
CCAPM
consumption CAPM
Allocate current wealth between current consumption and savings and investments (i.e. future consumption)
Assets are riskier if they co-vary positively with consumption growth -> they have higher equilibrium risk premiums
Risk premium of an asset as a function of its consumption risk:
A financial model that extends the concepts of the capital asset pricing model (CAPM) to include the amount that an individual or firm wishes to consume in the future. The CCAPM uses consumption measures, in terms of a consumption beta, in its calculation of a given investment's expected return.
In its simplest form, the CCAPM differs from the CAPM by only the beta coefficient used in the calculation. The beta for consumption attempts to measure the covariance between an investor's ability to consume goods and services from investments, and the return from a market index.
with CCAPM, what kind of premium would you require for assets that comove highly with consumption in order to buy them?
require a high premium.
security market line.
an overpriced security will plot where on the SML?
representation of CAPM. displays the expected rate of return of an individual security as a function of systematic, non diversifiable risk.
x axis represents risk (beta) and y axis reps expected return. market risk premium is determined from slope of the SML.
if risk vs expected return is plotted above SML, it is undervalued bc investor can expect greater return for the inherent risk.
market portfolio has beta of 1.
fairly prices security is on SML.
SML is a plot of individual portfolios
plots below the SML.An overpriced security will have a lower expected return than the SML would predict; therefore it will plot below the SML..
to correct a misplacing under CAPM, all investigators slightly tilt their portfolios and assign a higher weight on the underpriced stock, which means a higher price and Er down and return it to equilibrium
difference of CAPM and index model?
index model: model that relates stock returns to returns on both a broad market index and firm-specific factors.
capital asset pricing model is expected returns, index model is realized returns
low beta securities generally have negative/positive alphas
positive
weak emh vs semi-strong vs strong
Weak: stock prices reflect all information that can be derived by examining market trading data (e.g. prices, volume, etc.).
Semi-strong: stock prices reflect all publicly available information regarding past performance and prospects of the firm (i.e. including e.g. earning forecasts)
Strong: stock prices reflect all information, including private information and company insiders. very extreme.
if private info is traded and now everyone can observe that info, then private info becomes public. so private info become public. this is strong emh.
types of stock analysis
technical analysis: using past trading info (price and volume) to predict future prices. the weak form of EMH rules out its benefits. focuses on stock price patterns and on proxies for buy or sell pressure in the market.
fundamental analysis: uses economic and accounting info (earnings, dividends, expectations of future IRs) to predict prices. the semi strong form of EMH rules out its merits. focuses on the determinants of the underlying value of the firm, such as current profitability and growth prospects.
bc both types of analysis are based on public info, neither should generate excess profits if markets are operating efficiently.
response of stocks to new info over time
at -1 time, already some effects- from selected set of investors, possibly inside trading. then effects boom either positively or negatively, then flatten out.
active vs passive management
Active management
Involves security analysis, timing
Large vs. small investors
The EMH: no benefit from pursuing active management
Passive management
Involves the creation of an index fund, buy and hold Supported by the EMH
even if the market is efficient, there is still these roles for portfolio management:
selecting a well-diversified portfolio.
selecting investment policies with tax considerations.
matching a portfolio policy with the risk profile of the investor.
weak form tests of the EMH
Weak form tests (patterns in stock returns)
Momentum over short horizons: market indices have low serial correlation, but portfolios of best/worst recent performers display a momentum effect
Reversals over long horizons: overreaction to news, which is corrected subsequently by a reversal (negative serial correlation as a response to a short term momentum effect)
cumulative abnormal return
CAR: sum of all abnormal returns in the event window
predictors of broad market returns:
fama and french (aggregate returns are higher with higher dividend/price ratios.
campbell and shiller: earnings yield can predict market returns.
keim and stambaugh (bond spreads can predict market returns)
Is asset return predictability In violation of the EMH?
Or: these are proxies for variation of the market risk premium and thus don't provide evidence of market inefficiencyx
test of EMH: strong form tests.
inside information.
The ability of insiders to trade profitability in their own stock
Regulators usually require all insiders to register their trading activity
Evidence that following insider transactions (after they become public) generates no abnormal returns net of transaction costs
interpreting evidence of EMH:
Inefficiencies or risk premiums?
• On one hand these effects could be due to risk premiums associated with risk factors (e.g. Fama and French)
• Alternatively, they may be evidence of inefficient markets (e.g. systematic errors in the analysts' forecasts)
beta
measure of market risk.
smeasure of volatility, or systematic risk, of a security or portfolio in comparison to the market as a whole. it gives a sense of a stocks market risk compared to the greater market.
ensitivity of the expected excess asset returns to the expected excess market returns.
According to the CAPM, the risk premium a student taking Investments 3.4 expects to receive on any stock or portfolio increases positively with beta.
The market rewards systematic risk, which is measured by beta, and thus, the risk premium on a stock or portfolio varies directly with beta.
beta less than 1: security will be less volatile than the market. greater than 1 means price is more volatile than market. equal to 1 means security price will move with the market.
In a well-diversified portfolio, what risk is present?
Market, systematic, or nondiversifiable, risk is present in a diversified portfolio; the unsystematic risk has been eliminated.
unsystematic risk is negligible (so small it isn't worth being considered)
Company XYZ just announced yesterday that its first quarter sales were 35% higher than last year's first quarter. You observe that XYZ had an abnormal return of -2% yesterday. This suggests that
investors expected the sales increase to be larger than what was actually announced.
According to proponents of the efficient market hypothesis, the best strategy for a small investor with a portfolio worth €10,000 is probably to
invest in mutual funds. Individual investors tend to have relatively small portfolios and are usually unable to realize economies of size. The best strategy is to pool funds with other small investors and allow professional managers to invest the funds.
framing
a person may reject an investment when it is posed in terms of risk surrounding potential gains but may accept the same investment if it is posed in terms of risk surrounding potential losses
multifactor models
can be motivated by APT or CAPM extensions.
allow for more than one factor-thus introduce different sensitivities of assets to the separate sources of systematic risk.
factors may include unanticipated changes in GDP, IRs, inflation, etc.
estimate the loadings for each factor using multiple regression.
employs multiple factors in its computations to explain market phenomena and equilibrium asset prices. can be used to explain individual securities and portfolios of securities.
it may be possible to hedge some economic factors that affect future consumption risk with appropriate portfolios.
Tests of multifactor models suggest that industrial production, the risk premium on bonds and unanticipated inflation have significant explanatory power for security returns and it may be possible to hedge these risks if appropriate hedge portfolios can be constructed.
correlation coefficient
the covariance divided by the product of the standard deviations of the returns on each fund. denoted by greek letter Rho (p)
p(SB)= Cov(Rs,Rb)/ (stdevS*stdevB)
correlation can range from -1 to 1.
-1 = one assets returns varies perfectly inversely with the others. with -1, you predict 100% of the variability of one asset's return if you knew the return on the other asset.
correlation of 1: perfect positive correlation. imply R^2 of 100%.
0= indicates that the returns on the two assets are unrelated.
Consider the regression equation:
ri- rf= g0+ g1bi+ g2s2(ei) + eit
where: ri- rt= the average difference between the monthly return on stock i and the monthly risk-free rate; bi= the beta of stock I; s2(ei) = a measure of the nonsystematic variance of the stock i. If you estimated this regression equation and the CAPM was valid, you would expect the estimated coefficient g2 to be:
and the coefficient g1 to be:
0. If the CAPM is valid, the excess return on the stock is predicted by the systematic risk of the stock and the excess return on the market, not by the nonsystematic risk of the stock.
g1 would be the market risk premium which equals the avg difference btwn monthly return on market portfolio and monthly Rf rate
If an investor has a portfolio that has constant proportions in T-bills and the market portfolio, the portfolio's characteristic line will plot as a line with ___________; if the investor can time bull markets, the characteristic line will plot as a line with ___________.
constant slope, positive slope.
These characteristics are shown in Figure 24.5. If the proportions are constant the beta of the portfolio stays constant. If the investor switches the proportions in favor of the market portfolio to take advantage of bull markets the beta will increase during times of higher market risk premiums. This will cause the slope of the curve to increase.
Hedge fund performance may reflect significant compensation for ________ risk.
liquidity. Hedge fund performance may reflect significant compensation for liquidity risk - recall the LTCM example
A zero-coupon bond has a yield to maturity of 11% and a par value of $1,000. If the bond matures in 27 years, the bond should sell for a price of _______ today.
59.74
1000/1.11^27
FV/(1+i)^N
straight bond
callable bond
straight: has a coupon that is paid to bondholders periodically. issuer repays principal at maturity.
callable: debt security / bond that allows the issuer of the bond to retain the privilege of redeeming the bond at some point before the bond reaches its maturity.the issuer has an option which it pays for by offering a higher coupon rate.
The straight bond's price will be higher than the callable bond's price for low interest rates.
For low interest rates, the price difference is due to the value of the firm's option to call the bond at the call price. The firm is more likely to call the issue at low interest rates, so the option is valuable. At higher interest rates the firm is less likely to call and this option loses value. The prices converge for high interest rates. A graphical representation is shown in Figure 14.4.
forward rates
short rates
spot rate
Which of these are observable today?
forward: future yield on a bond. calculated using yield curve. imperfect forecasts.The forward rate for period n is the short rate that would satisfy a "break-even condition" equating the total returns on two n-period investment strategies. forward IR is a forecast of a future short rate.
The forward rate for period n is the short rate that would satisfy a "break-even condition"
equating the total returns on two n-period investment strategies.*
short: future evolution of IRs by describing the evolution of the short rate. The short rate for period n is the one-period interest rate that will prevail in period n
the rate for a given maturity (such as one year) at different points in time.
The short rate for period n is the one-period interest rate that will prevail in period n. *
spot rate: The price quoted for immediate settlement on a commodity, a security or a currency. The spot rate, also called "spot price," is based on the value of an asset at the moment of the quote
the rate that prevails today for a given maturity. The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods. geometric average of its component short rates.
The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n
periods*
Forward rates are the estimates of future short rates extracted from yields to maturity but they are not perfect forecasts because the future cannot be predicted with certainty; therefore they will usually differ
The first strategy is an
investment in an n-period zero-coupon bond. The second is an investment in an n-1 period
zero-coupon bond "rolled over" into an investment in a one-period zero. Spot rates and
forward rates are observable today, but because interest rates evolve with uncertainty, future
short rates are not.
duration of bond
duration of a bond normally increases with an increase in A. term to maturity.
The relationship between duration and term to maturity is a direct one; the relationship between duration and yield to maturity and to coupon rate is negative.
Duration (and thus price volatility) is lower when the coupon rates are higher.
. Holding other factors constant, the interest-rate risk of a coupon bond is lower when the bond's: A. term-to-maturity is higher.
B. coupon rate is lower.
C. yield to maturity is higher.
D. term-to-maturity is higher and coupon rate is lower. E. All of these are correct.
longer the maturity, the lesser/greater the IR risk.
yield to maturity is higher.
greater.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
Before expiration, the time value of a call option is equal to A. zero.
B. the actual call price minus the intrinsic value of the call.
C. the intrinsic value of the call.
D. the actual call price plus the intrinsic value of the call. E. None of these is correct.
actual call price minus the intrinsic value of the call.
The difference between the actual call price and the intrinsic value is the time value of the option, which should not be confused with the time value of money. The option's time value is the difference between the option's price and the value of the option were the option expiring immediately.
intrinsic value is set to equal to zero for out of the money or at the money options.
put option
call option
put: option to SELL assets at an agreed price on or before a particular date.
the holder will not exercise unless the asset price is less than the exercise price.
ex: if Fin Corp shares were to fall from 80 to 70, a put option with exercise price of 80 could be exercised to give a 10 payoff. the holder would purchase a share of 70 and simultaneously deliver it to the put option writer for the exercise price of 80.
call: option to buy assets at an agreed price on or before a particular date. gives holder the right to BUY asset at specified price
To the option holder, put options are worth ______ when the exercise price is higher; call options are worth ______ when the exercise price is higher.
B. more; less
The holder of the put would prefer to sell the asset to the writer at a higher exercise price. The holder of the call would prefer to buy the asset from the writer at a lower exercise price.
hedge ratio of an option
aka delta.
compares the value of a position protected through the use of a hedge with the size of the entire position itself. A hedge ratio may also be a comparison of the value of futures contracts purchased or sold to the value of the cash commodity being hedged
Quantifies the overall exposure of a portfolio of options to
a change in the price of the underlying
• The hedge ratio: the change in the price of an option for a
$1 increase in the price of the underlying
• Often referred to as the option's delta
If the hedge ratio for a stock call is 0.70, the hedge ratio for a put with the same expiration date and exercise price as the call would be _______. -.30
Call hedge ratio = N(d1); Put hedge ratio = N(d1) - 1; 0.7 - 1.0 = -0.3
clearinghouse
guarantees that a future contract will be fulfilled.
once two parties have agreed to enter the transaction, the clearinghouse becomes the buyer and seller of the contract and guarantees its completion.
• Clearing house says if somebody has obligation, such as writer of the option, that person might default. To make sure they don't, we will ask for some contribution called a margin requirement. Check whether value in account is large enough in order for person who is obligated to perform their obligation.
• Ex: you issue a call option. You are giving right to another party to exercise this right. If they do, you have obligation to deliver the underlying. This obligation calls for clearing house to ask you to give money. Daily we will track your portfolio to check for value of portfolio relative to strike price. If you start losing money we will ask you to provide more money to this account. Writer of the option has to pose this amount. Holder of the option does not have to do that.
sharpe ratio=
treynors measure=
jensons alpha=
=(return of portfolio - RD rate) / stdev
=(return of portfolio -Rf rate ) / beta
=return of portfolio - (Rf rate + beta x (return of the market - Rf rate)) =Return portfolio - CAPM
1. Which of the following statements is (are) true regarding the variance of a portfolio of two risky
securities?
A. The higher the coefficient of correlation between securities, the greater the reduction in the portfolio
variance.
B. There is a linear relationship between the securities' coefficient of correlation and the portfolio
variance.
C. The degree to which the portfolio variance is reduced depends on the degree of correlation between
securities.
D. A and B.
E. A and C.
C. The degree to which the portfolio variance is reduced depends on the degree of correlation between
securities.
the lower the correlation between the returns of the securities, the more portfolio risk is reduced
An investor who wishes to form a portfolio that lies to the right of the optimal risky portfolio on the
Capital Allocation Line must:
A. lend some of her money at the risk-free rate and invest the remainder in the optimal risky portfolio.
B. borrow some money at the risk-free rate and invest in the optimal risky portfolio.
C. invest only in risky securities.
D. such a portfolio cannot be formed.
B and C. the only way that an investor can create portfolios to the right of the capital allocation line is to create a borrowing portfolio (buying stocks on margin). in this case, the investor will not hold any of the risk-free security, but will hold only risky securities.
to find beta, using stdevs..
stdev squared of portfolio / stdev squared of market = beta squared
beta and returns equation =
The risk-free rate is 7 percent. The expected market rate of return is 15 percent. If you expect a stock
with a beta of 1.3 to offer a rate of return of 12 percent, you should
Rate of return < Rf rate + beta (Market rate of return - Rf rate)
sell short the stock because it is overpriced.
12% < 7% + 1.3 (15% - 7%)= 17.4%, therefore stock is overpriced and should be shorted.
systematic vs nonsystematic risk
nonsystematic=unsystematic=diversifiable risk.
uncertainty that comes with company you invest in. it can be reduced through diversification.ex: news specific to small number of stocks, like a sudden strike by employees of a company you have shares in.
systematic = market risk = undiversifiable risk.
uncertainty inherent to entire market / market segment. it consists of day to day volatility in stocks prices.
multifactor SML
if risk exposures are measured by a multi factor model, the expected rate of return of a security will be a sum of:
Rf rate, sensitivity to F1 risk (its beta) times the risk premium for bearing this risk.......
....the sensitivity of Fn risk (its beta) times risk premium for bearing the risk.
one theoretical motivation for using a multi factor SML is ICAPM and APT (arbitrage pricing theory)
APT.
how to solve an APT for expected return of a portfolio:
calculate variance for a well-diversified portfolio:
arbitrage pricing theory. law of one price.
an arbitrage opportunity is a portfolio with zero volatility and positive return. no arbitrage in an efficient market. if theres an arbitrage opportunity, everyone will slightly shift their portfolios towards this stock. everyone has same knowledge.
tells us why assets stay the way they are. why don't they deviate from certain asset pricing models?
non factor risk is diversified away, so only factor risk command a risk premium in market equilibrium. nonsystematic risk across firms cancels out in well diversified portfolios, so only systematic risk is related to expected returns.
an arbitrage opportunity arises when an investor can construct a sure=profit portfolio with zero net investment. ex: different prices of the asme security on different exchanges.
the law of one price: if two assets are equivalent, they should have the same market price.
in efficient markets, profitable arbitrage opportunities will quickly disappear
an arbitrage opprtunity arises when the disparity btwn 2+ security prices enables investors to construct a zero net investmtnet portfolio that will yield a sure profit.
ERi = Rf + (B1 x (Er1 - Rf )) + (b2 x (Er2 - Rf)
stdev ^2 = Beta of Portfolio^2 x stdev of first variable^2 + Beta of Portfolio^2 x stdev of second variable^2.
apply this formula to obtain stdev^2 then take square root.
stdev^2=(1/n)*stdev^2
CAPM vs APT
APT applies to well diversified portfolios and not
necessarily to individual stocks
• With APT it is possible for some individual stocks to be
mispriced - not lie on the SML
• APT is more general in that it gets to an expected return
and beta relationship without the assumption of the
market portfolio
• APT can be extended to multi factor models
CAPM depends on risk-return dominance. it assumes many small changes are required to bring the market back to eliquibrium. when an equilibrium price is violated many investors will make small portfolio changes, depending on their risk tolerance, until equilibrium is restored.
APT depends on a no arbitrage condition. it assumes a few large changes are required to bring the market back to equilibrium. under no-arbitrage argument of APT, each investor will take as large a position as possible so only a few investors must act to restore equilibrium.
Both the CAPM and the APT are market equilibrium models, which examine the factors that affect securities' prices. In equilibrium, there are no overpriced or underpriced securities. In both models, mispriced securities can be identified and purchased or sold as appropriate to earn excess profits.
The CAPM is based on the idea that there are large numbers of investors who are focused on risk-return dominance. Under the CAPM, when a mispricing occurs, many individual investors make small changes in their portfolios, guided by their degrees of risk aversion. The aggregate effect of their actions brings the market back into equilibrium. Under the APT, each investor wants an infinite arbitrage position in the mispriced asset. Therefore, it would not take many investors to identify the arbitrage opportunity and act to bring the market back to equilibrium.
implications for prices derived from APT are much stronger than those prices derived from CAPM.
fama-french 3 factor model
asset pricing model that expands on CAPM by adding size and value factors to the market risk factor.
value and small-cap stocks outperform markets on a regular basis.
CAPM vs fama french
CAPM uses one factor (market portfolio) to estimate expected return for an individual stock as compared to the returns of the market as a shoe.
fama french: three factors
1: market risk (market index),
2: company price-to-book ratio, aka book to market ratio HML
3:company size, firm size SMB return on small firms minus large firms
used for portfolio return estimations.
fama french: E(Ri)-Rf=ai + bi (E(Rm)-Rf) + SiE(SMB)+hiE(HML)
how to find variance in a stock model
variance = stdev^2.
so.... stdev^2 = B^2 x stdev^2
what is the volatility and risk premium of a risk free asset?
0
speculation
gamble
fair game
a positive risk premium distinguishes speculation from gambling.
investors taking on risk to earn a risk premium are **speculating.
speculation is undertaken despite the risk bc of a favorable risk return tradeoff.
in contrast, gamblers take on risk even without a risk premium
A risk-averse investor:
Accepts risk-free or speculative prospects with positive risk- premiums
Rejects portfolios that are fair games (or worse)
How to rank portfolios based on investor's preferences over their risk-return trade-off?
Assign a utility (welfare) score
consider case when investor has to allocate her portfolio btwn risk free money market securities and a risky one. return of the portfolio is:
risk free money market securities have return of Rf, risky one has return of Re
Rp= w*Re + (1-w)Rf
where "w" is the weight on the risky asset
markowitz portfolio selection models
mean variance frontier: minimize variance for each target level of expected return. draw efficient frontier.
optimal risky portfolio: search for the CAL with the highest reward-to-variabliity ratio. locate point P.
risk-riskless asset mix: solve for optimal mix btwn risk free and risky assets, which depends on individual investor preferences. locate point C.
this is similar:
What is the highest attainable utility level given the
investment opportunity set?
• Fix the level of risk aversion of the investor
• The tangent indifference curve to the CAL gives the highest
possible utility
• The tangent point gives the optimal portfolio
active management vs passive management
active: security analysis, timing. EMH says no benefit from pursuit active management.
passive: creation of an index fund, buy and hold. supported by EMH.
regardless, its important to select a well-diversified portfolio. select investment policies with tax considerations. match portfolio policy with risk profile of investor.
jagannathan and wang:
added human capital and cyclical variations in betas
glamour firms
characterized by recent good performance, high prices, and lower book-to-market ratios
jensens measure:
treynors measure:
average return above predicted from the CAPM
aP= Rp-(Rf+Bp(Rm-Rf))
treynor: excess return per unit of systematic risk.
Tp= Rp-Rf / Bp
= avg portfolio return - avg rf rate / portfolio beta
a more easily interpretable measure than the sharpe ratio
developed by modigliani and modigliani.
equates the volatility of the managed portfolio with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio. then performance can be compared by comparing returns:
M^2= Rp*-Rm
which measure to use? treynor, sharpe, or jensen?
if portfolio represents the entire investment for an individual:
if many alternatives are possible:
if allocating between an active portfolio and an index:
It depends on the investment assumptions
If the portfolio represents the entire investment for an individual: Sharpe Index compared to the Sharpe Index for the market
If many alternatives are possible, use the Jensen's alpha or the Treynor measure. The Treynor measure is potentially more complete because it adjusts better for risk.
If allocating between an active portfolio and an index: information ratio.
bull markets
bear markets
market in which share prices are rising, encouraging buying.
a market in which share prices are falling, encouraging selling
market timing. if weights on the risky assets change: shift to market portfolio in bear/bull markets.
shift to the risk-free asset in bear/bull markets.
market portfolio-bull.
risk-free asset-bear
betas of the portfolio: large in bull or bear markets? small in which?
large in bull markets.
small in bear markets.
style analysis
Returns-based style analysis is a statistical technique used in finance to deconstruct the returns of investment strategies using a variety of explanatory variables. The model results in a strategy's exposures to asset classes or other factors, interpreted as a measure of a fund or portfolio manager's style.
effective asset mix
style of the investor's overall portfolio (especially for multiple-managed portfolios
performance evaluation has two main problems:
many observations are needed for significant results.
shifting parameters when portfolios are actively managed makes accurate performance evaluation elusive
bogey portfolio
aka benchmark portfolio: used to evaluate a funds performance. the benchmark is an index that reflects the investment scope of the funds investment.
hedge funds strategies
directional:
bets that one sector or another will outperform other sectors.
nondirectional:
exploits temporary misalignments in security valuations. buys one type of security and sells another. strives to be market neutral. convergence vs relative value.
hedge funds vs mutual funds
...
hedge fund performance
hedge funds tend to outperform the market:
signifcant positive alphas. higher sharpe ratios.
reason?
liquidity (lock up periods, serial correlation of hedge fund returns as an indication of illiquid holdings).
survivorship bias.
changing factor loadings
measurement problems
Many hedge funds rack up fame through strategies that make money most of the time, but expose investors to rare but extreme losses (fat tails, ex: october 1987 crash, russian default)
LTCM
long term capital management- a large hedge fund led by nobel prize winning economists and renowned wall street traders that nearly collapsed the global financial system in 1998 as a result of high risk arbitrage trading strategies
big hedge fund due to high exposure to liquidity risk.
a hedge fund case: LTCM. engaged primarily in convergence trades.
Involves trading in securities that are mispriced with respect to one another
Buying underpriced ones and selling overpriced ones
In the long run prices of both securities converge or the spread between them narrows
Strategy designed to make significant profits in the long run
Consider a bond with a coupon rate of 6% paid semi- annually, face value of $1000, maturity of 10 years. Suppose that the interest rate is 5% annually
Then F=$1000, r=2.5% semi-annually, T=20 periods, coupon per 6-month period = 3% of $1000 face value, i.e. $30
• The price of the bond:
Pb= 30/0.025 x (1- (1/(1+0.025)^20) + 1000/(1+0.025^20) = 1077.95
accrued interest
The quoted price does not include the interest that has accrued since the last coupon payment date
• For bonds traded between coupon dates: the buyer pays the accrued interest
• E.g. for a bond with semi-annual coupon payments:
accrued interest = annual coupon payment/2 × Days since last coupon payment/days btwn coupon pmt
Invoice price = flat (quoted) price + accrued interest
YTM
realized yield if coupons are reinvested at an IR that is equal to the yield of the bond. if the reinvestment rate is above/below the yield of the bond, then the realized compound return will exceed/fall below it.
the IR that makes the PV of the bonds payments equal to its price. to find YTM, we solve the bond valuation equation for the IR r:
premium bonds vs dissent bonds. which have higher IRs? higher FV? higher coupon? higher maturity? higher price?
premium bonds have higher price and lower IR.
FV and coupon stay the same for both.
relationship btwn bond prices and yields?
inverse relationship
what is a bonds current yield?
bonds annual coupon pmt divided by the bond price.
the YTM is the bonds internal rate of return.
for bonds selling at a premium, coupon rate > current yield >YTM.
reversed for discount bonds
yield curve
graphic representation of the relationship btwn yield and maturity.
investors can form expectations of future IRs using the yield curve.
a line that plots the IRs, at a set point in time, of bonds having equal credit quality but differing maturity dates.
nelson-siegel function
...
theories of term structure
expecation hypothesis: zerio liquidity premium. upward sloping yield curve means that investors anticipate increase in IRs.
liquidity preference: investors demand a premium to hold long term bonds. liquidity premium is positive, which implies an upward sloping yield curve.
The yield curve is upward sloping. According to the expectations theory, an upward sloping curve indicates that investors anticipate an increase in interest rates. According to liquidity preference theory, an upward sloping curve does not necessarily imply an anticipation for an increase in interest rates.
an upward sloping yield curve is associated with the fact that the ___ rate for the coming period is lower/higher than the ___ yield.
forward rate higher than current yield
capital allocation line slope equation?
slope= (Er - Rf)/ stdev
A company whose stock is selling at a P/E ratio greater than the P/E ratio of a market index most likely
has ________.
a dividend yield less than that of the average firm.
profit per earnings ratio.
firms lower than avg dividend yields are usually growth firms, having higher P/E ratio than avg
P/E= (1-B)/(Er-BxROE)
A bond will sell at a discount when __________.
A. the coupon rate is greater than the current yield and the current yield is greater than yield to maturity
B. the coupon rate is greater than yield to maturity
C. the coupon rate is less than the current yield and the current yield is greater than the yield to
maturity
D. the coupon rate is less than the current yield and the current yield is less than yield to maturity
E. None of these is correct.
D: coupon rate is less than current yield is less tha YTM.
In order for the investor to earn more than the current yield the bond must be selling for a discount.
Yield to maturity will be greater than current yield as investor will have purchased the bond at discount
and will be receiving the coupon payments over the life of the bond.
how to calculate a forward rate?
forward rate= (1 + i)^N / (1+FwdRateYr1 x 1+FwdRateYr2 )
greater duration is made up what?
longer maturity and lower coupon percent
what factors effect the price of a stock option?
Rf rate, riskiness of stock, and time to expiration. all 3 directly related to price of the option. the expected rate of return on the stock does not affect the price of the option.
the intrinsic value of an at-the-money put option is equal to:
An option is at the money (ATM) if the strike price (the price at which a put or call option can be exercised) is the same as the current spot price of the underlying security. An at-the-money option has no intrinsic value, only time value. For example, with an "at the money" call stock option, the current share price and strike price are the same.
zero.
this option is part of contract.
intrinsic value is set to equal to zero for out of the money or at the money options.
. To hedge a short position in Treasury bonds, an investor most likely would
buy IR futures.
By taking the long position, the hedger is obligated to accept delivery of T-bonds at the contract
maturity date for the current futures price, which locks in the sales price for the bonds and guarantees
that the total value of the bond-plus-futures position at the maturity date is the futures price.
Suppose that the risk-free rates in the United States and in Japan are 5.25% and 4.5%, respectively.
The spot exchange rate between the dollar and the yen is $0.008828/yen. What should the futures price
of the yen for a one-year contract be to prevent arbitrage opportunities, ignoring transactions costs?
.008891dollars per yen.
0.008828 (1.0525/1.045)
give an expression for the systematic risk variance of two stocks:
nonsystematic risk component?
stdev of investment = (B^2 x stdev of market ^2) + (HML coeff^2 x stdev of hml^2)
stdev^2= 0.5 x stdev^2
Construct a portfolio out of the two stocks that has exposure of 1 to the HML factor. Give an
analytical expression of its weights.
W1+W2=1.
Coeff1xW1 + Coeff2xW2=1.
performance attribution procedures.
The portfolio management decision process typically involves three choices: (1) allocation of funds
across broad asset categories, such as stocks, bonds, and the money market; (2) industry (sector) choice
within each category; and (3) security selection within each sector. The returns resulting from each of
these decisions are measured against a benchmark return resulting from a passive, index-investment
approach. The excess returns (if any) resulting from these decisions over and above those earned from a
passive indexing strategy are attributed to the success of the portfolio manager
how to calculate expected return for two stocks, given 3 different states of probability (1,2,3) each with a probability and return amount.
how to calc stdev?
calculate correlation coefficient between two stocks?
If you invest 35% in stock A, and 65% in stock B, what would be your portfolio's expected
rate of return and standard deviation?
Er= P1xR1 + P2xR2 + P3xR3
stdev^2= P1x((R1-Er1)^2) + P2x(R2-Er2)^2 + P3x(R3-Er3)^2
Cov= ?corr?
ERp- WaxERa + WbxERb
stdev^2= (Wa^2 x stdeva^2 + Wb^2xstdevb^2 + 2 x Wa x Wb x stdeva x stdevb x corra,b
Discuss the differences in risk-taking behavior between investors who are risk averse, risk neutral, and
risk loving
The investor who is risk averse will take additional risk only if that risk-taking is likely to be rewarded
with a risk premium. This investor examines the potential risk-return trade-offs of investment
alternatives. The investor who is risk neutral looks only at the expected returns of the investment
alternative and does not consider risk; this investor will select the investment alternative with the
highest expected rate of return. The risk lover will engage in fair games and gambles; this investor
adjusts the expected return upward to take into account the "fun" of confronting risk.
Discuss duration. Include in your discussion what duration measures, how duration relates to maturity,
what variables affect duration, and how duration is used as a portfolio management tool (include some
of the problems associated with the use of duration as a portfolio management tool)
Duration is a measure of the time it takes to recoup one's investment in a bond, assuming that one
purchased the bond for, say, 1000 euro. Duration is shorter than term to maturity on coupon bonds as
cash flows are received prior to maturity. Duration equals term to maturity for zero-coupon bonds, as
no cash flows are received prior to maturity. Duration measures the price sensitivity of a bond with
respect to interest rate changes. The longer the maturity of the bond, the lower the coupon rate of the
bond, and the lower the yield to maturity of the bond, the greater the duration. Interest-rate risk
consists of two components: price risk and reinvestment risk. These two risk components move in
opposite direction; if duration equals horizon date, the two types of risk exactly offset each other,
resulting in zero net interest-rate risk. This portfolio management strategy is immunization. Some of the
problems associated with this strategy are: the portfolio is protected against one interest rate change
only; thus, once interest rates change, the portfolio must be rebalanced to maintain immunization;
duration assumes a horizontal yield curve (not the shape most commonly observed); duration also
assumes that any shifts in the yield curve are parallel (resulting in a continued horizontal yield curve); in
addition, the portfolio manager may have trouble locating acceptable bonds that produce immunized
portfolios; finally, both duration and horizon dates change with the mere passage of time, but not in a
lockstep fashion, thus rebalancing is required. Although immunization is considered a passive bond
portfolio management strategy, considerable rebalancing must occur, as indicated above. The portfolio
manager must consider the tradeoffs between the transaction costs and not being perfectly immunized
at all times.
formula for zero rate coupon bonds
...
protective put.
advantages?
consists of investing in stock and simultaneously purchasing a put option on the stock.
regardless of what happens to the price of the stock, you are guaranteed a payoff equal to the put option exercise price.
You are pricing a European call option using a binomial tree and the Black-Scholes formula. What is the
relationship between the two valuations when you increase/decrease the number of steps in the
binomial tree?
The Black-Scholes formula can be thought of the limiting case of a binomial tree. Increasing the number
of nods in the binomial tree brings the value of the option closer to that of the Black-Scholes formula.
This document is property of Vrije Universiteit Amsterdam
Please do not share or redistribute
16
I would further give 2 bonus points to students who mention that the price approximation using
binomial tree does not monotonically approach the Black-Scholes value when you increase the number
of steps, but rather oscillates around it (I did give you a hint that I will ask this question in class, but since
it is a very difficult one I would only award bonus points for it and not penalize you if you do not
mention this relationship).
an upward sloping yield curve is associated with fact that ........ (forward rate, short rate, current yield)
what explains this?
forward rate for coming period is higher than current yield
increase in forward rate is explained by:
Fn- ERn + liquidity premium
macaulays duration
weighted avg time until CFs are received, and is measured in years for bond duration
duration=
modified duration=
=D*= D/1+y.
it can be used to approximate the percentage change in price of a bond due to a change in the yield:
percent change in price= -D x (change in i / 1+i)
dollar duration:
measure price change for a given change in the yield
change in Price / change in Y = -D x P
the duration of a zero coupon bond is what?
for bonds with the same maturity, duration increase/decreases for higher coupons
for bonds with the same coupon rate, duration usually increases/decreases with time to maturity
the duration of a coupon bond is lower/higher for a lower yield to maturity
the duration of a perpetuity is:
equal to its maturity
decreases
increases
higher
: 1+y / y
pros and cons of duration?
pros: key measure of IR sensitivity. easy to compute and interpret.
cons: gives good approximations only for small yields. assumes parallel yield shifts.
applicable only for securities with fixed cash flows.
passive bond management strategies:
aim is to control the risk of a bond portfolio.
bond market indexing and immunization
bond index funds
contains thousands of issues many of which are infrequently traded. bond indexed turn over more than stock indexed as the bonds mature.
therefore, bond index funds only hold a representative sample of the bonds in the actual index.
bond indexed contain thousands of issues, many of which are infrequently traded
o bond indexed turn over more than stock indexes as the bonds mature
• index that tries to track all short term bonds that mature within 6 months- you must turnover a lot of securities every few months bc of maturity problem
o therefore, bond index funds hold only a representative sample of the bonds in the actual index
• indexes only hold a very small representative sample of all actual bonds
o instead of buying all the fixed income securities (this is hard to do), we can just buy a fraction of all of them so that we hold the same percent of the value of that portfolio, such as 12.1% for treasuries like in the chart
o treasury is 12.1% for less than 1 year
• follow a very broad index. Within this index, 12% of all o fthe securities mature in less than 1 year and are treasuries. If you want to buy all of these securities, that is hard bc they aren't heavily traded. They are 12% in the index. So in my portfolio, I will do 12% of value to these stocks.
• Same approach as this with equities
immunization. limits?
Immunization, also known as "multiperiod immunization," is a strategy that matches the durations of assets and liabilities, thereby minimizing the impact of interest rates on the net worth.
targeting net worth: matching duration of assets and liabilities.
targeting the investment horizon: match duration with the holding period. offsets price risk with reinvestment risk.
one problem with immunization is that duration assumes that if shifts in the yield curve occur, these shifts are parallel.
limits: immunization using duration matching provides approximate hedging against shifts in the yield curve.
duration matching immunizes the portfolio only against parallel shifts of the yield curve.
active bond management strategies
swapping strategies:
substitution swap
intermarket swap
rate anticipation swap
pure yield pickup
tax swap
horizon analysis:
Select a particular holding period and predict the yield curve at end
of period.
• Given a bond's time to maturity at the end of the holding period,
• its yield can be read from the predicted yield curve and the endof-period
price can be calculated.
leverage ratio:
helps determine bond ratings.
assets/equity.
debt to equity ratio.
liquidity ratios:
helps determine bond ratings.
current ratio (current assets/current liabilities)
quick ratio (current assets without inventories/current liabilities)
derivative contract
aka contingent claims, bc their payoffs depend on the value of other securities.
financial contract whose value is derived from the value of an underlying asset
main types of contracts: forwards, futures, options, swaps.
standardized contracts traded on exchanges or OTC trading
key ingredients of an options contract:
call option(BUY at specified price),
put option (SELL at specified price)
underlying asset, exercise/strike price, premium (price of the option), maturity (expiration of the option)
maturity of the option: right to exercise option within 1 year, 2 years, etc.
there are different types of options depending on the time the option can be exercised:
European: can be exercised only at expiration.
American: can be exercised at any time before expiration
Bermudan: exercise is restricted to certain periods during the life of the option
types of options in terms of the underlying.
which option is most liquid and most majority of trade volume?
SIFFI
stock options
index options (buy an option on an index such as AJAX-you don't buy al the stocks, but you buy an ETF share)
futures options (option to enter into a future contract such as entering future contract three months from now to buy oil three years from now
foreign currency options (most liquid option and majority of trade volume)
intterest rate options (grown in popularity over last 20 years. right to enter some contract in future at a given interest rate, such as giving a loan)
what is the largest exchange for trading derivatives?
CBOE chicago board of exchange
the clearing house
options clearing corporation (OCC):
effective buyer/writer of the option.
guaranteed contract performance.
margin required from the option writers to guarantee that they will fulfill contract obligations. (depends on the likelihood that the option will be exercised against the writer: moneyness of an option. and does the option holder need to post margin?)
moneyness of an option
In the money:
• Call option: Market price > Exercise price (you want to exercise your right in this case bc you can buy the underlying for this amount and sell it fro exercise price)
• Put option: Exercise price > Market price
Out of the money:
• Call option: Market price < Exercise price
• Put option: Exercise price < Market price
At the money
• Market price = Exercise price (you are indifferent between exercising and not exercising)
The intrinsic value of an out-of-the-money call option is equal to zero. The fact that the owner of the option can buy the stock at a price greater than the market price gives the contract an intrinsic value of zero, and the holder will not exercise.
payoff and profit of a call option=
payoff and profit of a put option=
for the holder:
payoff= max(0,spot price at expiration-strike price)
profit=pay off - option premium
for the holder:
payoff=max(0,strike price-spot price at expiration)
profit=pay off - option premium
covered call
invest in a stock and at the same time write a call option on the stock
straddle
a bet on volatility*
buying both a call and a put on a stock, with the same exercise price and expiration date.
Long call plus long put on a stock with the same strike
price and the same expiration date
• Used by investors who expect that the stock will move
considerably away from today's price, but are not certain
about the direction of the move
• Thus: a bet on volatility
spread
bullish spread
A combination of several calls or puts on the same stock with differing exercise prices of times to expiration
with different exercise prices or times to maturity
• Money spread: exercise prices differ
• Time spread: expiration dates differ
Bullish spread
• Long and short call at different exercise prices
• Pay-off either increases or is unaffected by price increases
• Investors benefit from stock price increases
• Alternatively, one may believe that one option is overpriced with
respect to another and take a spread position without aiming at a
bullish position in the stock
collar
A strategy designed to keep the value of the portfolio
between two bounds
• Limits upside potential
• Provides downside protection
• Good for targeting some wealth goal
A collar with a net outlay of approximately zero is an options strategy that combines a put and a call to lock in a price range for a security and uses the gains from sale of a call to purchase a put.the collar brackets the value of a portfolio between two bounds.
intrinsic value of an option?
time value of an option?
Intrinsic value of an option: the pay-off if immediate
exercise
• Call: stock price - exercise price
• Put: exercise price - stock price
• Intrinsic value adjustment: PV of the exercise price
Time value of an option = option price - intrinsic value
what determines the value of an option? how does it drive the value of a call option?
-Stock price
• Exercise price
• Volatility
• Time to expiration
• Interest rate
• Dividends
if you hold a call on a stock with exercise price of $80, and the stock is now selling at 90, you can exercise your option to purchase the stock at 80 and simultaneously sell the shares at the market price of 90, clearing $10 per share. yes it the shares sell below 80, you can sit on the option and do nothing, realizing no further gain or loss.
Stock price:If stock price increases, call option increases. Higher and higher chance that at expiration you will exercise, so value will go up
Exercise price:Two calls with different exercises- lower exercise price more chance that you will exervise call price. Opposite of stock price.
Volatility: Higher volatility = higher value bc higher hcnace stock price takes extreme value. Volatility is positively related to the call
Time to expiration:Positive- increases the time value of the call. More time to expiration- high chance there'll be an extremely positive event in the future
Interest rate: High IRs- PV of any future payment is relatively low. PV of the strike is relatively low. So expected payoff is relatively low. Negative relationship
Dividends: When dividends are paid out, stock prices go down. Value of stock price goes down with increased dividiends.
Beta? Expected returns? :Do I have beta listed? No. but beta is important to determine value. So why doesn't it matter now? Beta is a measure of riskiness of the underlying. Something similar to that is the volatility bullet. So instead of using beta (measures systematic risk) we value option by overall risk (measured by volatility instead of beta which just measures systematic risk)
Option pricing
• Binomial tree approach
two ways to price options:
binomial tree approach
black and scholes formula
how to construct the replicating portfolio?
how to determine the weights of the replicating portfolio constituents?
A replicating portfolio is a portfolio that replicates the payoff
of the derivative
How to determine the weights of the replicating portfolio
constituents?
• Buy H shares of the underlying stock and invest B in the risk-free
asset (rate rf)
• Solve for H and B so that the pay-off of the replicating portfolio is
equal to that of the option
• Then the cost of the replicating portfolio at t=0 should be equal to
the price of the option
the price of the call option is the:
risk-free discounted expected payoff using risk-neutral probabilities
the price of the option is the same as the one that we obtain using the replicating portfolio or the perfectly hedged portfolio
start solving a price tree how?
solve for the last nodes first
instead of calculating the replicating portfolio using the hedge ratio H, we can compute what instead?
the risk-neutral probabilities Q at each node, starting from the ones ta time T, and then use those to obtain the call price. this should yield the same result.
Whether we price options using the replicating portfolio or
risk-neutral probabilities, the same general rules apply:
Start by calculating the option pay-offs at expiration for all spot
price scenarios
• Move iteratively backwards one step at a time, calculating option
prices on each node
• Ultimately, you obtain the price today
The time discretization of the tree can be made finer by
increasing the number of time steps. However, the
binomial tree parameters (upward or downward move
factors) should be adjusted in a way so as to allow for the
same time T range of spot prices (an increase in the
number of steps and a decrease in the size of the steps
assumptions underlying the black-scholes formula?
The stock pays no dividends
• Adjust the stock price downwards: S0 - PV(dividends)
• The interest rate and the variance of the stock are
constant
• Stock prices are continuous (no jumps)
I) the risk-free interest rate is constant over the life of the option.
II) the stock price volatility is constant over the life of the option.
IV) there will be no sudden extreme jumps in stock prices
The risk-free rate and stock price volatility are assumed to be constant but the option value does not depend on the expected rate of return on the stock. The model also assumes that stock prices will not jump markedly.
option greeks: measurements of the risk involved in options:
all measurements of sensitivity of the call or the call's greeks.
Delta - aka hedge ratio. the sensitivity to the underlying instrument's price
• Gamma - the sensitivity of delta in response to price changes in the
underlying instrument. how much delta changes as stock prices change.
• Vega - the sensitivity to the underlying instrument's volatility
• Other greeks
• Theta - time decay of the option
• Rho - the sensitivity to changes in the interest rate
• Lambda - the percentage change in an option contract's price to
the percentage change in the option's underlying price
option delta
sensitivity of the price of the option towards changes in the underlying stock price
delta hedging
A portfolio of stocks and options that is hedged against
fluctuations of the price of the underlying
• Recall that:
• Assume that we buy δ shares of the stock and we write
one call option
• If the stock goes up/down by $1, then the option goes
up/down by δ. The effect is offset if we hold δ shares of
the stock delta hedging
• However, the position must be readjusted as δ changes
through time
• Delta hedging works well only for small price changes
the option's gamma
rate of change of delta with respect to the price of the underlying asset (sensitivity with respect to S)
the option's vega:
Volatility risk: the risk from unpredictable changes in
volatility
• While delta-neutral strategies eliminate the risk exposure
from fluctuations in the price of the underlying, the
volatility risk remains
• The option's vega: the sensitivity of an option price with
respect to volatility
when a distribution is negatively skewed...
standard deviation underestimates risk.
When borrowing and lending at a risk-free rate are allowed, which Capital Allocation Line (CAL) should the investor choose to combine with the efficient frontier?
I) The one with the highest reward-to-variability ratio.
II) The one that will maximize his utility.
III) The one with the steepest slope.
all three.
The optimal CAL is the one that is tangent to the efficient frontier. This CAL offers the highest reward-to- variability ratio, which is the slope of the CAL. It will also allow the investor to reach his highest feasible level of utility.
The standard deviation of a two-asset portfolio is a linear function of the assets' weights when
the assets have a correlation coefficient equal to one.
When there is a perfect positive correlation (or a perfect negative correlation), the equation for the portfolio variance simplifies to a perfect square. The result is that the portfolio's standard deviation is linear relative to the assets' weights in the portfolio.
statistical arbitrage
uses quantitative techniques and often automated trading systems to seek out many temporary misalignments among securities
the YTM on a bond is...
the discount rate that will set the PV of the payments equal to the bond price
Consider a 5-year bond with a 10% coupon that has a present yield to maturity of 8%. If interest rates remain constant, one year from now the price of this bond will be _______
lower.
bond is a premium bond as IRs have declined since the bond was issued. if IRs remain constant, the price of a premium bond declines as the bond approached maturity.
YTM is lower than coupon rate, so it is a premium bond.
given the bond described above, if interest were paid semi-annually (rather than annually), and the bond continued to be priced at $917.99, the resulting effective annual yield to maturity would be:
more than 10%.
if coupon frequencies increase, then duration decreases and we discount over a short time period (on average). Hence, to keep the same price, the yield has to go up.
inverted yield curve
occurs when short term rates are higher than long term rates.
it slopes downward.
The duration of a 20-year zero-coupon bond is
equal to 20.
duration of zero coupon bond equals bonds maturity
Speculators buying put options anticipate the value of the underlying asset will __________ and speculators selling call options anticipate the value of the underlying asset will _______.
decrease, decrease.
The buyer of the put option hopes the price will fall in order to exercise the option and sell the stock at a price higher than the market price. Likewise, the seller of the call option hopes the price will decrease so the option will expire worthless.
future contracts
profits from future contract:The amount that the holder of the long position gains must equal the amount that the holder of the short position loses.
the net profit on the contract is zero- it s a zero-sum game.
A futures contract is a legal agreement, generally made on the trading floor of a futures exchange, to buy or sell a particular commodity or financial instrument at a predetermined price at a specified time in the future. Futures contracts are standardized to facilitate trading on a futures exchange and, depending on the underlying asset being traded, detail the quality and quantity of the commodity
Credit risk in the swap market
is limited to the difference between the values of the fixed rate and floating rate obligations.
Swaps obligate two counterparties to exchange cash flows at one or more future dates. Swaps allow firms to restructure balance sheets, and the firm is obligated only for the difference between the fixed and floating rates.
w
=ERe-Rf / A
stdevE^2 weight for portfolio with risky and risk free assets. you've found the optimal risk.
discuss each variable and relationship. assume investor is risk averse.
The optimal proportion in the risky portfolio (the one containing the optimal mix of risky assets) is the one that maximizes the investor's utility. Utility is positively related to the risk premium [E(re)-rf]. This makes sense because the more expected return an investor gets, the happier she is. The variable "A" represents the degree of risk aversion. As risk aversion increases, "A" increases. This causes w
to decrease because we are dividing by a higher number. It makes sense that a more risk-averse investor would hold a smaller proportion of his complete portfolio in the risky asset and a higher proportion in the risk-free asset. Finally, the standard deviation of the risky portfolio is inversely related to w
. As the risky portfolio's risk increases, we are again dividing by a larger number, making w* smaller. This corresponds with the risk- averse investor's dislike of risk as measured by standard deviation.
How would you define an ideal situation of perfect foresight, (using a graphical interpretation). Hint: think of a derivative instrument.
perfect foresight is equivalent to holding a call option on the index portfolio.
Consider two perfectly negatively correlated risky securities X and Z. X has an expected rate of return of 10% and a standard deviation of 15%, and Z has an expected return of 5% and a standard deviation of 7%. You want to construct a portfolio out of the two securities that has as small variance as possible.
variance formula? what is the variance of the portfolio?
what proportions of X and Z should you hold in the portfolio?
What is the expected return of the portfolio?
What is the formula for the variance of the portfolio?
stdev^2= (WeightE
stdevE - WeightD
StdevD)^2
ii. What is the variance of the portfolio?
It is 0, since the two assets are perfectly negatively correlated.
iii. What proportions of X and Z should you hold in the portfolio?
w1=7/(15+7)=0.32; w2=1-w1=0.68
iv. What is the expected return of the portfolio?
w1
ER1 + w2
ER2=0.66
What does credit risk mean in the context of bond pricing? Which agencies assess the credit risk of bond issuers? Explain briefly the rating scheme employed by those agencies.
A credit risk is the risk of default on a debt that may arise from a borrower failing to make required payments. Credit risk is measured by credit rating agencies, most important of which are Moody's, Standard & Poor's, and Fitch. Each credit rating agencies assigns letter grades to the bonds of corporations / governments / municipalities to reflect their assessment of the safety of the entity. The top rating is AAA or Aaa. Those rated BBB (Baa) or above by the credit agencies are considered investment grade bonds, whereas lower-rated bonds are classified as speculative grade or junk bonds.
State the put-call parity. Why is it called in such a way? **Derive the put-call parity, using two options strategies that provide the same payoffs.
The put-call parity represents the proper relationship between put and call prices. More specifically,
C + (x/(1+Rf)^T) = S0 +P
call price + purchase price of the ZC bond = purchase price of the stock + put price
To derive the parity, consider two strategies:
- A protective put
- Buying a call option and a zero-coupon bond with a face value that equals the strike
price of the call, and the same maturity Now examine the pay-offs:
Because the two portfolios provide the same return, they must cost the same. Thus, we reach the put-call parity
You hold a portfolio of 56 million euro with a beta of 0.75 on the AEX. You expect that the AEX will drop by 2.5% over the next one year. You decide to hedge the AEX exposure of your portfolio with index futures with expiration of one year. The contract multiplier is 200 and the current value of the AEX is 420.
i. Do you buy or sell index futures in order to hedge your position?
ii. What is the projected loss on your portfolio (in euro), if you do not hedge?
iii. How much would each index future contract change in value for the projected 2.5% drop in the AEX?
iv. What is the hedge ratio, i.e. how many future contracts do you need for a perfect hedge?
Do you buy or sell index futures in order to hedge your position?
You sell index futures
ii. What is the projected loss on your portfolio (in euro), if you do not hedge?
Beta
projected loss (in %)
portfolio value=0.75
0.025
56*10^6=1.05 mil
iii. How much would each index future contract change in value for the projected 2.5% drop in the AEX?
AEX change
AEX starting level
Contract multiplier=0.025
420
200=2100
iv. What is the hedge ratio, i.e. how many future contracts do you need for a perfect hedge?
Projected loss in euro / change in the value of the futures = 1.05*10^6/2100=500 contracts
what is the optimal portfolio for an investor?
along the CAL if there is one risky and one risk free asset. if there is no risk free asset, there is no tangency portfolio that is best for all investors. so investors then choose from the efficient frontier aka indifference curve of risky assets.
portfolio of two risky assets:
(a stock fund and bond fund).
Expected return of portfolio:
Variance of portfolio:
Cov(Re,Rd)=
Erp= We
Ere + Wd
Erd
stdev^2= (We^2)
(stdeve^2) + (Wd^2)
(Stdevd^2) + 2
We
Wd*Cov(Re,Rd)
Cov= stdevE
stdevD
CorrelationDE
in case of perfect positive correlation, what are the diversification benefits?
no diversification benefits (correlationDE=1)
correlation for perfect negative correlation?
correlationD,E = -1
sharpe ratio is shown how on graph?
sharpe ratio= reward to volatility = CAL
passive strategy
investment policy that avoids security analysis.
choose a broad index fund or ETF and divide your savings between it and a money market fund.
simplest way to reduce risk in the risky portfolio?
shift funds from the risky portfolio to the risk-free asset. another method involved diversification of the risky portfolio
a risky investment portfolio (referred to as what?) can be characterized by its what ratio?
this ratio is the slope of what?
what lies on this slope/line?
investors prefer a steepless or steeper slopoing CAL, because of what?
aka the risky asset.
characterized by reward to volatility / sharpe ratio.
its the slope of the CAL, the line connecting the Rf asset to the risky asset.
all combos of risky and risk free asset lie on this line.
steep CAL is preferred bc it means higher expected returns for any level of risk
represents a set of portfolios that maximized expected return at each level of portfolio risk
efficient frontier aka indifference curve
book to market effect
tendency for investments in shares of firms with high ratios of book value to market value to generate abnormal returns
random walk
notion that stock price changes are random and unpredictable.
stock prices should follow a random walk. price changes should be random and unpredictable.
statistical research has shown that to a close approximation stock prices seem to follow a random walk with no discernible predictable patterns that investors can exploit. such findings are now taken to be evidence of market efficiency, that is, evidence that market prices reflect all currently available info. only new info will move stock prices, and this info is equally likely to be good or bad news.
durbin watson test
examines if there is correlation in residuals.
a number that tests for autocorrelation in the residuals from a statistical regression analysis. always between 0 and 4. 2=no autocorrelation in the sample. approaching 0=positive autocorrelation, approaching 4= negative autocorrelation
how to assess if markets are informational efficient?
use an event study- measure normal returns over estimated period.
run CAPM model for estimated period to find estimates at time T for a alpha and beta.
run the model and calculate normal return.
NR=AR-ER
TomTom
July 2013: company isn't doing as poorly as people thought. so you expect price of this stock to go up.
step 1: estimate market model.
estimate a and b.
A=0, b=1.23.
then calculate normal returns and compare them to actual returns.
at time 0, there had been relatively flat returns up to time 0.
graph shows increase in returns on TomTom over next few days.
stocks with low book ratio have high or low returns?
inefficiencies?
low book ratio is low returns.
high book ratio have high returns.
inefficiencies?
why would high BTM stocks have low returns if marketers are inefficient? investors may be overoptimistic about some stocks. so market value of stocks go up more than fundamentals would prescribe.
higher stocks are riskier?
stocks with high BTM ratio have higher returns and are riskier
f you have one source of systematic risk, what will happen when you start aggregating assets in a big portfolio?
systematic component will remain.
if you have a well diversified portfolio, sensitivity will start decreasing.
is there covariance between systematic and nonsystematic risk?
no
how to test CAPM?
o Testing CAPM
• Capm is relationship about expected returns
• First, select a sample from 1 to T
• We want to get the returns of n stocks (a large number of stocks)
• CAPM must hold for all stocks, not just one stock
• We need to find out systematic source of risk and risk-free rate in the economy
• We want to test if the relationship holds for all stocks. In a good model, the relationship will hold for all stocks
• Run some regressions- estimate Betas and Alphas. For every stock, run the regression. If there are 500 assets, you must run 500 regressions
• The risk can differ amongst stocks
• So once you get your data, estimate for every stock at every point in time a regressions
• We get Bi estimates
• Now calculate average excess return for every stock
• Y= "gamma"
• Ri-Rj=y0+y1b1
• We want to have a gamma that is 0, on average for all stocks
• But we also want gamma1 y1 to equal excess reutrns of the market
• So even if gamme is 0, gamma 1 might not equal average return of markets
o Test outcomes for the single factor model
• Predicted SML is different than actual SML linke, as shown on graph
o Example: 500 largest European stocks, 2005-2014, market return-avg of 500 stocks
• Monthly market return chart shows there is a high volatility period
• First-stage regression
• Alphas are 0 or slightly positive, positively skewed
• What does this show us? That the CAPM doesn't hold
• It is normal that if you estimate some alphas, even if the CAPM holds, there will be some different alphas
• Then a regression for the betas
o Distribution is centered around 1. Is this a surprise?
o Where should the mean be? Around 1. Average return for 500 regressions should be 1, bc youre using the overall return of the market
o Second-stage regression
• Rwgress the average return of every stock ...
• 500 stocks to do this to, giving you 500 average returns
• results from regression:
• .0009,.0067
• .0007,.0008
• these are always different than 0, bc there will always be some risk premium
• is this risk premium equal to the risk premiu of the market?
• Compare .0067 to average return on market (Rm=0.0077)
• Accept or reject CAPM? Accept the CAPM bc the alphas are 0, and the market risk premium is not different from that of the market.
• We don't have enough confidence to say .67 and .77 are different from each other, bc the 77 is volatile enough to say theyre difft from each other
what does it mean if you reject CAPM?
it means exposed portfolio you used is not efficient.
every test is a test on the efficient scale. so even if we don't find very strong evidence, the CAPM might still hold. this is the reason this is the most popular theoretical model***
if there is a dividend tomorrow and we buy stock today, do we get full divined? or only a fraction?
buy gets full dividend.
for premiums, which are greater than which? current yield, YTM, coupon rate
coupon rate > current yield > YTM.
current yield exceeds YTM bc YTm accounts for the built in capital loss on the bond. the bond bought at premium of 1276 will eventually fall in value to 1000 at maturity.
current yield: investments annual income (interest or dividends) divided by current price of security. this measure looks at current price of bond instead of its face value.
current yiel= annual coupon pmt / current market value of bond. based on purchase price
yield to maturity: total return anticipated on a bond if the bond is held until the end of its lifetime. YTM is expressed as an annual rate. its the IRR of an investment bond if the investor holds the bond until maturity. based on par value, purchase price, duration, and coupon rate.
coupon rates remain stable and don't change.
• If FV is 1000 and bond is selling at a premium, then the price will be high. So the current yield will be relatively lower.
• For discount bonds, reverse this.
• Current yield is only what you get over next year
o Ex: 2 year bond with FV 1000. At time 1 year we receive 100 in coupon payment. In year 2 we receive 1,100 (FV + final coupon payment)
• Coupon = 10% and yield = 10%
• YTM=realized rate of return here
• If reinvestment rate drops to 8%? We still get our 100 at end of year 1, and reinvest that 100 at 8%/ so we receive total of 108 from this coupon payment at end of year 2, plus 1100 from year 2. So here, the r=9.91%, lower than the YTM.
• The YTM may not be the actual return that we get.
price path of premium bond. premium increases or decreases over time. why?
decreases.
eventually at maturity, the bond will be at par. as you approach maturity date, there are less and less coupon payments. closer to maturity, fewer coupon payments. so over time, there is a convergence of premium and discount bonds to the par value. so right before maturity, premium and discount bonds should be all close to their par value.
are HPR and YTM always the same?
no, can be very different. HPR is the rate of return over given interval. YTM is an average return if bond is held until maturity and theres no reinvestment.
why are yield curves useful?
we can get info about future IRs from yield curves.
why do we want to know what IRs will be in the future? matters for market risk premium. and for what investors will do.
why do we have different rates? (forward, short, spot)
there is uncertainty in the future. forward rate gives an expectation.
**if you're a risk averse investor, would you like forward rates to be slightly higher or lower than expected rate in order to take account risk that IR might be different than your expectation?
...
when something is risky, what do you expect to get with it?
some sort of risk premium (liquidity premium)
three good ratios for determinants of ratings:
coverage ratio (comparing earnings to fixed costs. you expect company with high coverage ratio to have high rating).
leverage ratio.
liquidity ratio: compare assets to liabilities (assets/liabilitiyes). if you have more assets on your balance sheet, you are more likely to pay.
sinking funds
a fund formed by periodically setting aside money for the gradual repayment of a debt or replacement of a wasting asset.
when you issue a fixed income security, you pay a coupon payment. at end you pay the larger FV. at time of maturity there is a very large fixed payment for issuer. this could cause liquidity problems for issuer. so instead of paying large FV at end, issuer can start buying off some of debt so once bond matures, issuer pays relatively lower amount of money.
if you are worried issuer won't pay back everything, in case of default, the bond you had will be bought back first.
if you are worried that issuer won't pay back debt, you can impose this covenant saying that you have a max on amount of divined you can pay back to shareholders.
subordination of further debt
dividend restrictions
collateral
issuer pledges some assets to the bondholder, in case the issuer defaults on their obligation (equipment, building, etc)
promised YTM vs expected YTM
• YTM is not the actual yield that you get
• Promised YTM: what we infer from current prices
• Expected YTM: takes into account probabilitiy/likelihood that issuer might default on their obligation
what means " to hold the broader market"
indexing
this is specific for fixing securities. try to completely eliminate any IR risk in our portfolio
immunization.
Suppose you are managing APP fund. What is your biggest risk (for a pension fund)? Interest- IR risk bc you have huge liabilities with very long duration (30-40 years). Your assets cant match this duration. If theres a decrease in IR, present value of your liabilities is going to increase*. This is not easily offset by increase in asset value, bc liabilities have much longer duration
price risk
reinvestment risk
• Technique that targets net worth, matching duration of assets and liabilities
o When you invest in fixed income securities- problem with price risk (change in value of fixed income security), and problem with reinvestment risk (refer to fact that you receive coupon payments, but IR may change overtime, so affects rate at which you can reinvest your money)
• Targeting the investment horizon: match duration with the holding period
o Offsets price risk with reinvestment risk
swapping strategies
benefit from projected or real changes in IRs. A bond swap consists of selling one debt instrument in order to use the proceeds to purchase another debt instrument. Investors engage in bond swapping with the goal of improving their financial positions. Bond swapping can reduce an investor's tax liability, give an investor a higher rate of return or help an investor to diversify a portfolio. The pure yield pickup swap and the tax swap are two common bond-swapping strategies.
substitution swap
intermarket swap
rate anticipation swap
pure yield pickup
bond swapping strategies
substitution swap: long in some undervalued asset and short in overvalued asset with similar risk profile. An exchange that is carried out by trading a fixed-income security for a higher yielding bond with similar features. A substitution swap involves the swapping of one bond for another bond that has a higher yield, but has a similar coupon rate, maturity date, call feature, credit quality, etc. A substitution swap allows the investor (such as a firm) to increase returns without altering the terms or risk level of the security. Investors will participate in substitution swaps when they believe there is a temporary discrepancy in bond prices due to market disequilibrium.
intermarket swap: sub across sectors in overall market in terms of riskiness, with different yields on the market. A swap transaction meant to capitalize on a yield discrepancy between bond market sectors. Intermarket spread swaps are based upon expectations of yield spreads between different bond sectors or spots on the yield curve. By entering a swap, parties are able to gain exposure to the underlying bonds, without having to directly hold the securities.
rate anticipation swap: suppose you expect in future IR to go down. optimal thing to do? which bonds to start buying? long term bonds. A type of swap in which bonds are exchanged according to their current duration and predicted interest rate movements. A rate anticipation swap is often made in order to take advantage of more profitable bond opportunities. Rate anticipation swaps are speculative in nature, since they depend on the outcome of the expected interest rate change. Various bond types respond differently to rising or falling interest rates and those who participate in rate anticipation swaps generally choose bonds based on performance.
pure yield pickup: if you observe that the yield curve is upwards sloping, you simply start buying long term bonds to benefit from IR increase. A transaction in which bonds with lower returns are swapped for bonds with higher returns. With a pure yield pickup swap the sole purpose of the transaction is to increase yield, the new bonds will have a similar maturity and risk rating as the old bonds; only the coupon will differ.
horizon analysis
The analysis of a security or portfolio's total returns over a period of time, referred to as the investment horizon. Horizon analysis allows an investor to assess performance under different levels of risk, market yields and return expectations. This is referred to as scenario analysis. The horizon date chosen is dependent on the needs of the analyst, and can correspond to a business cycle or maturity date.
I will hold this bond for 2 years. I will calculate price today and price two years from now when I will sell the bond. Using IRs I expect from the market. I expect IRs to go down, so I will make a profit from holding this fixed income security. Trying to see what IRs will be in the future.
what is good/special about derivatives?
their payoffs can be higher and nonlinear to underlying asset.
very bad payoffs if things are bad.
contingent claims
aga derivatives.
is a call option an obligation?
no, it is an option. if it was an obligation, it would be called a future.
exercise price
aka strike price.
pay a premium to get this asset.
price at which you agree to give that option to the holder. give holder right to buy the underlying asset at pre specified exercised/strike price
investing in options versus the stock market: 3 options:
invest entirely in stock. this is risky. using no derivatives. it stock goes below 100, your rate of return -100. you will get nothing.
call options with 6 month to maturity. strike price of 100.
1000 in call option and rest in T bills. safest option.
what is a perfectly hedged position?
value is independent of value of stock price at year end
two ways to calculate value of our call right now:
1) calculate replicating portfolio where we had a fraction invested in stock and in riskless bonds
2) discounting payoffs in two stages of world using risk neutral probabilities. when apply binomial tree approach, we don't know actual/true probabilities. we can assume. replicating portfolio approach says nothing about two different states of the world.
in case of call, do we want underlying to have high or low value?
high, in case of call. for higher returns.
why do we buy future contract instead of buying underlying today? why postpone?
benefit if there was a price increase (speculation)
three options for pricing of options:
replication portfolio, risk neutral probabilities, black scholes formula
what is the yield to maturity on a bond
the discount rate that will set the present value of the payments equal to the bond price.
NOT based on assumption that any payments received are reinvested at the coupon rate.
the intrinsic value of an out of the money call option is equal to what?
zero- the fact that the owner of the option can buy the stock at a price greater than the market price gives the contract an intrinsic value of zero, and the holder will not exercise
stdev in APT model?
stdev^2= (1/n)*(stdev^2)
how to calculate interest rate for a bond?
(Face Value / Price ) ^1/N
graph of a call option?
graph of a put option?
horizontal line shows the cost of the option (if it costed $14, then the horizontal line is at -$14, and then slopes diagonally for a profit
downward sloping line from 0 to the exercise price of 80. then horizontal flat line from 80 to the right
how to make a binomial tree and definition:
an option valuation model predicated on the assumption that stock prices can move to only two values over any short time period. binomial model requires a computer to be useful in actual trading.
start with stock price. you buy the stock at 100. then the two outcomes are either 1.2 increase (120) or 0.9 decrease (90).
so you have drawn:
100-> 120 or 90 for stock price.
your exercise price is 110. so at 120, you will gain 10. and at 90, you will profit 0.
so you draw this second chart as: C (call)-> 10 or 0 for call option value.
when to use binomial model vs black-scholes
both are option valuations.
while binomial model we have described is extremely flexible, it requires a computer to be useful in actual trading.
an option-pricing formula would be far easier to use than the tedious algorithm involved in the binomial mode. this formula can be derived if one if willing to make just two more assumptions: that both the risk free interest rate and stock price volatility are constant over the life of the option.
black scholes formula
values an option that uses the stock price, risk free interest rate, the time to expiration, and the standard decimation of the stock return.
used to price european-style call options
standard deviation of portfolio with one risky and one risk free asset?
stdevP=W*stdevRisky asset
weight of whole portfolio = the weight of the risky asset * stdev risky asset
you took a short position in three S&P 500 futures contracts at a price of 90 (the contract multiplier is 250) and closed the position when the index futures was 885, you incurred:
a gain of $11,250
900-885=15 x250 x3
one reason swaps are desirable is that
they offer participants easy ways to restructure their balance sheets.
for example, a firm can change a floating rate obligation into a fixed rate obligation and vice versa
. The buyer of an American call option on a non-dividend paying stock will
A. always exercise the call as soon as it is in the money.
B. only exercise the call when the stock price exceeds the previous high.
C. never exercise the call early.
D. buy an offsetting put whenever the stock price drops below the strike price.
E. None of these is correct
C: an american call option buyer will not exercise early if the stock does not pay dividends; exercising forfeits the time value. rather, the option buyer will sell the option to collect both the intrinsic value and the time value
Construct a portfolio out of the three stocks that has exposure of 1 to the Size factor
and an exposure of 0.5 to the Value factor. Provide the system of equations to be used
to solve for the weights. You do not need to find the explicit solution for the weights.
The weights solve the following system of equations:
𝑤1+𝑤2+𝑤3 =1
𝛾1𝑤1+𝛾2𝑤2+𝛾3𝑤3 =1
𝛿1𝑤1+𝛿2𝑤2+𝛿3𝑤3 =0.5
how to compute the prices, duration, and the modified duration of bonds.
Consider the data on the following three coupon bonds:
Bond Maturity Coupon Yield Face Value
A 2 0.07 0.02 100
B 3 0.06 0.03 100
C 4 0.04 0.04 100
i. Compute the prices, duration and the modified duration of the three bonds
P= coupon/(1+YTM) + (FV+YTM)/(1+YTM)^N
For bond A:
PA = 7/1.02 + 107/1.02^2 =109.71
DA = 1
7/1.02/109.71 + 2
107/1.02^2/109.71 = 1.94
DA* = 1.94/(1+0.02) = 1.90
You can easily follow the steps for the other two bonds
Price Duration Modified Duration
Bond A 109.71 1.94 1.90
Bond B 108.49 2.84 2.76
Bond C 100.00 3.78 3.63
Discuss the relationship between option prices and a) volatility of the underlying stock, and b)
the exercise price
The greater the volatility of the underlying stock, the greater the option premium; the more
volatile the stock, the more likely it is that the option will become more valuable (e. g., move
from an out of the money to an in the money option, or become more in the money). For call
options, the lower the exercise price, the more valuable the option, as the option owner can
buy the stock at a lower price. For a put option, the lower the exercise price, the less valuable the option, as the owner of the option may be required to sell the stock at a lower than market
price
;